PATENT DOCUMENT

Abstract:
A computer implemented method, system and computer program product for estimating an offset of at least one sensor is disclosed. The computer implemented method, system and computer program product comprise calculating combinations of running sums to define for computing/or one or more slopes for the sensor. At least one of the one or more slopes provides an indication of the sensor offset changes with temperature.

Full Description:
FIELD OF THE INVENTION 
     The present invention relates generally to sensor systems and more specifically to estimating an offset on sensor systems of a device as a function of temperature. 
     BACKGROUND 
     Sensor systems are utilized in a plurality of environments to determine the orientation of a device. The sensor systems can oftentimes include any combination of gyroscopes, accelerometers and magnetometers to provide the orientation information. It is known that an offset of a sensor can be estimated as a function of temperature. However traditionally the computation of offset estimation can require a significant amount of memory. Accordingly what is desired is a system and method that provides the offset estimation computation but uses significantly lees memory bandwidth than conventional solutions. The method and system should be simple, easily implemented and adaptable to existing environments. The present invention addresses such a need. 
     SUMMARY OF THE INVENTION 
     A computer implemented method, system and computer program product for estimating an offset of at least one sensor is disclosed. The computer implemented method, system and computer program product comprise calculating combinations of running sums to define for computing /or one or more slopes for the sensor. At least one of the one or more slopes provides an indication of the sensor offset changes with temperature. 
     Other aspects and advantages of the present invention will become apparent from the following detailed description, taken in conjunction with the accompanying drawings, illustrating by way of example the principles of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a system that utilizes an offset estimation algorithm in accordance with an embodiment. 
         FIG. 2  is a flow chart that illustrates utilizing the offset with the temperature in accordance with the present invention. 
         FIG. 3  is a flow chart that illustrates the offset estimation process in accordance with the present invention. 
         FIG. 4A  illustrates the variables to be utilized for offset estimation for a three element gyroscope. 
         FIG. 4B  shows a plot of the data, and the resulting curve fit. 
     
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     The present invention relates generally to sensor systems and more specifically to estimating an offset on the sensor systems of a device as a function of temperature in such sensor systems. The following description is presented to enable one of ordinary skill in the art to make and use the invention and is provided in the context of a patent application and its requirements. Various modifications to the preferred embodiment and the generic principles and features described herein will be readily apparent to those skilled in the art. Thus, the present invention is not intended to be limited to the embodiment shown but is to be accorded the widest scope consistent with the principles and features described herein. 
     A system and method in accordance with the present invention provides for temperature compensation via the estimation of an offset for motion sensors such as gyroscopes, accelerometers and magnetometers, when the sensors are used in environments where there is limited memory. Such environments include, for example, mobile devices such as, cell phones, tablet PCs, portable electronics and the like that are designed for mobility. Often the offset of sensors in such environments can be estimated as a function of temperature. The simplest function that can be used for the offset estimate is a line that represents a plurality of data points of temperature. To compute the best fit line, a history of the offset and corresponding temperatures must be recorded. Typically, keeping track of all the offsets and their corresponding temperatures utilizes a significant amount of the memory resources of the mobile device. 
     A system and method in accordance with the present invention provides for the estimation of an offset due to temperature that utilizes significantly less memory resources than known techniques. To describe the features of the offset estimation methodology refer now to the following description in conjunction with the accompanying figures. 
       FIG. 1  is a block diagram of a system that utilizes an offset estimation algorithm in accordance with an embodiment.  FIG. 1  is a block diagram of a system  10  that utilizes an offset estimation algorithm  100  in accordance with an embodiment. The system  10  includes a device  11 . The device  11  may include any type of mobile device including but not limited to, a cell phone, a tablet PC, or other type of portable electronic device. The device  11  receives input data from a plurality of sensors such as one or more magnetometers  12 , one or more accelerometers  14  and one or more gyroscopes  16 . Although the sensors  12 ,  14  and  16  are shown external to the device  11 , one of ordinary skill in the art readily recognizes that those sensors could be internal to the device  11  and that would be within the spirit and scope of the present invention. 
     The device  11  includes memory  18  for storing data. Data  20  stores data from the magnetometer  12 , the accelerometer  14  and the gyroscope  16 . Memory  18  also includes an offset estimation algorithm  100  in accordance with the present invention. A processor  24  executes the algorithm  100  which operates on the data received from the sensors  12 - 16 . The processor  24  provides the executed data to a sensor fusion  19 . The sensor fusion  19  provides orientation information of the device. 
     A system and method in accordance with the present invention provides a temperature compensation mechanism which utilizes offset estimation for sensors such as gyroscopes, accelerometers and magnetometers, in devices that have limited memory. Often the offset for sensors can be estimated as a function of temperature. The simplest function that can be used estimate the offset is a line. To compute the best fit line a history of the offset and corresponding temperatures must be recorded. Keeping track of all of the offsets and corresponding temperatures utilizes a significant amount of memory. An offset estimation algorithm in accordance with the present invention estimates the offset in a manner that utilizes significantly less memory. 
     A system that utilizes an offset estimation algorithm in accordance with the present invention can take the form of an entirely hardware implementation, an entirely software implementation, or an implementation containing both hardware and software elements. In one implementation, the offset estimation algorithm is implemented in software, which includes, but is not limited to, application software, firmware, resident software, microcode, etc. 
     Furthermore, the offset estimation algorithm can take the form of a computer program product accessible from a computer-usable or computer-readable medium providing program code for use by or in connection with a computer or any instruction execution system. For the purposes of this description, a computer-usable or computer-readable medium can be any apparatus that can contain, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device. 
     The medium can be an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system (or apparatus or device) or a propagation medium. Examples of a computer-readable medium include a semiconductor or solid state memory, magnetic tape, a removable computer diskette, a random access memory (RAM), a read-only memory (ROM), a rigid magnetic disk, and an optical disk. Current examples of optical disks include DVD, compact disk-read-only memory (CD-ROM), and compact disk-read/write (CD-R/W). To describe the features of the present invention in more detail, refer now to the following description in conjunction with the accompanying Figures. 
     An equation of a line representing a graph or chart of temperature versus offset can be expressed as:
 
 y=mx+b,  where m is the slope and b is the offset (y-intercept).
 
     To compute the slope between 2 points (x1,y1) and (x2,y2) is simply: 
     
       
         
           
             
               
                 
                   slope 
                   = 
                   
                     
                       
                         y 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                       - 
                       
                         y 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     
                       
                         x 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                       - 
                       
                         x 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     To compute the average slope over a set of points, the following equation can be used: 
     
       
         
           
             
               
                 
                   slope 
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           
                             i 
                             + 
                             1 
                           
                         
                         N 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             ( 
                             
                               
                                 y 
                                 j 
                               
                               - 
                               
                                 y 
                                 i 
                               
                             
                             ) 
                           
                           
                             ( 
                             
                               
                                 x 
                                 j 
                               
                               - 
                               
                                 x 
                                 i 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     The temperature will be the x values in the equation and the offset will be the y values. All of the temperature and offset values are affected by noise. Instead of taking the average, a weighted average with weight factors based upon how far apart the points are in x (or temperature) is computed. Weighting by the distance in the x dimension gives the same answer as a minimum squared error through a set of points. More emphasis on the data points that are far apart is what is desired. Accordingly the distance to the third power along the x dimension is used. Using the distance leads to the equation: 
     
       
         
           
             
               
                 
                   slope 
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             
                               i 
                               + 
                               1 
                             
                           
                           N 
                         
                         ⁢ 
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   y 
                                   j 
                                 
                                 - 
                                 
                                   y 
                                   i 
                                 
                               
                               ) 
                             
                             · 
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     j 
                                   
                                   - 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                               3 
                             
                           
                           ] 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             
                               i 
                               + 
                               1 
                             
                           
                           N 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 x 
                                 j 
                               
                               - 
                               
                                 x 
                                 i 
                               
                             
                             ) 
                           
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     The above equation is still difficult to compute, so next the identity is used: 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           
                             i 
                             + 
                             1 
                           
                         
                         N 
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             ( 
                             
                               
                                 b 
                                 j 
                               
                               - 
                               
                                 b 
                                 i 
                               
                             
                             ) 
                           
                           · 
                           
                             ( 
                             
                               
                                 a 
                                 j 
                               
                               - 
                               
                                 a 
                                 i 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                       
                   
                   ⁢ 
                   
                       
                     
                       
                         [ 
                         
                             
                         
                         ⁢ 
                         
                           N 
                           · 
                           
                               
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   b 
                                   i 
                                 
                                 · 
                                 
                                   a 
                                   i 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       - 
                       
                         
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 b 
                                 i 
                               
                               ) 
                             
                           
                           ] 
                         
                         · 
                         
                           [ 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 a 
                                 i 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   4 
                 
               
             
           
         
       
     
     Then the slope equation is rewritten such that the identity can be applied that is described Equation 4. 
     
       
         
           
             
               
                 
                   slope 
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                             N 
                           
                           ⁢ 
                           
                             
                               〚 
                               
                                 
                                   
                                     ( 
                                     
                                       x 
                                       j 
                                     
                                     ) 
                                   
                                   3 
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       x 
                                       i 
                                     
                                     ) 
                                   
                                   3 
                                 
                               
                               ] 
                             
                             · 
                             
                               ( 
                               
                                 
                                   y 
                                   j 
                                 
                                 - 
                                 
                                   y 
                                   i 
                                 
                               
                               ) 
                             
                           
                         
                       
                       - 
                       
                         3 
                         · 
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       x 
                                       j 
                                     
                                     · 
                                     
                                       y 
                                       j 
                                     
                                   
                                   - 
                                   
                                     
                                       x 
                                       i 
                                     
                                     · 
                                     
                                       y 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         j 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                                 〛 
                               
                             
                             + 
                             
                               3 
                               · 
                               
                                 [ 
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         i 
                                       
                                       - 
                                       
                                         x 
                                         j 
                                       
                                     
                                     ) 
                                   
                                   · 
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               x 
                                               i 
                                             
                                             ) 
                                           
                                           2 
                                         
                                         · 
                                         
                                           y 
                                           i 
                                         
                                       
                                       - 
                                       
                                         
                                           
                                             ( 
                                             
                                               x 
                                               j 
                                             
                                             ) 
                                           
                                           2 
                                         
                                         · 
                                         
                                           y 
                                           j 
                                         
                                       
                                     
                                     〛 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             
                               i 
                               + 
                               1 
                             
                           
                           N 
                         
                         ⁢ 
                         
                           [ 
                           
                             
                               4 
                               · 
                               
                                 ( 
                                 
                                   
                                     x 
                                     j 
                                   
                                   - 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         j 
                                       
                                       ) 
                                     
                                     3 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     3 
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 3 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       
                                         ( 
                                         
                                           x 
                                           j 
                                         
                                         ) 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         ( 
                                         
                                           x 
                                           i 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         j 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                                 〛 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
     
     Applying the identity in Equation 4 to Equation 5 yields the following Equation 6: 
     
       
         
           
             
               
                 
                   slope 
                   = 
                   
                     
                       
                         
                           
                             
                               〚 
                               
                                 N 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     [ 
                                     
                                       
                                         
                                           ( 
                                           
                                             x 
                                             i 
                                           
                                           ) 
                                         
                                         3 
                                       
                                       · 
                                       
                                         y 
                                         i 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               ] 
                             
                             - 
                             
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     3 
                                   
                                 
                                 ] 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       y 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 〛 
                               
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               3 
                               · 
                               
                                 〚 
                                 
                                   N 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       [ 
                                       
                                         
                                           
                                             ( 
                                             
                                               x 
                                               i 
                                             
                                             ) 
                                           
                                           3 
                                         
                                         · 
                                         
                                           y 
                                           i 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                                 ] 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         i 
                                       
                                       · 
                                       
                                         y 
                                         i 
                                       
                                     
                                     ) 
                                   
                                 
                                 〛 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               3 
                               · 
                               
                                 〚 
                                 
                                   N 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       [ 
                                       
                                         
                                           
                                             ( 
                                             
                                               x 
                                               i 
                                             
                                             ) 
                                           
                                           3 
                                         
                                         · 
                                         
                                           y 
                                           i 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     [ 
                                     
                                       
                                         
                                           ( 
                                           
                                             x 
                                             i 
                                           
                                           ) 
                                         
                                         2 
                                       
                                       · 
                                       
                                         y 
                                         i 
                                       
                                     
                                     ] 
                                   
                                 
                                 〛 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               4 
                               〚 
                               
                                 N 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     [ 
                                     
                                       
                                         
                                           ( 
                                           
                                             x 
                                             i 
                                           
                                           ) 
                                         
                                         3 
                                       
                                       ⁢ 
                                       
                                         x 
                                         i 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               ] 
                             
                             - 
                             
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     3 
                                   
                                 
                                 ] 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       x 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 〛 
                               
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               3 
                               · 
                               
                                 〚 
                                 
                                   N 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           x 
                                           i 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     3 
                                   
                                 
                                 ] 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                                 〛 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
               
             
           
         
       
     
     Next, equation 6 is simplified to provide equation 7 below: 
     
       
         
           
             
               
                 
                   slope 
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 3 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           x 
                                           i 
                                         
                                         · 
                                         
                                           y 
                                           i 
                                         
                                       
                                       ) 
                                     
                                     · 
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         N 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           ( 
                                           
                                             x 
                                             i 
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                 
                               
                               - 
                               
                                 3 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       x 
                                       i 
                                     
                                     · 
                                   
                                 
                               
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 N 
                               
                               ⁢ 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   · 
                                   
                                     y 
                                     i 
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 N 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   y 
                                   i 
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     3 
                                   
                                 
                               
                             
                             + 
                             
                               N 
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   N 
                                 
                                 ⁢ 
                                 
                                   [ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         ( 
                                         
                                           x 
                                           i 
                                         
                                         ) 
                                       
                                       3 
                                     
                                     · 
                                     
                                       y 
                                       i 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           3 
                           · 
                           
                             
                               [ 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   N 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       x 
                                       i 
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               ] 
                             
                             2 
                           
                         
                         - 
                         
                           4 
                           · 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                             ⁢ 
                             
                               
                                 x 
                                 i 
                               
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   N 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       x 
                                       i 
                                     
                                     ) 
                                   
                                   3 
                                 
                               
                             
                           
                         
                         + 
                         
                           N 
                           · 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   x 
                                   i 
                                 
                                 ) 
                               
                               4 
                             
                           
                         
                       
                       ⁢ 
                       
                           
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   7 
                 
               
             
           
         
       
     
     Equation 7 is written in a manner that provides for the calculation of eight running sums (x i y i , x i   2 , x i , x i   2 y i , y i , x i   3 , x i   3 y i , x i   4 ). As a result, it is not necessary to have to keep track of the past values. The eight running sums and N are updated with each new set of points that are utilized to fit to a curve or line. To avoid overflow and loss of sensitivity all of the running sums can be divided by a factor and also the number of points, N, can be divided by the same factor. The division can be based upon threshold for the sums or number of points and/or based upon reaching a particular spread of values. In addition it may be beneficial to keep track of the maximum and minimum temperature range. The curve fit can only be used if the temperature is within a particular range. 
     The y-intercept for the line still needs to be computed. Given 2 points (x1,y1) and (x2,y2) the y-intercept would be: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         y 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           · 
                           x 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                       - 
                       
                         x 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           · 
                           y 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     ) 
                   
                   
                     ( 
                     
                       
                         x 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                       - 
                       
                         x 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     ) 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
     
     To compute the average y-intercept for a range of points, the following Equation 9 would be used: 
     
       
         
           
             
               
                 
                   intercept 
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             
                               i 
                               + 
                               1 
                             
                           
                           N 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 y 
                                 i 
                               
                               · 
                               
                                 x 
                                 j 
                               
                             
                             - 
                             
                               
                                 x 
                                 i 
                               
                               · 
                               
                                 y 
                                 j 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             
                               i 
                               + 
                               1 
                             
                           
                           N 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               x 
                               j 
                             
                             - 
                             
                               x 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   9 
                 
               
             
           
         
       
     
     As was done for the slope, the 3 rd  power of the distance along the x-axis needs to be weighted. The following Equation 10 results: 
     
       
         
           
             
               
                 
                   intercept 
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             
                               i 
                               + 
                               1 
                             
                           
                           N 
                         
                         ⁢ 
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   
                                     y 
                                     i 
                                   
                                   · 
                                   
                                     x 
                                     j 
                                   
                                 
                                 - 
                                 
                                   
                                     x 
                                     i 
                                   
                                   · 
                                   
                                     y 
                                     j 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     j 
                                   
                                   - 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                               3 
                             
                           
                           ] 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             
                               i 
                               + 
                               1 
                             
                           
                           N 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 x 
                                 j 
                               
                               - 
                               
                                 x 
                                 i 
                               
                             
                             ) 
                           
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   10 
                 
               
             
           
         
       
     
     Which can be rewritten as Equation 1 
     
       
         
           
             
               
                 
                   intercept 
                   = 
                   
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   j 
                                   = 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                                 N 
                               
                               ⁢ 
                               
                                 [ 
                                 
                                   
                                     3 
                                     · 
                                     
                                       [ 
                                       
                                         
                                           
                                             x 
                                             j 
                                           
                                           · 
                                           
                                             y 
                                             j 
                                           
                                         
                                         - 
                                         
                                           
                                             x 
                                             i 
                                           
                                           · 
                                           
                                             y 
                                             i 
                                           
                                         
                                       
                                       ) 
                                     
                                     · 
                                     
                                       [ 
                                       
                                         
                                           
                                             ( 
                                             
                                               x 
                                               j 
                                             
                                             ) 
                                           
                                           3 
                                         
                                         - 
                                         
                                           
                                             ( 
                                             
                                               x 
                                               i 
                                             
                                             ) 
                                           
                                           3 
                                         
                                       
                                       〛 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           x 
                                           j 
                                         
                                         ) 
                                       
                                       3 
                                     
                                     · 
                                     
                                       y 
                                       j 
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         ( 
                                         
                                           x 
                                           i 
                                         
                                         ) 
                                       
                                       3 
                                     
                                     · 
                                     
                                       y 
                                       i 
                                     
                                   
                                 
                                 ] 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     x 
                                     j 
                                   
                                   - 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               3 
                               · 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 〚 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           x 
                                           j 
                                         
                                         ) 
                                       
                                       2 
                                     
                                     · 
                                     
                                       y 
                                       j 
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         ( 
                                         
                                           x 
                                           i 
                                         
                                         ) 
                                       
                                       2 
                                     
                                     · 
                                     
                                       y 
                                       i 
                                     
                                   
                                 
                                 ] 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         j 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                                 〛 
                               
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 
                                   y 
                                   i 
                                 
                                 - 
                                 
                                   y 
                                   j 
                                 
                               
                               ) 
                             
                             · 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       x 
                                       i 
                                     
                                     ) 
                                   
                                   4 
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       x 
                                       j 
                                     
                                     ) 
                                   
                                   4 
                                 
                               
                               〛 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   j 
                                   = 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                                 N 
                               
                               ⁢ 
                               
                                 [ 
                                 
                                   
                                     4 
                                     · 
                                     
                                       ( 
                                       
                                         
                                           x 
                                           j 
                                         
                                         - 
                                         
                                           x 
                                           i 
                                         
                                       
                                       ) 
                                     
                                     · 
                                     
                                       [ 
                                       
                                         
                                           
                                             ( 
                                             
                                               x 
                                               j 
                                             
                                             ) 
                                           
                                           3 
                                         
                                         - 
                                         
                                           
                                             ( 
                                             
                                               x 
                                               i 
                                             
                                             ) 
                                           
                                           3 
                                         
                                       
                                       ] 
                                     
                                   
                                   - 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               3 
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         j 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                                 ] 
                               
                             
                             · 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       x 
                                       j 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       x 
                                       i 
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               〛 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   11 
                 
               
             
           
         
       
     
     The double sum identity above in Equation 4 is applied to Equation 11 for the following results: 
     
       
         
           
             
               
                 
                   intercept 
                   = 
                   
                     
                       
                         
                           
                             
                               3 
                               · 
                               
                                 〚 
                                 
                                   N 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ⌈ 
                                       
                                         
                                           
                                             ( 
                                             
                                               x 
                                               i 
                                             
                                             ) 
                                           
                                           4 
                                         
                                         · 
                                         
                                           y 
                                           i 
                                         
                                       
                                       ⌉ 
                                     
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                     3 
                                   
                                 
                                 ] 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         i 
                                       
                                       · 
                                       
                                         y 
                                         i 
                                       
                                     
                                     ) 
                                   
                                 
                                 〛 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               〚 
                               
                                 
                                   [ 
                                   
                                     N 
                                     · 
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         N 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ⌈ 
                                         
                                           
                                             
                                               ( 
                                               
                                                 x 
                                                 i 
                                               
                                               ) 
                                             
                                             4 
                                           
                                           · 
                                           
                                             y 
                                             i 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                   
                                   ] 
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         N 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         x 
                                         i 
                                       
                                     
                                     ) 
                                   
                                   · 
                                   
                                     [ 
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         N 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ⌈ 
                                         
                                           
                                             
                                               ( 
                                               
                                                 x 
                                                 i 
                                               
                                               ) 
                                             
                                             3 
                                           
                                           · 
                                           
                                             y 
                                             i 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               ] 
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   3 
                                   · 
                                   
                                     〚 
                                     
                                       N 
                                       · 
                                       
                                         
                                           ∑ 
                                           
                                             i 
                                             = 
                                             1 
                                           
                                           N 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ⌈ 
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   x 
                                                   i 
                                                 
                                                 ) 
                                               
                                               4 
                                             
                                             · 
                                             
                                               y 
                                               i 
                                             
                                           
                                           ⌉ 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 - 
                                 
                                   
                                     [ 
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         N 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           ( 
                                           
                                             x 
                                             i 
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     ] 
                                   
                                   · 
                                   
                                     [ 
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         N 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ⌈ 
                                         
                                           
                                             
                                               ( 
                                               
                                                 x 
                                                 i 
                                               
                                               ) 
                                             
                                             2 
                                           
                                           · 
                                           
                                             y 
                                             i 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                     〛 
                                   
                                 
                               
                               ] 
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               
                                 〚 
                                 
                                   N 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ⌈ 
                                       
                                         
                                           
                                             ( 
                                             
                                               x 
                                               i 
                                             
                                             ) 
                                           
                                           4 
                                         
                                         · 
                                         
                                           y 
                                           i 
                                         
                                       
                                       ⌉ 
                                     
                                   
                                 
                                 ] 
                               
                               - 
                               
                                 
                                   [ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           x 
                                           i 
                                         
                                         ) 
                                       
                                       4 
                                     
                                   
                                   ] 
                                 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       y 
                                       i 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           
                             [ 
                             
                               
                                 4 
                                 〚 
                                 
                                   N 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       [ 
                                       
                                         
                                           
                                             ( 
                                             
                                               x 
                                               i 
                                             
                                             ) 
                                           
                                           3 
                                         
                                         · 
                                         
                                           x 
                                           i 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 ] 
                               
                               - 
                               
                                 
                                   [ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           x 
                                           i 
                                         
                                         ) 
                                       
                                       3 
                                     
                                   
                                   ] 
                                 
                                 · 
                                 
                                   [ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         x 
                                         i 
                                       
                                       ) 
                                     
                                   
                                   〛 
                                 
                               
                               - 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 3 
                                 · 
                                 
                                   〚 
                                   
                                     N 
                                     · 
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         N 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           ( 
                                           
                                             x 
                                             i 
                                           
                                           ) 
                                         
                                         4 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 
                                   [ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           x 
                                           i 
                                         
                                         ) 
                                       
                                       3 
                                     
                                   
                                   ] 
                                 
                                 · 
                                 
                                   [ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           x 
                                           i 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   〛 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   12 
                 
               
             
           
         
       
     
     Which reduces to: 
     
       
         
           
             
               
                 
                   intercept 
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             y 
                             i 
                           
                           · 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   x 
                                   i 
                                 
                                 ) 
                               
                               4 
                             
                           
                         
                       
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             x 
                             i 
                           
                           · 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       x 
                                       i 
                                     
                                     ) 
                                   
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                   Equation 
                   ⁢ 
                   
                       
                   
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                   13 
                 
               
             
           
         
       
     
       FIG. 2  is a simple flow chart that illustrates calculating gyroscope offset in accordance with an embodiment. First, running sums of temperature  106  and offset  102  are provided, via step  104 . In an embodiment, the running sum of the slope is calculated via step  110 . In another embodiment, the running sum of the intercept is calculated via step  108 . Step  108  may be optional. The gyroscope offset is calculated utilizing the calculated slope and/or intercept, via step  112 . To describe the calculation of the gyroscope offset in more detail refer now to the following description in conjunction with the accompanying figures. 
       FIG. 3  is a flow chart that illustrates the offset estimation process in accordance with the present invention. First, a new temperature (X) and its associated offset are received, via step  302 . Then, new running sums of temperature and offset (S′) are calculated using the new temperature (X), stored offset (Y) and old running sums of temperature and offset (S), via step  304 . Next, a count of new temperature values (N) is incremented, via step  305 . Then, one or more slopes are calculated using the new running sums of temperature and offset (S′), via step  306 . Thereafter, a new offset (Y′) is calculated and stored, via step  308 . In addition to calculating one or more slopes, it is important in some circumstances to also calculate one or more intercepts to provide a more accurate offset value estimation. 
     Implementation 
     With the calculations defined in  FIG. 3 , implementation details can now be carried out.  FIG. 4A  illustrates the variables to be utilized for offset estimation for a three element gyroscope. For example, given a three element gyroscope offset and a temperature, the temperature would be the x variable, and the gyroscope offset would be the y variable in each of the three axes.  FIG. 4A  illustrates running sums to be calculated: for N  401  which is the number of data points, for a first column  402  which represents temperature, for a second column  404  which represent the temperature and the gyroscope offset along a first axis (x); for a third column  406  which represent the temperature and the gyroscope offset along a second axis (y); and for a fourth column  408  which represent the running sum for temperature and the gyroscope offset along a third axis (z). Therefore for each access, running sums are calculated for a total of 17 variables. The equation for slope and intercept can be applied after N is 2 or ore. 
     It may be desired to limit the value of N for numerical stability. One embodiment would be to limit N to 3000, and after N reaches 3000, then all the running sums and the counter N, are multiplied by 0.9. Scaling by 0.9 has an effect of weighting new data more and to prevent numerical stability issues that may arise if the running sums to continue to grow. Another method to limit the effect of numerical stability would be to make sure the temperature has changed by a minimum amount since the last data storage. A change of 1 or 2 degrees Celsius should be sufficient before adding the data to the data set of running sums. Storing the previous temperature would require 1 more persistent data storage variable. 
     To use the results to perform temperature compensation, the temperature is measured. Then the gyroscope would be corrected by subtracting out the estimated offset, where slope and intercept is determined by the math with the running sums in it above.
 
Gyro_Corrected=Gyro_Measured−Slope*Temperature−Intercept
 
Extension to Piece-Wise Linear
 
     The core algorithm can be extended for piecewise linear. For this embodiment, an accelerometer is considered where a piecewise linear fit is done where the piecewise breakpoint is at 25 degrees C.  FIG. 4B  shows a plot of the data, and the resulting curve fit. 
     Just as above, for the gyroscope example embodiment, where 17 running sums and 1 count of the number of terms for 3 axes were stored, a similar method can also be used to reduce the number of terms to store. However, because 2 pieces are required for a linear fit the number of terms would double to 34 terms. If the temperature is forced to move by a set amount such as at least 1 or 2 degrees C. then 35 persistent storage terms would be needed for a 2 piecewise term and 3 axes. When storing the data, for this embodiment, if the temperature was above 25 C, then the data would be applied to one set of running sums and number of data points. If the temperature was below 25 C, then the data would be applied to the other set of running sums and number of data points. 
     Although the present invention has been described in accordance with the embodiments shown, one of ordinary skill in the art will readily recognize that there could be variations to the embodiments and those variations would be within the spirit and scope of the present invention. Accordingly, many modifications may be made by one of ordinary skill in the art without departing from the spirit and scope of the appended claims.