Patent Publication Number: US-6714940-B2

Title: Systems, methods, and computer program products to rank and explain dimensions associated with exceptions in multidimensional data

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     In co-pending and pending application Ser. No. 09/998,960, entitled “Systems, Methods, and Computer Program Products to Interpret, Explain, and Manipulate Exceptions in OLAP Multidimensional Data,” filed on Nov. 15, 2001, by Bhooshan Kelkar et al., assigned to the assignee of the present invention, and incorporated herein in its entirety by this reference, there is described a method of interpreting, explaining, and manipulating exceptions in multidimensional data. Although not limited thereto, the present invention employs such a method in one of its preferred embodiments. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention is directed to the field of computer-based multidimensional data modeling. It is more particularly directed to ranking and explaining dimensions associated with exceptions in multidimensional data on a computer system 
     2. Description of the Background Art 
     On-Line Analytical Processing (OLAP) is a computing technique for summarizing, consolidating, viewing, analyzing, applying formulae to, and synthesizing data according to multiple dimensions. OLAP software enables users, such as analysts, managers and executives, to gain insight into performance of an enterprise through rapid access to a wide variety of data “views” or “dimensions” that are organized to reflect the multidimensional nature of the enterprise performance data. An increasingly popular data model for OLAP applications is the multidimensional database (MDDB), which is also known as the “data cube.” OLAP data cubes are often used by a data analyst for interactive exploration of performance data for finding regions of anomalies in the data, which are also referred to as “exceptions” or “deviations.” Problem areas and new opportunities associated with the enterprise arc often identified when an anomaly in the enterprise data is located. 
     An exception represents the degree of surprise associated with data that is included in an OLAP data cube. An exception may be defined by means of example. Given a two-dimensional data cube having “p” values along a first dimension “A,” and “q” values along a second dimension “B,” the element or quantity corresponding to the ith value of dimension A and jth value of dimension B is denoted as, “y ij .” To estimate the exception, y ij , in this data cube, an expected value, “ŷ ij ,” of y ij  is calculated as a function, “f( ),” of three terms: (1) a term “μ” that denotes a trend that is common to all y values of the cube, (2) a term “α i ” that denotes special trends along the ith row with respect to the rest of the cube, and (3) a term “β j ” that denotes special trends along the jth column with respect to the rest of the cube. The residual difference “r ij ” between the expected value ŷ ij =f(μ,α i ,β j ) and the actual value y ij  represents the relative importance of the exception, y ij , based on its position in the cube. 
     By means of further explanation, when a data cube has three dimensions, for example, with dimension, “C,” being the third dimension, the expected value ŷ ijk  is calculated by taking into account not only the kth value of the third dimension, but also the three values corresponding to the pairs (i,j) in the AB plane, (i,k) in the AC plane and (j,k) in the BC plane. The expected value ŷ ijk  is then expressed as a function of seven terms as: 
     
       
           ŷ   ijk   =f (μ,α i ,β j ,γ k ,(αβ) ij ,(αγ) ik ,(γβ) kj ),  (1) 
       
     
     where (αβ) ij  denotes the contribution of the ijth value in the AB plane, (αγ) ik  denotes the contribution of jkth value in the AC plane, and (γβ) kj  denotes the contribution of the kjth value in the BC plane. In general, for any k-dimensional cube, the y value can be expressed as the sum of the coefficients corresponding to each of the 2 k −1 levels of aggregations or group-bys of the cube. The “coefficient” represents a component that provides information used in making predictions about the expected value of ŷ and a “group-by” represents different combinations of the dimensions associated with the multidimensional cube. In the present example, group-bys include “AB” and “ABC.” Therefore, a coefficient is a group-by component that contributes to predictability of a cell in a multidimensional cube. The coefficient model may be used to make predictions about the expected value of an exception. 
     By means of example, a three-dimensional cube will be considered. The function, f( ) can take several forms or models, such as an additive form, where function f( ) is a simple addition of all its arguments, and a multiplicative form, where function f( ) is a product of its arguments. It will be appreciated by those skilled in the art that the multiplicative form can be transformed to the additive form by performing a logarithm on the original data values. For a multiplicative model, the y ijk  values denote the log of the original y-values of the cube. The log is used to remove bias associated with the distribution. That is, taking the log will tend to normalize the distribution. The choice of the best form of the function depends on the particular class of data, and is preferably selected by a user having understanding and experience with the data at hand. For example, the distribution of the data is one of the factors that may be used to determine the best form of the function. 
     The final form of Equation One as shown in Equation Two is, 
     
       
           y   ijk   =ŷ   ijk   +r   ijk =μ+α i +β j +γ k +(αβ) ij +(αγ) ik +(γβ) kj ,  (2) 
       
     
     where r ijk  is the residual difference between the expected value ŷ ikj  and the actual value y ijk . The relative importance of an exception is based on the value of its residual. That is, the higher the value of the residual, the higher the importance of the exception. 
     There are several ways of deriving values of the coefficients of Equation Two. One way of deriving coefficients is shown in U.S. Pat. No. 6,094,651. The approach is a mean-based solution where the coefficients are estimated by taking the logs of all the relevant numbers and then the mean of the previous result. Taking the log will distribute the numbers so that the effect of large differences in the values of the cells is reduced. When the mean is derived a trend may be observed. In general, the coefficient corresponding to any group-by, “G,” is recursively determined, according to the mean-based solution, by subtracting the coefficients from group-bys that are at a smaller level of detail than, G, from the average y value at G. 
     The mean-based approach for calculating the coefficients is not particularly robust in the presence of extremely large numbers that are outliers. An “outlier” represents data that is related to a coefficient that deviates from the trend of the data by a significant amount. There are statistical methods for deciding when to keep or discard these suspected outlier data points. A number of well-known alternative approaches for handling large outliers can be used, such as the Median Polish Method and the Square Combining Method, disclosed by D. Hoaglin et al.,  Exploring Data Tables, Trends and Shape,  Wiley Series in Probability, 1988, and incorporated by reference herein. These two alternative approaches are based on using a “median” instead of “mean” for calculating the coefficients. Nevertheless, these alternative approaches have an associated high computational cost. Consequently, the mean-based approach is preferred for most OLAP data sets because significantly large outliers are uncommon in most data sets. 
     The method for determining a residual, “r ijk ,” may be determined from Equation Two as shown in Equation Three. 
     
       
           r   ijk   =|y   ijk   −ŷ   ijk |  (3) 
       
     
     The greater the value of r ijk , the more likely that the cell in the multidimensional data for which an expected value is being calculated is an exception in the data model. However, the residual value may need to be standardized for a meaningful comparison of multidimensional data. A “standardized residual value” is calculated as shown in Equation Four. 
     
       
           sr=|y   ijk   −ŷ   ijk |/σ iljk   (4) 
       
     
     The step of standardization is performed because the magnitude of the residual may appear to be significantly larger than the other values considered. Considering the magnitude of the residual alone can be misleading because the residual should be evaluated in relation to the data in the neighboring cells. Normalization of the data is achieved by applying a standard deviation to the process. It will be appreciated by those skilled in the art that there are many methods of calculating a standard deviation associated with data in the multidimensional cube. The standardized residual can then be used to rank the exceptions that are found. The higher the value of the standardized residual, the higher is the rank. The first exception in a decreasingly sorted array of exceptions will have the highest value of the standardized residual. A residual approach however is limited since the user views actual data and not the residual details, therefore the interpretation and explanation of an exception is not always obvious to the user. 
     The process of determining and analyzing a multidimensional cube exception is quantitative, while the analyst would like to use a qualitative approach. The information that is viewed in a quantitative approach, such as the coefficient approach, can be overwhelming. This happens because the number of possible two-dimensional or three-dimensional views that spawn three or two dimensions from the OLAP multidimensional sub-cube increases steeply. The number of possible three-dimensional views for N dimensions are (N)*(N−1)*(N−2)/6 and two-dimensional views are (N)*(N−1)/2. For example, if we have a cube with 7 dimensions, then the number of views for the end-user to analyze, “C,” are C=(7*6)/2=21 for two-dimensional views, and C=(7*6*5)/6=35 for three-dimensional views. In the absence of any formal way to focus on a few dimensions, the analyst has to view all thirty-five or twenty-one views to identify the best view for an exception, which makes it difficult to interpret and explain an exception. 
     There is a solution described in U.S. Pat. No. 6,094,651 that addresses exceptions and uses the concept of maximal terms. In general, a coefficient approach is limited since large coefficients are typically associated with smaller dimensional terms and the explanations are often too broad, spanning more data than necessary. This method looks at two-dimensional or three-dimensional views. However, the views are used for analysis rather than to examine individual dimensions. This limits the amount of data that is used to determine trends in the data. 
     Missing values are not considered in current exception-related solutions. For example, in a dimension in which a portion of the cells have no value the coefficient approach or the residual approach may generate information that indicates a more significant exception than actually exists. 
     Another limitation in the current solutions is that linguistic labels are not often assigned to an exception. The information is presented in a quantitative manner instead of a qualitative manner. This limits data analysts from easy access to information about the value of the data that is presented. Therefore, it is difficult for data analysts to make use of the available multidimensional data. 
     When an OLAP data cube has a large number of dimensions, such are ten or more, it is likely that the contribution to the exception graph may have many dimensions having contributions only slightly different. Then ranking these dimensions purely based on their density and numerical contribution may not provide enough information for an optimal display of the results. 
     Yet another limitation of the current solutions is that they lack a more formal and efficient way of assisting the data analyst with the view of simplifying the analysis of results of exception exploration. 
     From the foregoing it will be apparent that there is still a need to improve the interpretation, explanation, and manipulation of exceptions in multidimensional data on a computer system. 
     SUMMARY OF THE INVENTION 
     An embodiment of the present invention relates to systems, methods, and computer products that rank and explain dimensions associated with exceptions in multidimensional data on a computer system. The present invention is related to the field of computer-based multidimensional data modeling often used by data analysts. The present invention assists the data analyst by providing a simplified view of the multidimensional data that enables analysis of the important results of data exception exploration. 
     The preferred embodiment of the present invention operates with an exception-solver module and enables determination and analysis of an exception in a multidimensional data cube by a qualitative approach. The overwhelming amount of information that may be viewed in a quantitative approach is reduced to the important information, according to the present invention. More particularly, the preferred embodiment of the present invention enables ranking and explanation of dimensions associated with selected exceptions in multidimensional data. 
     Further, the preferred embodiment of the present invention incorporates the effect of weighting factors associated with the importance of the data along with an analysis of the numerical contribution from each dimension. That is the preferred embodiment of the present invention assigns weighting factors associated with the importance of a dimension. The weighting factors may be based on data mining results or may be obtained from the user. When the weighting factors are based on data mining results, the preferred embodiment of the present invention resolves results obtained from a plurality of data mining runs to arrive at a set of weighting factors for different dimensions. 
     The preferred embodiment of the present invention also provides the framework necessary to assign linguistic meaning to relative dimensions associated with each exception. This enables data analysts to obtain information about the value of the data that is presented in a more formal and efficient manner. The information about the dimension may also be presented visually. 
     An embodiment of the present invention is achieved by systems, methods, and computer products that rank and explain dimensions associated with exceptions that are selected from multidimensional data. The method comprises (a) associating at least one dimension with each selected exception; (b) obtaining weighting factors for certain associated dimensions by either (i) taking information from the user, (ii) or performing data mining runs for the data, or (iii) invoking the results of pre-computed data mining results on the data; (c) ranking the dimensions that are associated with each selected exception by numerical importance; (d) identifying the dimensions that need to be re-evaluated for possible re-ranking related to the exception of the multidimensional data; (e) re-ranking the dimensions based on the weighting factors; and (f) then visually or linguistically presenting the corrected contributions for selection of the best and most important two-dimensional or three-dimensional view for the exception. 
     Other aspects and advantages of the present invention will become apparent from the following detailed description, taken in conjunction with the accompanying drawings, illustrating by way of example the principles of the invention. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     In the following detailed description and in the several figures of the drawings, like elements are identified with like reference numerals. 
     FIG. 1A is a block diagram that illustrates the present invention; 
     FIG. 1B is a block diagram that further illustrates the present invention; 
     FIG. 2 is a block diagram that illustrates the exception-solver module and the OLAP data cube; 
     FIG. 3 is a block diagram of an OLAP data cube that is suitably configured for operation with the present invention; 
     FIG. 4A is a flow diagram that illustrates the method of present invention; 
     FIG. 5A is a block diagram that illustrates an example of dimension-wise contributions on an absolute scale; 
     FIG. 5B is a block diagram that illustrates an example of relative contributions; and 
     FIG. 6 is a block diagram of a computer system suitably configured for employment of the present invention. 
    
    
     DESCRIPTION OF THE INVENTION 
     As shown in the drawings and for purposes of illustration, the preferred embodiment of the invention novelly interprets and explains exceptions in multidimensional data on a computer system. The present invention assists the data analyst by providing a simplified view of the multidimensional data that incorporates the effect of weighting factors associated with the importance of the data along with an analysis of the numerical contribution from each dimension and the data density associated with each dimension. Existing systems have not been able to efficiently and adequately interpret and explain exceptions from selected multidimensional data that includes weighting factors representing the importance of the multidimensional data. 
     The overwhelming amount of information that may be viewed in a quantitative approach is reduced to the important information, according to the present invention. More particularly, the preferred embodiment of the present invention assigns weighting factors associated with the importance of a dimension. The weighting factors may be based on data mining results or may be obtained from the user. 
     The preferred embodiment of the present invention advantageously incorporates the weighting information with an analysis of the numerical contribution from each dimension and the data density associated with each dimension. This incorporation of a variety of information useful in ranking the importance of a contribution to an exception expands the information available to the data analyst over solutions of the past. Further, the operation of the present invention enables the data analyst to rearrange the dimensions for selection of two-dimensional or three-dimensional views of an exception better than in the past. This enables the data analyst to efficiently decide on the best and most important view of a dimension. 
     As shown in FIG.  1 A and in element  100 , the preferred embodiment of the present invention may operate in a client-server computer system configuration. Therefore, a client computer system  104  may communicate with a server computer system  102  during the operation of the present invention. The exception-solver module  120  operates in either the client  104  or the server  102  to perform the preferred embodiment of the present invention. For example, information may be communicated to either the server  102  or the client  104  via the user interface  117 . Through such communication threshold information may be established and may subsequently be used by the exception-solver module  120  to manipulate data  108 , such as multidimensional data  110 , according to the operation of the present invention. The user interface  117  may communicate with the preferred embodiment of the present invention, either via batch input  119  or user input  118 . 
     Further, an OLAP data cube  106  may be configured in the memory  658  of either the client  104  or the server  102 . Alternatively, the OLAP data cube  106  may be configured in computer storage such as that of a disk  122 . Typically, the OLAP data cube  106  is configured in computer storage of a disk  122  associated with a client  104 . The terms “OLAP data cube” and “data cube” will be used interchangeably herein. Element  658  is described with reference to FIG.  6 . 
     FIG. 1B, and in element  130 , illustrates the preferred embodiment of the present invention operating in a computer system. Therefore, the OLAP data cube  106  is built on a database  140  and exceptions  214  are found in the OLAP data cube  106 . The Absolute Scale Contribution to Exception Before Re-ranking  132  shows the absolute contribution  218  of dimensions  216  for an exception  214  before the operation of the present invention. The weighting factors  224  may be defined by a variety of mining techniques, represented herein by elements  134  and  136 . Therefore, mining technique_ 1   134  and mining technique_ 2   136  operate on the same database  140  and pass the results of the data mining runs to the exception-solver module  120 . The database  140 , and mining technique_ 1   134  and mining technique_ 2   136  may be stored on the disk  122 . The exception-solver module  120  then uses the Absolute Scale Contribution to Exception Before Re-ranking  132  and the results from multiple data mining runs herein represented by elements  134  and  136  to re-rank the dimensional contributions  218  to the exception  214 . The re-ranking is shown in element  138 . Thus, the data analyst is better placed to select the dimensions for two-dimensional or three-dimensional views. Elements  214 ,  216 ,  218 , and  224  are described with reference to FIG.  2 . 
     As shown in FIG. 2, the OLAP data cube  106  includes the following elements: the exception  214 , the dimension  216 , and the contribution  218 . The dimension  216  may be represented as a row or column in an OLAP data cube  106  and is organized to reflect the multidimensional nature of the enterprise performance data  108 . The exception-solver module  120  interprets contributions  218  associated with the dimension  216 . The exception  214  represents regions of anomalies in the multidimensional data  110  and at least one dimension  216  is associated with each selected exception  214 . 
     The exception-solver module  120  includes elements used in the preferred embodiment of the present invention. The exception-solver module  120  is typically program code that may be embodied as a computer program  642  (as shown in FIG.  6 ). The exception-solver module  120  includes weighting factors  224  that are associated with the importance of a dimension  216 . The exception-solver module  120  also uses data mining run results  222  that associate relative importance to dimensions  216 . 
     The exception-solver module  120  includes a linguistic interpretation of dimensional contributions, as shown in element  212 . The linguistic interpretation framework  211  is used by the preferred embodiment of the present invention to determine the linguistic interpretation  212 . The linguistic interpretation  212  is used to describe relationships among the important multidimensional data  110  to the data analyst. Therefore, sorted dimensions  216  may be represented by a linguistic interpretation  216  via the linguistic interpretation framework  211 . The exception-solver module  120  also includes a visual interpretation of dimensional contributions, as shown in element  220 , that visually represents the normalized density-corrected contribution  210 . 
     As shown in FIG. 3, an OLAP data cube  106  is suitably configured for operation with the present invention. Therefore, by means of explanation, an example of the operation of the present invention is described. The dimension  216  is herein represented by dimension “i”  312 , dimension “j”  304 , and dimension “k”  306 . Further dimension “j” herein represents market data  302 , dimension “k”  306  herein represents product data  308 , and dimension “i”  312  herein represents year data  310 . An example in the multidimensional cube  106  of an exception  214  is a three-dimensional cell. Further, the three-dimensional cell may be shown to have dimensions  216  or views. Here, there are three views: view “i*j”  332 , view “j*k”  334 , and view “i*k”  330 . 
     FIG.  4  and element  401  illustrate the preferred method of the present invention that re-ranks dimensions  216  associated with exceptions  214  in multidimensional data  110 . The dimension re-ranking may be based on data mining results or on user input  118 . Initially, as shown in element  403 , at least one dimension  216  is associated with each selected exception  214 . That is, at least one dimension  216  is identified. An identified dimension  216  may be generational, or non-generational if no generational structure dimension  216  is associated with a first generation dimension  216 . The user may define an exception  214  that may include any combination of dimensions  216 . Elements  214  and  216  are described with reference to FIG.  2 . 
     Then, the dimension information is used to obtain weighting factors  224 . The weighting factors  224  may be obtained from the user, such as a data analyst, as shown in element  420 . Alternatively, the weighting factors  224  may be obtained by using the results of data mining run results  222  for data  108 , such as zero-level data  108 , as shown in element  421 . Those skilled in the art will appreciate that zero-level data is non-aggregated data  108  that has not been manipulated by data mining. In yet another alternative, the weighting factors  224  for dimensions may be obtained by invoking the results of pre-computed data mining run results  222  on data, such as zero-level data  108 , as shown in element  422 . While zero-level data  108  is more efficiently manipulated by the preferred embodiment of the present invention than higher-level data  108 , the present invention may be practiced on other types of data  108  and is not limited to zero-level data  108 . Element  108  is described with reference to FIG. 1, and elements  222  and  224  are described with reference to FIG.  2 . 
     If data mining is used to determine the weighting factors  224  and if multiple data mining runs  222  are used, discrepancies between the multiple results are resolved, as shown in element  423 . Linear programming methods may be used to resolve the multiple results. The dimensions  216  that are associated with each selected exception  214  are ranked by numerical importance, as shown in element  405 . After the weighting factors  224  are obtained, the dimensions that need to be re-evaluated are identified, as shown in element  424 . The need for re-evaluation is for possible re-ranking of the dimension  216  as it is related to the exception  214 . 
     Then, as shown in element  424 , the dimensions  216  that need to be re-evaluated for possible re-ranking related to the exception  214  are identified. Then, as shown in element  426 , the dimensions  216  are re-ranked based on the weighting factors  224  and other numerical exception information. The re-ranked information is presented so that the data analyst may select the best two-dimensional or three-dimensional view of the exception, as shown in element  428 . 
     By means of example, FIG.  5 A and element  502  illustrate absolute scale numerically derived contributions  218  to exceptions  214 . In the present example, the exception  214  relates to a five-dimensional analysis for sales data  108 . The five dimensions intersecting at this exception  214  include: “Market,” as shown in element  506 , “Age,” as shown in element  508 , “Income,” as shown in element  510 , “Population,” as shown in element  512 , and “Time,” as shown in element  514 . By means of explanation of this example, let the exception be at Market equals San Francisco, Age equals 39, Income equals 45,000, Population equals 700,000, and Time equals 1999. Elements  214 ,  216 , and  218  are described with reference to FIG.  2 . 
     In the present example, the importance of dimensions  216  associated with the exception  214  is calculated according to the operation that is described in U.S. patent application Ser. No. 09/998,955. The contributions  218  associated with the dimensions  216  are numerically analyzed and the associated dimensions  216  are arranged according to descending importance as shown in Table 1. Therefore, from the rearranged dimensions based only on numerical analysis, the best view is Market*Age. However, the relationships between the following dimensions  216  are within a predetermined confidence limit that triggers possible re-arrangement of the dimensions: the dimension “Age”  508 , the dimension “Income”  510 , and the dimension “Population”  512 . In this example the confidence limit for an exception contribution, as shown in element  218 , is 0.05. For example, the “contribution from Age” minus the “contribution from Income” equals 0.01, which is less than 0.05. Thus, the importance values associated with three dimensions  216  are very close and therefore there are three candidates for rearrangement: the dimension “Age”  508 , the dimension “Income”  510 , and the dimension “Population”  512   
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Dimensions Arranged by Numerical Calculation 
               
            
           
           
               
               
               
               
               
               
            
               
                 Rearranged 
                   
                   
                   
                   
                   
               
               
                 dimensions 
                 Market 
                 Age 
                 Income 
                 Population 
                 Time 
               
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 Cd 
                 −.0365 
                 0.22 
                 0.21 
                 0.185 
                 −0.02 
               
               
                 Contribution 
                 0.365 
                 0.22 
                 0.21 
                 0.185 
                 0.02 
               
               
                 |Cd| 
               
               
                   
               
            
           
         
       
     
     In the present example, the user has no input for the relative importance among the dimensions  216 . While user input  118  is ascribed the highest priority, since there is no user input  118 , the weighting factors  224  for the dimensions  216  are obtained from data mining run results  222 . The information that is generated by techniques based on Principle Component Analysis (PCA) data mining is used by those skilled in the art for numerical data analysis of multidimensional data  110 . PCA is used to discover and possibly reduce the dimensionality of a set of multidimensional data  110 . Therefore, via PCA, the “Income” dimension, as shown in element  510 , is determined to be more important than the “Population” dimension, as shown in element  512 . More particularly, as a result of the PCA data mining technique, the importance of the “Income” dimensions, as shown in element  510 , is determined to be about 1.5 times that of the “Population” dimension, as shown in element  512 . Elements  110  and  118  are described with reference to FIG. 1, and elements  222  and  224  are described with reference to FIG.  2 . 
     The results obtained from a second data mining technique, clustering, indicate that the “Population” dimension, as shown in element  512 , is more important than the “Age” dimension, as shown in element  508 . More particularly, the “Population” dimension, as shown in element  512  is about twice the importance of the “Age” dimension, as shown in element  508 . Clustering is a generic term that represents a technique that attempts to group multidimensional data  110  on the basis of similarity in the multidimensional data  110 . 
     A linear programming method is used in the present example to resolve discrepancies between the multiple data mining run results  222 , as discussed with reference to element  423 . The linear programming method is enlisted when the values of contributions  218  associated with dimensions  216  from data mining run results  222  are within a confidence level. Therefore, the following variables are used to represent importance: X 1  represents the importance of the dimension “Income”  510 , X 2  represents the importance of the dimension “Population”  512 , and X 3  represents the importance of the dimension “Age”  508 . The standard objective function is: Maximum (X 1 +X 2 +X 3 ) or Minimum (−X 1 −X 2 −X 3 ). Element  423  is described with reference to FIG.  4 . 
     The constraints used in the analysis include the following relationships. Given a confidence limit “d1,” the PCA result implies that zero is less than or equal to (2*X 1 −3*X 2 ). Also (2*X 1 −3*X 2 ) is less than or equal to d 1 . Also, given a confidence limit of “d 2 ,” the Clustering result implies that zero is less than or equal to (2*X 1 −3*X 2 ). Also (2*X 1 −3*X 2 ) is less than or equal to d 2 . Those skilled in the art will appreciate that d 1  and d 2  should be set to sufficiently small values to ensure an accurate analysis, such as five percent. 
     A boundary condition used in the analysis is that the values of X 1 , X 2 , and X 3  must be between zero and one. Also, the sum of (X 1 +X 2 +X 3 ) must be less than or equal to one and greater than or equal to zero. 
     For the purposes of explanation, data  108  associated with each dimension  216  is shown in Table 2. Further, Table 2 shows the results for the importance of the following dimensions as a result of the data mining run results  222  that have been resolved by linear programming: the dimension “Income”  510 , the dimension “Population”  512 , and the dimension “Age”  508 . Therefore, the “Income” dimension, as shown in element  510  and having an importance value of 0.525, is determined to be more important than the “Population” dimension, as shown in element  512  and having an importance value of 0.333. Also, the “Age” dimension, as shown in element  508  has an importance value of 0.142. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Importance of Dimension from Data Mining Analysis 
               
            
           
           
               
               
               
            
               
                   
                 Dimension 
                 Importance 
               
               
                   
                   
               
               
                   
                 Income 
                 0.525 
               
               
                   
                 Population 
                 0.333 
               
               
                   
                 Age 
                 0.142 
               
               
                   
                   
               
            
           
         
       
     
     FIG.  5 B and element  520  illustrate an example of the operation of the present invention. The absolute contribution  518  is identified in FIG.  5 B. Therefore, based on linear programming resolution of data mining results the following dimensions  216  are listed in decreasing order of importance: the dimension “Income”  510 , the dimension “Population”  512 , and the dimension “Age”  508 . The importance of dimensions  216 , derived from numerical contribution, and in decreasing order follows: the dimension “Age”  508 , the dimension “Income”  510 , and the dimension “Population”  512 . Therefore, since the data mining run results  222  take precedence over the numerical calculation, when the value of the dimensions  216  are within a defined confidence level, the visual representation of the information presented to the data analyst, in decreasing order, is rearranged as follows: the dimension “Market”  506 , the dimension “Income”  510 , the dimension “Population”  512 , the dimension “Age”  508 , and the dimension “Time”  514 . Therefore, even though the dimension “Income”  510  has an absolute numerical contribution  218  that is lower than that for the dimension “Age”  508 , the best view of the exception is represented by Market*Income. 
     FIG. 6 is a block diagram of a computer system  600 , suitable for employment of the present invention. System  600  may be implemented on a general-purpose microcomputer, such as one of the members of the IBM Personal Computer family, or other conventional work-station or graphics computer devices, or mainframe computers. In its preferred embodiment, system  600  includes a user interface  617 , a user input device  607 , a display  615 , a printer  620 , a processor  655 , a read only memory (ROM)  650 , a data storage device  122 , such as a hard drive, a random access memory (RAM)  640 , and a storage media interface  635 , all of which are coupled to a bus  625  or other communication means for communicating information. Although system  600  is represented herein as a standalone system, it is not limited to such, but instead can be part of a networked system. For example, the computer system  600  may be connected locally or remotely to fixed or removable data storage devices  122  and data transmission devices  645 . Further, the computer system  100 , the server computer system  102 , and the client computer system  104  also could be connected to other computer systems via the data transmission devices  645 . Elements  100 ,  102 , and  104  are described with reference to FIG.  1 . 
     The RAM  640 , the data storage device  122  and the ROM  650 , are memory components  658  that store data  108  and instructions for controlling the operation of processor  655 , which may be configured as a single processor or as a plurality of processors. The processor  655  executes a program  642  to perform the methods of the present invention, as described herein. 
     While the program  642  is indicated as loaded into the RAM  640 , it may be configured on a storage media  630  for subsequent loading into the data storage device  122 , the ROM  650 , or the RAM  640  via an appropriate storage media interface  635 . Storage media  630  can be any conventional storage media such as a magnetic tape, an optical storage media, a compact disk, or a floppy disk. Alternatively, storage media  630  can be a random access memory  640 , or other type of electronic storage, located on a remote storage system. 
     Generally, the computer programs and operating systems are all tangibly embodied in a computer-readable device or media, such as the memory  658 , the data storage device  122 , or the data transmission devices  645 , thereby making an article of manufacture, such as a computer program product, according to the invention. As such, the terms “computer program product” as used herein are intended to encompass a computer program  642  accessible from any computer readable device or media. 
     Moreover, the computer programs  642  and operating systems are comprised of instructions which, when read and executed by the computer system  100 , the server computer system  102 , and the client computer system  104 , cause the computer system  100 , the server computer system  102 , and the client computer system  104  to perform the steps necessary to implement and use the present invention. Under control of the operating system, the computer programs  642  may be loaded from the memory  658 , the data storage device  122 , or the data transmission devices  645  into the memories  658  of the computer system  100 , the server computer system  102 , and the client computer system  104  for use during actual operations. Those skilled in the art will recognize many modifications may be made to this configuration without departing from the scope of the present invention. 
     The user interface  617  is an input device, such as a keyboard or speech recognition subsystem, for enabling a user to communicate information and command selections to the processor  655 . The user can observe information generated by the system  600  via the display  615  or the printer  620 . The user input device  607  is a device such as a mouse, track-ball, or joy-stick, which allows the user to manipulate a cursor on the display  615  for communicating additional information and command selections to the processor  655 . 
     When operating in accordance with one embodiment of the present invention, the system  600  ranks and explains dimensions associated with exceptions in multidimensional data  110 . The processor  655  and the program  642  collectively operate as a module for ranking and explanation of dimensions associated with exceptions in multidimensional data  110 . It will be appreciated that the present invention offers many advantages over prior art techniques. Element  110  is described with reference to FIG.  1 . 
     The present invention is typically implemented using one or more computer programs  642 , each of which executes under the control of an operating system and causes the computer system  100 , the server computer system  102 , and the client computer system  104  to perform the desired functions as described herein. Thus, using the present specification, the invention may be implemented as a machine, process, method, system, or article of manufacture by using standard programming and engineering techniques to produce software, firmware, hardware or any combination thereof. 
     It should be understood that various alternatives and modifications can be devised by those skilled in the art. However, these should not be viewed as limitations upon the practice of these teachings, as those skilled in the art, when guided by the foregoing teachings, may derive other suitable characteristics of a similar or different nature. The present invention is intended to embrace all such alternatives, modifications and variances that fall within the scope of the appended claims 
     TRADEMARKS 
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