Cosma Shalizi

Statistics 36-350: Data Mining

Fall 2009

Important update, December 2011

If you are looking for the latest version of this class, it is 36-462, taught by Prof. Tibshirani in the spring of 2012. 36-350 is now the course number for Introduction to Statistical Computing.

Data mining is the art of extracting useful patterns from large bodies of data; finding seams of actionable knowledge in the raw ore of information. The rapid growth of computerized data, and the computer power available to analyze it, creates great opportunities for data mining in business, medicine, science, government, etc. The aim of this course is to help you take advantage of these opportunities in a responsible way. After taking the class, when you're faced with a new problem, you should be able to (1) select appropriate methods, and justify their choice, (2) use and program statistical software to implement them, and (3) critically evaluate the results and communicate them to colleagues in business, science, etc.

Data mining is related to statistics and to machine learning, but has its own aims and scope. Statistics is a mathematical science, studying how reliable inferences can be drawn from imperfect data. Machine learning is a branch of engineering, developing a technology of automated induction. We will freely use tools from statistics and from machine learning, but we will use them as tools, not things to study in their own right. We will do a lot of calculations, but will not prove many theorems, and we will do even more experiments than calculations.

Outline, Notes, Readings

This is a rough outline of the material; details may change depending on time and class interests.
  1. Searching by similarity: Searching by content (texts, images, genes, ...); attributes, representations and definitions of similarity and distance; choice of representation; multi-dimensional scaling; classifications; image search and invariants; user feedback; evaluating searches
  2. Information: information and uncertainty; classes and attributes; interactions among attributes
  3. Clustering: supervised and unsupervised learning; categorization; unsupervised category-learning, a.k.a. clustering; k-means clustering; hierarchical clustering; geometry of clusters; what makes a good cluster?
  4. Data-reduction and feature-enhancement: Standardizing data; using principal components to eliminate attributes; using factor analysis to eliminate attributes; limits and pitfalls of PCA and factor analysis; nonlinear dimensionality reduction: local linear embedding, diffusion maps
  5. Regression Review of linear regression; transformations to linearity; the truth about linear regression; local linear regression; polynomial regression; kernel regression; additive models; other non-parametric methods
  6. Prediction: Evaluating predictive models; over-fitting and capacity control; regression trees; classification trees; combining predictive models; forests; how to gamble if you must
  7. Classification: Supervised categorization; linear classifiers; logistic regression; the kernel trick; base rates, Neyman-Pearson classifiers, ROC curves
  8. Distributions: Histograms and the fundamental theorem of statistics; kernel density estimation; conditional density estimation; relative distributions; mixture models, probabilistic clustering, the EM algorithm; clustering with confidence; large numbers of rare events
  9. Modeling interventions: Estimating causal impacts without experiments; matching; graphical causal models and Tetrad.
  10. Waste and Abuse: when data mining will fail: bad data, wrong data, insufficient data, overwhelming false positives, impossible problems, attacking the wrong problem; when data mining is evil; some failures

Lecture Notes and Supplementary Readings

Note: Solutions are temporarily off-line, since some of the exercises are being re-used in this semester's class. Feel free to write me directly, however.

  1. Introduction to the course (24 August) What is data mining? how is it used? where did it come from? Some themes.
  2. Information retrieval and searching by similarity (26 August) Finding data by content. Approaches we will neglect: metadata, coding. Textual features. The bag-of-words representation. Vector representations of documents. Measures of similarity and the importance of normalization. Our first stab at classification. Example with the New York Times Annotated Corpus.
  3. IR continued (28 August). The trick to searching: queries are documents. Search evaluation: precision, recall, precision-recall curves; error rates. Classification: nearest neighbors and prototypes; classifier evaluation by mis-classification rate and by confusion matrices. Inverse document frequency weighting. Visualizing high-dimensional data by multi-dimensional scaling. Miscellaneous topics: stemming, incorporating user feedback.

    Homework 1, due 4 September: assignment, R, data

  4. Page Rank (31 August). Links as pre-existing feedback. How to exploit link information? The random walk on the graph; using the ergodic theorem. Eigenvector formulation of page-rank. Combining page-rank with textual features. Other applications. Further reading on information retrieval.
  5. Image Search, Abstraction and Invariance (2 September). Similarity search for images. Back to representation design. The advantages of abstraction: simplification, recycling. The bag-of-colors representation. Examples. Invariants. Searching for images by searching text. An example in practice. Slides for this lecture.
  6. Information Theory I (4 September). Good features help us guess what we can't represent. Good features discriminate between different values of unobserved variables. Quantifying uncertainty with entropy. Quantifying reduction in uncertainty/ discrimination with mutual information. Ranking features based on mutual information. Examples, with code, of informative words for the Times. Code.
    Supplementary reading: David P. Feldman, Brief Tutorial on Information Theory, chapter 1

    Homework 2, due 11 September: assignment

  7. Information Theory II (9 September). Dealing with multiple features. Joint entropy, the chain rule for entropy. Information in multiple features. Conditional information, chain rule for information, conditional independence. Interactions, positive and negative, and redundancy. Greedy feature selection with low redundancy. Example, with code, of selecting words for the Times. Sufficient statistics and the information bottleneck. Code.
    Supplementary reading; Aleks Jakulin and Ivan Bratko, "Quantifying and Visualizing Attribute Interactions", arxiv:cs.AI/0308002
  8. Categorization; Clustering I (11 September). Dividing the world up into categories. Classification: known categories with labeled examples. Taxonomy of learning problems (supervised, unsupervised, semi-supervised, feedback, ...). Clustering: discovering unknown categories from unlabeled data. Benefits of clustering, with an digression on where official classes come from. Basic criterion for good clusters: lots of information about features from little information about cluster. Practical considerations: compactness, separation, parsimony, balance. Doubts about parsimony and balance. The k-means clustering algorithm, or unlabeled prototype classification: analysis, geometry, search. Appendix: geometric aspects of the prototype and nearest-neighbor method.

    Homework 3, due 18 September: assignment

  9. Clustering II (14 September). Distances between partitions; variation-of-information distance. Hierarchical clustering by agglomeration and its varieties. Picking the number of clusters by merging costs. Performance of different clustering methods on various doodles. Why we would like to pick the number of clusters by predictive performance, and why it is hard to do at this stage. Reifying clusters.
  10. Transformations: Rescaling and Low-Dimensional Summaries (16 September). Improving on our original features. Re-scaling, standardization, taking logs, etc., of individual features. Forcing things to be Gaussian considered harmful. Low-dimensional summaries by combining features. Exploiting geometry to eliminate redundancy. Projections on to linear subspaces. Searching for structure-preserving projections.
  11. Principal Components I (18 September). Principal components are the directions of maximum variance. Derivation of principal components as the best approximation to the data in a linear subspace. Equivalence to variance maximization. Avoiding explicit optimization by finding eigenvalues and eigenvectors of the covariance matrix. Example of principal components with cars; how to tell a sports car from a minivan. The standard recipe for doing PCA. Cautions in interpreting PCA. Data-set used in the notes.

    Homework 4, due 25 September: assignment

  12. Principal Components II (21 September). PCA + information retrieval = latent semantic indexing; why LSI is a Good Idea. PCA and multidimensional scaling.
  13. Factor Analysis (23 and 25 September). From PCA to factor analysis by adding noise. Roots of factor analysis in causal discovery: Spearman's general factor model and the tetrad equations. Problems with estimating factor models: more unknowns than equations. Solution 1, "principal factors", a.k.a. estimation through heroic feats of linear algebra. Solution 2, maximum likelihood, a.k.a. estimation through imposing distributional assumptions. The rotation problem: the factor model is unidentifiable; the number of factors may be meaningful, but the individual factors are not.
  14. The Truth about PCA and Factor Analysis (28 September) PCA is data reduction without any probabilistic assumptions about where the data came from. Picking number of components. Faking predictions from PCA. Factor analysis makes stronger, probabilistic assumptions, and delivers stronger, predictive conclusions --- which could be wrong. Using probabilistic assumptions and/or predictions to pick how many factors. Factor analysis as a first, toy instances of a graphical causal model. The rotation problem once more with feeling. Factor models and mixture models. Factor models and Thomson's sampling model: an outstanding fit to a model with a few factors is actually evidence of a huge number of badly measured latent variables. Final advice: it all depends, but if you can only do one, try PCA. R code for the Thomson sampling model.
  15. Nonlinear Dimensionality Reduction I: Locally Linear Embedding (5 and 7 October). Failure of PCA and all other linear methods for nonlinear structures in data; spirals, for example. Approximate success of linear methods on small parts of nonlinear structures. Manifolds: smoothly curved surfaces embedded in higher-dimensional Euclidean spaces. Every manifold looks like a linear subspace on a sufficiently small scale, so we should be able to patch together many small local linear approximations into a global manifold. Local linear embedding: approximate each vector in the data as a weighted linear combination of its k nearest neighbors, then find the low-dimensional vectors best reconstructed by these weights. Turning the optimization problems into linear algebra. Coding up LLE. A spiral rainbow. R.
  16. Nonlinear Dimensionality Reduction II: Diffusion Maps (9 October). Making a graph from the data; random walks on this graph. The diffusion operator, a.k.a. Laplacian. How the Laplacian encodes the shape of the data. Eigenvectors of the Laplacian as coordinates. Connection to page-rank. Advantages when data are not actually on a manifold. Example.

    Pre-midterm review (12 October): highlights of the course to date; no handout.
    MIDTERM (14 October): exam

    Homework 5, due 23 October: assignment

  17. Regression I: Basics (19 October). Guessing a real-valued random variable; why expectation values are mean-square optimal point forecasts. The regression function; why its estimation must involve assumptions beyond the data. The bias-variance decomposition and the bias-variance trade-off. First example of improving prediction by introducing variance. Ordinary least squares linear regression as smoothing. Other linear smoothers: k-nearest-neighbors and kernel regression. How much should we smooth? R, data for running example
  18. Regression II: The Truth About Linear Regression (21 October). Linear regression is optimal linear (mean-square) prediction; we do this because we hope a linear approximation will work well enough over a small range. What linear regression does: decorrelate the input features, then correlate them separately with the response and add up. The extreme weakness of the probabilistic assumptions needed for this to make sense. Difficulties of linear regression; collinearity, errors in variables, shifting distributions of inputs, omitted variables. The usual extra probabilistic assumptions and their implications. Why you should always looking at residuals. Why you generally shouldn't use regression for causal inference. How to torment angels. Likelihood-ratio tests for restrictions of nice models.
  19. Regression III: Extending Linear Regression (23 October). Weighted least squares. Heteroskedasticity: variance is not the same everywhere. Going to consult the oracle. Weighted least squares as a solution to heteroskedasticity. Nonparametric estimation of the variance function. Local polynomial regression: local constants = kernel regression, local linear regression, higher-order local polynomials. Lowess = locally-linear smoothing for scatter plots. The oracles fall silent.

    Homework 6, due Friday, 30 October: assignment, data set

  20. Evaluating Predictive Models (26 and 28 October). In-sample, out-of-sample and generalization loss or error; risk as expected loss on new data. Under-fitting, over-fitting, and examples with polynomials. Methods of model selection and controlling over-fitting: empirical risk minimization, penalization, constraints/sieves, formal learning theory, cross-validation. Limits of generalization. R for creating figures.
  21. Smoothing Methods in Regression (30 October). How much smoothing should we do? Approximation by local averaging. How much smoothing we should do to find the unknown curve depends on how smooth the curve really is, which is unknown. Adaptation as a partial substitute for actual knowledge. Cross-validation for adapting to unknown smoothness. Application: testing parametric regression models by comparing them to nonparametric fits. The bootstrap principle. Why ever bother with parametric regressions? R code for some of the examples.

    Homework 7, due Friday, 6 November: assignment

  22. Additive Models (2 November). A nice feature of linear models: partial responses, partial residuals, and backfitting estimations. Additive models: regression curve is a sum of partial response functions; partial residuals and the backfitting trick generalize. Parametric and non-parametric rates of convergence. The curse of dimensionality for unstructured nonparametric models. Additive models as a compromise, introducing bias to reduce variance. Example with the data from homework 6.
  23. Classification and Regression Trees (4 and 6 November). Prediction trees. A classification tree we can believe in. Prediction trees combine simple local models with recursive partitioning; adaptive nearest neighbors. Regression trees: example; a little math; pruning by cross-validation; more R mechanics. Classification trees: basics; measuring error by mis-classification; weighted errors; likelihood; Neyman-Pearson classifiers. Uncertainty for trees.

    Homework 8, due 5 pm on Monday, 16 November: assignment

  24. Combining Models 1: Bagging and Model Averaging (9 November)
  25. Combining Models 2: Diversity and Boosting (11 November)
  26. Linear Classifiers (16 November). Geometry of linear classifiers. The perceptron algorithm for learning linear classifiers. The idea of "margin".
  27. Logistic Regression (18 November). Attaching probabilities to linear classifiers: why would we want to? why would we use the logistic transform to do so? More-than-binary logistic regression. Maximizing the likelihood; Newton's method for optimization. Generalized linear models and generalized additive models; testing GLM specifications with GAMs.
  28. Support Vector Machines (20 November). Turning nonlinear problems into linear ones by expanding into high-dimensional feature spaces. The dual representation of linear classifiers: weight training points, not features. Observation: in the dual representation, only inner products of vectors matter. The kernel trick: kernel functions let us compute inner products in feature spaces without computing the features. Some bounds on the generalization error of linear classifiers based on "margin" and the number of training points with non-zero weight ("support vectors"). Learning support vector machines by trading in-sample performance against bounds on over-fitting.

    Homework 9, due at 5 pm on Monday, 30 November: assignment

  29. Density Estimation (23 November). Histograms as distribution estimates. Glivenko-Cantelli, "the fundamental theorem of statistics". Histograms as density estimates; selecting density estimates by cross-validation. Kernel density estimates. Why kernels are better than histograms. Curse of dimensionality again. Hint at alternatives to kernel density estimates.
  30. Mixture Models, Latent Variables and the EM Algorithm (30 November). Compressing and restricting density estimates. Mixtures of limited numbers of distributions. Mixture models as probabilistic clustering; finally an answer to "how many clusters?" The EM algorithm as an iterative way of maximizing likelihood with latent variables. Analogy to k-means. More theory of the EM algorithm. Applications: density mixtures, signal processing/state estimation, mixtures of regressions, mixtures of experts; topic models and probabilistic latent semantic analysis. A glance at non-parametric mixture models.
  31. Graphical Causal Models (2 December). Distinction between causation and association, and between causal and probabilistic prediction. Some examples. Directed acyclic graphs and causal models. The Markov property. Conditional independence via separation. Faithfulness.
  32. Causal Inference (4 December). Estimating causal effects; control for confounding. Discovering causal structure: the SGS algorithm and its variants. Limitations.

    Take-home final exam, due 15 December: assignment; data sets: expressdb_cleaned (20 Mb), HuIyer_TFKO_expression (20 Mb). With great thanks to Dr. Timothy Danford.

Previous versions of the course

With old notes, can be found here.

Course Mechanics

Details on grading, exams, etc., can be found in the full syllabus.

Blackboard

Homework, solutions, grades, class announcements, etc., will be handled through Blackboard; let me know if you cannot access the course site.

Textbook

Our required textbook is Principles of Data Mining by Hand, Mannila and Smyth. It should be at the campus bookstore already, but you can also buy it online (I like Powell's), or directly from MIT Press.

Berk's Statistical Learning from a Regression Perspective (Powell's; publisher) is an optional book, which covers some topics (mostly from the second half of the course) in greater detail.

R

R is a free, open-source software package/programming language for statistical computing. (It is a dialect of a language called S, whose commercial version is S-plus.) You can expect at least one assignment every week which uses R. If you do not have ready access to a computer which can run it, contact me at once.

Here are some resources for learning R:

You should also read the page of Minimal Advice on Programming.