We construct @math kernel, which generalizes the classical Gaussian RBF kernel to the case of incomplete data. We model the uncertainty contained in missing attributes making use of data distribution and associate every point with a conditional probability density function. This allows to embed incomplete data into the function space and to define a kernel between two missing data points based on scalar product in @math . Experiments show that introduced kernel applied to SVM classifier gives better results than other state-of-the-art methods, especially in the case when large number of features is missing. Moreover, it is easy to implement and can be used together with any kernel approaches with no additional modifications.
The most common approach to learning from incomplete data is known as deterministic imputation @cite_11 . In this two-step procedure, the missing features are filled first, and only then a standard classifier is applied to the complete data @cite_18 . Although the imputation-based techniques are easy to use for practitioners, they lead to the loss of information which features were missing and do not take into account the reasons of missingness. To preserve the information of missing attributes, one can use an additional vector of binary flags, indicating which coordinates were missing.
@cite_18 @cite_11
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Preface.PART I: OVERVIEW AND BASIC APPROACHES.Introduction.Missing Data in Experiments.Complete-Case and Available-Case Analysis, Including Weighting Methods.Single Imputation Methods.Estimation of Imputation Uncertainty.PART II: LIKELIHOOD-BASED APPROACHES TO THE ANALYSIS OF MISSING DATA.Theory of Inference Based on the Likelihood Function.Methods Based on Factoring the Likelihood, Ignoring the Missing-Data Mechanism.Maximum Likelihood for General Patterns of Missing Data: Introduction and Theory with Ignorable Nonresponse.Large-Sample Inference Based on Maximum Likelihood Estimates.Bayes and Multiple Imputation.PART III: LIKELIHOOD-BASED APPROACHES TO THE ANALYSIS OF MISSING DATA: APPLICATIONS TO SOME COMMON MODELS.Multivariate Normal Examples, Ignoring the Missing-Data Mechanism.Models for Robust Estimation.Models for Partially Classified Contingency Tables, Ignoring the Missing-Data Mechanism.Mixed Normal and Nonnormal Data with Missing Values, Ignoring the Missing-Data Mechanism.Nonignorable Missing-Data Models.References.Author Index.Subject Index. Part 1. A Gentle Introduction to Missing Data. The Concept of Missing Data. The Prevalence of Missing Data. Why Data Might Be Missing. The Impact of Missing Data. What's Missing in the Missing Data Literature? A Cost-Benefit Approach to Missing Data. Missing Data - Not Just for Statisticians Anymore. Part 2. Consequences of Missing Data. Three General Consequences of Missing Data. Consequences of Missing Data on Construct Validity. Consequences of Missing Data on Internal Validity. Consequences on Causal Generalization. Summary. Part 3. Classifying Missing Data. "The Silence That Betokens". The Current Classification System: Mechanisms of Missing Data. Expanding the Classification System. Summary. Part 4. Preventing Missing Data by Design. Overall Study Design. Characteristics of the Target Population and the Sample. Data Collection and Measurement. Treatment Implementation. Data Entry Process. Summary. Part 5. Diagnostic Procedures. Traditional Diagnostics. Dummy Coding Missing Data. Numerical Diagnostic Procedures. Graphical Diagnostic Procedures. Summary. Part 6. The Selection of Data Analytic Procedures. Preliminary Steps. Decision Making. Summary. Part 7. Data Deletion Methods for Handling Missing Data. Data Sets. Complete Case Method. Available Case Method. Available Item Method. Individual Growth Curve Analysis. Multisample Analyses. Summary. Part 8. Data Augmentation Procedures. Model-Based Procedures. Markov Chain Monte Carlo. Adjustment Methods. Summary. Part 9. Single Imputation Procedures. Constant Replacement Methods. Random Value Imputation. Nonrandom Value Imputation: Single Condition. Nonrandom Value Imputation: Multiple Conditions. Summary. Part 10. Multiple Imputation. The MI Process. Summary. Part 11. Reporting Missing Data and Results. APA Task Force Recommendations. Missing Data and Study Stages. TFSI Recommendations and Missing Data. Reporting Format. Summary. Part 12. Epilogue.