1. Machine LearningThe Art and Science of Algorithms that Make Sense of DataPeter A. FlachIntelligent Systems Laboratory, University of Bristol, United KingdomDecember 29, 2013

2. These slides accompany the above book published by Cambridge University Press in 2012, andare made freely available for teaching purposes (the copyright remains with the author, however).The material is divided in four difficulty levels A (basic) to D (advanced); this PDF includes allmaterial up to level B, and advanced material indicated by ? up to D.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 2 / 540 3. Table of contents I1 The ingredients of machine learningTasks: the problems that can be solved with machine learningLooking for structureModels: the output of machine learningGeometric modelsProbabilistic modelsLogical modelsGrouping and gradingFeatures: the workhorses of machine learningTwo uses of featuresFeature construction and transformation2 Binary classification and related tasksClassificationAssessing classification performanceVisualising classification performancecs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 3 / 540 4. Table of contents IIScoring and rankingAssessing and visualising ranking performanceTurning rankers into classifiersClass probability estimationAssessing class probability estimatesTurning rankers into class probability estimators3 Beyond binary classificationHandling more than two classesMulti-class classificationMulti-class scores and probabilitiesRegressionUnsupervised and descriptive learningPredictive and descriptive clusteringOther descriptive models4 Concept learningThe hypothesis spacecs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 4 / 540 5. Table of contents IIILeast general generalisationInternal disjunctionPaths through the hypothesis spaceMost general consistent hypothesesUsing first-order logic ?Learnability ?5 Tree modelsDecision treesRanking and probability estimation treesSensitivity to skewed class distributionsTree learning as variance reductionRegression treesClustering trees6 Rule modelsLearning ordered rule listscs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 5 / 540 6. Table of contents IVRule lists for ranking and probability estimationLearning unordered rule setsRule sets for ranking and probability estimationA closer look at rule overlap ?Descriptive rule learningRule learning for subgroup discoveryAssociation rule mining7 Linear modelsThe least-squares methodMultivariate linear regressionRegularised regression ?Using least-squares regression for classification ?The perceptron: a heuristic learning algorithm for linear classifiersSupport vector machinesSoft margin SVMcs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 6 / 540 7. Table of contents VObtaining probabilities from linear classifiersGoing beyond linearity with kernel methods ?8 Distance-based modelsNeighbours and exemplarsNearest-neighbour classificationDistance-based clusteringK-means algorithmClustering around medoidsSilhouettesHierarchical clusteringFrom kernels to distances ?9 Probabilistic modelsThe normal distribution and its geometric interpretationsProbabilistic models for categorical dataUsing a naive Bayes model for classificationcs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 7 / 540 8. Table of contents VITraining a naive Bayes modelDiscriminative learning by optimising conditional likelihood ?Probabilistic models with hidden variablesExpectation-Maximisation ?Gaussian mixture models ?Compression-based models ?10 FeaturesKinds of featureCalculations on featuresCategorical, ordinal and quantitative featuresStructured featuresFeature transformationsThresholding and discretisationNormalisation and calibration11 Model ensemblescs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 8 / 540 9. Table of contents VIIBagging and random forestsBoostingBias, variance and margins12 Machine learning experimentsWhat to measureHow to measure itHow to interpret itInterpretation of results over multiple data setscs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 9 / 540 10. Assassinating spam e-mailSpamAssassin is a widely used open-source spam filter. It calculates a score foran incoming e-mail, based on a number of built-in rules or tests inSpamAssassins terminology, and adds a junk flag and a summary report to thee-mails headers if the score is 5 or more.-0.1 RCVD_IN_MXRATE_WL RBL: MXRate recommends allowing[123.45.6.789 listed in sub.mxrate.net]0.6 HTML_IMAGE_RATIO_02 BODY: HTML has a low ratio of text to image area1.2 TVD_FW_GRAPHIC_NAME_MID BODY: TVD_FW_GRAPHIC_NAME_MID0.0 HTML_MESSAGE BODY: HTML included in message0.6 HTML_FONx_FACE_BAD BODY: HTML font face is not a word1.4 SARE_GIF_ATTACH FULL: Email has a inline gif0.1 BOUNCE_MESSAGE MTA bounce message0.1 ANY_BOUNCE_MESSAGE Message is some kind of bounce message1.4 AWL AWL: From: address is in the auto white-listFrom left to right you see the score attached to a particular test, the testidentifier, and a short description including a reference to the relevant part of thee-mail. As you see, scores for individual tests can be negative (indicatingevidence suggesting the e-mail is ham rather than spam) as well as positive. Theoverall score of 5.3 suggests the e-mail might be spam.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 10 / 540 11. Example 1, p.2 Linear classificationSuppose we have only two tests and four training e-mails, one of which is spam(see Table 1). Both tests succeed for the spam e-mail; for one ham e-mail neithertest succeeds, for another the first test succeeds and the second doesnt, and forthe third ham e-mail the first test fails and the second succeeds.It is easy to see that assigning both tests a weight of 4 correctly classifies thesefour e-mails into spam and ham. In the mathematical notation introduced inBackground 1 we could describe this classifier as 4x1 4x2 5 or(4,4) (x1,x2) 5.In fact, any weight between 2.5 and 5 will ensure that the threshold of 5 is onlyexceeded when both tests succeed. We could even consider assigning differentweights to the tests as long as each weight is less than 5 and their sumexceeds 5 although it is hard to see how this could be justified by the trainingdata.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 11 / 540 12. Table 1, p.3 Spam filtering as a classification taskE-mail x1 x2 Spam? 4x1 4x21 1 1 1 82 0 0 0 03 1 0 0 44 0 1 0 4The columns marked x1 and x2 indicate the results of two tests on four different e-mails.The fourth column indicates which of the e-mails are spam. The right-most columndemonstrates that by thresholding the function 4x1 4x2 at 5, we can separate spamfrom ham.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 12 / 540 13. Figure 1, p.5 Linear classification in two dimensions++ +++++ +x1x0x2wThe straight line separates the positives from the negatives. It is defined by w xi t ,where w is a vector perpendicular to the decision boundary and pointing in the directionof the positives, t is the decision threshold, and xi points to a point on the decisionboundary. In particular, x0 points in the same direction as w, from which it follows thatw x0 jjwjj jjx0jj t (jjxjj denotes the length of the vector x).cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 13 / 540 14. Background 1, p.4 Homogeneous coordinatesIt is sometimes convenient to simplify notation further by introducing an extraconstant variable x0 1, the weight of which is fixed to w0 t .The extended data point is then x (1,x1, . . . ,xn) and the extended weightvector is w (t ,w1, . . . ,wn), leading to the decision rule w x 0 and thedecision boundary w x 0.Thanks to these so-called homogeneous coordinates the decision boundarypasses through the origin of the extended coordinate system, at the expense ofneeding an additional dimension.t note that this doesnt really affect the data, as all data points and the realdecision boundary live in the plane x0 1.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 14 / 540 15. Important point to rememberMachine learning is the systematic study of algorithms and systems that improvetheir knowledge or performance with experience.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 15 / 540 16. Figure 2, p.5 Machine learning for spam filteringE-mails Data Spam?SpamAssassintestsLinear classifierweightsLearn weightsTraining dataAt the top we see how SpamAssassin approaches the spam e-mail classification task:the text of each e-mail is converted into a data point by means of SpamAssassinsbuilt-in tests, and a linear classifier is applied to obtain a spam or ham decision. At thebottom (in blue) we see the bit that is done by machine learning.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 16 / 540 17. Example 2, p.6 OverfittingImagine you are preparing for your Machine Learning 101 exam. Helpfully,Professor Flach has made previous exam papers and their worked answersavailable online. You begin by trying to answer the questions from previouspapers and comparing your answers with the model answers provided.Unfortunately, you get carried away and spend all your time on memorising themodel answers to all past questions. Now, if the upcoming exam completelyconsists of past questions, you are certain to do very well. But if the new examasks different questions about the same material, you would be ill-prepared andget a much lower mark than with a more traditional preparation.In this case, one could say that you were overfitting the past exam papers andthat the knowledge gained didnt generalise to future exam questions.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 17 / 540 18. A Bayesian classifier IBayesian spam filters maintain a vocabulary of words and phrases potentialspam or ham indicators for which statistics are collected from a training set.t For instance, suppose that the word Viagra occurred in four spam e-mailsand in one ham e-mail. If we then encounter a new e-mail that contains theword Viagra, we might reason that the odds that this e-mail is spam are4:1, or the probability of it being spam is 0.80 and the probability of it beingham is 0.20.t The situation is slightly more subtle because we have to take into accountthe prevalence of spam. Suppose that I receive on average one spame-mail for every six ham e-mails. This means that I would estimate the oddsof an unseen e-mail being spam as 1:6, i.e., non-negligible but not very higheither.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 18 / 540 19. A Bayesian classifier IIt If I then learn that the e-mail contains the word Viagra, which occurs fourtimes as often in spam as in ham, I need to combine these two odds. As weshall see later, Bayes rule tells us that we should simply multiply them: 1:6times 4:1 is 4:6, corresponding to a spam probability of 0.4.In this way you are combining two independent pieces of evidence, oneconcerning the prevalence of spam, and the other concerning the occurrence ofthe word Viagra, pulling in opposite directions.The nice thing about this Bayesian classification scheme is that it can berepeated if you have further evidence. For instance, suppose that the odds infavour of spam associated with the phrase blue pill is estimated at 3:1, andsuppose our e-mail contains both Viagra and blue pill, then the combined oddsare 4:1 times 3:1 is 12:1, which is ample to outweigh the 1:6 odds associatedwith the low prevalence of spam (total odds are 2:1, or a spam probability of0.67, up from 0.40 without the blue pill).cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 19 / 540 20. A rule-based classifiert if the e-mail contains the word Viagra then estimate the odds of spam as4:1;t otherwise, if it contains the phrase blue pill then estimate the odds of spamas 3:1;t otherwise, estimate the odds of spam as 1:6.The first rule covers all e-mails containing the word Viagra, regardless ofwhether they contain the phrase blue pill, so no overcounting occurs. Thesecond rule only covers e-mails containing the phrase blue pill but not the wordViagra, by virtue of the otherwise clause. The third rule covers all remaininge-mails: those which neither contain neither Viagra nor blue pill.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 20 / 540 21. Figure 3, p.11 How machine learning helps to solve a taskTaskLearning problemFeaturesDomainobjectsData OutputModelLearningalgorithmTraining dataAn overview of how machine learning is used to address a given task. A task (red box)requires an appropriate mapping a model from data described by features to outputs.Obtaining such a mapping from training data is what constitutes a learning problem (bluebox).cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 21 / 540 22. Important point to rememberTasks are addressed by models, whereas learning problems are solved bylearning algorithms that produce models.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 22 / 540 23. Important point to rememberMachine learning is concerned with using the right features to build the rightmodels that achieve the right tasks.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 23 / 540 24. 1. The ingredients of machine learningWhats next?1 The ingredients of machine learningTasks: the problems that can be solved with machine learningLooking for structureModels: the output of machine learningGeometric modelsProbabilistic modelsLogical modelsGrouping and gradingFeatures: the workhorses of machine learningTwo uses of featuresFeature construction and transformationcs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 24 / 540 25. 1. The ingredients of machine learningImportant point to rememberModels lend the machine learning field diversity, but tasks and features give itunity.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 25 / 540 26. 1. The ingredients of machine learning 1.1 Tasks: the problems that can be solved with machine learningWhats next?1 The ingredients of machine learningTasks: the problems that can be solved with machine learningLooking for structureModels: the output of machine learningGeometric modelsProbabilistic modelsLogical modelsGrouping and gradingFeatures: the workhorses of machine learningTwo uses of featuresFeature construction and transformationcs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 26 / 540 27. 1. The ingredients of machine learning 1.1 Tasks: the problems that can be solved with machine learningTasks for machine learningThe most common machine learning tasks are predictive, in the sense that theyconcern predicting a target variable from features. .t Binary and multi-class classification: categorical targett Regression: numerical targett Clustering: hidden targetDescriptive tasks are concerned with exploiting underlying structure in the data.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 27 / 540 28. 1. The ingredients of machine learning 1.1 Tasks: the problems that can be solved with machine learningExample 1.1, p.15 Measuring similarityIf our e-mails are described by word-occurrence features as in the textclassification example, the similarity of e-mails would be measured in terms ofthe words they have in common. For instance, we could take the number ofcommon words in two e-mails and divide it by the number of words occurring ineither e-mail (this measure is called the Jaccard coefficient).Suppose that one e-mail contains 42 (different) words and another contains 112words, and the two e-mails have 23 words in common, then their similarity wouldbe 234211223 23130 0.18. We can then cluster our e-mails into groups, suchthat the average similarity of an e-mail to the other e-mails in its group is muchlarger than the average similarity to e-mails from other groups.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 28 / 540 29. 1. The ingredients of machine learning 1.1 Tasks: the problems that can be solved with machine learningLooking for structure IConsider the following matrix:0BBBBBBB@1 0 1 00 2 2 20 0 0 11 2 3 21 0 1 10 2 2 31CCCCCCCAImagine these represent ratings by six different people (in rows), on a scale of 0to 3, of four different films say The Shawshank Redemption, The UsualSuspects, The Godfather, The Big Lebowski, (in columns, from left to right). TheGodfather seems to be the most popular of the four with an average rating of 1.5,and The Shawshank Redemption is the least appreciated with an average ratingof 0.5. Can you see any structure in this matrix?cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 29 / 540 30. 1. The ingredients of machine learning 1.1 Tasks: the problems that can be solved with machine learningLooking for structure II0BBBBBBB@1 0 1 00 2 2 20 0 0 11 2 3 21 0 1 10 2 2 31CCCCCCCA0BBBBBBB@1 0 00 1 00 0 11 1 01 0 10 1 11CCCCCCCA 0@1 0 00 2 00 0 11A 0@1 0 1 00 1 1 10 0 0 11At The right-most matrix associates films (in columns) with genres (in rows):The Shawshank Redemption and The Usual Suspects belong to twodifferent genres, say drama and crime, The Godfather belongs to both, andThe Big Lebowski is a crime film and also introduces a new genre (saycomedy).t The tall, 6-by-3 matrix then expresses peoples preferences in terms ofgenres.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 30 / 540 31. 1. The ingredients of machine learning 1.1 Tasks: the problems that can be solved with machine learningLooking for structure IIIt Finally, the middle matrix states that the crime genre is twice as importantas the other two genres in terms of determining peoples preferences.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 31 / 540 32. 1. The ingredients of machine learning 1.1 Tasks: the problems that can be solved with machine learningTable 1.1, p.18 Machine learning settingsPredictive model Descriptive modelSupervised learning classification, regression subgroup discoveryUnsupervised learning predictive clustering descriptive clustering,association rule discoveryThe rows refer to whether the training data is labelled with a target variable, while thecolumns indicate whether the models learned are used to predict a target variable orrather describe the given data.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 32 / 540 33. 1. The ingredients of machine learning 1.2 Models: the output of machine learningWhats next?1 The ingredients of machine learningTasks: the problems that can be solved with machine learningLooking for structureModels: the output of machine learningGeometric modelsProbabilistic modelsLogical modelsGrouping and gradingFeatures: the workhorses of machine learningTwo uses of featuresFeature construction and transformationcs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 33 / 540 34. 1. The ingredients of machine learning 1.2 Models: the output of machine learningMachine learning modelsMachine learning models can be distinguished according to their main intuition:t Geometric models use intuitions from geometry such as separating(hyper-)planes, linear transformations and distance metrics.t Probabilistic models view learning as a process of reducing uncertainty,modelled by means of probability distributions.t Logical models are defined in terms of easily interpretable logicalexpressions.Alternatively, they can be characterised by their modus operandi:t Grouping models divide the instance space into segments; in each segmenta very simple (e.g., constant) model is learned.t Grading models learning a single, global model over the instance space.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 34 / 540 35. 1. The ingredients of machine learning 1.2 Models: the output of machine learningFigure 1.1, p.22 Basic linear classifier++ +++++ +pnw=pn(p+n)/2The basic linear classifier constructs a decision boundary by half-way intersecting theline between the positive and negative centres of mass. It is described by the equationw x t , with w pn; the decision threshold can be found by noting that (pn)/2 ison the decision boundary, and hence t (pn) (pn)/2 (jjpjj2 jjnjj2)/2, wherejjxjj denotes the length of vector x.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 35 / 540 36. 1. The ingredients of machine learning 1.2 Models: the output of machine learningFigure 1.2, p.23 Support vector machine++ +++++ +wThe decision boundary learned by a support vector machine from the linearly separabledata from Figure 1. The decision boundary maximises the margin, which is indicated bythe dotted lines. The circled data points are the support vectors.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 36 / 540 37. 1. The ingredients of machine learning 1.2 Models: the output of machine learningTable 1.2, p.26 A simple probabilistic modelViagra lottery P(Y spamjViagra,lottery) P(Y hamjViagra,lottery)0 0 0.31 0.690 1 0.65 0.351 0 0.80 0.201 1 0.40 0.60Viagra and lottery are two Boolean features; Y is the class variable, with values spamand ham. In each row the most likely class is indicated in bold.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 37 / 540 38. 1. The ingredients of machine learning 1.2 Models: the output of machine learningDecision ruleAssuming that X and Y are the only variables we know and care about, theposterior distribution P(Y jX) helps us to answer many questions of interest.t For instance, to classify a new e-mail we determine whether the wordsViagra and lottery occur in it, look up the corresponding probabilityP(Y spamjViagra, lottery), and predict spam if this probability exceeds0.5 and ham otherwise.t Such a recipe to predict a value of Y on the basis of the values of X andthe posterior distribution P(Y jX) is called a decision rule.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 38 / 540 39. 1. The ingredients of machine learning 1.2 Models: the output of machine learningExample 1.2, p.26 Missing values ISuppose we skimmed an e-mail and noticed that it contains the word lottery butwe havent looked closely enough to determine whether it uses the word Viagra.This means that we dont know whether to use the second or the fourth row inTable 1.2 to make a prediction. This is a problem, as we would predict spam if thee-mail contained the word Viagra (second row) and ham if it didnt (fourth row).The solution is to average these two rows, using the probability of Viagraoccurring in any e-mail (spam or not):P(Y jlottery) P(Y jViagra 0, lottery)P(Viagra 0)P(Y jViagra 1, lottery)P(Viagra 1)cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 39 / 540 40. 1. The ingredients of machine learning 1.2 Models: the output of machine learningExample 1.2, p.26 Missing values IIFor instance, suppose for the sake of argument that one in ten e-mails containthe word Viagra, then P(Viagra 1) 0.10 and P(Viagra 0) 0.90. Usingthe above formula, we obtainP(Y spamjlottery 1) 0.65 0.900.40 0.10 0.625 andP(Y hamjlottery 1) 0.35 0.900.60 0.10 0.375. Because theoccurrence of Viagra in any e-mail is relatively rare, the resulting distributiondeviates only a little from the second row in Table 1.2.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 40 / 540 41. 1. The ingredients of machine learning 1.2 Models: the output of machine learningLikelihood ratioAs a matter of fact, statisticians work very often with different conditionalprobabilities, given by the likelihood function P(XjY ).t I like to think of these as thought experiments: if somebody were to sendme a spam e-mail, how likely would it be that it contains exactly the wordsof the e-mail Im looking at? And how likely if it were a ham e-mail instead?t What really matters is not the magnitude of these likelihoods, but their ratio:how much more likely is it to observe this combination of words in a spame-mail than it is in a non-spam e-mail.t For instance, suppose that for a particular e-mail described by X we haveP(XjY spam) 3.5 105 and P(XjY ham) 7.4 106, thenobserving X in a spam e-mail is nearly five times more likely than it is in aham e-mail.t This suggests the following decision rule: predict spam if the likelihood ratiois larger than 1 and ham otherwise.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 41 / 540 42. 1. The ingredients of machine learning 1.2 Models: the output of machine learningImportant point to rememberUse likelihoods if you want to ignore the prior distribution or assume it uniform,and posterior probabilities otherwise.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 42 / 540 43. 1. The ingredients of machine learning 1.2 Models: the output of machine learningExample 1.3, p.28 Posterior oddsP(Y spamjViagra 0, lottery 0)P(Y hamjViagra 0, lottery 0)0.310.69 0.45P(Y spamjViagra 1, lottery 1)P(Y hamjViagra 1, lottery 1)0.400.60 0.67P(Y spamjViagra 0, lottery 1)P(Y hamjViagra 0, lottery 1)0.650.35 1.9P(Y spamjViagra 1, lottery 0)P(Y hamjViagra 1, lottery 0)0.800.20 4.0Using a MAP decision rule we predict ham in the top two cases and spam in thebottom two. Given that the full posterior distribution is all there is to know aboutthe domain in a statistical sense, these predictions are the best we can do: theyare Bayes-optimal.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 43 / 540 44. 1. The ingredients of machine learning 1.2 Models: the output of machine learningTable 1.3, p.29 Example marginal likelihoodsY P(Viagra 1jY ) P(Viagra 0jY )spam 0.40 0.60ham 0.12 0.88Y P(lottery 1jY ) P(lottery 0jY )spam 0.21 0.79ham 0.13 0.87cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 44 / 540 45. 1. The ingredients of machine learning 1.2 Models: the output of machine learningExample 1.4, p.30 Using marginal likelihoodsUsing the marginal likelihoods from Table 1.3, we can approximate the likelihoodratios (the previously calculated odds from the full posterior distribution areshown in brackets):P(Viagra 0jY spam)P(lottery 0jY spam)0.600.79 0.62 (0.45)P(Viagra 0jY ham)P(lottery 0jY ham)0.880.87P(Viagra 0jY spam)P(Viagra 0jY ham)P(lottery 1jY spam)P(lottery 1jY ham)0.600.880.210.13 1.1 (1.9)P(Viagra 1jY spam)P(Viagra 1jY ham)P(lottery 0jY spam)P(lottery 0jY ham)0.400.120.790.87 3.0 (4.0)P(Viagra 1jY spam)P(Viagra 1jY ham)P(lottery 1jY spam)P(lottery 1jY ham)0.400.120.210.13 5.4 (0.67)We see that, using a maximum likelihood decision rule, our very simple modelarrives at the Bayes-optimal prediction in the first three cases, but not in thefourth (Viagra and lottery both present), where the marginal likelihoods areactually very misleading.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 45 / 540 46. 1. The ingredients of machine learning 1.2 Models: the output of machine learningFigure 1.3, p.31 The Scottish classifier4164200 2 4 6PeterlotteryHam Spam264200 2 4 6PeterlotteryHam Spam6420Ham Spam0 2 4 6Peterlottery 3(top) Visualisation of two marginal likelihoods as estimated from a small data set. Thecolours indicate whether the likelihood points to spam or ham. (bottom) Combining thetwo marginal likelihoods gives a pattern not unlike that of a Scottish tartan.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 46 / 540 47. 1. The ingredients of machine learning 1.2 Models: the output of machine learningFigure 1.4, p.32 A feature treeViagralottery=0=1spam: 20ham: 5spam: 20ham: 40=0=1spam: 10ham: 5spam: 10ham: 5!Viagra"!lottery"0 10 1spam: 20ham: 5spam: 20ham: 40(left) A feature tree combining two Boolean features. Each internal node or split islabelled with a feature, and each edge emanating from a split is labelled with a featurevalue. Each leaf therefore corresponds to a unique combination of feature values. Alsoindicated in each leaf is the class distribution derived from the training set. (right) Afeature tree partitions the instance space into rectangular regions, one for each leaf. Wecan clearly see that the majority of ham lives in the lower left-hand corner.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 47 / 540 48. 1. The ingredients of machine learning 1.2 Models: the output of machine learningExample 1.5, p.33 Labelling a feature treet The leaves of the tree in Figure 1.4 could be labelled, from left to right, asham spam spam, employing a simple decision rule called majority class.t Alternatively, we could label them with the proportion of spam e-mailoccurring in each leaf: from left to right, 1/3, 2/3, and 4/5.t Or, if our task was a regression task, we could label the leaves withpredicted real values or even linear functions of some other, real-valuedfeatures.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 48 / 540 49. 1. The ingredients of machine learning 1.2 Models: the output of machine learningFigure 1.5, p.33 A complete feature treeViagralottery=0=1lotteryspam: 20ham: 40=0=1spam: 10ham: 5=0spam: 20ham: 4spam: 0ham: 1=1spam: 0ham: 1spam: 10ham: 5!Viagra"!lottery"0 10 1spam: 20ham: 4spam: 20ham: 40(left) A complete feature tree built from two Boolean features. (right) The correspondinginstance space partition is the finest partition that can be achieved with those twofeatures.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 49 / 540 50. 1. The ingredients of machine learning 1.2 Models: the output of machine learningExample 1.6, p.34 Overlapping rulesConsider the following rules:if lottery 1 then Class Y spamif Peter 1 then Class Y hamAs can be seen in Figure 1.6, these rules overlap for lottery 1 ^ Peter 1, forwhich they make contradictory predictions. Furthermore, they fail to make anypredictions for lottery 0 ^ Peter 0.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 50 / 540 51. 1. The ingredients of machine learning 1.2 Models: the output of machine learningFigure 1.6, p.35 Overlapping rules!Peter"!lottery"0 10 1The effect of overlapping rules in instance space. The two rules make contradictorypredictions in the top right-hand corner, and no prediction at all in the bottom left-handcorner.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 51 / 540 52. 1. The ingredients of machine learning 1.2 Models: the output of machine learningFigure 1.7, p.37 Mapping machine learning models1086420246810TreesRulesnaive BayesLinear ClassifierLinear RegressionkNNLogistic RegressionKmeansSVMGMMAssociations10 5 0 5 10 15 20A map of some of the models that will be considered in this book. Models that sharecharacteristics are plotted closer together: logical models to the right, geometric modelson the top left and probabilistic models on the bottom left. The horizontal dimensionroughly ranges from grading models on the left to grouping models on the right.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 52 / 540 53. 1. The ingredients of machine learning 1.2 Models: the output of machine learningFigure 1.8, p.38 ML taxonomygradinglogicala bita lotgeometricsupervisedyesnot somuchnaiveBayesassociationrulesnoyestrees &rulesnotcom-pletelysupervisedyesgroupingGMMnoyesSVMnolinearclassifierssomesupervisedK-meansnoyesk-NNA taxonomy describing machine learning methods in terms of the extent to which theyare grading or grouping models, logical, geometric or a combination, and supervised orunsupervised. The colours indicate the type of model, from left to right: logical (red),probabilistic (orange) and geometric (purple).cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 53 / 540 54. 1. The ingredients of machine learning 1.3 Features: the workhorses of machine learningWhats next?1 The ingredients of machine learningTasks: the problems that can be solved with machine learningLooking for structureModels: the output of machine learningGeometric modelsProbabilistic modelsLogical modelsGrouping and gradingFeatures: the workhorses of machine learningTwo uses of featuresFeature construction and transformationcs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 54 / 540 55. 1. The ingredients of machine learning 1.3 Features: the workhorses of machine learningExample 1.7, p.39 The MLM data setSuppose we have a number of learning models that we want to describe in termsof a number of properties:t the extent to which the models are geometric, probabilistic or logical;t whether they are grouping or grading models;t the extent to which they can handle discrete and/or real-valued features;t whether they are used in supervised or unsupervised learning; andt the extent to which they can handle multi-class problems.The first two properties could be expressed by discrete features with three andtwo values, respectively; or if the distinctions are more gradual, each aspectcould be rated on some numerical scale. A simple approach would be tomeasure each property on an integer scale from 0 to 3, as in Table 1.4. Thistable establishes a data set in which each row represents an instance and eachcolumn a feature.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 55 / 540 56. 1. The ingredients of machine learning 1.3 Features: the workhorses of machine learningTable 1.4, p.39 The MLM data setModel geom stats logic group grad disc real sup unsup multiTrees 1 0 3 3 0 3 2 3 2 3Rules 0 0 3 3 1 3 2 3 0 2naive Bayes 1 3 1 3 1 3 1 3 0 3kNN 3 1 0 2 2 1 3 3 0 3Linear Classifier 3 0 0 0 3 1 3 3 0 0Linear Regression 3 1 0 0 3 0 3 3 0 1Logistic Regression 3 2 0 0 3 1 3 3 0 0SVM 2 2 0 0 3 2 3 3 0 0Kmeans 3 2 0 1 2 1 3 0 3 1GMM 1 3 0 0 3 1 3 0 3 1Associations 0 0 3 3 0 3 1 0 3 1The MLM data set describing properties of machine learning models.cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 56 / 540 57. 1. The ingredients of machine learning 1.3 Features: the workhorses of machine learningExample 1.8, p.41 Two uses of featuresSuppose we want to approximate y cosx on the interval 1 x 1. A linearapproximation is not much use here, since the best fit would be y 0. However,if we split the x-axis in two intervals 1 x 0 and 0 x 1, we could findreasonable linear approximations on each interval. We can achieve this by usingx both as a splitting feature and as a regression variable (Figure 1.9).cs.bris.ac.uk/ flach/mlbook/ Machine Learning: Making Sense of Data December 29, 2013 57 / 540 58. 1. The ingredients of machine learning 1.3 Features: the workhorses of machine learningFigure 1.9, p.41 A small regression treex = 2x+1