Bioinformatics II Theoretical Bioinformatics and Machine Learning Part 1 Institute of Bioinformatics Johannes Kepler University, Linz, Austria
Course 6 ECTS 4 SWS VO (class) 3 ECTS 2 SWS UE (exercise) Basic Course of Master Bioinformatics Class: Mo 15:30-17:00 (S3 318) and Thu 15:30-17:00 (S3 318) Exercise: Fr 11:00-12:45 (S3 318) VO: final exam (oral if few students subscribe) UE: weekly homework (evaluated) Other Courses of the Masters in Bioinformatics: Struc. BI and Gene Analysis: Fr 08:30-11:00 (SI 048) Infor. Systems: 5./12./19.03.2013 8:30-11:45 (S3 047) Exercise: Thu 8:30 10:00 (S3 048) Intro. to R (instead of Math. Modeling I): We 15:30-17:00 (S3 057) Alg. Disc. Meth.: Thu 13:45-15:15 (HS 12) Population Genetics: Thu 10:15-11:45 (S3 318)
Outline 1 Introduction 2 Basics of Machine Learning 3 Theoretical Background of Machine Learning 4 Support Vector Machines 5 Error Minimization and Model Selection 6 Neural Networks 7 Bayes Techniques 8 Feature Selection 9 Hidden Markov Models 10 Unsupervised Learning: Projection Methods and Clustering **11 Model Selection **12 Non-parametric methods:decision trees and k-nearest neighbors **13 Graphical Models / Belief networks / Bayes Networks
Outline 1 Introduction 2 Basics of Machine Learning 2.1 Machine Learning in Bioinformatics 2.2 Introductory Example 2.3 Supervised and Unsupervised Learning 2.4 Reinforcement Learning 2.5 Feature Extraction, Selection, and Construction 2.6 Parametric vs. Non-Parametric Models 2.7 Generative vs. descriptive Models 2.8 Prior and Domain Knowledge 2.9 Model Selection and Training 2.10 Model Evaluation, Hyperparameter Selection, and Final Model
Outline 3 Theoretical Background of Machine Learning Criteria 3.2 Generalization error 3.3 Minimal Risk for a Gaussian Classification Task 3.4 Maximum Likelihood 3.6 Statistical Learning Theory
Outline 4 Support Vector Machines 4.1 Support Vector Machines in Bioinformatics 4.2 Linear Separable Problems 4.3 Linear SVM 4.4 Linear SVM for Non-Linear Separable Problems 4.5 Average Error Bounds for SVMs 4.6 nu-svm 4.7 Non-Linear SVM and the Kernel Trick 4.8 Example: Face Recognition 4.9 Multiclass SVM 4.10 Support Vector Regression 4.11 One Class SVM 4.12 Least Square SVM 4.13 Potential Support Vector Machine 4.14 SVM Optimization and SMO 4.15 Designing Kernels for Bioinformatic Applications 4.16 Kernel Principal Component Analysis 4.17 Kernel Discriminant Analysis 4.18 Software
Outline 5 Error Minimization and Model Selection 5.1 Search Methods and Evolutionary Approaches 5.2 Gradient Descent 5.3 Step-size Optimization 5.4 Optimization of the Update Direction 5.5 Levenberg-Marquardt Algorithm 5.6 Predictor Corrector Methods for R(w) = 0 5.7 Convergence Properties 5.8 On-line Optimization
Outline 6 Neural Networks 6.1 Neural Networks in Bioinformatics 6.2 Motivation of Neural Networks 6.3 Linear Neurons and Perceptron 6.4 Multi Layer Perceptron 6.5 Radial Basis Function Networks 6.6 Reccurent Neural Networks
Outline 7 Bayes Techniques 7.1 Likelihood, Prior, Posterior, Evidence 7.2 Maximum A Posteriori Approach 7.3 Posterior Approximation 7.4 Error Bars and Confidence Intervals 7.5 Hyper-parameter Selection: Evidence Framework 7.6 Hyper-parameter Selection: Integrate Out 7.7 Model Comparison 7.8 Posterior Sampling
Outline 8 Feature Selection 8.1 Feature Selection in Bioinformatics 8.2 Feature Selection Methods 8.3 Microarray Gene Selection Protocol 9 Hidden Markov Models 9.1 Hidden Markov Models in Bioinformatics 9.2 Hidden Markov Model Basics 9.3 Expectation Maximization for HMM: Baum-Welch Algorithm 9.4 Viterby Algorithm 9.5 Input Output Hidden Markov Models 9.6 Factorial Hidden Markov Models 9.7 Memory Input Output Factorial Hidden Markov Models 9.8 Tricks of the Trade 9.9 Profile Hidden Markov Models
Outline 10 Unsupervised Learning: Projection Methods and Clustering 10.1 Introduction 10.2 Principal Component Analysis 10.3 Independent Component Analysis 10.4 Factor Analysis 10.5 Projection Pursuit and Multidimensional Scaling 10.6 Clustering
Literature ML: Duda, Hart, Stork; Pattern Classification; Wiley & Sons, 2001 NN: C. M. Bishop; Neural Networks for Pattern Recognition, Oxford Univ. Press, 1995 SVM: Schölkopf, Smola; Learning with kernels, MIT Press, 2002 SVM: V. N. Vapnik; Statistical Learning Theory, Wiley & Sons, 1998 Statistics: S. M. Kay; Fundamentals of Statistical Signal Processing, Prent. Hall, 1993 Bayes Nets: M. I. Jordan; Learning in Graphical Models, MIT Press, 1998 ML: T. M. Mitchell; Machine Learning, Mc Graw Hill, 1997 NN: R. M. Neal, Bayesian Learning for Neural Networks, Springer, 1996 Feature Selection: Guyon, Gunn, Nikravesh, Zadeh; Feature Extraction - Foundations and Applications, Springer, 2006 BI: Schölkopf, Tsuda, Vert ; Kernel Methods in Computational Biology, MIT, 2003
Chapter 1 Introduction
Introduction 1 Introduction part of curriculum master of science in bioinformatics 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation many fields in bioinformatics are based on machine learning - sequencing data: RNA-Seq, copy numbers - microarrays: data preprocessing, gene selection, prediction - DNA data: alternative splicing, nucleosome position, gene regulation methods: neural networks, support vector machines, kernel approaches, projection method, belief networks goals: noise reduction, feature selection, structure extraction, classification / regression, modeling
Introduction 1 Introduction Examples: - cancer treatment outcomes / microarrays 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation - classification of novel protein sequences into structural or functional classes - dependencies between DNA markers (SNP - single nucleotide polymorphisms) and diseases (schizophrenia, autism, multiple sclerosis) only the most prominent machine learning techniques Goals: - how to chose appropriate methods from a given pool - understand and evaluate the different approaches - where to obtain and how to use them - adapt and modify standard algorithms
Chapter 2 Basics of Machine Learning
Basics of Machine Learning 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge deductive: programmer must understand the problem and find a solution and implement it inductive: solution to a problem is found by a machine which learns inductive is data driven: biology, chemistry, biophysics, medicine, and other fields in life sciences possess a huge amount of data learning: automatically finds structures in the data algorithms that automatically improve a solution with more data 2.9 Model Selection 2.10 Model Evaluation
Basics of Machine Learning 1 Introduction Machine Learning: 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge classification and regression (prediction) structure extraction (clustering, components) compression (redundancy reduction) visualization filtering (feature selection) data modeling (generative models) 2.9 Model Selection 2.10 Model Evaluation
Machine Learning in Bioinformatics 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation gene recognition microarray data: normalization protein structure and function classification alternative splice site recognition prediction of nucleosome positions single nucleotide polymorphism (SNP) and diseases copy numbers and diseases chromatin structure and methylation and diseases
Introductionary Example 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge Example from ``Pattern Classification'', Duda, Hart, and Stork, 2001, John Wiley \& Sons, Inc. salmons must be distinguished from sea bass given images automated system to separate fishes in a fish-packing company Given: a set of pictures with known fishes, the training set Goal: in the future, automatically separate images of salmon from images of sea bass, that is generalization 2.9 Model Selection 2.10 Model Evaluation
Introductionary Example 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Introductionary Example 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction First step: preprocessing and feature extraction Preprocessing: contrast / brightness correction, segmentation, alignment Features: length of the fish, lightness Length: 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation optimal decision boundary: minimal misclassifications
Introductionary Example 1 Introduction Lightness: 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation Different features may be differently suited for the problem Misclassifcations are weighted equally (otherwise new optimal boundary
Introductionary Example 1 Introduction Width of the fishes: 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation width may only be suited in combination with other features Hypothesis: Lightness changes with age, width indicates age
Introductionary Example 1 Introduction 2 Basics optimal lightness: nonlinear function of the width that is optimal boundary is a nonlinear curve 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation new fish at?, we would guess salmon but system fails: low generalization, one outlier sea bass changed the curve
Introductionary Example 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised one sea bass has lightness and width typically for salmon complex boundary curve also catches this outlier and assign surrounding space to sea bass future examples in this region will be wrongly classified 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation decision boundary with high generalization
Introductionary Example 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation we selected the features which are best suited bioinformatics applications: number of features is large selecting the best feature by visual inspections is impossible certain cancer type must be chosen from 30,000 human genes feature selection is important: machine selects the features construct new features from the old ones: feature construction question of cost: how expensive is a certain error measurement noise: features classification noise: what errors of human labeling are to expect first example of too complex model overspecialized to training data
Supervised and Unsupervised Learning 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation in our fish example an expert characterized the data by labeling them supervised learning : desired output (target) for each object is given unsupervised learning : no desired output per object supervised: error value on each object classification / regression / time series analysis fish example: classification salmon vs. see bass regression predict age of the fish time series prediction growth from past unsupervised: - cumulative error over all objects (entropy, statistical independence, information content, etc.) - probability of model producing the data: likelihood - principal component analysis (PCA), independent component analysis (ICA), factor analysis, projection pursuit, clustering (k-means), mixture models, density estimation, hidden Markov models, belief networks
Supervised and Unsupervised Learning 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection projection: representation of objects, down-project feature vectors, PCA: orthogonal maximal data variation components, ICA: statistically mutual independent components, factor analysis: PCA with noise density estimation: density model of observed data clustering: extract clusters regions data accumulation (typical data) clustering and (down-)projection: feature construction, compact representation of the data, non-redundant, noise removal 2.10 Model Evaluation
Supervised and Unsupervised Learning 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Supervised and Unsupervised Learning 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Supervised and Unsupervised Learning 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Supervised and Unsupervised Learning 1 Introduction Isomap: method for down-projecting data 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Supervised and Unsupervised Learning 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Supervised and Unsupervised Learning 3. Basics of Machine Leaning 3.1 Bioinformatics Original: 3.2 Example 3.3 Un-/Supervised 3.4 Reinforcement Mixtures: 3.5 Feature Extract. 3.6 Non-/Parametric 3.7 Generat. / des. 3.8 Prior Knowl. Demixed by ICA: 3.9 Model Selection 3.10 Model Evaluat. 3.11 Error Bounds 3.12 Support Vector Machines 3.12.1 SVM / Bioinf. 3.12.2 Linear Separable 3.12.3 Linear SVM 3.12.4 Nonli. Sepa. 3.12.5 Example 3.12.6 Software
Supervised and Unsupervised Learning 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Supervised and Unsupervised Learning 1 Introduction ICA: on images 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Supervised and Unsupervised Learning 1 Introduction ICA: on video components 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Reinforcement Learning 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation Not considered because not relevant for bioinformatics: reinforcement learning: - model produces output sequence - reward or a penalty at sequence end or during the sequence (no target output) neither supervised nor unsupervised learning model: policy learning: world model or value function two learning techniques : direct policy optimization vs. policy / value iteration (world model) exploitation / exploration trade-off: better to learn or to gain reward methods: Q-learning, SARSA, Temporal Difference (TD), Monte Marlo estimation
Feature Extraction, Selection, and Construction 1 Introduction 2 Basics 2.1 Bioinformatics our example salmon - sea bass: features must be extracted fmri brain images and EEG measurements: 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Feature Extraction, Selection, and Construction 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Feature Extraction, Selection, and Construction 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation Feature Selection: features are directly measured huge number of features: microarray 30,000 genes other measurements with many features: peptide arrays, protein arrays, mass spectrometry, SNPs many features not related to the task (genes relevant for cancer)
Feature Extraction, Selection, and Construction 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Feature Extraction, Selection, and Construction 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Feature Extraction, Selection, and Construction 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Feature Extraction, Selection, and Construction 1 Introduction 2 Basics features without target correlation may be helpful feature with highest target correlation may be a suboptimal selection 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Feature Extraction, Selection, and Construction 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge Feature Construction: combine features to a new features - PCA or ICA - averaging out kernel methods map another space where new features are used example: sequence of amino acids may be presented by - occurrence vector - certain motifs - their similarity to other sequences 2.9 Model Selection 2.10 Model Evaluation
Parametric vs. Non-Parametric Models 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation important step in machine learning is to select a model class parametric models: each parameter vector represents a model - neural networks, where the parameter are the synaptic weights - support vector machines learning: paths through the parameter space disadvantages: - different parameterizations of the same function - model complexity and class via the parameters nonparametric models: model is locally constant / superimpositions - k-nearest-neighbor (k is hyperparameter not adjusted) - kernel density estimation - decision tree constant models (rules) must be a priori selected that is hyperparameters must be fixed (k, kernel width, splitting rules)
Generative vs. descriptive Models 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised descriptive model: additional description or another representation of the data projection methods (PCA, ICA) 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation generative models: model should produce the distribution observed for the real world data points describing or representing random components which drive the process prior knowledge about the world or desired model predict new states of the data generation process (brain, cell)
Prior and Domain Knowledge 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation reasonable distance measures for k-nearest-neighbor construct problem-relevant features extract appropriate features from the raw data bioinformatics: distances based on alignment - string-kernel - Smith-Waterman-kernel - local alignment kernel - motif kernel bioinformatics: secondary structure prediction with recurrent networks 3.7 amino acid period of a helix in the input bioinformatics: knowledge about the microarray noise (log-values) bioinformatics: 3D structure prediction of proteins disulfidbonds
Model Selection and Training 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation Goal: select model with highest generalization performance, that is with the best performance on future data, from the model class model selection is training is learning model which best explains or approximates the training set remember: salmon vs. sea bass the model which perfectly explains the training data had low generalization performance overfitting : model is fitted (adapted) to special training characteristics - noisy measurements - outliers - labeling errors
Model Selection and Training 1 Introduction 2 Basics underfitting : training data cannot be fitted well enough trade-off between underfitting and overfitting 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Model Selection and Training 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection overfitting bounded: model class (k in k-nearest-neighbor, number of units in neural networks, maximal weights, etc.) model class often chosen a priori Sometimes model class can be adjusted during training structural risk minimization model selection parameters may influence the model complexity - nonlinearity of neural networks is increased during training - model selection procedure cannot find complex models hyperparameters: parameters controlling the model complexity 2.10 Model Evaluation
Model Evaluation, Hyperparameter Selection, and Final Model 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation how to select the hyperparameters? ( number of features) kernel density estimation (KDE): best hyperparameter (the kernel width) can be computed under certain assumptions n-fold cross-validation for hyperparameter selection: - training set is divided into n parts - n runs where in the i-th run part i is used for test - average error over all runs for all hyperparameter combinations - chose parameter combination with smallest average error cross-validation error approximates generalization error, but - cross validation training sets are overlapping - points from the withhold fold are predicted with the same model so that an outlier would have multiple influence on the result leave-one-out cross validation: only one data point is removed assumption: trainings size is not important (one fold is removed)
Model Evaluation, Hyperparameter Selection, and Final Model 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive How to estimate the performance of a model? n-fold cross validation, but - another k-fold cross validation on each training set to select the hyperparameters - also feature selection and feature ranking must be done for each training set, i.e. for each fold well know error: feature selection on all data and then cross-validation - from equal relevant features the ones which are relevant also on the test fold are ranked higher 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation
Model Evaluation, Hyperparameter Selection, and Final Model 1 Introduction 2 Basics 2.1 Bioinformatics 2.2 Example 2.3 Un-/Supervised 2.4 Reinforcement 2.5 Feature Extraction 2.6 Non-/Parametric 2.7 Generative descriptive 2.8 Prior Knowledge 2.9 Model Selection 2.10 Model Evaluation Comparing models type I and type II error: - Type I: wrongly detect a difference - Type II: miss a difference methods for testing the performance: - paired t-test: > multiply dividing the data into test and training set > to many type I errors - k-fold cross-validated paired t-test: fewer type I errors than p. t-test - McNemar's test: type I and type II errors well estimated - 5x2CV (5 times two fold cross-validation): comparable to McNemar > two fold: many test points, no overlapping training other criteria: - space and time complexity - above for training and for testing (practical use) - training time oft not relevant (wait a week to make money) - faster test, then averaging over many runs is possible
Chapter 3 Theoretical Background of Machine Learning
Theoretical Background of Machine Learning 3.6.2.1 Complexity: quality criteria goal for model selection / learning approximations unsupervised learning: Maximum Likelihood concepts: bias and variance, efficient estimator, Fisher information supervised learning considered in an unsupervised framework: error model
Theoretical Background of Machine Learning 3.6.2.1 Complexity: does learning from examples help in the future? empirical risk minimization (ERM) complexity is restricted and dynamics fixed learning helps : more training examples improve the model converges to the best model for all future data convergence is fast complexity of a model class: VC-dimension (Vapnik-Chervonenkis) structural risk minimization (SRM): complexity and model quality bounds on the generalization error
Model Quality Criteria 3.6.2.1 Complexity: learning equivalent to model selection quality criteria: future data is optimally processed other concepts: visualization, modeling, data compression Kohonen networks: no scalar quality criterion (potential function) advantage quality criteria: - comparison of different models - quality during learning known supervised quality criteria: rate of misclassifications or squared error unsupervised criteria: - likelihood - ratio of between and within cluster distance - independence of the components - information content - expected reconstruction error
Generalization Error 3.6.2.1 Complexity: Now: supervised performance of a model on future data: generalization error error on one example: loss or error expected loss: risk or generalization error
Definition of the Generalization Error/Risk 3.6.2.1 Complexity: Training set: Label or target value: Simple: and Training set: Matrix notation for training inputs: Vector notation for labels: Matrix notation for training set:
Definition of the Generalization Error/Risk 3.6.2.1 Complexity: The loss function quadratic loss: zero-one loss: Generalization error:
Definition of the Generalization Error/Risk 3.6.2.1 Complexity: y is a function of x (target function: y = f(x)) plus noise: Now the risk can be computed as
Definition of the Generalization Error/Risk 3.6.2.1 Complexity:
Definition of the Generalization Error/Risk 3.6.2.1 Complexity: The noise-free case is simplifies to:
Empirical Estimation of the Generalization Error 3.6.2.1 Complexity: p(z) is unknown especially p(y x) risk cannot be computed practical applications: approximation of the risk model performance estimation for the user
Test Set 3.6.2.1 Complexity: Test set approximation: expectation can be approximated using with test set:
Cross-Validation 3.6.2.1 Complexity: not enough data for test set (needed for training) cross-validation Cross-validation folds:
Cross-Validation 3.6.2.1 Complexity: n-fold cross-validation (here 5-fold):
Cross-Validation 3.6.2.1 Complexity: cross-validation is an almost unbiased estimator for the generalization error: Generalization error on trainings size without one fold l l/n can be estimated by crossvalidation on training data l by n-fold crossvalidation
Cross-Validation 3.6.2.1 Complexity: advantage: test examples only once used (better than multiple dividing the data into training and test set) disadvantage: - training sets are overlapping - one fold on same model test examples dependent - these dependencies cv has high variance (one outlier influences all estimates) special case: leave-one-out cross-validation (LOO-CV) - l-fold cross-validation, where each fold is one example - test examples to not use the same model - training sets are maximal overlapping
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity: Class y = 1 data points are drawn according to and class y = -1 according to where the Gaussian has density
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity: Linear transformations of Gaussians lead to Gaussians
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity: probability of observing a point from class y=1 at x: probability of observing a point from class y=-1 at x: Conditional probability: probability of observing a point at x: y is integrated out - here summed out
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity: two-dimensional classification task data for each class from a Gaussian (black: class 1, red: class -1) optimal discriminant functions are two hyperbolas
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity: Bayes rule for probability of x belonging to class y = 1:
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity: Risk: Loss function contributions:
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity: Optimal discriminant (see later) function: at each position x take smallest value The minimal risk is
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity:
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity:
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity: discriminant function g: g(x)>0 then x is assigned to y = 1 g(x)<0 then x is assigned to y = -1 classification functions : optimal discriminant functions (minimal risk): or
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity: For Gaussians:
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity: 1D 3D 2D
Minimal Risk for a Gaussian Classification Task 3.6.2.1 Complexity:
Maximum Likelihood 3.6.2.1 Complexity: One of the major objectives if learning generative models It has certain theoretical properties Theoretical concepts like efficient estimator or biased estimator are introduced Even supervised methods can be viewed as special case of maximum likelihood
Loss for Unsupervised Learning 3.6.2.1 Complexity: First we consider different loss functions which are used for unsupervised learning Generative approaches maximum likelihood Projection methods Parameter estimation low information loss plus desired property difference of estimated parameter vector to the optimal parameter vector
Projection Methods 3.6.2.1 Complexity: data projection into another space with desired requirements
Projection Methods 3.6.2.1 Complexity: Principal Component Analysis (PCA): projection to a low dimensional space under maximal information conservation Independent Component Analysis (ICA): projection into a space with statistically indpendent components (factorial code) often characteristics of a factorial distribution are optimized: - maximal entropy (given variance) - cummulants or prototype distributions should be matched: - product of special super-gaussians Projection Pursuit : components are maximally non-gaussian
Generative Models 3.6.2.1 Complexity: generative model : model simulates the world and produces the same data as the world
Generative Models 3.6.2.1 Complexity: data generation process is probabilistic: underlying distribution generative model attempts at approximation this distribution loss function the distance between model output distribution and the distribution of the data generation process Examples: Factor Analysis, Latent Variable Models, Boltzmann Machines, Hidden Markov Models
Parameter Estimation 3.6.2.1 Complexity: parameterized model known task: estimate actual parameters loss: difference between true and estimated parameter evaluate estimator: expected loss
Mean Squared Error, Bias, and Variance 3.6.2.1 Complexity: Theoretical concepts of parameter estimation training data: where simply (the matrix of training data) true parameter vector: estimate of :
Mean Squared Error, Bias, and Variance 3.6.2.1 Complexity: unbiased estimator: on average (over training set) the true parameter is obtained bias: variance: mean squared error (MSE, different to supervised loss): expected squared error between the estimated and true parameter
Mean Squared Error, Bias, and Variance 3.6.2.1 Complexity: Only zero depends on
Mean Squared Error, Bias, and Variance 3.6.2.1 Complexity: Averaging reduces variance each of the subsets has examples which gives examples in total Average is where Unbiased: Variance:
Mean Squared Error, Bias, and Variance 3.6.2.1 Complexity: averaging: training sets are independent, therefore covariance between them vanishes Minimal Variance Unbiased (MVU) estimator: construct from all unbiased estimators the one with minimal variance MVU estimator does not always exist methods to check whether a given estimator is a MVU
Fisher Information Matrix, Cramer-Rao Lower Bound, and Efficiency 3.6.2.1 Complexity: We will find a lower bound for the variance of an unbiased estimator: Cramer-Rao Lower Bound (that is a lower bound for the MSE) We need the Fisher information matrix :
Fisher Information Matrix, Cramer-Rao Lower Bound, and Efficiency 3.6.2.1 Complexity: If satisfies then the Fisher information matrix is Fisher information: information of observation about parameter upon which the parameterized density function of depends
Fisher Information Matrix, Cramer-Rao Lower Bound, and Efficiency 3.6.2.1 Complexity:
Fisher Information Matrix, Cramer-Rao Lower Bound, and Efficiency 3.6.2.1 Complexity: efficient estimator: reaches the CRLB (efficiently uses the data) MVU estimator can be efficient but need not dashed: CRLB
Fisher Information Matrix, Cramer-Rao Lower Bound, and Efficiency 3.6.2.1 Complexity: dashed: CRLB
Maximum Likelihood Estimator 3.6.2.1 Complexity: MVU estimator is unknown or does not exist Maximum Likelihood Estimator (MLE) MLE can be applied to a broad range of problems MLE approximates the MVU estimator for large data sets MLE is even asymptotically efficient and unbiased MLE does everything right and this efficiently (enough data)
Maximum Likelihood Estimator 3.6.2.1 Complexity: The likelihood of the data set : probability of the model to produce the data iid (independent identical distributed) data: Negative log-likelihood:
Maximum Likelihood Estimator 3.6.2.1 Complexity: likelihood is based on finite many densities values which have zero measure: problem? assume instead of (region around ) MLE popular: - simple use - properties the volume element
Properties of Maximum Likelihood Estimator 3.6.2.1 Complexity: MLE: invariant under parameter change asymptotically unbiased and efficient asymptotically optimal consistent for zero CRLB
MLE is Invariant under Parameter Change 3.6.2.1 Complexity:
MLE is Asymptotically Unbiased and Efficient 3.6.2.1 Complexity: The maximum likelihood estimator is asymptotically unbiased: The maximum likelihood estimator is asymptotically efficient:
MLE is Asymptotically Unbiased and Efficient 3.6.2.1 Complexity:
MLE is Asymptotically Unbiased and Efficient 3.6.2.1 Complexity: practical applications: finite examples MLE performance unknown Example: general linear model MLE is which is efficient and MUV Note the noise covariance must be known where
MLE is Consistent for Zero CRLB 3.6.2.1 Complexity: consistent: for large training sets the estimator approaches the true value (difference to unbiased variance decreases) Later more formal definition for consistency as Thus, the MLE is consistent if the CRLB is zero
Expectation Maximization 3.6.2.1 Complexity: likelihood can be optimized by gradient descent methods likelihood cannot be computed analytically: -- hidden states -- many-to-one output mapping -- non-linearities
Expectation Maximization 3.6.2.1 Complexity: hidden variables, latent variables, unobserved variables likelihood is determined by all mapped to
Expectation Maximization 3.6.2.1 Complexity: Expectation Maximization (EM) algorithm: -- joint probability is easier to compute than likelihood -- estimate by Jensen's inequality
Expectation Maximization 3.6.2.1 Complexity: EM algorithm is an iteration between E -step and M -step:
Expectation Maximization 3.6.2.1 Complexity: After E-step: Proof: Kullback-Leibler divergence: Zero for:
Expectation Maximization 3.6.2.1 Complexity: EM increases the lower bound in both steps beginning of the M-step: E-step does not change the parameters EM algorithm: -- hidden Markov models -- mixture of Gaussians -- factor analysis -- independent component analysis
Noise Models 3.6.2.1 Complexity: connecting unsupervised and supervised learning quality measure noise on the targets apply maximum likelihood
Noise Models 3.6.2.1 Complexity: Gaussian target noise linear model log-likelihood:
Noise Models 3.6.2.1 Complexity: minimize least square criterion linear least square estimator derivative with respect to : Setting the derivative to zero (Wiener-Hopf equations):
Gaussian Noise 3.6.2.1 Complexity: Noise covariance matrix gives the noise for each measure In most cases we have the same noise for each observation: We obtain minimal value: : pseudo inverse or Moore-Penrose inverse
Laplace Noise and Minkowski Error 3.6.2.1 Complexity: Laplace noise assumption: More general Minkowski error: gamma function
Laplace Noise and Minkowski Error 3.6.2.1 Complexity:
Binary Models 3.6.2.1 Complexity: noise considerations do not hold for binary target classification not treated
Cross-Entropy 3.6.2.1 Complexity: classification problem with K classes: Likelihood:
Cross-Entropy 3.6.2.1 Complexity: The log-likelihood: loss function: cross entropy (Kullback-Leibler)
Logistic Regression 3.6.2.1 Complexity: a function g mapping x onto R can be transformed into a probability:
Logistic Regression 3.6.2.1 Complexity: If follows: log-likelihood: maximum likelihood maximizes
Logistic Regression 3.6.2.1 Complexity: derivative of the log-likelihood: similar to the derivative of the quadratic loss function in the regression: instead of
Statistical Learning Theory 3.6.2.1 Complexity: Does learning help for future tasks? Explains a model which explains the training data also new data? Yes, if complexity is bounded VC-dimension as complexity measure statistical learning theory : bounds for the generalization error (future) bounds comprise training error and complexity structural risk minimization minimizes both terms simultaneously
Statistical Learning Theory 3.6.2.1 Complexity: statistical learning theory: -- (1) the uniform law of large numbers (empirical risk minimization) -- (2) complexity constrained models (structural risk minimization) error bound on the mean squared error: bias-variance formulation -- bias is training error = empirical risk -- variance is model complexity high complexity more models more solutions large variance
Error Bounds for a Gaussian Classification Task 3.6.2.1 Complexity: We revisit the Gaussian classification task
Error Bounds for a Gaussian Classification Task 3.6.2.1 Complexity: Gaussian assumption: Chernoff bound: maximizing Bhattacharyya bound: with respect to
Empirical Risk Minimization 3.6.2.1 Complexity: empirical risk minimization (ERM) principle states: if the training set is explained by the model then the model generalizes to future examples restrict the complexity of the model class empirical risk minimization (ERM): minimize error on training set
Complexity: Finite Number of Functions 3.6.2.1 Complexity intuition why complexity matters complexity is just the number M of functions in model class difference training error (empirical risk) and test error (risk ) empirical risk: finite set of functions worst case (learning chooses unknown function):
Complexity: Finite Number of Functions 3.6.2.1 Complexity : union bound distance of average and expectation: Chernoff inequality (for each j ) where is the empirical mean of the true value for trials we obtain complexity term
Complexity: Finite Number of Functions 3.6.2.1 Complexity
Complexity: Finite Number of Functions 3.6.2.1 Complexity should converge to zero as l increases, therefore
Complexity: VC-Dimension 3.6.2.1 Complexity we want apply the previous bound for infinite function classes idea: on training set only finite number of functions is different example: all discriminant functions g giving the same classification function sign g(.) parametric models g(.;w) with parameter vector w Does minimizing the parameter on the training set convergence to the best solution with increasing training set? empirical risk minimization (ERM): consistent or not? do we select better models with larger training sets?
Complexity: VC-Dimension 3.6.2.1 Complexity parameter which minimizes the empirical risk for l training examples: ERM is consistent if convergence in probability Empirical risk and expected risk converge to minimal risk
Complexity: VC-Dimension 3.6.2.1 Complexity
Complexity: VC-Dimension 3.6.2.1 Complexity ERM is strictly consistent if for all holds (convergence in probability) Instead of strictly consistent we write consistent maximum likelihood is consistent for a set of densities if
Complexity: VC-Dimension 3.6.2.1 Complexity Under what conditions is the ERM consistent? New concepts and new capacity measures: -- points to be shattered -- annealed entropy -- entropy (new definition) -- growth function -- VC-dimension Possibilities to label the input data shattering the input data by binary labels complexity of a model class: number different labelings how many points can be shattered
Complexity: VC-Dimension 3.6.2.1 Complexity Note, that each x is placed in a circle around its position independent of the other x. Therefore each constellation represents a set with non-zero probability mass.
Complexity: VC-Dimension 3.6.2.1 Complexity number of points a function class can shatter: VC-dimension (later) function class shattering coefficient: (# labeling class can shatter) entropy of a function class: annealed entropy of a function class: growth function of a function class: Jensen supremum
Complexity: VC-Dimension 3.6.2.1 Complexity ERM fast rate of convergence (exponential convergence):
Complexity: VC-Dimension 3.6.2.1 Complexity theorems valid for a given probability measure on the observations probability measure enters the formulas via the expectation
Complexity: VC-Dimension 3.6.2.1 Complexity VC (Vapnik-Chervonenkis) dimension for which holds If the maximum does not exists: is the largest integer VC-dimension is the maximum number of vectors that can be shattered by the function class
Complexity: VC-Dimension 3.6.2.1 Complexity
Complexity: VC-Dimension 3.6.2.1 Complexity function class with finite VC-dim.: consistent and converges fast -- Linear functions in d-dimensional of the input space: -- Nondecreasing nonlinear one-dimensional functions -- Nonlinear one-dimensional functions:
Complexity: VC-Dimension 3.6.2.1 Complexity -- Neural Networks: M are the number of units, W is the number of weights, e is the base of the natural logarithm (Baum & Haussler 89, Shawe-Taylor & Anthony 91) inputs restricted to Bartlett & Williamson (1996)
Error Bounds 3.6.2.1 Complexity idea of deriving the error bounds: set of distinguishable functions cardinality given by trick of two half-samples and their difference ( symmetrization ): therefore in the following we use 2 l l example used for complexity definition and l for empirical error minimal possible risk:
Error Bounds 3.6.2.1 Complexity
Error Bounds 3.6.2.1 Complexity complexity measure depend on the ratio The bound above is from Anthony and Bartlett whereas an older bound from Vapnik is complexity term decreases with zero empirical risk then the bound on the risk decreases with Later: expected risk decreases with
Error Bounds 3.6.2.1 Complexity bound on the risk bound is similar to the bias-variance formulation -- bias corresponds to empirical risk -- variance corresponds to complexity
Error Bounds 3.6.2.1 Complexity In many practical cases the bound is not useful: not tight However in many practical cases the minimum of the bound is close to the minimum of the test error
Error Bounds 3.6.2.1 Complexity regression: instead of the shattering coefficient covering number ( covering of the functions with distance epsilon) growth function is then: bounds on the generalization error: where
Structural Risk Minimization 3.6.2.1 Complexity The Structural Risk Minimization (SRM) principle minimizes the guaranteed risk that is a bound on the risk instead of the empirical risk alone
Structural Risk Minimization 3.6.2.1 Complexity nested set of function classes: where class possesses VC-dimension and
Structural Risk Minimization 3.6.2.1 Complexity Example for SRM: minimum description length - sender transmits a model (once) and the inputs and errors - receiver has to recover the labels goal: minimize transmission costs (description length) Is the SRM principle consistent? SRM is consistent!! asymptotic rate of convergence: where How fast does it converge? is the minimal risk of the function class
Structural Risk Minimization 3.6.2.1 Complexity If the optimal solution belongs to some class convergence rate is then the
Margin as Complexity Measure 3.6.2.1 Complexity VC-dimension: restrictions on the class of function most famous: zero isoline of the discriminant function has minimal distance (margin) to all training data points which are contained in a sphere with radius R
Margin as Complexity Measure 3.6.2.1 Complexity
Margin as Complexity Measure 3.6.2.1 Complexity linear discriminant functions classification function scaling w and b does not change classification function classification function: one representative discriminant function canonical form w.r.t. the training data X:
Margin as Complexity Measure 3.6.2.1 Complexity
Margin as Complexity Measure 3.6.2.1 Complexity If at least one data point exists for which the discriminant function is positive and at least one data point exists for which it is negative, then we can optimize b and rescale in order to obtain the smallest This gives the tightest bound and smallest VC-dimension After optimizing b and rescaling we have points for which
Margin as Complexity Measure 3.6.2.1 Complexity
Margin as Complexity Measure 3.6.2.1 Complexity After this optimization: the distance of and to the boundary function is