Patent Publication Number: US-2019197411-A1

Title: Characterizing model performance using global and local feature contributions

Description:
BACKGROUND 
     Field 
     The disclosed embodiments relate to statistical model performance. More specifically, the disclosed embodiments relate to techniques for performing hybrid characterization of model performance using global and local feature contributions. 
     Related Art 
     Analytics may be used to discover trends, patterns, relationships, and/or other attributes related to large sets of complex, interconnected, and/or multidimensional data. In turn, the discovered information may be used to gain insights and/or guide decisions or actions related to the data. For example, business analytics may be used to assess past performance, guide business planning, and/or identify actions that may improve future performance. 
     To glean such insights, large data sets of features may be analyzed using regression models, artificial neural networks, support vector machines, decision trees, naïve Bayes classifiers, and/or other types of statistical models. The discovered information may then be used to guide decisions and/or perform actions related to the data. For example, the output of a statistical model may be used to guide marketing decisions, assess risk, detect fraud, predict behavior, and/or customize or optimize use of an application or website. 
     However, significant time, effort, and overhead may be spent on feature selection during creation and training of statistical models for analytics. For example, a data set for a statistical model may have thousands to millions of features, including features that are created from combinations of other features, while only a fraction of the features and/or combinations may be relevant and/or important to the statistical model. At the same time, training and/or execution of statistical models with large numbers of features typically require more memory, computational resources, and time than those of statistical models with smaller numbers of features. Excessively complex statistical models that utilize too many features may additionally be at risk for overfitting. 
     At the same time, statistical models are commonly associated with a tradeoff between interpretability and performance. For example, a linear regression model may include coefficients that identify the relative weights or importance of features in the model but does not perform well with complex problems. Conversely, a nonlinear model such as a random forest or gradient boosted trees can be trained to perform well with a variety of problems but typically operates in a way that is not easy to understand. 
     Consequently, creation and use of statistical models in analytics may be facilitated by mechanisms for efficiently and effectively performing feature selection and interpretation for the statistical models. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  shows a schematic of a system in accordance with the disclosed embodiments. 
         FIG. 2  shows a system for processing data in accordance with the disclosed embodiments. 
         FIG. 3  shows a flowchart illustrating the processing of data in accordance with the disclosed embodiments. 
         FIG. 4  shows a computer system in accordance with the disclosed embodiments. 
     
    
    
     In the figures, like reference numerals refer to the same figure elements. 
     DETAILED DESCRIPTION 
     The following description is presented to enable any person skilled in the art to make and use the embodiments, and is provided in the context of a particular application and its requirements. Various modifications to the disclosed embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be applied to other embodiments and applications without departing from the spirit and scope of the present disclosure. Thus, the present invention is not limited to the embodiments shown, but is to be accorded the widest scope consistent with the principles and features disclosed herein. 
     The data structures and code described in this detailed description are typically stored on a computer-readable storage medium, which may be any device or medium that can store code and/or data for use by a computer system. The computer-readable storage medium includes, but is not limited to, volatile memory, non-volatile memory, magnetic and optical storage devices such as disk drives, magnetic tape, CDs (compact discs), DVDs (digital versatile discs or digital video discs), or other media capable of storing code and/or data now known or later developed. 
     The methods and processes described in the detailed description section can be embodied as code and/or data, which can be stored in a computer-readable storage medium as described above. When a computer system reads and executes the code and/or data stored on the computer-readable storage medium, the computer system performs the methods and processes embodied as data structures and code and stored within the computer-readable storage medium. 
     Furthermore, methods and processes described herein can be included in hardware modules or apparatus. These modules or apparatus may include, but are not limited to, an application-specific integrated circuit (ASIC) chip, a field-programmable gate array (FPGA), a dedicated or shared processor that executes a particular software module or a piece of code at a particular time, and/or other programmable-logic devices now known or later developed. When the hardware modules or apparatus are activated, they perform the methods and processes included within them. 
     The disclosed embodiments provide a method, apparatus, and system for processing data. As shown in  FIG. 1 , the system may be a data-processing system  102  that analyzes one or more sets of input data (e.g., input data 1  104 , input data x  106 ). More specifically, data-processing system  102  may create and train one or more statistical models  110  for analyzing input data related to users, organizations, applications, job postings, purchases, electronic devices, network devices, images, audio, video, websites, content, sensor measurements, and/or other categories. Statistical models  110  may include, but are not limited to, regression models, artificial neural networks, support vector machines, decision trees, random forests, boosted gradient trees, naïve Bayes classifiers, Bayesian networks, deep learning models, hierarchical models, and/or ensemble models. 
     Analysis performed by data-processing system  102  may be used to discover relationships, patterns, and/or trends in the data; gain insights from the input data; and/or guide decisions or actions related to the data. For example, data-processing system  102  may use statistical models  110  to generate output  118  that includes scores, classifications, recommendations, estimates, predictions, and/or other inferences or properties. Output  118  may be inferred or extracted from features  114  in the input data, including primary features and/or derived features that are generated from primary features and/or other derived features. 
     For example, the primary features may include profile data, user activity, sensor data, and/or other data that is extracted directly from fields or records in the input data. The primary features may be aggregated, scaled, combined, bucketized, and/or otherwise transformed to produce derived features, which in turn may be further combined or transformed with one another and/or the primary features to generate additional derived features. After output is generated from one or more sets of primary and/or derived features, the output may be queried and/or used to improve revenue, interaction with the users and/or organizations, use of the applications and/or content, and/or other metrics associated with the input data. 
     In one or more embodiments, features  114  are obtained and/or used with an online professional network or other community of users that is used by a set of entities to interact with one another in a professional, social, and/or business context. The entities may include users that use the online professional network to establish and maintain professional connections, list work and community experience, endorse and/or recommend one another, search and apply for jobs, and/or perform other actions. The entities may also include companies, employers, and/or recruiters that use the online professional network to list jobs, search for potential candidates, provide business-related updates to users, advertise, and/or take other action. 
     As a result, features  114  may include member features, company features, and/or job features. The member features include attributes from the members&#39; profiles with the online professional network, such as each member&#39;s title, skills, work experience, education, seniority, industry, location, and/or profile completeness. The member features also include each member&#39;s number of connections in the social network, the member&#39;s tenure on the social network, and/or other metrics related to the member&#39;s overall interaction or “footprint” in the online professional network. The member features further include attributes that are specific to one or more features of the online professional network, such as a classification of the member as a job seeker or non-job-seeker. 
     The member features may also characterize the activity of the members with the online professional network. For example, the member features may include an activity level of each member, which may be binary (e.g., dormant or active) or calculated by aggregating different types of activities into an overall activity count and/or a bucketized activity score. The member features may also include attributes (e.g., activity frequency, dormancy, total number of user actions, average number of user actions, etc.) related to specific types of social or online professional network activity, such as messaging activity (e.g., sending messages within the social network), publishing activity (e.g., publishing posts or articles in the social network), mobile activity (e.g., accessing the social network through a mobile device), job search activity (e.g., job searches, page views for job listings, job applications, etc.), and/or email activity (e.g., accessing the social network through email or email notifications). 
     The company features include attributes and/or metrics associated with companies. For example, company features for a company may include demographic attributes such as a location, an industry, an age, and/or a size (e.g., small business, medium/enterprise, global/large, number of employees, etc.) of the company. The company features may further include a measure of dispersion in the company, such as a number of unique regions (e.g., metropolitan areas, counties, cities, states, countries, etc.) to which the employees and/or members of the online professional network from the company belong. 
     A portion of company features may relate to behavior or spending with a number of products, such as recruiting, sales, marketing, advertising, and/or educational technology solutions offered by or through the online professional network. For example, the company features may also include recruitment-based features, such as the number of recruiters, a potential spending of the company with a recruiting solution, a number of hires over a recent period (e.g., the last 12 months), and/or the same number of hires divided by the total number of employees and/or members of the online professional network in the company. In turn, the recruitment-based features may be used to characterize and/or predict the company&#39;s behavior or preferences with respect to one or more variants of a recruiting solution offered through and/or within the online professional network. 
     The company features may also represent a company&#39;s level of engagement with and/or presence on the online professional network. For example, the company features may include a number of employees who are members of the online professional network, a number of employees at a certain level of seniority (e.g., entry level, mid-level, manager level, senior level, etc.) who are members of the online professional network, and/or a number of employees with certain roles (e.g., engineer, manager, sales, marketing, recruiting, executive, etc.) who are members of the online professional network. The company features may also include the number of online professional network members at the company with connections to employees of the online professional network, the number of connections among employees in the company, and/or the number of followers of the company in the online professional network. The company features may further track visits to the online professional network from employees of the company, such as the number of employees at the company who have visited the online professional network over a recent period (e.g., the last 30 days) and/or the same number of visitors divided by the total number of online professional network members at the company. 
     One or more company features may additionally be derived features that are generated from member features. For example, the company features may include measures of aggregated member activity for specific activity types (e.g., profile views, page views, jobs, searches, purchases, endorsements, messaging, content views, invitations, connections, recommendations, advertisements, etc.), member segments (e.g., groups of members that share one or more common attributes, such as members in the same location and/or industry), and companies. In turn, the company features may be used to glean company-level insights or trends from member-level online professional network data, perform statistical inference at the company and/or member segment level, and/or guide decisions related to business-to-business (B2B) marketing or sales activities. 
     The job features describe and/or relate to job listings and/or job recommendations within the online professional network. For example, the job features may include declared or inferred attributes of a job, such as the job&#39;s title, industry, seniority, desired skill and experience, salary range, and/or location. One or more job features may also be derived features that are generated from member features and/or company features. For example, the job features may provide a context of each member&#39;s impression of a job listing or job description. The context may include a time and location (e.g., geographic location, application, website, web page, etc.) at which the job listing or description is viewed by the member. In another example, some job features may be calculated as cross products, cosine similarities, statistics, and/or other combinations, aggregations, scaling, and/or transformations of member features, company features, and/or other job features. 
     In one or more embodiments, data-processing system  102  includes functionality to characterize a performance  108  of statistical models  110  using features  114  inputted into statistical models  110 . In addition, performance  108  may include the global behavior of each statistical model across a set of predictions, scores, and/or inferences made by the statistical model, as well as the local behavior of each statistical model with respect to a specific prediction, score, and/or inference. As shown in  FIG. 2 , a system for processing data (e.g., data-processing system  102  of  FIG. 1 ) may include an analysis apparatus  202  and a management apparatus  204 . Each of these components is described in further detail below. 
     Analysis apparatus  202  performs processing related to characterizing the performance or operation of a statistical model  206 . For example, analysis apparatus  202  may obtain statistical model  206  as a regression model, artificial neural network, naïve Bayes classifier, Bayesian network, clustering technique, decision tree, random forest, gradient boosted tree, support vector machine, and/or other type of machine learning model or technique. Output  214  of statistical model  206  may be used to perform prediction, classification, scoring, recommendation, estimation, and/or other tasks. For example, statistical model  206  may generate scores that represent propensities of users in performing an action and/or of customers in purchasing a product. 
     As shown in  FIG. 2 , statistical model  206  may be trained and/or executed using feature values  208  from multiple features  222 - 224 . For example, features  222 - 224  used by statistical model  206  to predict a spending behavior and/or churn risk of a customer with a product may include demographic attributes, historic spending behavior, product usage attributes, and/or other attributes that characterize the customer and/or the customer&#39;s behavior with respect to the product. 
     Features  222 - 224  and/or feature values  208  may be stored in a database, data store, distributed filesystem, messaging service, and/or another type of data repository  234 . During training, statistical model  206  may be fit to training data that includes a subset of feature values  208  from data repository  234 . Statistical model  206  may then be validated and/or tested using validation and/or test data that include one or more additional subsets of feature values  208 . Finally, statistical model  206  may be applied to unseen data containing new and/or remaining subsets of feature values  208  in data repository  234  to generate output  214  that infers properties associated with features  222 - 224 . 
     In one or more embodiments, the system of  FIG. 2  includes functionality to perform analysis and interpretation of the performance of statistical model  206  using a hybrid approach that combines global contributions  230  and local contributions  232  of features  222 - 224  and/or feature values  208  inputted into statistical model  206 . Global contributions  230  may represent the global effects of individual features  222 - 224  on multiple output  214  values from statistical model  206 , while local contributions  232  may represent the local effects of features  222 - 224  on individual output  214  values from statistical model  206 . For example, global contributions  230  may characterize the effects of features  222 - 224  on multiple predictions of customer spending or churn risk from statistical model  206  (e.g., all predictions from statistical model  206  or predictions for a given subset of customers). On the other hand, a different set of local contributions  232  may be generated for each prediction to identify specific features  222 - 224  that affect that prediction. 
     First, analysis apparatus  202  uses multiple sets of feature values  208  and the corresponding output  214  values from statistical model  206  to build a linear model  212  that estimates values of output  214  based on the corresponding feature values  208 . For example, analysis apparatus  202  may use feature values  208  inputted into statistical model  206  to train an additive linear model  212  so that the output of linear model  212  estimates output  214  of statistical model  206 . 
     Second, analysis apparatus  202  uses coefficients  220  of linear model  212 , sets of feature values  208  inputted into statistical model  206 , and/or measures of feature importance associated with statistical model  206  to characterize the performance or output  214  of statistical model  206 . In particular, analysis apparatus  202  uses measures of feature importance from statistical model  206  to assess the relative global contributions  230  of features  222 - 224  toward output  214  of statistical model  206 . For example, statistical model  206  may have the following linear representation: 
         y=β   1   x   1 +β 2   x   2 + . . . +β n   x   n +β 0  
 
     In the above equation, x 1 , x 2 , . . . , x n  represent features  222 - 224  inputted into statistical model  206  and linear model  212 , and β 1 , β 2 , . . . β n  represent coefficients (e.g., coefficients  220 ) associated with features  222 - 224 . The coefficients may be obtained directly from a linear statistical model  206 , or the coefficients may include weights representing the relative importance of the features in nonlinear models, such as random forest impurity decreases. Thus, a feature with a higher coefficient and/or weight may have a greater linear effect on output  214  than a feature with a lower coefficient and/or weight. 
     Analysis apparatus  202  also combines coefficients  220  and feature values  208  to assess the relative local contributions  232  of features  222 - 224  toward individual values of output  214 . Continuing with the previous example, analysis apparatus  202  may determine local contributions  232  for a given output  214  value from statistical model  206  using the following: 
         c   i =β i   x   i  
 
     In the above equation, each local contribution c i  may be calculated by multiplying the corresponding coefficient β 1  with the corresponding feature value x i  used to produce the output value. 
     After global contributions  230  are obtained from measures of feature importance in statistical model  206  and local contributions  232  are determined using coefficients  220  of linear model  212  multiplied with the corresponding feature values  208  inputted into statistical model  206 , management apparatus  204  selects subsets  218  of features  222 - 224  associated with significant global contributions  230  and significant local contributions  232  toward output  214  for use in characterizing the performance of statistical model  206 . For example, management apparatus  204  may rank features  222 - 224  in descending order of global contributions  230  (i.e., coefficients  220  of linear model  212 ) and select a first subset of features with the highest global contributions  230  from the ranking. Similarly, management apparatus  204  may rank features  222 - 224  in descending order of local contributions  232  for a given set of feature values  208  inputted into statistical model  206  (i.e., coefficients  220  multiplied by feature values  208 ) and select a second subset of features with the highest local contributions  232  from the ranking. 
     In one or more embodiments, management apparatus  204  selects subsets  218  of features  222 - 224  associated with significant global contributions  230  and significant local contributions  232  based on one or more parameters  216  used to determine the number of features to be included in each subset. First, management apparatus  204  may select subsets  218  of highest-ranked features in global contributions  230  and/or local contributions  232  based on one or more parameters  216  that specify proportions associated with each subset. For example, management apparatus  204  may obtain a user-generated and/or fixed parameter that specifies that 30% of features  222 - 224  used to characterize the performance of statistical model  206  have high global contributions  230  to output  214 . Thus, 10 features that are selected as factors for characterizing a given output  214  of statistical model  206  may include three features with the highest global contributions  230  to output  214  (i.e., the highest coefficients  220  in linear model  212 ). 
     Second, management apparatus  204  may select subsets  218  of features  222 - 224  associated with significant global contributions  230  and significant local contributions  232  based on one or more parameters  216  for placing features in either subset and/or selecting one subset before another. For example, parameters  216  may specify that five features with high global contributions  230  be selected before 10 features with high local contributions  232 . Thus, a feature that is associated with both a high global contribution and a high local contribution for a given output  214  of statistical model  206  may be included in the five features with high global contributions  230  and omitted from the remaining 10 features with high local contributions  232 . In turn, the remaining 10 features with high local contributions  232  may omit features that are already included in the five features with the highest global contributions  230  to output  214 . 
     Management apparatus  204  then outputs the selected subsets  218  of features  222 - 224  and attributes associated with the selected features  222 - 224  for use in characterizing the behavior, performance, and/or output  214  of statistical model  206 . For example, management apparatus  204  may output feature names, feature namespaces, feature sources, feature values  208 , coefficients  220 , products of coefficients  220  and feature values  208 , and/or other data associated with subsets  218  of features  222 - 224  selected to have the highest global contributions  230  and local contributions  232  toward a given output  214  value generated by statistical model  206 . The output may be included in a table, chart, spreadsheet, visualization, file, message, notification, and/or other human-readable form. 
     Management apparatus  204  may also output statistics and/or metrics associated with subsets  218  of features  222 - 224  with high global contributions  230  and local contributions  232  toward a given output  214  value generated by statistical model  206 . For example, management apparatus  204  may calculate the proportional “weight” of each feature with a high global contribution by dividing the coefficient for the feature from linear model  212  by the sum of all coefficients  220  from linear model  212 . Management apparatus  204  may then output the weight in lieu of or in addition to the coefficient of the feature. In another example, management apparatus  204  may determine a percentile or quantile associated with a feature value for a feature with a high local contribution toward a given output  214  value from statistical model  206 . Management apparatus  204  may then output the percentile or quantile to assist in the interpretation of the feature value&#39;s position in a distribution of feature values for the feature. 
     In turn, administrators, developers, data scientists, researchers, and/or other users associated with developing, maintaining, and/or using features  222 - 224  and/or statistical model  206  may use output from management apparatus  204  to interpret and/or understand scores, predictions, inferences, and/or other behavior or output  214  from statistical model  206 . For example, the users may use the outputted subsets  218  of features  222 - 224  and/or the associated attributes to better understand and/or validate the behavior of statistical model  206 . The users may also, or instead, identify key feature values for an entity that affect a corresponding output  214  value from statistical model  206  (e.g., a certain type of behavior that increases a customer&#39;s risk of churning from a product) and use insights associated with the feature values to guide decisions or actions for maintaining or adjusting output  214  for the entity (e.g., interacting with the customer in a way that reduces that type of behavior and/or the customer&#39;s risk or churning). 
     By using a single linear model  212  and feature values  208  of features  222 - 224  inputted into statistical model  206  to characterize the behavior or performance of statistical model  206 , the system of  FIG. 2  may interpret specific output  214  values of statistical model  206  with respect to individual sets of feature values  208  used to generate those output values. On the other hand, conventional model-interpretation techniques may interpret the local behavior of a statistical model by fitting a separate linear model to each set of feature values and corresponding output value generated from the set of feature values, which may incur significant computational overhead and fail to scale with the amount of output  214  generated by statistical model  206 . Consequently, the system may improve the computational efficiency associated with characterizing local statistical model behavior while providing additional context for understanding global statistical model behavior, thereby improving technologies and computer systems used in developing and using statistical models and features. 
     Those skilled in the art will appreciate that the system of  FIG. 2  may be implemented in a variety of ways. First, analysis apparatus  202 , management apparatus  204 , and/or data repository  234  may be provided by a single physical machine, multiple computer systems, one or more virtual machines, a grid, one or more databases, one or more filesystems, and/or a cloud computing system. Analysis apparatus  202  and management apparatus  204  may additionally be implemented together and/or separately by one or more hardware and/or software components and/or layers. 
     Second, feature values  208 , output  214 , coefficients  220 , global contributions  230 , local contributions  232 , and/or other data used by the system may be stored, defined, and/or transmitted using a number of techniques. For example, the system may be configured to accept features from different types of repositories, including relational databases, graph databases, data warehouses, filesystems, and/or flat files. The system may also obtain and/or transmit feature names, feature namespaces, feature sources, feature values  208 , output  214 , coefficients  220 , global contributions  230 , local contributions  232 , and/or other data used to characterize the performance of statistical model  206  in a number of formats, including database records, property lists, Extensible Markup language (XML) documents, JavaScript Object Notation (JSON) objects, and/or other types of structured data. 
       FIG. 3  shows a flowchart illustrating the processing of data in accordance with the disclosed embodiments. In one or more embodiments, one or more of the steps may be omitted, repeated, and/or performed in a different order. Accordingly, the specific arrangement of steps shown in  FIG. 3  should not be construed as limiting the scope of the embodiments. 
     Initially, a linear model is built using multiple sets of feature values of features inputted into a statistical model and multiple output values from the statistical model (operation  302 ). For example, the linear model may be trained using the feature values to estimate the output of the statistical model. Next, measures of feature importance from the statistical model are obtained as global contributions of features represented by the feature values toward the output of the statistical model (operation  304 ). As a result, a larger measure of feature importance (e.g., linear model coefficient, feature weight, random forest impurity decrease, etc.) may represent a higher contribution of the corresponding feature toward the output of the statistical model, and a smaller measure of feature importance may represent a lower contribution of the corresponding feature toward the output generated by the statistical model. 
     The coefficients are also combined with a set of feature values of the features to obtain a set of local contributions of the features toward the output of the statistical model (operation  306 ). For example, the local contributions may represent the effect of a specific set of feature values on a specific output value generated by the statistical model. As a result, the local contribution of each feature may be determined by multiplying a variable feature value of the feature by the corresponding fixed coefficient from the linear model. 
     Rankings of the features by the local contributions and global contributions are generated (operation  308 ), and subsets of features with the highest local and global contributions are selected from the rankings based on one or more parameters (operation  310 ). For example, the subsets of features may be selected based on a parameter that specifies a percentage or proportion (e.g., 30%, ⅕ th , 0.25, etc.) of the number of features to be included in one or both subsets. In another example, the subsets of features may be selected based on a parameter that places a feature in the first or second subset and/or specifies selection of one subset of features before the other. As a result, a feature that is ranked highly in both the local and global contributions may be selected for inclusion in the first subset and excluded from the second subset, allowing the second subset to include features that are not highly ranked with respect to contributions in the first subset. 
     The selected subsets are then outputted with attributes of features in the subsets (operation  312 ). For example, feature names and/or feature values of the features may be outputted, along with values of the corresponding local and global contributions and/or metrics (e.g., proportions, percentiles, etc.) associated with the contributions or feature values. The outputted data may be used to characterize both the local and global behavior, performance, or output of the statistical model. 
     Model performance may continue to be characterized (operation  314 ) with respect to different sets of feature values and the corresponding output values. For example, the performance of the statistical model may be characterized for each of a set of predictions, scores, or inferences made by the statistical model. If the performance of the statistical model is to be characterized with remaining sets of feature and output values, a different set of local contributions is calculated for each set of feature values (operation  306 ), and subsets of features with the highest local and global contributions are selected from the corresponding rankings (operations  308 - 310 ). The selected subsets of features and the corresponding attributes are then outputted (operation  312 ). Operations  306 - 312  may be repeated until the performance of the statistical model is characterized with respect to all relevant sets of feature values. 
       FIG. 4  shows a computer system  400  in accordance with the disclosed embodiments. Computer system  400  includes a processor  402 , memory  404 , storage  406 , and/or other components found in electronic computing devices. Processor  402  may support parallel processing and/or multi-threaded operation with other processors in computer system  400 . Computer system  400  may also include input/output (I/O) devices such as a keyboard  408 , a mouse  410 , and a display  412 . 
     Computer system  400  may include functionality to execute various components of the present embodiments. In particular, computer system  400  may include an operating system (not shown) that coordinates the use of hardware and software resources on computer system  400 , as well as one or more applications that perform specialized tasks for the user. To perform tasks for the user, applications may obtain the use of hardware resources on computer system  400  from the operating system, as well as interact with the user through a hardware and/or software framework provided by the operating system. 
     In one or more embodiments, computer system  400  provides a system for processing data. The system may include an analysis apparatus and a management apparatus, one or more of which alternatively be termed or implemented as a module, mechanism, or other type of system component. The analysis apparatus obtains a set of coefficients from a linear model that uses a set of features inputted into a statistical model to estimate an output of the statistical model. Next, the analysis apparatus combines the set of coefficients with a set of feature values of the features to obtain a set of local contributions of the features toward the output of the statistical model. The management apparatus then outputs a first subset of the features with highest local contributions toward the output of the statistical model for use in characterizing a local performance of the statistical model. The management apparatus also outputs a second subset of the features with highest coefficients from the linear model for use in characterizing a global performance of the statistical model. 
     In addition, one or more components of computer system  400  may be remotely located and connected to the other components over a network. Portions of the present embodiments (e.g., analysis apparatus, management apparatus, data repository, etc.) may also be located on different nodes of a distributed system that implements the embodiments. For example, the present embodiments may be implemented using a cloud computing system that uses a linear model and different sets of feature values to characterize the local and global performance of a remote statistical model. 
     The foregoing descriptions of various embodiments have been presented only for purposes of illustration and description. They are not intended to be exhaustive or to limit the present invention to the forms disclosed. Accordingly, many modifications and variations will be apparent to practitioners skilled in the art. Additionally, the above disclosure is not intended to limit the present invention.