# Statistical Models in Data Mining

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1 Statistical Models in Data Mining Sargur N. Srihari University at Buffalo The State University of New York Department of Computer Science and Engineering Department of Biostatistics 1 Srihari

2 Flood of Data New York Times, January 11, 2010 Video and Image Data Unstructured Structured and Unstructured (Text) Data 2 Srihari

3 Large Data Sets are Ubiquitous 1.Due to digital data acquisition and storage technology Business Supermarket transactions Credit card usage records Telephone call details Government statistics Scientific Images of astronomical bodies Molecular databases Medical records 2. Automatic data production leads to need for automatic data consumption 3. Large databases mean vast amounts of information 4. Difficulty lies in converting data to useful knowledge 3 Srihari

4 Data Mining Definition Analyze Observational Data to find unsuspected relationships and Summarize data in novel ways that are understandable and useful to data owner Unsuspected Relationships non-trivial, implicit, previously unknown Ex of Trivial: Those who are pregnant are female Summarize as Patterns and Models (usually probabilistic) Linear Equations, Rules, Clusters, Graphs, Tree Structures, Recurrent Patterns in Time Series Extracting useful information from large data sets Usefulness: meaningful: lead to some advantage, usually economic Analysis: Automatic/Semi-automatic Process (Extraction of knowledge) Srihari

5 Reasons for Uncertainty 1. Data may only be a sample of population to be studied Uncertain about extent to which samples differ from each other 2. Interest is in making a prediction about tomorrow based on data we have today 3. Cannot observe some values and need to make a guess 5 Srihari

6 Dealing with Uncertainty Several Conceptual bases 1. Probability 2. Fuzzy Sets 3. Rough Sets Probability Theory vs Probability Calculus Probability Calculus is well-developed Generally accepted axioms and derivations Probability Theory has scope for perspectives 6 Lack theoretical backbone and the wide acceptance of probability Mapping real world to what probability is

7 Frequentist vs Bayesian Frequentist Probability is objective It is the limiting proportion of times event occurs in identical situations Bayesian An idealization since all customers are not identical Subjective probability Explicit characterization of all uncertainty including any parameters estimated from the data Frequently yield same results 7 Srihari

8 Data Mining vs Statistics Observational Data Objective of data mining exercise plays no role in data collection strategy E.g., Data collected for Transactions in a Bank Experimental Data Collected in Response to Questionnaire Efficient strategies to Answer Specific Questions In this way it differs from much of statistics For this reason, data mining is referred to as secondary data analysis 8 Srihari

9 Statistics vs Data Mining Size of data set (large in data mining) Eyeballing not an option (terabytes of data) Entire dataset rather than a sample Many variables Curse of dimensionality Make predictions Small sample sizes can lead to spurious discovery: Superbowl winner conference correlates to stock market (up/down)

10 Multidisciplinary terminology Structured Data Training Set Unstructured Data Information Retrieval Machine Learning Pattern Recognition Records Database Data Mining Statistics Samples Table Visualization Artificial Intelligence Expert Systems Data Points Instances 10 Leading Conference known as Knowledge Discovery and Data Mining Srihari

11 Data Mining Tasks Not so much a single technique Idea that there is more knowledge hidden in the data than shows itself on the surface Any technique that helps to extract more out of data is useful Five major task types: 1. Exploratory Data Analysis (Visualization: boxplots, charts) 2. Descriptive Modeling (Density estimation, Clustering) Model 3. Predictive Modeling (Classification and Regression) building 4. Discovering Patterns and Rules (Association rules) 5. Retrieval by Content (Retrieve items similar to pattern of interest) 11 Srihari

12 12 Clustering Old Faithful (Hydrothermal Geyser in Yellowstone) 272 observations Duration (mins, horiz axis) vs Time to next eruption (vertical axis) Simple Gaussian unable to capture structure Linear superposition of two Gaussians is better Gaussian has limitations in modeling real data sets Gaussian Mixture Models give very complex densities p( x) = π N( x µ, Σ k = 1 π k are mixing coefficients that sum to one One dimension Three Gaussians in blue Sum in red K k k k )

13 Global Model 13 Models and Patterns High level global description of data set Make statement about any point in d-space E.g., prediction, clustering It takes a large sample perspective Summarizing data in convenient, concise way Local Patterns Make statement about restricted regions of d-space E.g.: if x > thresh1 then Prob (y > thresh2) = p Departure from run of data Identify members with unusual properties Outliers in a database

14 Models for Prediction: Regression and Classification Predict response variable from given values of others Response variable y given predictor variables x 1,.., x d When y is quantitative the task is known as regression When y is categorical, it is known as classification learning or supervised classification 14

15 Statistical Models for Regression and Classification Generative Naïve Bayes Mixtures of multinomials Mixtures of Gaussians Hidden Markov Models (HMM) Bayesian networks Markov random fields Discriminative Logistic regression SVMs Traditional neural networks Nearest neighbor Conditional Random Fields (CRF) Gaussian Processes 15

16 National Academy of Sciences: Keck Center 16

17 Regression Problem: Carbon Dioxide in Atmosphere 400? CO Concentration ppm Year

18 Regression Single input variable 18 Linear Models Polynomial y(x,w) = w 0 +w 1 x+w 2 x 2 + =Σ w i x i Several variables linear combination of non-linear (basis) functions Bayesian Linear Regression Classification y(x,w) = w 0 + M 1 j=1 w j φ j (x) = w T φ(x) Logistic Regression (with sigmoid or soft-max) y(x,w) = σ[w T φ(x)]

19 Neural Network Function Overall function M D y k (x,w) = σ (2) w kj h (1) (1) (2) w ji x i + w j 0 + w k 0 j=1 i=1 Where w is the set of all weights and bias parameters nonlinear functions from inputs {x i } to outputs {y k } Note presence of both σ and h functions If σ is identity for regression If σ is sigmoid for two-class classification If σ is softmax for multi-classification 19

20 Gaussian Processes for Regression Radically different viewpoint not involving weight parameters Functions are drawn from a Gaussian where each data point is a function Gaussian Kernel k(x,x') = exp( x x' 2 /2σ 2 ) Exponential Kernel k(x,x') = exp( θ x x' ) Ornstein-Uhlenbeck process for Brownian motion 20

21 Gaussian Process with Two Samples Let y be a function (curve) of a one-dimensional variable x We take two samples y 1 and y 2 corresponding to x 1 and x 2 Assume they have a bivariate Gaussian distribution Each point from this distribution y 2 y 1 x 1 x 2 has an associated probability It also defines a function y(x) Assuming that two points are enough to define a curve y 2 More than two points will be needed to define a curve Which leads to a higher dimensional probability distribution 21 y 1

22 Gaussian Process Regression Generalize multivariate Gaussian to infinite variables (over all values of input x) A Gaussian distribution is fully specified by a mean vector µ and covariance matrix Σ f = (f 1,..f n ) T ~ N (µ,σ) indexes i =1,..n A Gaussian process is fully specified by a mean function m(x) and covariance function k(x,x ) f (x) ~ GP(m(x), k(x,x )) indexes x Kernel function k appears in place of covariance matrix Both express similarity of two in multidimensional space

23 Gaussian Process Fit to CO 2 data

24 Dual Role of Probability and Statistics in Data Analysis Generative Model of data allows data to be generated from the model Inference allows making statements about data 24

25 2. Nature of Data Sets Structured Data set of measurements from an environment or process Simple case n objects with d measurements each: n x d matrix d columns are called variables, features, attributes or fields 25

26 Unstructured Data 1. Structured Data Well-defined tables, attributes (columns), tuples (rows) 2. Unstructured Data World wide web Documents (HTML/XML) and hyperlinks HTML: tree structure with text and attributes embedded at nodes XML pages use metadata descriptions Text Documents Document viewed as sequence of words and punctuations Mining Tasks» Text categorization» Clustering Similar Documents» Finding documents that match a query Image Databases 26

27 Retrieval by Content User has pattern of interest and wishes to find that pattern in database, Ex: Text Search Estimate the relative importance of web pages using a feature vector whose elements are derived from the Query-URL pair Image Search Search a large database of images by using content descriptors such as color, texture, relative position 27 Srihari

28 Representations of Text Documents Boolean Vector Document is a vector where each element is a bit representing presence/absence of word A set of documents can be represented as matrix (d,w) where document d and word w has value 1 or 0 (sparse matrix) Vector Space Representation Each element has a value such as no. of occurrences or frequency A set of documents represented as a document-term matrix 28

29 Vector Space Example Document-Term Matrix t1 database t2 SQL t3 index t4 regression t5 likelihood t6 linear d ij represents number of times that term appears in that document 29

30 Image Retrieval Results Crime scene mark and their closest matches

31 Conclusion Data mining objective is to make discoveries from data We want to be as confident as we can that our conclusions are correct Nothing is certain Fundamental tool is probability Universal language for handling uncertainty Allows us to obtain best estimates even with data inadequacies and small samples 31 Srihari

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