Mixture model

About: Mixture model is a research topic. Over the lifetime, 18155 publications have been published within this topic receiving 588317 citations.

Deciding on the Number of Classes in Latent Class Analysis and Growth Mixture Modeling: A Monte Carlo Simulation Study

Abstract: Mixture modeling is a widely applied data analysis technique used to identify unobserved heterogeneity in a population. Despite mixture models' usefulness in practice, one unresolved issue in the application of mixture models is that there is not one commonly accepted statistical indicator for deciding on the number of classes in a study population. This article presents the results of a simulation study that examines the performance of likelihood-based tests and the traditionally used Information Criterion (ICs) used for determining the number of classes in mixture modeling. We look at the performance of these tests and indexes for 3 types of mixture models: latent class analysis (LCA), a factor mixture model (FMA), and a growth mixture models (GMM). We evaluate the ability of the tests and indexes to correctly identify the number of classes at three different sample sizes (n = 200, 500, 1,000). Whereas the Bayesian Information Criterion performed the best of the ICs, the bootstrap likelihood ratio test ...

Adaptive background mixture models for real-time tracking

Abstract: A common method for real-time segmentation of moving regions in image sequences involves "background subtraction", or thresholding the error between an estimate of the image without moving objects and the current image. The numerous approaches to this problem differ in the type of background model used and the procedure used to update the model. This paper discusses modeling each pixel as a mixture of Gaussians and using an on-line approximation to update the model. The Gaussian, distributions of the adaptive mixture model are then evaluated to determine which are most likely to result from a background process. Each pixel is classified based on whether the Gaussian distribution which represents it most effectively is considered part of the background model. This results in a stable, real-time outdoor tracker which reliably deals with lighting changes, repetitive motions from clutter, and long-term scene changes. This system has been run almost continuously for 16 months, 24 hours a day, through rain and snow.

Fitting a mixture model by expectation maximization to discover motifs in biopolymers.

Abstract: The algorithm described in this paper discovers one or more motifs in a collection of DNA or protein sequences by using the technique of expectation maximization to fit a two-component finite mixture model to the set of sequences Multiple motifs are found by fitting a mixture model to the data, probabilistically erasing the occurrences of the motif thus found, and repeating the process to find successive motifs The algorithm requires only a set of unaligned sequences and a number specifying the width of the motifs as input It returns a model of each motif and a threshold which together can be used as a Bayes-optimal classifier for searching for occurrences of the motif in other databases The algorithm estimates how many times each motif occurs in each sequence in the dataset and outputs an alignment of the occurrences of the motif The algorithm is capable of discovering several different motifs with differing numbers of occurrences in a single dataset

Applied Missing Data Analysis

Abstract: Part 1. An Introduction to Missing Data. 1.1 Introduction. 1.2 Chapter Overview. 1.3 Missing Data Patterns. 1.4 A Conceptual Overview of Missing Data heory. 1.5 A More Formal Description of Missing Data Theory. 1.6 Why Is the Missing Data Mechanism Important? 1.7 How Plausible Is the Missing at Random Mechanism? 1.8 An Inclusive Analysis Strategy. 1.9 Testing the Missing Completely at Random Mechanism. 1.10 Planned Missing Data Designs. 1.11 The Three-Form Design. 1.12 Planned Missing Data for Longitudinal Designs. 1.13 Conducting Power Analyses for Planned Missing Data Designs. 1.14 Data Analysis Example. 1.15 Summary. 1.16 Recommended Readings. Part 2. Traditional Methods for Dealing with Missing Data. 2.1 Chapter Overview. 2.2 An Overview of Deletion Methods. 2.3 Listwise Deletion. 2.4 Pairwise Deletion. 2.5 An Overview of Single Imputation Techniques. 2.6 Arithmetic Mean Imputation. 2.7 Regression Imputation. 2.8 Stochastic Regression Imputation. 2.9 Hot-Deck Imputation. 2.10 Similar Response Pattern Imputation. 2.11 Averaging the Available Items. 2.12 Last Observation Carried Forward. 2.13 An Illustrative Simulation Study. 2.14 Summary. 2.15 Recommended Readings. Part 3. An Introduction to Maximum Likelihood Estimation. 3.1 Chapter Overview. 3.2 The Univariate Normal Distribution. 3.3 The Sample Likelihood. 3.4 The Log-Likelihood. 3.5 Estimating Unknown Parameters. 3.6 The Role of First Derivatives. 3.7 Estimating Standard Errors. 3.8 Maximum Likelihood Estimation with Multivariate Normal Data. 3.9 A Bivariate Analysis Example. 3.10 Iterative Optimization Algorithms. 3.11 Significance Testing Using the Wald Statistic. 3.12 The Likelihood Ratio Test Statistic. 3.13 Should I Use the Wald Test or the Likelihood Ratio Statistic? 3.14 Data Analysis Example 1. 3.15 Data Analysis Example 2. 3.16 Summary. 3.17 Recommended Readings. Part 4. Maximum Likelihood Missing Data Handling. 4.1 Chapter Overview. 4.2 The Missing Data Log-Likelihood. 4.3 How Do the Incomplete Data Records Improve Estimation? 4.4 An Illustrative Computer Simulation Study. 4.5 Estimating Standard Errors with Missing Data. 4.6 Observed Versus Expected Information. 4.7 A Bivariate Analysis Example. 4.8 An Illustrative Computer Simulation Study. 4.9 An Overview of the EM Algorithm. 4.10 A Detailed Description of the EM Algorithm. 4.11 A Bivariate Analysis Example. 4.12 Extending EM to Multivariate Data. 4.13 Maximum Likelihood Software Options. 4.14 Data Analysis Example 1. 4.15 Data Analysis Example 2. 4.16 Data Analysis Example 3. 4.17 Data Analysis Example 4. 4.18 Data Analysis Example 5. 4.19 Summary. 4.20 Recommended Readings. Part 5. Improving the Accuracy of Maximum Likelihood Analyses. 5.1 Chapter Overview. 5.2 The Rationale for an Inclusive Analysis Strategy. 5.3 An Illustrative Computer Simulation Study. 5.4 Identifying a Set of Auxiliary Variables. 5.5 Incorporating Auxiliary Variables Into a Maximum Likelihood Analysis. 5.6 The Saturated Correlates Model. 5.7 The Impact of Non-Normal Data. 5.8 Robust Standard Errors. 5.9 Bootstrap Standard Errors. 5.10 The Rescaled Likelihood Ratio Test. 5.11 Bootstrapping the Likelihood Ratio Statistic. 5.12 Data Analysis Example 1. 5.13 Data Analysis Example 2. 5.14 Data Analysis Example 3. 5.15 Summary. 5.16 Recommended Readings. Part 6. An Introduction to Bayesian Estimation. 6.1 Chapter Overview. 6.2 What Makes Bayesian Statistics Different? 6.3 A Conceptual Overview of Bayesian Estimation. 6.4 Bayes' Theorem. 6.5 An Analysis Example. 6.6 How Does Bayesian Estimation Apply to Multiple Imputation? 6.7 The Posterior Distribution of the Mean. 6.8 The Posterior Distribution of the Variance. 6.9 The Posterior Distribution of a Covariance Matrix. 6.10 Summary. 6.11 Recommended Readings. Part 7. The Imputation Phase of Multiple Imputation. 7.1 Chapter Overview. 7.2 A Conceptual Description of the Imputation Phase. 7.3 A Bayesian Description of the Imputation Phase. 7.4 A Bivariate Analysis Example. 7.5 Data Augmentation with Multivariate Data. 7.6 Selecting Variables for Imputation. 7.7 The Meaning of Convergence. 7.8 Convergence Diagnostics. 7.9 Time-Series Plots. 7.10 Autocorrelation Function Plots. 7.11 Assessing Convergence from Alternate Starting Values. 7.12 Convergence Problems. 7.13 Generating the Final Set of Imputations. 7.14 How Many Data Sets Are Needed? 7.15 Summary. 7.16 Recommended Readings. Part 8. The Analysis and Pooling Phases of Multiple Imputation. 8.1 Chapter Overview. 8.2 The Analysis Phase. 8.3 Combining Parameter Estimates in the Pooling Phase. 8.4 Transforming Parameter Estimates Prior to Combining. 8.5 Pooling Standard Errors. 8.6 The Fraction of Missing Information and the Relative Increase in Variance. 8.7 When Is Multiple Imputation Comparable to Maximum Likelihood? 8.8 An Illustrative Computer Simulation Study. 8.9 Significance Testing Using the t Statistic. 8.10 An Overview of Multiparameter Significance Tests. 8.11 Testing Multiple Parameters Using the D1 Statistic. 8.12 Testing Multiple Parameters by Combining Wald Tests. 8.13 Testing Multiple Parameters by Combining Likelihood Ratio Statistics. 8.14 Data Analysis Example 1. 8.15 Data Analysis Example 2. 8.16 Data Analysis Example 3. 8.17 Summary. 8.18 Recommended Readings. Part 9. Practical Issues in Multiple Imputation. 9.1 Chapter Overview. 9.2 Dealing with Convergence Problems. 9.3 Dealing with Non-Normal Data. 9.4 To Round or Not to Round? 9.5 Preserving Interaction Effects. 9.6 Imputing Multiple-Item Questionnaires. 9.7 Alternate Imputation Algorithms. 9.8 Multiple Imputation Software Options. 9.9 Data Analysis Example 1. 9.10 Data Analysis Example 2. 9.11 Summary. 9.12 Recommended Readings. Part 10. Models for Missing Not at Random Data. 10.1 Chapter Overview. 10.2 An Ad Hoc Approach to Dealing with MNAR Data. 10.3 The Theoretical Rationale for MNAR Models. 10.4 The Classic Selection Model. 10.5 Estimating the Selection Model. 10.6 Limitations of the Selection Model. 10.7 An Illustrative Analysis. 10.8 The Pattern Mixture Model. 10.9 Limitations of the Pattern Mixture Model. 10.10 An Overview of the Longitudinal Growth Model. 10.11 A Longitudinal Selection Model. 10.12 Random Coefficient Selection Models. 10.13 Pattern Mixture Models for Longitudinal Analyses. 10.14 Identification Strategies for Longitudinal Pattern Mixture Models. 10.15 Delta Method Standard Errors. 10.16 Overview of the Data Analysis Examples. 10.17 Data Analysis Example 1. 10.18 Data Analysis Example 2. 10.19 Data Analysis Example 3. 10.20 Data Analysis Example 4. 10.21 Summary. 10.22 Recommended Readings. Part 11. Wrapping Things Up: Some Final Practical Considerations. 11.1 Chapter Overview. 11.2 Maximum Likelihood Software Options. 11.3 Multiple Imputation Software Options. 11.4 Choosing between Maximum Likelihood and Multiple Imputation. 11.5 Reporting the Results from a Missing Data Analysis. 11.6 Final Thoughts. 11.7 Recommended Readings.