Patent Publication Number: US-8977588-B1

Title: Detection of physiological changes

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS 
     This application claims priority from U.S. Provisional Application No. 61/470,390, filed Mar. 31, 2011, which is incorporated herein by reference in its entirety. This application is related to U.S. patent application Ser. No. 13/208,039, titled “State Change Detection,” now U.S. Pat. No. 8,725,676, U.S. patent application Ser. No. 13/208/071, titled “Security Threat Detection,” now U.S. Pat. No. 8,578,494, and U.S. patent application Ser. No. 13/208,099, titled “State Change Detection Using Multi-Dimensional Phase Space,” now U.S. Pat. No. 7,725,677, all of which were filed on Aug. 11, 2011, all of which also claim priority to U.S. Provisional Application No. 61/470,390, and all of which are incorporated herein by reference in their entireties. 
    
    
     BACKGROUND 
     The present disclosure relates generally to the field of detecting changes in physiological systems. 
     Many physiological changes are detected in a post-emergent process a considerable time after the change was originally manifested. Many physiological changes manifest themselves early through small event evidence, such as neuronal firing changes prior to epileptic seizures. However, detection of these small events is a difficult task, and the medical community lacks accurate and portable equipment to detect the early small events. 
     SUMMARY 
     One embodiment of the disclosure relates to a method of detecting changes in a physiological system. The method comprises embedding time series data relating to the physiological system within a reconstructed phase space. The method further comprises partitioning the reconstructed phase space into a plurality of regions. The method further comprises generating a first matrix having a plurality of cells. The first matrix comprises a row and a column for each of the plurality of regions. A value stored in each cell is based on a probability that the system will transition from a first region associated with the cell to a second region associated with the cell and a rate of separation of trajectories of the embedded data within at least one of the first region and the second region. The first matrix is generated using a first set of the time series data that is associated with a normal operating condition of the physiological system. The method further comprises generating a second matrix based on a second set of the time series data and comparing the first matrix and the second matrix to detect an anomalous condition of the physiological system. 
     Another embodiment relates to a system comprising an electronic processor configured to embed time series data relating to a physiological system within a reconstructed phase space and partition the reconstructed phase space into a plurality of regions. The electronic processor is further configured to generate a first matrix having a plurality of cells. The first matrix comprises a row and a column for each of the plurality of regions. A value stored in each cell is based on a probability that the system will transition from a first region associated with the cell to a second region associated with the cell and a rate of separation of trajectories of the embedded data within at least one of the first region and the second region. The first matrix is generated using a first set of the time series data that is associated with a normal operating condition of the physiological system. The electronic processor is further configured to generate a second matrix based on a second set of the time series data and to compare the first matrix and the second matrix to detect an anomalous condition of the physiological system. 
     In some embodiments, the electronic processor is configured to compare the first matrix and the second matrix by: (1) calculating a state change parameter representing a degree of state change between a first state associated with the first matrix and a second state associated with the second matrix; and (2) comparing the state change parameter to a threshold value, wherein the electronic processor is further configured to perform an action when the state change parameter exceeds the threshold value. In some such embodiments, the action performed by the electronic processor when the state change parameter exceeds the threshold value comprises enacting one or more steps configured to counteract the anomalous condition. In some such embodiments, the action performed by the electronic processor when the state change parameter exceeds the threshold value comprises transmitting a signal to one or more machines, wherein the one or more machines, after receiving the signal, are configured to administer a substance that is configured to counteract the anomalous condition, wherein an amount of the substance administered is based on the state change parameter. 
     Another embodiment relates to one or more computer-readable media having instructions stored thereon that are executable by one or more processors to execute a method. The method comprises embedding time series data within a reconstructed phase space. The time series data comprises physiological data relating to a physiological system of a person. The method further comprises partitioning the reconstructed phase space into a plurality of regions and generating a matrix having a plurality of cells. The matrix comprises a row and a column for each of the plurality of regions. A value stored in each cell is based on a probability that the system will transition from a first region associated with the cell to a second region associated with the cell and a rate of separation of trajectories of the embedded data within at least one of the first region and the second region. The method further comprises generating an image based on the matrix and transmitting the image to a display. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The disclosure will become more fully understood from the following detailed description, taken in conjunction with the accompanying figures, wherein like reference numerals refer to like elements, in which: 
         FIG. 1A  is a flow diagram of a process for embedding data within a reconstructed phase space according to an exemplary embodiment; 
         FIGS. 1B and 1C  are Henon maps associated with a reconstruction of an example phase space using different embedding dimensions according to an exemplary embodiment; 
         FIG. 1D  is a general block diagram of a control system configured to analyze a system according to an exemplary embodiment; 
         FIG. 2  is a surface plot of a sample ergodic transition matrix according to an exemplary embodiment; 
         FIG. 3  is a graph illustrating electrical pulses associated with the beating of a human heart according to an exemplary embodiment; 
         FIG. 4  is a phase space plot of the electrical pulses shown in  FIG. 3  according to an exemplary embodiment; 
         FIGS. 5 and 6  are recurrence plots of sample systems according to exemplary embodiments; 
         FIGS. 7 and 8  are ergodicity plots of the sample systems associated with  FIGS. 5 and 6 , respectively, according to exemplary embodiments; 
         FIG. 9  is a contour plot of a sample ergodic transition matrix of normality for a single computer system with no users according to an exemplary embodiment; 
         FIG. 10  is a contour plot of a sample ergodic transition matrix of normality for a single computer system with one or more users according to an exemplary embodiment; 
         FIG. 11  is a contour plot of a sample ergodic transition matrix of normality for a single computer system that has one or more users and is connected to a network according to an exemplary embodiment; 
         FIG. 12  is a contour plot of the absolute differences between the ergodic transition matrix of normality associated with the contour plot shown in  FIG. 9  (e.g., for a system having no users) and the ergodic transition matrix of normality associated with the contour plot shown in  FIG. 10  (e.g., for a system having users) according to an exemplary embodiment; 
         FIG. 13  is a contour plot of a sample ergodic transition matrix of normality for an intrusion detection dataset according to an exemplary embodiment; 
         FIG. 14  is a contour plot of a sample ergodic transition matrix of normality for the Henon Map according to an exemplary embodiment; 
         FIG. 15  is a contour plot of a sample ergodic transition matrix for a second Henon Map dataset having a single data point changed from the dataset associated with  FIG. 14  according to an exemplary embodiment; 
         FIG. 16  is a contour plot of the absolute differences between the ergodic transition matrix of normality associated with the contour plot shown in  FIG. 14  and the ergodic transition matrix associated with the contour plot shown in  FIG. 15  according to an exemplary embodiment; 
         FIG. 17  is a phase space plot of the Henon Map according to an exemplary embodiment; 
         FIG. 18  is a contour plot of a sample ergodic transition matrix for the Henon Map associated with  FIG. 17  according to an exemplary embodiment; 
         FIG. 19  is an ergodicity plot of the Henon Map associated with  FIGS. 17 and 18  according to an exemplary embodiment; 
         FIG. 20  is a phase space plot of the Henon Map with additive noise according to an exemplary embodiment; 
         FIG. 21  is a contour plot of the absolute differences between an ergodic transition matrix of normality associated with the noisy Henon Map of  FIG. 20  and the ergodic transition matrix of a single changed iteration of the noisy Henon Map according to an exemplary embodiment; 
         FIG. 22  is a block diagram of a processing circuit configured to detect changes in the state of a system according to an exemplary embodiment; 
         FIG. 23  is a graph illustrating electrical pulses associated with the beating of a human heart according to an exemplary embodiment; 
         FIG. 24  is a phase space plot of the electrical pulses shown in  FIG. 23  according to an exemplary embodiment; 
         FIG. 25  is an ergodicity plot for the phase space shown in  FIG. 24  according to an exemplary embodiment; 
         FIG. 26  is a contour plot of a sample ergodic transition matrix associated with the data shown in  FIG. 24  according to an exemplary embodiment; 
         FIG. 27  is a contour plot of an ergodict transition matrix for a sample arrhythmic heart pattern according to an exemplary embodiment; 
         FIG. 28  is an ergodicity plot for sample data associated with arterial fibrillation in a heart according to an exemplary embodiment; 
         FIG. 29  is an ergodicity plot for sample data associated with a normal sinus pattern in a heart according to an exemplary embodiment; and 
         FIG. 30  is an ergodicity plot for sample data associated with ventricular arrhythmia in a heart according to an exemplary embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Before turning to the figures, which illustrate the exemplary embodiments in detail, it should be understood that the application is not limited to the details or methodology set forth in the description or illustrated in the figures. It should also be understood that the terminology is for the purpose of description only and should not be regarded as limiting. As discussed below, the systems and methods can be utilized in a number of control devices for various types of applications or analyzed systems. 
     Referring generally to the figures, systems and methods for detecting system state changes and/or anomalies are disclosed according to various exemplary embodiments. Exemplary embodiments disclosed herein generate a representation of the normality of a system (e.g., the operation of a system under normal conditions, which is defined based on each individual system or application) under consideration and monitor data (e.g., streaming data) for divergence from that normality. Data (e.g., time series data) regarding the system under consideration is embedded within a reconstructed phase space which may be created through the calculation of a time delay metric and an embedding dimension metric. 
     The phase space is then partitioned into a grid or other partitioned structure (e.g., an N×N grid such that regions of phase space are equally divided into square cells of size 1/N×1/N, where N may be a predetermined or user-defined parameter). The embedded phase space may be plotted by assigning to each point in the phase space the centroid, or center point, of the grid cell representing the region of phase space of which that point is a member. Each centroid can be considered a region of phase space, or a state of the dynamical system. This plot is referred to herein as the Ergodicity plot of the embedding phase space. 
     The Ergodicity plot may then be used to construct a transition matrix that can in turn be used to detect system changes as new data is provided to (e.g., streamed into) the systems and/or processes. The system can utilize a control system or algorithm (e.g., implemented in software) to create the transition matrix and/or detect the system state changes. Such transition matrixes are referred to herein as ergodic transition matrixes (ETMs). An ETM of normality may be generated representing a normal state of the system being considered. Subsequent ETMs may be generated using new data as it is encountered and/or provided to the algorithm. Changes in the state of the system (e.g., changes above a specified threshold) can be detected by comparing the subsequent ETMs to the ETM of normality for the system. Various exemplary embodiments disclosed herein have a very small computational footprint and enable highly effective detection of small changes and/or anomalies in a chaotic system in real time or near-real time. Various embodiments may also be tunable with user defined parameters to assist in reducing false positive rates in different applications. 
     Recent research has indicated that many of the physiological systems in humans are chaotic in nature. The brain&#39;s neuronal firing patterns are a good example of this chaotic nature. The normal heart beat has proven to be chaotic as well. It has also been demonstrated that, in patients with abnormal heart conditions or heart disease, there is less chaos present than in those whose hearts are normal. 
     Various exemplary embodiments of the present disclosure relate to systems and methods for accurate early detection of physiological changes that can be indicative of future larger medical events such as a heart attack or epileptic seizures. An understanding of normality in many different scenarios for a subject and physiological system being monitored may be developed, for example, using ETMs. Streaming data from sensors related to the physiological system under consideration may be monitored, and any system shifts or anomalies may be reported and/or addressed. 
     Referring now to  FIG. 1A , a flow diagram of a process  100  for embedding data within a reconstructed phase space is shown according to an exemplary embodiment. Time series data relating to a system being analyzed is received and embedded within a reconstructed phase space. The time series data may be data relating to one or more aspects of the system being analyzed that is measured at regular intervals. Time series data includes a series or sequence of data points, each of which is measured at and/or associated with a particular point in time. In some embodiments, time series data may be measured at regular periodic time intervals (e.g., 10 milliseconds, one second, 30 seconds, one minute, 15 minutes, one hour, one day, etc.). The data measured and/or received may be sensor data and may represent various conditions relating to the state of the system. For example, in one embodiment, the data (e.g., electrocardiogram data) may represent electrical signals associated with the beating of a human heart used, to monitor the state of the heart. 
     The time series data is used to reconstruct a phase space representation of various states of the system. According to an exemplary embodiment, reconstruction of the phase space may be done using several metrics, including a time delay metric and a minimum embedding dimension. The time delay metric represents a time delay or time lag used to construct a new set of coordinates in the newly reconstructed phase space. The minimum embedding dimension represents a minimum number of dimensions that may be used to reconstruct, or unfold, the new phase space (e.g., the metric phase space) back to a multivariate phase space that is representational of the original phase. Once the time delay metric and minimum embedding dimension have been determined, a metric representing the rate of separation of trajectories in the new embedded phase space may be determined. In some embodiments, this metric may be a maximum Lyapunov exponent of the nonlinear system. Exemplary methods of constructing the embedded phase space are provided in U.S. patent application Ser. No. 13/051,672, filed Mar. 18, 2011, titled “Prediction Method,” listing as inventor Joseph J. Engler, which is incorporated herein by reference in its entirety. In some embodiments, the embedded phase space may be constructed in a manner similar to that proposed by Takens in “Detecting Strange Attractors in Turbulence.” Takens, F., “Detecting Strange Attractors in Turbulence,” in  Dynamical Systems and Turbulence: Lecture Notes in Mathematics .: Springer, 1981, Vol. 898, pp. 366-381. 
     At step  105 , a time delay metric, or delay factor, for use in embedding time series data within the reconstructed phase space is determined. The time delay metric represents a time delay or time lag used to construct a new set of coordinates in the newly reconstructed phase space. In some embodiments, a time delay metric z may be calculated to be the first value for which an autocorrelation function (e.g., representing a cross-correlation of the time series data signal with itself as a function of time separation) falls below 1/e. The autocorrelation function may be represented as follows: 
     
       
         
           
             
               r 
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             = 
             
               
                 
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                     = 
                     
                       τ 
                       + 
                       1 
                     
                   
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                 ⁢ 
                 
                   
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     In the above,  x  is the mean of the time series data, x t  is the value of the data point x at time t, and r is the autocorrelation coefficient. 
     At step  110 , a minimum embedding dimension for the reconstructed phase space may be determined. The minimum embedding dimension represents a minimum number of dimensions that may be used to reconstruct, or unfold, the new phase space (e.g., the metric phase space) back to a multivariate phase space that is representational of the original phase. A scalar time series for a given variable of the nonlinear system may be represented by the expression x(n i )={x 0 , x 0+τ , x 0+2τ , . . . , x 0+(d-1)τ }. The time delay factor Z may be calculated from the autocorrelation of the variable as described above with respect to step  105 . The symbol d in the scalar time series expression for the variable x above is the minimal embedding dimension for which the scalar time series is most representational of the original phase space. 
     In some embodiments, a False Nearest Neighbors (“FNN”) methodology may be used to calculate the minimum embedding dimension d. A FNN methodology is a procedure for obtaining the minimum embedding dimension for phase space reconstruction based on the neighborhood of points whose distance is below a given threshold ε. Consider a point x(n) and a point x r  (n)εU(x(n)) which is the r th  nearest neighbor of x(n) in the neighborhood of points around x(n). The distance between these two points in d dimensional space may be calculated as follows: 
     
       
         
           
             
               
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                 d 
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     In the above, τ is the time delay factor from the autocorrelation function. In some embodiments, all points within the neighborhood of x(n) (e.g., within the threshold distance ε) may be calculated. 
     The embedding dimension is then increased from dimension d to dimension d+1 such that, by time delay embedding, the (d+1)th coordinate is added to each of the vectors x(n). After the dimension is increased, the distance of each point x′(n)εU(x(n)) to x(n) is again calculated. Those points whose distance to x(n) is larger than c are removed from the neighborhood U(x(n)). The removed points are considered false nearest neighbors to x(n) since they were nearest neighbors in the smaller embedding dimension but are not nearest neighbors to x(n) when the embedding dimension is expanded. In some embodiments, the proper embedding dimension d is determined to be the smallest embedding dimension d in which no false nearest neighbors are discovered. In other embodiments, the proper embedding dimension d may be determined to be the embedding dimension d in which the number of false nearest neighbors discovered is less than a certain maximum threshold of false nearest neighbors. 
     The operation of the FNN method may be illustrated with reference to  FIGS. 1B and 1C .  FIG. 1B  illustrates a Henon map associated with a reconstruction of an exemplary phase space using only an x variable. For d=1, the reconstructed scalar phase space is a line. In d=1 dimension embedding space two points a and b are illustrated as being close to a point c. These points are nearest neighbors of c. However, when the embedding dimension d is increased to d=2, as illustrated in  FIG. 1C , point a remains close to point c but point b is no longer in the neighborhood of point c. Point b was a false neighbor of point c in d=1 dimension embedding space due to the constriction of the embedding space. 
     Referring again to  FIG. 1A , at step  115 , a maximum Lyapunov exponent of the system is calculated. The maximum Lyapunov exponent is a measure of the rate of separation (e.g., divergence or convergence) of trajectories or orbits in the reconstructed phase space (e.g., pseudo-periodic cycles of the system in which the system visits nearby points in the reconstructed phase space). Two trajectories x(n 1 ) and y(n 1 ) in the reconstructed phase space may be represented as follows:
 
 x ( n   i )={ x   0   ,x   0+τ   ,x   0+2τ   , . . . ,x   0+(d-1)τ }
 
 y ( n   i )={ y   0   ,y   0+τ   ,y   0+2τ   , . . . ,y   0+(d-1)τ }
 
     The distance between x(n 1 ) and y(n 1 ) may be represented by a value ε (e.g., a small value). The distance between the trajectories at some time Δn in the future may be represented by the following: 
     
       
         
           
             
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     The maximum Lyapunov exponent λ of the trajectories for the selected variable may be determined as follows: 
     
       
         
           
             
               
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     The above represents the maximum Lyapunov exponent λ of the trajectories as n approaches ∞. In some embodiments, a more accurate maximum Lyapunov exponent may be calculated to determine the degree of non-linearity with the system by using a time series analysis of the Lyapunov exponent. Consider a time series T and a neighborhood U(x(n 1 )) about a point x(n 1 )εT, such that each point y n1 εU(x(n 1 )) has a distance from x(n 1 ) of some small value ε. The Lyapunov exponent of the time series T can be calculated as follows: 
     
       
         
           
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     For λ&lt;0, the trajectories are periodic and the system is not chaotic. For λ=0, the system approaches a marginally stable point. If λ&gt;0, the two trajectories are diverging from one another at an exponential rate, the system is chaotic, and the degree of non-linearity of the system containing x(n 1 ) and y(n 1 ) may be estimated exponentially by the maximum Lyapunov exponent λ. For a given system, there are as many Lyapunov exponents as there are dimensions in the phase space of the system. 
     Referring still to  FIG. 1A , at step  120 , the reconstructed phase space may be generated. The reconstructed phase space may be generated by selecting one variable of the system and generating a set of d dimensional vectors of the form {right arrow over (x)}={x t , x t+τ , . . . , x t+dτ } where τ is the time delay calculated at step  105  and d is the minimum embedding dimension calculated at step  110 . The Lyapunov exponent calculated at step  115  is indicative of the rate of divergence or convergence of the nearby trajectories in the embedded phase space. In some embodiments, this divergence or convergence may be visible in a plot of the embedded phase space. In other embodiments, higher dimension phase spaces may not be plotted. 
     Referring now to  FIG. 1D , a block diagram of a control system  150  is shown according to an exemplary embodiment. System  150  may be used, for, example, to implement process  100  and/or various other exemplary methods described herein. A sensor  160  is configured to measure or receive data from a system  155  that is being analyzed. Control system  150  is configured to receive the captured sensor data (e.g., time series data) and generate the reconstructed phase space. In some embodiments, system  150  may be configured to generate an Ergodicity plot (EP) and/or ergodic transition matrix (ETM), as described in detail in the present disclosure. System  150  may be configured to output data (e.g., data relating to the reconstructed phase space, EP, and/or ETM) to a user interface  165  for viewing by a user and/or back to system  155  for use in making changes to system  155 . 
     Once the data has been embedded in the reconstructed phase space, one or more transition matrixes can be generated and used (e.g., in conjunction with an ergodicity plot) to understand mathematically the regions of phase space which are visited by the orbits of the chaotic system. One example of a transition probability matrix that represents the transition probabilities of a system is the Markov Chain. Chung, K.,  Markov Chains with Stationary Transition Probabilities,  2nd ed. New York: Springer, 1967. In some embodiments, the transition matrixes utilized herein, which may be referred to as ergodic transition matrixes (ETMs), differ from a Markov model, for example, through the use of a local Lyapunov exponent of the region represented by a member of a partitioned phase space to generate a transition measure rather than a transition probability. 
     To construct an ETM, the embedded phase space may be partitioned into a grid of partitions (e.g., equivariant partitions). In one embodiment, the partition may result in a 
               1     n       ×     1     n             
square matrix. For each cell, m i,j  of the matrix, the probability, ρ, that the orbit transitions from the ith region of the partitioned phase space to the jth region may be calculated. The local Lyapunov exponent for each partitioned region may also be calculated using methods similar to those above and using data only from the local region for which the Lyapunov exponent is being calculated. The ergodic transition measure etm(m i,j ) for the cell m i,j  may be determined based on the determined probability and local Lyapunov exponent (e.g., the Lyapunov exponent associated with the ith region) and may be calculated as follows:
 
 etm ( m   i,j )=ρλ
 
     The following Table 1 illustrates a small portion of a sample ETM (mean monthly temperature for Cedar Rapids, Iowa, USA): 
     
       
         
           
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
             
            
               
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                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0.000779 
                 0.000779 
                 0 
                 0 
                 0 
                 0 
               
               
                   
               
            
           
         
       
     
     Each ETM may contain both a row and a column for each local region in the phase space (e.g., grid). The value in each cell is a function (e.g., product) of (1) the probability that the orbit will transition from the first region (e.g., the region in the row) of the partitioned phase space to the second region (e.g., the region in the column) and (2) the rate of separation (e.g., divergence or convergence) of trajectories or orbits in the local region (e.g., the row region) of the reconstructed phase space. As can be seen in Table 1, the ergodic transition matrix is often a sparse matrix. The sparsity of the ETM helps enable state change detection, as discussed in further detail below, that can be performed with limited computational resources and in real time and near-real time computing applications. 
       FIG. 2  illustrates a surface plot  200  of the monthly mean temperature ETM according to an exemplary embodiment. Axes  205  and  210  represent coordinates (e.g., columns and rows) in the ETM. Axis  215  represents the value of the ergodic transition measure (e.g., the measure stored in a particular row and column of the ETM) stored in the cells of the ETM. Surface plot  200  represents the entire ETM for which a portion is shown in Table 1. Inspection of surface plot  200  illustrates the general sparsity of the entire ETM. 
     The ergodic transition matrix is highly effective at representing the manners by which the chaotic system traverses the embedding phase space. Further, the partitioning of the embedding phase space can be viewed as the generation of system states where each member of the partition represents a specific system state. Then, the ETM can be viewed as representational of the traversal of system states by the chaotic system and can be used for system state change and anomaly detection. 
     In some embodiments, the ETM may be a concise and manageable representation of the ergodicity of the chaotic system, or likelihood that states will or will not recur. As such, it contains useful information pertaining to the normal operating conditions of the chaotic system under consideration. It is possible to utilize the ETM to determine when the chaotic system is experiencing small state changes which could indicate the early occurrence of anomalous features (e.g. changes in the system states which indicate system degradation, possibly to failure). 
     Consider a chaotic system which is currently operating without anomalies or outliers and is not experiencing changes in the system states. Such a normal operating state for a system such as the human heart, for example, may be electrical pulses associated with a normal, healthy heartbeat. In an industrial control system example, a normal operating state may be a state in which the system is operating in a normal manner without any active alarms or slowdown in production. The attractor representing the chaotic system which is operating without anomolies and is not experiencing state changes is representational of the system&#39;s normal operating conditions. As such, it is possible to form the ergodicity plot (EP) and ergodic transition matrix (ETM) from the system&#39;s attractor of normality. The ETM generated from the system&#39;s attractor, and EP, forms a baseline pattern of trajectory traversal within the embedding phase space and may be used as a benchmark by which future traversals may be measured. This type of ETM may be defined as the ETM of normality, ETM Norm . It should be noted that, as with all machine learning techniques, the larger the training, or in this case normal, dataset available, within a reasonable limit, the more accurate the model will be. Thus, a sufficient sized sample of the normal operating states is desirous for the algorithm. The size of the dataset required to establish a sufficient ETM of normality depends upon the system under consideration and the level of acceptable accuracy of the model. 
     Given the ETM of Normality, ETM Norm , for the system under consideration, it is possible to stream new data into the algorithm and compare the ETM of the new data to the ETM Norm . To facilitate the comparison, a buffer may be maintained, in a moving window method, of the new streaming data, as single data points do not facilitate the generation of an ETM. The size of this buffer may depend on the system being analyzed. In some embodiments, a sufficient buffer size may consist of 4-5 complete orbits of the chaotic system. The ETM of the streaming data may be created each time a complete (mean) orbit has been added to the buffer. As the new orbit is added to the buffer, the previous first orbit of the buffer may be removed to facilitate maintaining a constant buffer size. The absolute difference between the ETM for the buffer, ETM Buf , and the ETM Norm  represents the degree of system state change, Δs, as follows:
 
Δ s=|ETM   Norm   −ETM   Buf |
 
     The degree of system state change, Δs, can be used to warn of systems that may be experiencing shifts in their operating states, as these phenomena will appear as differences between the buffer ETM and the ETM of normality. It is possible to set a threshold, ε, based on the degree of state change metric, Δs, such that a warning is given for state changes Δs&gt;ε. The degree of system state change metric may be sensitive to even small changes in the system operation and is therefore highly effective for state change detection. The use of the alarm threshold affords the tuning of the state change detection algorithm to reduce false positive rates (e.g., if the threshold is set too low, an increased number of false positives may result, and if the threshold is set too high, anomalies may be missed). The buffer of the detection algorithm forms a moving window as new data is streamed into the system. Therefore, it is possible to continue to compare new data as it is encountered. Additionally, it is possible to evolve the ETM Norm  to include recently encountered data, such as may be desired for systems that may break include a break in period, or period of time before a normal state of operation of the system is encountered. In such systems it may be more desirable to consider the initial data as transient to the actual ETM Norm . The evolution of the ETM Norm  and determination of the appropriate timing and amount of training data to determine the ETM Norm  is system dependent. 
     Due to the compact size of the ETM (e.g., as a real valued matrix), exemplary embodiments disclosed herein could be used in a real time monitoring system with little computational overhead. The ETM Norm  may be only a matrix of state change metrics, double precision, which may be easily stored in resident memory for even the smallest of systems. The buffer ETM may be generated with little overhead as well, making embodiments disclosed herein appropriate for real or near real-time detection systems. 
     In various embodiments, systems and methods disclosed herein may be used to monitor physiological systems and processes for potential anomalies. For example, small, early physiological changes can be indicative of future larger medical events such as a heart attack, stroke, or epileptic seizures. Using an ETM-based state change detection method, data relating to a physiological system may be monitored, and small changes may be detected that may be indicative of early stages of an anomalous condition. For example, in some embodiments, epileptic seizures may be detected as early as two or more hours prior to the event through small changes taking place in the patient&#39;s neuronal firing. By detecting medical events such as heart attacks or seizures early, measures may be taken to counteract the events and reduce the damage that may be caused by the events, in some instances saving lives. While embodiments are described herein with reference to physiological systems of the human body, systems and methods described herein may also be applied to physiological processes of other organisms, such as animals (e.g., by a veterinarian). In various embodiments, systems for monitoring physiological processes may be implemented on any type of computing platform (e.g., general purpose or special purpose). 
     Physiological data may be collected from one or more sensors configured to measure particular types of physiological data, such as electrical activity. For example, data relating to the operation of the human heart may be received from an electrocardiogram (ECG) device configured to measure electrical activity of the heart over a period of time. Data relating to the operation of the human brain, which may be used to analyze and/or predict seizures, may be received from an electroencephalogram (EEG) device. In some embodiments, the devices from which physiological data is received may be intended to be used in a relatively stationary fashion (e.g., at a hospital). In other embodiments, the devices may be configured to be portable. For example, in some embodiments, the sensors and measurement equipment may be configured to be worn by a user (e.g., as part of a biometric suit or harness) while moving and performing various tasks. In some embodiments, various sensors and/or measurement equipment may be inserted or embedded inside the body. 
     An ETM Norm  for a physiological system may be determined based on physiological data relating to a particular user. As discussed in detail in the present disclosure, the accuracy of the ETM Norm  may be improved through the use of training data. In some embodiments, data may be collected specific to a particular user and used to generate the ETM Norm . In some embodiments, a database of multiple ETMs that have been developed for various body types may be maintained, and an initial ETM Norm  may be selected for a user based on his or her physical characteristics. The determined ETM Norm  may be further refined using further data collected for the subject over time. 
     Once the ETM Norm  has been determined, additional physiological data may be streamed from the patient and used to develop further ETMs. The new ETMs may be compared to the ETM Norm  to determine whether a substantial change in the chaotic behavior of the physiological system is detected (e.g., if a state change parameter associated with the difference between the ETM and the ETM Norm  has exceeded a threshold). If a substantial change has occurred, it may be indicative of the early stages of a medical event or anomalous condition such as a heart attack or seizure. Actions may be undertaken to counteract the medical event, such as providing an alert (e.g., visual alert on a display, aural alert, etc.) that an anomalous condition has been detected (e.g., such that a doctor can address the condition) and/or administering drugs or other substances meant to counteract or fight the medical event. In some embodiments, the substance may be administered automatically by a machine. In some embodiments, an amount of the substance may be based at least in part on the state change parameter and/or the difference between the state change parameter and the threshold. In various embodiments, alerts may be provided on any type of device configured to provide information to a user, such as a LCD or CRT monitor, television, cellular phone, tablet, computer display, pager, etc. 
       FIGS. 3 and 4  can be used to illustrate the operation of an ETM-based state change detection method in the context of monitoring the human heart with a chaotic detection algorithm according to exemplary embodiments (the human heart provides an analogy to acoustic sensing systems).  FIG. 3  illustrates a graph  300  showing electrical pulses associated with the beating of a human heart.  FIG. 4  illustrates a phase space plot  400  of the electrical pulses shown in  FIG. 3 . A confusion matrix illustrating the accuracy of the chaotic detection algorithm is shown in Table 2: 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                   
                 True Anomaly 
                 False Anomaly 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 True Anomaly 
                 31 
                 1 
               
               
                   
                 False Anomaly 
                 0 
                 982,156 
               
               
                   
                   
               
            
           
         
       
     
     The confusion matrix indicates the number of true anomalies detected in the upper left cell (31), the number of false anomalies detected in the upper right cell (1), and the number of anomalies that were missed in the lower-left cell (0). The confusion matrix shown in Table 2 is exemplary and may vary according to various exemplary embodiments. As can be seen in Table 2, the ETM-based state change detection method may have a high degree of accuracy at detecting physiological anomalies and small system state shifts while maintaining low false positive rates. 
     One application of the exemplary ETM-based state change detection methods disclosed herein is in the area of cybersecurity (e.g., security of computing and information systems). Cybersecurity has risen in criticality to the level of a national security interest. Personal, corporate, military and government systems are now affected by the implications of potential attack. As such, a great amount of effort and research support has been funneled to the domain. Recently, the U.S. Defense Advances Research Projects Agency (DARPA) opened two requests for proposal (RFPs) centered on this domain. The U.S. Department of Homeland Security has also opened a RFP in the domain of cybersecurity, and the U.S. Department of Energy has indicated the need for security on the cyber systems which control the nation&#39;s power grids. Existing cybersecurity systems may be incapable of adapting rapidly enough to changes in the attack profiles due to their use of list based detection. For example, some systems utilize blacklists (e.g., lists indicating locations or resources that cannot be accessed due to known security risks) or whitelists (e.g., lists containing locations or resources that are allowed to be accessed as they are believed to pose little security risks, such that all other resources may be restricted). Further, some cybersecurity systems may result in a large number of false positives. 
     The exemplary ETM-based state change detection methods disclosed herein may be used to accurately detect cyber attacks on computing systems. Single or multiple computing systems may be monitored for any system shifts or anomalies that occur in the system. Use of such ETM-based state change detection methods may allow for more accurate detection and/or a lower number of false positives as compared to list-based detection systems. ETM-based state change detection methods may also allow for threats to be detected as they occur, rather than waiting until the threats are encountered and updating a list based on the threats. Accordingly, the damage caused by such threats may be reduced. While various embodiments are described here with respect to cybersecurity applications, it should be noted that such embodiments may be equally applicable to different applications and contexts as well. 
     In some instances, such as with data derived from an experimental time series rather than exact equations of motion, the embedding phase space can prove to be of little added assistance to the visualization of the dynamics of the system in general. This may be due, in part, to the “bird&#39;s nest” quality of the phase space often experienced in visualization of traditional embedding phase space for such data. Exemplary embodiments disclosed herein may be used to determine or model normality within deterministically chaotic systems and to better represent the dynamics in a course-grained fashion for those scenarios where the normal embedding phase space is of little assistance. 
     Various methodologies have been presented in the domain of anomaly detection and/or the analysis of specific equations of motion. See, e.g., Chandola, V., Banerjee, A., and Kumar, V., “Anomaly Detection: A Survey,” Vol. 41, No. 3, 2009; Aydin, I., Karakose, M., and Akin, E., “Chaotic-based Hybrid Negative Selection Algorithm and its Application in Fault and Anomaly Detection,” Vol. 37, No. 7, 5285-5294 2010; Geisel, T. and Thomae, S., “Anomalues Diffusion in Intermittent Chaotic Systems,” Vol. 52, No. 22, pp. 1936-1939, 1984; Xiong, W., Hu, H., Yang, Y., and Wang, Q., “Anomaly Detection of Network Traffic based on the Largest Lyapunov Exponent,” in 2 nd Intl Conf on Advanced Computer Control , Shenyang, 2010, pp. 521-585; Sheu, L., Chen, H., Chen, J., and Tam, L., “Chaotic Dynamics of the Fractionally Damped Duffing Equation,” Vol. 32, No. 4, pp. 1459-1468, 2007; Chen, J. and Chen, W., “Chaotic Dynamics of the Fractionally Damped van der Pol Equation,” Vol. 35, No. 1, pp. 188-198, 2008; Sprott, J., “A Simple Chaotic Delay Differential Equation,” Vol. 366, No. 4-5, pp. 397-402, 2007; Tykierko, M., “Using Invariants to Change Detection in Dynamical Systems with Chaos,” Vol. 237, No. 1, pp. 6-13, 2008; Chakraborty, S., Sarkar, S., and Ray, A., “Symbolic Identification and Anomaly Detection in Complex Dynamical Systems,” in 2008  American Control Conference , Seattle, Wash., 2008, pp. 2792-2797; Rao, C., Ray, A., Sarkar, S., and Yasar, M., “Review and Comparative Evaluation of Symbolic Dynamic Filtering for Detection of Anomaly Patterns,” Vol. 3, No. 2, pp. 101-114, 2009; Hively, L., Protopopescu, V., and Munro, N., “Epilepsy Forewarning via Phase Space Dissimilarity,” Vol. 22, pp. 402-409, 2005. The methodologies are largely silent on the specific detection and analysis of normal operating states of deterministically chaotic systems. This is particularly true for analysis of experimental data of a chaotic system. 
     A new visualization technique is provided that represents a significant advance over current visualization techniques for the general understanding of the dynamics of deterministically chaotic systems. Consider a system exhibiting deterministic chaos. According to some embodiments, the mathematical space that may be best used for understanding the dynamics of this system may be that of embedding phase space. Due to the nature of the system, as deterministically chaotic, there exists a wholesale revisiting of the entire phase space by the chaotic behavior. See, e.g., Sprott, J., “A Simple Chaotic Delay Differential Equation,” Vol. 366, No. 4-5, pp. 397-402, 2007. This revisiting of previous regions of the phase space is a representation of the property of ergodicity and closely follows the Poincaré Recurrence Theorem. 
     Due to this ergodicity, there exists an ability to understand the manners&#39; by which a chaotic system revisits certain regions of phase space. Eckmann et al. utilized the property of ergodicity to create recurrence plots (RP), described below. See Eckmann, J., Kamphorst, S., and Ruelle, D., “Recurrence Plots of Dynamical Systems,” Vol. 4, No. 9, pp. 973-977, 1987. The RP describes the timing of revisitation of regions of the phase space, especially for larger dimensional systems. The purpose of RP is to indicate, for a given moment in time, the times at which a phase space trajectory revisits the same area in the phase space as the point under consideration. Taken as a matrix, the RP can indicate the wholesale revisiting of the phase space. 
     Consider a point  x  (i) on the orbit describing a dynamical system in d-dimensional embedding phase space, for i=1, . . . , N. The RP is an array of pixels, in a N×N square, where a pixel is colored at (i,j) if x(j) is within some small distance ε of x(i). The RP should be viewed as a binary N×N matrix where element (i,j) is assigned 1 if x(j) is within some small distance c of x(i), otherwise it is assigned 0. This matrix is fairly symmetric with respect to the identity diagonal; however, this symmetry is not a necessary condition of the RP. The RP indicates times in which points revisit the embedding phase space in an ergodic fashion. This can be used to determine system dynamics such as quasiperiodicity. Diagonal lines parallel with a central diagonal of the RP matrix indicate periodicity within the dynamic system, and the longer these diagonal lines are, the greater the periodic component of the system. 
     Consider a periodic system, x t =sin(t), and a chaotic system, x t =sin(2πx t-1 ).  FIG. 5  illustrates a recurrence plot associated with the periodic system.  FIG. 6  illustrates a recurrence plot associated with the chaotic system. The recurrence plot does not offer a clear understanding of the normal activity of visiting regions of phase space, rather just what times points are in similar regions. It is difficult to determine clearly from the RP which regions of phase space are being revisited and at what frequency these regions are visited. Further, the RP does not give a clear indication of normal local dynamics of the system. Certain regions of chaotic systems are more divergent while other regions tend to converge. See, e.g., Eckhardt, B. and Yao, D., “Local Lyapunov Exponents in Chaotic Systems,” Vol. 65, No. 1-2, pp. 100-108, 1993. This information is not clearly portrayed in the RP. 
     A new ergodicity plot (EP) using square or fractal partitioning may be used to better represent the course grained dynamics of the system. Through an imposed fractal partitioning, coupled with the rate of divergence, the ergodicity plot affords a clearer understanding of the dynamics under consideration. 
     The EP of a system allows for better visualization of the regions of phase space being dynamically visited by the system and the states in which the dynamical system finds itself at given times.  FIGS. 7 and 8  illustrate the EP for the periodic and chaotic systems (with N=10 in  FIG. 7  and N=5 in  FIG. 8 ) modeled in the recurrence plots shown in  FIGS. 5 and 6 , respectively, according to exemplary embodiments. From  FIGS. 7 and 8 , it is clear that the systems revisit specific regions of the phase space in a recurring fashion, a fact that was not abundantly clear in the RP of the systems. Additionally, the ergodicity plot indicates the states that are traversed during the ergodic revisiting of previous states. 
     The exemplary ETM-based state change detection methods disclosed herein may be applied to detection of threats in a cybersecurity context. Specifically, such methods may be used to characterize the normality of computing systems (e.g., single computing systems, systems including multiple computing devices, etc.) with different event parameters. An ETM of normality in a cybersecurity context may be generated, for example, when the computing system or systems being analyzed are not under a cyber attack (e.g., malware or spam attack). Separate ETMs of normality may be generated for each different event parameter or condition (e.g., standalone system with no users, standalone system with users, single system with users and network connectivity, etc.). A partial ETM of normality for a single computing system with no users, according to an exemplary embodiment, is provided in Table 3: 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
             
            
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0  
                 0 
                 0 
                 0 
                 0 
                 0 
                 0.307692 
                 0.538462 
                 0.076923 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0.666667 
                 0.333333 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0.008734 
                 0.917031 
                 0.004367 
               
               
                   
               
            
           
         
       
     
     As with the partial ETM of normality shown in Table 1, the sparsity of the sample ETM of normality in the cybersecurity context is evident upon inspection of Table 3.  FIGS. 9 ,  10 , and  11  are contour plots of sample ergodic transition matrixes of normality for a single computing system with no users, a single computing system with users, and a single computing system with users and network connectivity, respectively, according to exemplary embodiments. The values on the horizontal and vertical axes (e.g., axes  805  and  810 ) represent columns and rows in the ETMs of normality. The relative values at particular cells of the ETM of normality are represented by the size and/or darkness of each point in the contour plot, with darker points representing higher values (e.g., as shown in scale  815 ). 
     As can be seen in  FIGS. 9 through 11 , there are distinct differences in the ETM of normality for the three variants of the single pc system. The ETM of normality for these systems, in one embodiment, is based on a 15×15 partition of the phase space and the actual ETM being stored in a comma delimited file occupies approximately 100 Kb of disk space. Therefore, the ETM is highly effective at representing the dynamics of the cyber system in an extremely compact footprint. 
     A mathematical understanding of the difference between a single pc system without users and one with users is accessible through the ETM of normality for each of these systems. Calculating the absolute difference between the ETM of normality for a non-user system and the ETM of normality of a user system generates a matrix which describes, mathematically, the differences in the dynamics of the two systems.  FIG. 12  illustrates this point though the contour plot of the absolute differences between the ETM of Normality for the single pc system without users (e.g., as shown in  FIG. 9 ) and the ETM of Normality for the single pc system with users (as shown in  FIG. 10 ) according to an exemplary embodiment. 
     Referring now to  FIG. 13 , another contour plot is shown illustrating an ETM of normality for an intrusion detection dataset (e.g., a DARPA intrusion detection dataset consisting of Internet Protocol (IP) traffic capture) according to an exemplary embodiment. The ETM of normality was generated using a first week&#39;s worth of data, which was reported as clean and free of anomalies. The second week&#39;s dataset was used as the testing set, streamed into the ETM-based anomaly detection algorithm. The buffer window for the dataset was generated repeatedly and anomalies were detected. The anomalies detected were then compared with a set of “tagged” anomalies in the intrusion detection dataset. A confusion matrix illustrating the results is shown in Table 4: 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 4 
               
               
                   
                   
               
               
                   
                   
                 True Anomaly 
                 False Anomaly 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 True Anomaly 
                 42 
                 8 
               
               
                   
                 False Anomaly 
                 1 
                 12,432,000 
               
               
                   
                   
               
            
           
         
       
     
     As can be seen from Table 4, the ETM anomaly detection algorithm correctly identified 42 of the 43 tagged anomalies within the IP traffic. The ETM anomaly detection algorithm also erroneously identified eight data points as anomalous. As discussed above, in some embodiments, user-adjustable parameters may be used to reduce false positive rates. 
     In some embodiments, the ETM-based state change detection methods described with respect to various exemplary embodiments may be modified for use with a phase space having an embedding dimension of d, where d may be any dimension (e.g., greater than two, such as three, five, ten, etc.). The time-delay reconstructed phase space may be embedded in a d dimensional hypercube (e.g., a d dimensional analogue of a square, with d=2, or a cube, with d=3), forming a new space, termed a bounding space, from which an ergodic transition matrix of the ergodic orbits of the system may be generated and used to detect changes in the state of the system. 
     In some embodiments, a d dimensional hypercube may be created, where d is the embedding dimension of the time-delay phase space. The hypercube contains the time-delay phase space as well as regions which may not be visited by the system under consideration. This hypercube is termed the bounding space of the ergodic orbits of the system. A partition (e.g., possibly a fractal partition) is imposed on the bounding space, and the data points within the bounding space are mapped to the centroid of the cell of the partition which contains that point. This mapping forms the course grained representation of the ergodicity of the system under consideration. The transition from one partition cell to another is viewed as a traversal of the ergodic orbits of the system in a course grained context, or a transition between system states for the chaotic system. These transitions can be transformed numerically into a sparse two dimensional matrix which is used to detect changes in the chaotic system in a computationally simple manner, as described above with respect to various exemplary embodiments. 
     Let Ψ be the time-delay embedding phase space of a chaotic system, S (e.g., in a Euclidean space), with time-delay and embedding dimension d. Let s i (t)εS for I=1, 2, . . . , n be a discrete measurement of S at a given time t. Due to the chaotic nature of S, there exists an inherent ergodicity of S over Ψ. As such, orbits are formed in a quasi-periodic fashion in S, which are used for monitoring of the system for any deviation from norm of S. Let E be an n-dimensional hypercube which contains Ψ, as well as other points not necessarily in the system S. Thus, E forms the bounding space of Ψ. Then, we can impose on E a, possibly fractal, partition P and map the points e i1, i2, . . . , in εP (where i j  is the jth cell of the ith dimension of the hypercube) to the centroid of the cell p i1, i2, . . . , in εP which contains the point e i1, i2, . . . , in . 
     The use of the hypercube defined bounding space affords the detection of both parametric and population-based changes to the system. The detection of parametric changes is represented by changes in the time-delay phase space. See, e.g., Crutchfield, J. and Young, K., “Inferring Statistical Complexity,” Vol. 63, No. 2, pp. 105-108, 1989. The ε-machine of Crutchfield and Young may not detect population-based changes in which the population (time-series data) has experienced a system shift while maintaining similar probabilities of transition. This type of deviation may be captured in the hypercube defined bounding space. 
     The partition, P, imposed on the bounding space is useful for the creation of a sparse matrix of transitions of the ergodic orbits of the system. From P, we can form a 2 dimensional matrix M of size 
               ∏     k   =   1     n     ⁢           ⁢       i   k     ×     ∏     k   =   1     n             
i k , where i k  is the size of the kth dimension of the hypercube E. We can then assign to each cell, m i,j  εM, the probability, ρ, of transitioning from e i1, i2, . . . , in  to e j1, j2, . . . jn  with consideration given to the exponential divergence or convergence of the system at the transition from e i1, i2, . . . , in  to e j1, j2, . . . , jn . The probability value assigned to cell, m i,j  εM may be simply the number of times in E the transition from e i1, i2, . . . , in  to e j1, j2, . . . , jn , is encountered, normalized by the total number of single step ergodic transitions in E with respect to the local divergence or convergence of the system at the transition point (which may be represented by the local Lyapunov exponent, λ). In some embodiments, the probability value may be calculated as follows:
 
     
       
         
           
             
               m 
               
                 i 
                 , 
                 j 
               
             
             = 
             
               
                 ρ 
                 ⁡ 
                 
                   ( 
                   
                     
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                         ⋯jn 
                       
                     
                   
                    
                 
                 
                    
                   
                     ⅇ 
                     trans 
                   
                    
                 
               
             
           
         
       
     
     The resulting matrix may be referred to as the ETM of E. Given an ETM M t0  for the hypercube E containing the time-delay embedding of the system S at time t 0  and an ETM M t1  for the hypercube E containing the time-delay embedding of the system Sat time t 1 , the deviation, δ, of M t1  from M t0  can be calculated as follows: 
     
       
         
           
             
               δ 
               ⁡ 
               
                 ( 
                 
                   
                     M 
                     
                       t 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                   
                   , 
                   
                     M 
                     
                       t 
                       ⁢ 
                       
                           
                       
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                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
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               ⁢ 
               
                   
               
               ⁢ 
               
                  
                 
                   
                     
                       m 
                       
                         i 
                         , 
                         j 
                       
                     
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                       ( 
                       
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                       m 
                       
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                         , 
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                  
               
             
           
         
       
     
     The deviation of the system represents the amount of shift, possibly as a direct outlier, experienced in the system S between times t 0  and t 1 . Given a user defined threshold, ε, the deviations may be used to monitor the system to detect system shifts or anomalies which arise through the deviation, δ, of M t1  from M t0 . This ETM deviation methodology can operate as a detection system such that an anomaly or shift would be reported for conditions in which δ&gt;ε. 
     The creation of the ETM forms a sparse two dimensional matrix (e.g., due to the size of the bounding hypercube in relation to the time-delay phase space). As such, the computation of the deviations, δ, between matrices is computationally simple. In various embodiments, the ETM detection methodology may be used in a real-time or near real-time environment to perform system shift and anomaly detection even in systems where the minimum embedding dimension is greater than two dimensions. 
     In one exemplary embodiment, the operation of exemplary systems and methods described herein may be demonstrated using a standard Henon Map (shown below, with a=1.4 and b=0.3) to which small changes are imposed on the system:
 
 x   n+1   =y   n +1 −ax   n   2  
 
 y   n+1   =bx   n  
 
     The ETM of normality for the Henon Map may be created through the use of the first 4500 iterations of the map, with the first 500 of these iterations removed as possible transients to ensure the map is on the attractor. The partition of the hypercube used in this exemplary embodiment is a square 20×20 partition. Deviations are manually imposed on the Henon Map in the next 4000 iterations through a change made to the x variable at a single, randomly selected, interval. The imposed deviation results in small (e.g., less than 10%) additive noise on the single instance of the x variable of the system. This single instance of additive noise propagates throughout the remainder of the iterations due to sensitivity on initial conditions. The deviation described here results in changes to the population of the system rather than parametric changes. The ETM for this second dataset is then created and compared to the ETM of normality. 
       FIGS. 14 and 15  illustrate contour plots associated with the ETM of normality and the ETM of the second dataset, respectively, according to exemplary embodiments. It may be difficult to visualize from  FIGS. 14 and 15  that there is a detectable change in the datasets. To facilitate observation of the change, the absolute differences between the ETM of normality and the ETM of the second dataset for the Henon Map are shown in the contour plot of  FIG. 16 . 
     As can be seen in  FIG. 16 , the ETM-based anomaly and shift detection methodology applied in this exemplary embodiment is capable of detecting small changes to the system. The contour plot of  FIG. 16  indicates that the system did experience a change in its population values. An analysis of these absolute differences resulted in a total absolute difference of 0.214, with a standard deviation of 9.85×10 −4  for the differences, a median difference of 6.97×10 −4 , and an average difference of 1.34×10 −6 . The average transition probability difference indicates that the ETM detection methodology applied in the exemplary embodiment is capable of detecting even extremely small shifts in the system (e.g., a population-based shift). 
     Referring now to  FIGS. 17 through 19 , further exemplary embodiments illustrating the Henon Map are shown.  FIG. 17  illustrates a phase space plot  1700  of the Henon Map.  FIG. 18  illustrates a contour plot associated with the ETM of normality for the Henon Map illustrated in  FIG. 17 .  FIG. 19  illustrates an ergodicity plot associated with the Henon Map illustrated in  FIG. 17 . Inspection of  FIGS. 17 through 19  illustrates the relationship between the ergodicity plot and the other types of plots that may be used to represent the Henon Map and provides an understanding of how the ergodicity plot is used to illustrate state transitions in the Henon Map. 
     The Henon Map discussed above with respect to  FIGS. 14 through 16  is a standard Henon Map that is free from noise. Noise is present in many applicational situations. In some systems, noise reduction techniques may be employed prior to attempted system analysis and/or anomaly detection. See, e.g., Schreiber, T., “Extremely Simple Nonlinear Noise-Reduction Method,” Vol. 47, No. 4, pp. 2401-2404, 1993; Grassberger, P., Hegger, R., Kantz, H., Schaffrath, C., and Schreiber, T., “On Noise Reduction Methods for Chaotic Data,” Vol. 3, No. 2, pp. 127-142, 1993; Lang, M., Gua, H., Odegard, J., Burrus, C., and Wells, R., “Noise Reduction using an Undecimated Discrete Wavelet Transform,” Vol. 3, No. 1, pp. 10-12, 1996. 
     Referring now to  FIGS. 20 and 21 , it is illustrated according to exemplary embodiments that an ETM-based state change detection method may be used to accurately detect changes even in the presence of noise. The Henon Map reflected in  FIGS. 20 and 21  is identical to that discussed with respect to  FIGS. 14 through 16  with the exception of the addition of random Gaussian noise, with zero mean and a small standard deviation, added at each iteration.  FIG. 20  illustrates the time-delay phase space plot  2000 , in two dimensions, of the first 4500 iterations, with the first 500 removed as transients, of the Henon Map with added noise. 
     From  FIG. 20 , it can be seen that the ergodicity of the noisy Henon Map is similar to the standard Henon Map and that the overall attractor topology is maintained, with minimal variations due to the additive noise. The ETM of normality for the noisy Henon Map was created as described above for the standard Henon Map. As before, a single, randomly selected, iteration of the second 4000 iterations was changed, which then impacted the remaining iterations and an ETM of that dataset was created.  FIG. 21  illustrates, with a contour plot, the absolute differences of these two ETMs, effectively illustrating that the ETM detection methodology is capable of detection of system shifts, and anomalies, even in the presence of noise. 
     The contour plot of the absolute differences of the transition probabilities of the two ETMs for the noisy Henon Map indicates that the system did experience a change in its population values. An analysis of these absolute differences resulted in a total absolute difference between the two ETMs of 0.237, a standard deviation of 9.56×10 −4  for the differences, a median difference of 7.50×10 −4 , and an average difference of 1.48×10 −6 . The average difference also indicates that the ETM detection applied in this exemplary embodiment is capable of detecting extremely small shifts in the system, even in the presence of noisy data. 
     Referring now to  FIG. 22 , a block diagram of a processing circuit  2200  that may be used to implement one or more of the exemplary methods described herein is shown according to an exemplary embodiment. Processing circuit  2200  includes at least one processor  2205 , which may be any general purpose or special purpose processor (e.g., FPGA, CPLD, ASIC, etc.). In some embodiments, processor  2205  may be chosen based on the type of application with which processing circuit  2200  is being utilized. Processing circuit  2200  also includes a memory  2210  that may be any type of computer or machine-readable media (e.g., hard drive, flash memory, ROM, RAM, EEPROM, CD, DVD, etc.). Processing circuit  2200  may also include an input interface  2215  for receiving input data relating to the system or process being analyzed and an output interface  2220  for outputting signals (e.g., data related to an EP and/or ETM). In some embodiments, processing circuit  2200  may be networked to one or more other processing circuits. In some embodiments, processing circuit  2200  and one or more other processing circuits may be employed to perform various operations (e.g., in a distributed computing environment). 
     Memory  2210  may be configured to store one or more modules having instructions that are executable by processor  2205 . For example, memory  2210  may include a phase space embedding module  2225  configured to generate a reconstructed phase space and embed time series data for a system being predicted within the reconstructed phase space. Once the reconstructed phase space has been generated and the data has been embedded in the reconstructed phase space, an ETM generation module  2230  may be configured to generate one or more ETMs and/or EPs (e.g., according to various exemplary embodiments described herein). 
     As discussed above, various exemplary embodiments relate to systems and methods for monitoring the state of physiological systems. Referring now to  FIGS. 23 through 30 , systems and methods relating to monitoring the condition of the human heart based on electrocardiogram (ECG) data are shown and described according to various exemplary embodiments. While the physiological system discussed with reference to  FIGS. 23 through 30  is the circulatory system and the operation of the heart, various features discussed may also be relevant to monitoring other types of physiological systems, such as the brain (e.g., for strokes or epileptic seizures). 
     The use of electrocardiograms (ECG) has become one of the most important diagnostic tools a cardiologist has at their disposal. Signals within an ECG provide two major types of information. By measuring time intervals in the ECG, the duration of the electrical wave crossing the heart can be determined. See, e.g., Anuradha, B., &amp; Veera Reddy, V. (2008), ANN For Classification of Cardiac Arrhythmias 3(3). It is possible to determine whether the electrical activity is normal or abnormal. Second, by measuring the amount of electrical activity through the heart muscle, a cardiologist may be able to discover if parts of the heart are too large or are overworked. 
     Systems have been developed that are capable of automatically classifying an ECG signal as normal or arrhythmic. See, e.g., Palodeto, V., &amp; Marques, J. (2006), Methodology for Classification and Analysis of Neonate and Adult ECG,  Proc. of the IFMBE World Congress on Medical Physics and Biomedical Engineering , pp. 1214-1217; Yaghouby, F., &amp; Ayatollahi, A. (2009), An Arrhythmia Classification Method Based on Selected Features of Heart Rate Variability Signal and Support Vector Machine-based Classifier,  Proc. of the IFMBE World Congress on Medical Physics and Biomedical Engineering , pp. 128-1931, Munich; Acharya, U., Bhat, P., Iyengar, S., Roa, A., &amp; Dua, S. (2003), Classification of Heart Rate Data using Artificial Neural Network and Fuzzy Equivalence Relation,  Pattern Recognition,  36, 61-68; Ramirez, E., Castillo, O., &amp; Soria, J. (2010), Hybrid System for Cardica Arrhythmia Classification with Fuzzy K-Nearest Neighbors and Multi-layer Perceptrons combined by a Fuzzy Inference System,  The  2010  Intl. Joint Conf on Neural Networks , pp. 1-6, Barcelona. Many of the systems, such as neural networks and support vector machines, are from in the stochastic domain. There may be inherent drawbacks to the implementation of such methods. Stochastic methods such as artificial neural networks and support vector machines are known as black box methodologies; they may be accurate, but it is difficult to understand or describe which features of the data are influencing the classifications. Further, these methods can be computationally intense relative to the size of the dataset. These drawbacks may be reduced or eliminated by using a method outside of the stochastic domain to classify ECG signals. 
     One non-stochastic approach to classifying ECG signals is to consider the ECG signal as a time series which is representative of an ergodic, deterministically chaotic, system. Research has been conducted illustrating the degree of deterministic chaos involved in the beating of the human heart. For example Bernaola-Galvan et al discussed the invariance of scale, often attributed to fractality, in the human heart. See, e.g., Bernaola-Galvan, P., Ivanov, P., Nunes Amaral, L., &amp; Stanley, H. (2001), Scale Invariance in the Nonstationarity of Human Heart Rate,  Physical Review Letters,  87(16). Guzzetti et al reported chaotic metrics that were discovered in the HRV signal of normal and heart-transplanted patients. See, e.g., Guzzetti, S., Signorini, M., Cogliati, C., Mezzetti, S., Porta, A., Cerutti, S., et al. (1996), Non-linear Dynamics and Chaotic Indices in Heart Rate Variability of Normal Subjects and Heart-Transplanted Patients,  Cardiovascular Research,  31(3), 441-446. Goldberg et al described human physiology from the standpoint of chaos and fractals. See, e.g., Goldberg, A., Rigney, D., &amp; West, B. (1990), Chaos and Fractals in Human Physiology,  Scientific American,  262(2), 42-49. 
     Various exemplary systems and methods that utilize such a chaos-based approach may consider the ergodicity and sensitivity to initial conditions of the chaotic human heart in performing classification. Such systems and methods may avoid the inherent black box problems of stochastic methods through an ability to visualize heart rhythms in an embedding phase space. Further, the methods may be computationally simple, allowing for the possibility of embedding such a system in a piece of hardware which is limited in its computational power. 
     The ECG data may be considered as time-series data and embedded in a vector phase space according to any of the various exemplary methods disclosed herein (see, e.g., process  100 ). In some embodiments, the raw signal from an ECG dataset may be considered to be a scalar time series. To recreate the attractor in the vector phase space for these data, a single channel of the ECG data may be selected to use as the scalar time-series and a time-delay embedding process may be applied on that variable. According to the Takens Delay Embedding theory, as long as the single variable selected is of importance to the overall dataset it will represent the true dynamics of the system, regardless of which variable it is. The exemplary embodiments illustrated in  FIGS. 23 through 30  utilize the MLII channel from the ECG data available in the MIT-BIH data repository, although various ECG-related exemplary embodiments may utilize any type of ECG data.  FIG. 23  presents a sample graph  2300  of the MLII data and  FIG. 24  displays a vector phase space plot  2400  for this data according to exemplary embodiments. The parameters chosen to recreate the vector phase space plot  2400  in  FIG. 24  were a time delay of 5 and an embedding dimension of 3. 
     In the exemplary phase space plot  2400 , there exists a definite quasi-periodicity about the data. While it may be possible to visualize the quasi-periodicity in the raw scalar data, it is not clear which regions of phase space where repeatedly revisited in the ergodic system. The embedding phase space allows for better resolution in regards to the systems ergodicity. Further, phase space plot  2400  illustrates that there may be information to be gained from the heart rate variability (HRV) of this dataset in relation to the entire dataset. 
     Once the ECG data has been embedded in the phase space, the data can be used to form an ergodic transition matrix (ETM) according to various exemplary methods described herein. A bounding space for the phase space can be formed and split into a plurality of cells. By mapping the phase space points to the bounding space partition centroids, an ergodicity plot can be formed.  FIG. 25  illustrates an ergodicity plot  2500  for the phase space shown in  FIG. 24  according to an exemplary embodiment. Ergodicity plot  2500  is based on a 15 by 15 square partition of phase space plot  2400 . One or more ETMs may also be formed based on the data embedded in the phase space and the bounding space as described in detail herein with respect to various exemplary embodiments. 
     The ETMs may be used in classification of the ECG signals to determine anomalies in the operation of the heart. Due to the inherent determinism of the normally beating human heart, normal rhythm patterns may be established. Deviations from these patterns can be considered as anomalous, with some types referred to as arrhythmic. It is possible to classify rhythm patterns into various classes based upon the deviation from normality. ETMs can be highly sensitive to changes in a chaotic system and may be used to effectively classify cardiac rhythms based on ECG data.  FIG. 26  illustrates a contour plot  2600  corresponding to an ETM of a heart rhythm that is known to be normal according to an exemplary embodiment.  FIG. 27  illustrates a contour plot  2700  corresponding to an ETM of a heart rhythm that is known to be arrhythmic according to an exemplary embodiment. Comparison of plots  2600  and  2700  reveals that there are distinct differences between the two ETMs that may be used to effectively classify the cardiac rhythms. 
     In order to classify the heart rhythms, the chaotic signal may be collected into files. In the case of ECG data, each file may represent a human heart rhythm for a given period of time. The transformation to embedding phase space may be performed on the data in each file. In some embodiments, the parameters for the transformation to embedding phase space may be the same for each file and the most appropriate embedding parameters may be selected with regard to all of the files rather than each file separately. Once the transformation to embedding phase space has been performed, an ETM may be created for each of the files. 
     Classification of the datasets may be performed using the ETMs and a k-nearest neighbor classifier. The ETMs may become inputs to the classifier. The k-nearest neighbor classifier may utilize a training set for generation of the classification schema. This task may be accomplished through the creation of ETMs for a selection of representative files for each class label. In some embodiments, an equal number of files from each class may be used to generate the training ETMs. This may avoid a class imbalance issue that could cause problems later in the classification process. In some embodiments, the ETMs may be stored as files with the class label attached to the file name. As new data is acquired, classification may be performed through comparison of the ETM of the new data to the training set of ETMs that was previously generated. 
     In some embodiments, comparison of the ETM of newly acquired data to the set of training ETMs may be performed using the absolute Euclidean difference between the ETM of the new data and each of the training ETMs. The possibility exists for a single training ETM to be a very good match for the ETM of the new data but not be from a matching class, thus misclassifying the new data. To avoid such a scenario, in some embodiments, the mean absolute difference for each class may be used as the classifying metric. The generation of the mean absolute difference for a given class, according to one exemplary embodiment, is as given in the following equation, where C is the set of training ETMs in the given class, Ei is the ith training ETM in C, E is the ETM of the new data, and N is the partition size used to generate the ETMs: 
     
       
         
           
             
               μ 
               C 
             
             = 
             
               
                 1 
                 
                    
                   C 
                    
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     i 
                     ∈ 
                     C 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         0 
                       
                       N 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         N 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 E 
                                 jk 
                               
                               - 
                               
                                 E 
                                 jk 
                                 i 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The ETM for the new data may receive a class label of the class whose mean absolute difference is the smallest, as expressed in the following:
 
class=argmin iεC μ C  
 
     According to various exemplary embodiments, the classification methods presented herein may be computationally simple (e.g., given that an initial training set has been generated). The generation of an ETM from newly acquired data may also be computationally simple. Additionally, a substantial pitfall of the stochastic methods may be avoided—the black box effect. In various embodiments, analysts performing classification using the ETM methodology can readily display the attractor, Ergodicity Plot and/or plots of the ETM to explain the class label assignment, providing a transparency that may be beneficial (e.g., from an explanation and visualization standpoint). 
     A sample implementation of systems and methods disclosed herein in the context of classification of heart rhythms is now provided with reference to  FIGS. 28 through 30  according to exemplary embodiments. The description of various features of the exemplary embodiments are provided for the purposes of illustration and example only; the types of data, data format, substantive data, results, etc. may vary according to various other exemplary embodiments. 
     The data used in the sample implementation was ECG data from the MIT-BIH data repository. The data consists of ECG recordings of approximately 30 minutes in length. Each file was assigned one of three class labels, based upon the types of rhythms contained in the file. The three classes used were arterial fibrillation, ventricular arrhythmia, or normal sinus. Each file included the timestamp of a row in the file and two measurements for that time—MLII and V1. For the creation of the embedding phase space, the MLII data was used as the scalar signal. 
     Preprocessing of the data was performed to obtain the best training set possible. The data contained files with heart rhythms which were paced. These files were not used as training data to avoid erroneous class labeling. Additionally, files which contained arrhythmias not attributable to Arterial Fibrillation or a form of Ventricular Arrhythmia were not used. In total, 38 files were used from the MIT-BIH dataset. Of these files, 8 represented signals which contained Arterial Fibrillation, 17 files represented normal sinus rhythm signals, and 13 files represented signals which contained some form of Ventricular Arrhythmia. A training set of ETMs representing 6 files from each class was generated.  FIGS. 28 through 30  display ergodicity plots for a sample file from each class label according to exemplary embodiments. Specifically,  FIG. 28  includes an ergodicity plot  2800  for a sample file corresponding to arterial fibrillation,  FIG. 29  includes an ergodicity plot  2900  for a sample file corresponding to a normal sinus pattern, and  FIG. 30  includes an ergodicity plot  3000  for a sample file corresponding to ventricular arrhythmia, according to exemplary embodiments. There are distinct differences between the three ergodicity plots  2800 ,  2900 , and  3000 . This corresponds to differences in the ETMs representing each class. 
     Classification of the MIT-BIH dataset was performed in two sessions. The first session classified the raw ECG signal using ETM-based methods as described above according to various exemplary embodiments. The second session classified the HRV of the ECG signals using the same ETM-based methods. Each of the sessions used two classification strategies as well. In one strategy, the ETM training set was generated using 6 files from each class, as described above. The remaining files were then used as test files. The second strategy iteratively used all but a single file as training files, with the remaining file being used as the test file. The second strategy is similar to a 10 fold cross validation that is often performed in data mining. See, e.g., Witten, I., &amp; Frank, E. (2005),  Data Mining: Practical Machine Learning Tools and Techniques , San Francisco: Morgan Kaufmann. 
     The raw ECG signal (MLII data) was used to generate the ETMs for the first session of classification. ETMs were generated for all 38 data files. The first classification strategy produced an overall classification accuracy of 67.22% when classifying the data into the three classes discussed above. The sensitivity and specificity of the exemplary algorithm used in the present sample implementation can be described by considering all class labels which are not “Normal Sinus” as an arrhythmic signal. When this consideration is given, the exemplary algorithm achieved the sensitivity and specificity as given in Table 5. Table 6 displays the classification accuracy for each of the individual classes using the exemplary algorithm. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 5 
               
               
                   
                   
               
             
            
               
                   
                 Sensitivity 
                 0.663 
               
               
                   
                 Specificity 
                 0.79 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
             
               
                   
                 TABLE 6 
               
               
                   
                   
               
               
                   
                 Class 
                 Accuracy 
               
               
                   
                   
               
             
            
               
                   
                 Arterial Fibrillation 
                   75% 
               
               
                   
                 Normal Sinus 
                   76% 
               
               
                   
                 Ventricular 
                 50.67% 
               
               
                   
                 Arrhythmia 
               
               
                   
                   
               
            
           
         
       
     
     The second classification strategy, using all but one file as training data, produced an overall classification accuracy of 63.16% when classifying the data into the three classes. Again, sensitivity and specificity of the exemplary algorithm can be described as for the first classification strategy. When this consideration is given, the exemplary algorithm achieved the sensitivity and specificity given in Table 7. Table 8 displays the classification accuracy for each of the individual classes using the exemplary algorithm. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 7 
               
               
                   
                   
               
             
            
               
                   
                 Sensitivity 
                 0.642 
               
               
                   
                 Specificity 
                 0.765 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
             
               
                   
                 TABLE 8 
               
               
                   
                   
               
               
                   
                 Class 
                 Accuracy 
               
               
                   
                   
               
             
            
               
                   
                 Arterial Fibrillation 
                   75% 
               
               
                   
                 Normal Sinus 
                 76.47% 
               
               
                   
                 Ventricular 
                 38.46% 
               
               
                   
                 Arrhythmia 
               
               
                   
                   
               
            
           
         
       
     
     The results reported in the literature most often reflect classification performed on HRV rather than the raw ECG data. Comparison of the results of the ETM algorithm with those in the literature may be more easily accomplished using HRV data. 
     The raw ECG signal (MLII data) was used in the second classification session as well. This time, however, the signals were filtered to represent only R to R intervals, also known as HRV. There is some dispute regarding the existence of deterministic chaos within HRV. This is not of concern overall to the exemplary ETM methods utilized with the sample implementation, as even a non-chaotic signal can be transformed into a vector space such as embedding phase space. The raw ECG signal was transformed into HRV. That data was then transformed into the embedding phase space and into ETMs for classification. 
     Similar to the first classification session, the second session utilized the same two classification strategies. The first strategy produced an overall classification accuracy of 96.3%, which was an improvement over the first classification session. Table 9 displays the sensitivity and specificity for the second classification session with the first classification strategy. Table 10 displays the classification accuracy for each of the individual classes. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 9 
               
               
                   
                   
               
             
            
               
                   
                 Sensitivity  
                 1.0 
               
               
                   
                 Specificity 
                 96.59 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
             
               
                   
                 TABLE 10 
               
               
                   
                   
               
               
                   
                 Class 
                 Accuracy 
               
               
                   
                   
               
             
            
               
                   
                 Arterial Fibrillation 
                   100% 
               
               
                   
                 Normal Sinus 
                 93.75% 
               
               
                   
                 Ventricular 
                 95.15% 
               
               
                   
                 Arrhythmia 
               
               
                   
                   
               
            
           
         
       
     
     The second classification strategy produced an overall classification accuracy of 94.6% when classifying the data into the three classes. Again, sensitivity and specificity of the exemplary algorithm can be described as for the first classification strategy. When this consideration is given, the exemplary algorithm achieved the sensitivity and specificity given in Table 11. Table 12 displays the classification accuracy for each of the individual classes using the exemplary algorithm. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 11 
               
               
                   
                   
               
             
            
               
                   
                 Sensitivity  
                 1.0 
               
               
                   
                 Specificity 
                 93.75 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
             
               
                   
                 TABLE 12 
               
               
                   
                   
               
               
                   
                 Class 
                 Accuracy 
               
               
                   
                   
               
             
            
               
                   
                 Arterial Fibrillation 
                   100% 
               
               
                   
                 Normal Sinus 
                 93.75% 
               
               
                   
                 Ventricular 
                  92.3% 
               
               
                   
                 Arrhythmia 
               
               
                   
                   
               
            
           
         
       
     
     The ETM methodology using the HRV dataset produced a substantially higher overall accuracy and more desirable sensitivity and specificity metrics. The main reason for this discrepancy is the amount of noise that is present in the raw ECG signal compared to that in the HRV signal. Some amount of noise is generally present in any sensed system, and less noise will generally result in a better analysis and classification. 
     Various ETM-based methods may be highly accurate when used to classify physiological data, particularly when classifying HRV data. The accuracy of ETM-based methods may be comparable to that of stochastic classification methods, and ETM-based methods may overcome various disadvantages of stochastic methods such as those discussed herein. 
     The sample implementation provided above utilized two classification strategies for both classification sessions. The results of these strategies consistently indicated that, in the particular exemplary embodiments discussed with reference to  FIGS. 28 through 30 , there was better classification accuracy when a smaller training set was used over the use of the all-but-one strategy. This is due to the variability of the ETMs within the individual classes. The smaller training sets contained less variability than the larger ones, and the classifier had less difficulty determining the class label of new data. There may be a balance between too few and too many training files to achieve the most favorable results in various embodiments. 
     The exemplary algorithm discussed with respect to  FIGS. 28 through 30  exhibited high classification accuracies for normal sinus rhythm and arterial fibrillation as compared to those experienced for the Ventricular Arrhythmia class. The main reason behind this is that there were a number of conditions which combined to make the Ventricular Arrhythmia class whereas the Normal Sinus and Arterial Fibrillation classes were single condition classes. Conditions such as ventricular bigeminy, ventricular trigeminy, and ventricular tachycardia were included in the ventricular arrhythmia class. These conditions were combined in order to increase the class size to a workable level. The arterial fibrillation exhibited the highest classification accuracy. This may be due to the limited number of examples for arterial fibrillation. The amount of variability in the ETMs of the arterial fibrillation class was less than the variability in the other two classes. However, as discussed above, the exemplary embodiments discussed with respect to  FIGS. 28 through 30  are only some examples associated with one possible implementation of ETM-based methods disclosed herein, and different results may be obtained using other exemplary embodiments. 
     According to one exemplary embodiment, a method of detecting a change in the state of a system is provided. The method comprises embedding time series data relating to the system within a reconstructed phase space. The method further comprises partitioning the reconstructed phase space into a plurality of regions. The method further comprises generating a matrix having a plurality of cells, wherein the matrix comprises a row and a column for each of the plurality of regions, and wherein a value stored in each cell is based on a probability that the system will transition from a first region associated with the cell to a second region associated with the cell and a rate of separation of trajectories of the embedded data within at least one of the first region and the second region. In some embodiments, the method may comprise generating a first matrix based on a first dataset associated with a normal operating condition of the system, generating a second matrix based on a second dataset, and detecting a change in a state of the system by comparing the first matrix and the second matrix. 
     According to another exemplary embodiment, a method of detecting a potential security threat on a computing system is provided. The method comprises embedding time series data relating to the computing system within a reconstructed phase space. The method further comprises partitioning the reconstructed phase space into a plurality of regions. The method further comprises generating a first matrix having a plurality of cells, wherein the matrix comprises a row and a column for each of the plurality of regions, and wherein a value stored in each cell is based on a probability that the system will transition from a first region associated with the cell to a second region associated with the cell and a rate of separation of trajectories of the embedded data within at least one of the first region and the second region. The first matrix is generated using a first dataset associated with a normal operating condition of the computing system in which the computing system is not under attack from a security threat. The method further comprises generating a second matrix based on a second dataset and detecting the potential security threat on the computing system by comparing the first matrix and the second matrix. In some embodiments, detecting the potential security threat comprises determining when a difference between the first matrix and the second matrix exceeds a threshold. 
     According to another exemplary embodiment, a method of detecting a change in the state of a system is provided. The method comprises embedding time series data relating to the system within a reconstructed phase space having a D-dimensional embedding dimension. The method further comprises enclosing the reconstructed phase space in a D-dimensional hypercube. The method further comprises partitioning the hypercube into a plurality of regions. The method further comprises generating a matrix having a plurality of cells, wherein the matrix comprises a row and a column for each of the plurality of regions, and wherein a value stored in each cell is based on a probability that the system will transition from a first region associated with the cell to a second region associated with the cell and a rate of separation of trajectories of the embedded data within at least one of the first region and the second region. In some embodiments, the method may comprise generating a first matrix based on a first dataset associated with a normal operating condition of the system, generating a second matrix based on a second dataset, and detecting a change in a state of the system by comparing the first matrix and the second matrix. 
     The disclosure is described above with reference to drawings. These drawings illustrate certain details of specific embodiments that implement the systems and methods and programs of the present disclosure. However, describing the disclosure with drawings should not be construed as imposing on the disclosure any limitations that may be present in the drawings. The present disclosure contemplates methods, systems and program products on any machine-readable media for accomplishing its operations. The embodiments of the present disclosure may be implemented using an existing computer processor, or by a special purpose computer processor incorporated for this or another purpose or by a hardwired system. No claim element herein is to be construed under the provisions of 35 U.S.C. §112, sixth paragraph, unless the element is expressly recited using the phrase “means for.” Furthermore, no element, component or method step in the present disclosure is intended to be dedicated to the public, regardless of whether the element, component or method step is explicitly recited in the claims. 
     As noted above, embodiments within the scope of the present disclosure include program products comprising machine-readable media for carrying or having machine-executable instructions or data structures stored thereon. Such machine-readable media can be any available media which can be accessed by a general purpose or special purpose computer or other machine with a processor. By way of example, such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium (e.g., non-transitory medium) which can be used to carry or store desired program code in the form of machine-executable instructions or data structures and which can be accessed by a general purpose or special purpose computer or other machine with a processor. Combinations of the above are also included within the scope of machine-readable media. Machine-executable instructions comprise, for example, instructions and data which cause a general purpose computer, special purpose computer, or special purpose processing machine to perform a certain function or group of functions. 
     Embodiments of the disclosure are described in the general context of method steps which may be implemented in one embodiment by a program product including machine-executable instructions, such as program code, for example, in the form of program modules executed by machines in networked environments. Generally, program modules include routines, programs, objects, components, data structures, etc., that perform particular tasks or implement particular abstract data types. Machine-executable instructions, associated data structures, and program modules represent examples of program code for executing steps of the methods disclosed herein. The particular sequence of such executable instructions or associated data structures represent examples of corresponding acts for implementing the functions described in such steps. 
     Embodiments of the present disclosure may be practiced in a networked environment using logical connections to one or more remote computers having processors. Logical connections may include a local area network (LAN) and a wide area network (WAN) that are presented here by way of example and not limitation. Such networking environments are commonplace in office-wide or enterprise-wide computer networks, intranets and the Internet and may use a wide variety of different communication protocols. Those skilled in the art will appreciate that such network computing environments will typically encompass many types of computer system configurations, including personal computers, hand-held devices, multi-processor systems, microprocessor-based or programmable consumer electronics, network PCs, servers, minicomputers, mainframe computers, and the like. Embodiments of the disclosure may also be practiced in distributed computing environments where tasks are performed by local and remote processing devices that are linked (either by hardwired links, wireless links, or by a combination of hardwired or wireless links) through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices. 
     An exemplary system for implementing the overall system or portions of the disclosure might include a general purpose computing device in the form of a computer, including a processing unit, a system memory, and a system bus that couples various system components including the system memory to the processing unit. The system memory may include read only memory (ROM) and random access memory (RAM). The computer may also include a magnetic hard disk drive for reading from and writing to a magnetic hard disk, a magnetic disk drive for reading from or writing to a removable magnetic disk, and an optical disk drive for reading from or writing to a removable optical disk such as a CD ROM or other optical media. The drives and their associated machine-readable media provide nonvolatile storage of machine-executable instructions, data structures, program modules, and other data for the computer. 
     It should be noted that although the flowcharts provided herein show a specific order of method steps, it is understood that the order of these steps may differ from what is depicted. Also two or more steps may be performed concurrently or with partial concurrence. Such variation will depend on the software and hardware systems chosen and on designer choice. It is understood that all such variations are within the scope of the disclosure. Likewise, software and web implementations of the present disclosure could be accomplished with standard programming techniques with rule based logic and other logic to accomplish the various database searching steps, correlation steps, comparison steps and decision steps. It should also be noted that the word “component” as used herein and in the claims is intended to encompass implementations using one or more lines of software code, and/or hardware implementations, and/or equipment for receiving manual inputs. 
     The foregoing description of embodiments of the disclosure have been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure to the precise form disclosed, and modifications and variations are possible in light of the above teachings or may be acquired from practice of the disclosure. The embodiments were chosen and described in order to explain the principals of the disclosure and its practical application to enable one skilled in the art to utilize the disclosure in various embodiments and with various modifications as are suited to the particular use contemplated.