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-parametric Density Estimation\nFrom Density Estimation to NN Classifiers\nLiterature\nOur First Learning Algorithms: Single-Gaussian Classifier and Na¨ıve Bayes\nProblems with the Single-Gaussian Bayes Classifier\nObvious Problem 1:\n▶Highly unlikely that high-dimensional data will obey a single normal\ndistribution with...
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−1\n2 (x−µi)T Σ−1\ni\n(x−µi)\nwith its own mean vector µi and covariance matrix Σi.\n23 / 72" }, { "id": "lecture-03-p024", "sourceId": "lecture-03", "lectureId": "lecture-03", "page": 24, "title": "24 / 72", "text": "Motivation\nParametric Density Estimation\nNon-parametric ...
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: need to estimate O(nd2) parameters\n(square d × d covariance matrices for n classes)\n▶Problematic when relatively little training data\n▶Problematic when large number d of features\n(e.g., for document classification – see below ...)\nA Simpler Variant: The Na¨ıve Bayes Classifier\n▶Insight: source of the complexity p...
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)\n⇒Do estimation in d one-dimensional spaces, instead of a d-dimensional one\n27 / 72", "keywords": [ "estimation", "density", "classi", "the", "follow", "joint", "distribution", "decomposes", "bayes", "motivation", "parametr...
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features are pairwise\nindependent\n▶For each class ωk and each feature j, estimate the parameters of the single\nfeature distribution p(xj | ωk) via Maximum Likelihood\nClassification of a new instance x:\nω∗= arg min\ni\nRλ(αi | x)\nwhere P(ωk | x) = P(ωk) Q\ni p(xi|ωk)\np(x)\n28 / 72" }, { "id": "lectur...
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ve Bayes Independence Assumption:\n▶Estimate d one-dimensional distributions P(xj | ωi) for each class ωi\n▶Each requires O(k) values1\n▶Need to estimate only O(nkd) values!\n(if n is the number of classes)\n⇒Can help prevent overfitting ...\n1k −1, to be precise. The last value is redundant because the distribution mus...
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agency releases)\n▶sentiment classification (Twitter tweets, user comments on YouTube, ...)\n▶identification of offensive user postings in online forums\n2Nowadays, there are more complex and powerful document classification models, based on\nstatistical modeling and deep neural networks\n31 / 72" }, { "id":...
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independence assumption is clearly violated:\n▶Consider news categories Ω= {Politics, Sports, ...}:\nP(green | party, Politics) > P(green | Politics)\nIt’s much more likely to find the word “green” in a politics article that has the\nword “party” in it (as opposed to any political article)\n▶Positive correlation between...
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shapes", "rarely", "encountered", "real", "non-parametric" ], "retrievalText": "Density Estimation. ▶Common parametric shapes rarely fit densities encountered in real data. Motivation\nParametric Density Estimation\nNon-parametric Density Estimation\nFrom Density Estimation to...
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density", "estimation", "the", "nearest", "motivation", "parametric", "non-parametric", "classi", "ers", "literature", "neighbour", "via" ], "retrievalText": "Density Estimation. 36 / 72. Motivation\nParametric Density E...
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would happen to lie in area A?\nP(x ∈A)\n= ?\nBasic Idea:\n▶Use training set D (has been drawn randomly from the ‘world’)\n▶If |D| = N and k of the training instances xi ∈D are located within A\n⇒\nestimate ˆP(x ∈A) = k\nN\n38 / 72" }, { "id": "lecture-03-p039", "sourceId": "lecture-03", "lect...
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"experiment" ], "retrievalText": "Density Estimation. 40 / 72. Motivation\nParametric Density Estimation\nNon-parametric Density Estimation\nFrom Density Estimation to NN Classifiers\nLiterature\nNearest Neighbour Density Estimation\nA Simple Experiment\nStep 1: Define the ‘world’\n▶Define true (hidden) probab...
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/ 72. Motivation\nParametric Density Estimation\nNon-parametric Density Estimation\nFrom Density Estimation to NN Classifiers\nLiterature\nNearest Neighbour Density Estimation\nA Simple Experiment\nStep 3: Try to recover hidden distribution via NN Estimation\n▶Estimate density for each point on a grid over the entire fe...
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nGiven:\n▶training set D = {xi} of size N and dimensionality d\n▶query point x for which to estimate p(x)\n▶number of required nearest neighbours k (parameter)\nDo:\n▶Calculate radius h of neighbourhood within which the k nearest neighbours\nof x are located: h = distance from x to farthest of its k neighbours\n▶Estima...
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nn", "motivation", "parametric", "non-parametric", "classi", "ers", "literature", "simple", "experiment", "step" ], "retrievalText": "Density Estimation. 48 / 72. Motivation\nParametric Density Estimation\nNon-parametric Density Estimat...
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Motivation\nParametric Density Estimation\nNon-parametric Density Estimation\nFrom Density Estimation to NN Classifiers\nLiterature\nk-NN Density Estimation\nNeeded: Distance Metric D(x, y)\nMost common solution: Euclidean Distance\nD(x, y) =\nsX\ni\n(xi −yi)2\nwhere xi ... value of the ith feature of instance x\nProble...
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function) K(xi, ·) of its own\n▶Kernel function is centered on xi and decays with distance from xi\n▶Density estimate over entire space is just the normalised sum of the little\nkernel functions\n▶Density estimate at a point x is P\nxi∈D K(xi, x)\nMotivation:\n▶All training examples contribute to density estimate every...
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n1\n(σ2√\n2π)d e−D(x,xi)2\n2σ2\n54 / 72", "keywords": [ "density", "estimation", "kernel", "function", "and", "algorithm", "estimate", "motivation", "parametric", "non-parametric", "classi", "ers" ], "retri...
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, { "id": "lecture-03-p057", "sourceId": "lecture-03", "lectureId": "lecture-03", "page": 57, "title": "57 / 72", "text": "Motivation\nParametric Density Estimation\nNon-parametric Density Estimation\nFrom Density Estimation to NN Classifiers\nLiterature\nKernel Density Estimation...
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/ 72", "keywords": [ "density", "estimation", "kernel", "motivation", "parametric", "non-parametric", "classi", "ers", "literature", "simple", "experiment", "step" ], "retrievalText": "Density Estimation. 5...
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around x\n▶ki of these k belong to class ωi\n⇒Estimate for probability density p(x):\np(x) ∝k/N\n⇒Estimate for p(x ∧ωi):\np(x ∧ωi) ∝ki/N\nNow consider Bayes’ rule:\nP(ωi | x) = p(x|ωi)P (ωi)\np(x)\n... and the identity\np(x ∧ωi) = p(x|ωi)P(ωi)\n⇒\nP(ωi | x) = p(x | ωi)P(ωi)\np(x)\n= p(x ∧ωi)\np(x)\nˆ∝ki/N\nk/N = ki\nk\...
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D, count the class occurrences within\nthese, and predict the class that the majority of these neighbours belong to.\n62 / 72", "keywords": [ "the", "classi", "density", "estimation", "and", "neighbours", "training", "set", "function", ...
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into Voronoi cells, each labelled by the category of the\ntraining point it contains. In three dimensions, the cells are three-dimensional, and the\ndecision boundary resembles the surface of a crystal. (From [Duda et al., 2001].)\n64 / 72", "keywords": [ "the", "density", "estimation", ...
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nParametric Density Estimation\nNon-parametric Density Estimation\nFrom Density Estimation to NN Classifiers\nLiterature\nThe Curse of Dimensionality\nThe “Curse of Dimensionality”\nProblem:\n▶Quality of class prediction depends on ‘density’ of training instances around\nquery point x (there should be a sufficient number...
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\n▶Data set D of size N in d-dimensional unit hypercube H\n(i.e., hypercube with side length = 1.0):\nd numeric features with range [0, 1]\n▶Cubic neighbourhood A with side length h and volume hd\n▶To contain k points, average neighbourhood must occupy\na fraction k/N of the entire volume of H (which is 1.0)\n⇒hd = k/N...
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\n⇒Need for feature selection\nThe problem affects all distance-/neighbour-based methods for prediction.\n68 / 72" }, { "id": "lecture-03-p069", "sourceId": "lecture-03", "lectureId": "lecture-03", "page": 69, "title": "▶the Single-Gaussian and the Na¨ıve Bayes Classifiers", "...
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-parametric Density Estimation\nFrom Density Estimation to NN Classifiers\nLiterature\nThe Curse of Dimensionality\nWhat to expect in the exam: A few typical questions ...\nParametric Density Estimation:\nConsider the situation depicted below, as also discussed in the lecture.\nThe top graph shows a dataset D consisting...
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\nCambridge, MA: MIT Press.\n72 / 72", "keywords": [ "density", "estimation", "classi", "literature", "motivation", "parametric", "non-parametric", "ers", "duda", "hart", "and", "stork" ], "retrievalText": ...
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-04-p003", "sourceId": "lecture-04", "lectureId": "lecture-04", "page": 3, "title": "Cross-Validation", "text": "The General Setting\nEmpirical Evaluation of Classifiers\nOverfitting & Model Selection\nEvaluation Measures\nCost-sensitive Learning\nLiterature\nOutline\n1 The General Setting\n...
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according to the underlying distribution p\n▶The training examples x ∈D have been labelled with class labels ωi ∈Ω\nby some external expert entity.\n▶The training examples have been drawn independently and all come from\nthe same distribution p1\n▶The distribution p(X, Ω) will not change over time (or between training ...
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", "measures" ], "retrievalText": "Classification: General Issues. 6 / 56. The General Setting\nEmpirical Evaluation of Classifiers\nOverfitting & Model Selection\nEvaluation Measures\nCost-sensitive Learning\nLiterature\nClassification Learning: The General Scenario\n6 / 56" }, { "id": "...
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"general", "see", "later", "slides", "alternative", "criteria", "ers", "error" ], "retrievalText": "Classification: General Issues. (see later slides for alternative evaluation criteria). The General Setting\nEmpirical Evaluation of Classifiers\nOverfitt...
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Learning\nLiterature\nThe Resubstitution Estimate\nAccuracy Estimation (1): The Resubstitution Estimate\nResubstitution Estimate: Procedure\n▶Learn classifier c from training set D\n▶Then: apply c to training data D and count number of correct ‘predictions’\n▶Result: information about the classifier’s accuracy on the tra...
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Holdout Testing:\n▶Randomly re-shuffle and split N times ⇒gives N estimates\n▶Can compute mean, range, standard deviation of accuracy estimates\n11 / 56" }, { "id": "lecture-04-p012", "sourceId": "lecture-04", "lectureId": "lecture-04", "page": 12, "title": "▶Test ci on test set Fi ...
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, "predicting", "evaluation", "classi", "most", "frequent" ], "retrievalText": "Classification: General Issues. = the prediction accuracy one would achieve by always predicting the class. The General Setting\nEmpirical Evaluation of Classifiers\nOverfitting & Model Sele...
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nLiterature\nOverfitting vs. Generalisation\nGeneralisation:\n▶Given training set will usually not contain all possible situations (events) that\ncould ever occur\n▶Purpose of a learned model is to predict correct class for new cases\n▶Model must be more general than simple summary of training set!\nOverfitting:\n▶Set D ...
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\nCost-sensitive Learning\nLiterature\nBias vs. Variance\nBias\nGoal:\n▶Avoid learning an overfitting model\n▶Force learner to generalise\nApproach:\n▶Put constraints on models (decision boundaries) allowed to the learner:\n▶Hard constraints: Strictly restrict the class C of models (e.g., only permit\nlinear boundaries,...
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vs. complexity, generalisation vs. overfitting,\nbias vs. variance\n⇒Task of system designer/experimenter:\nfind a good place in this continuum, relative to given problem (data set)\n⇒No analytical basis for decision; our only guide is the data ...\n19 / 56", "keywords": [ "the", "bias", "er...
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n▶How good is the final selected classifier c∗?\nHow well will it perform on new data?\nProblem:\n▶Cannot use Dval to estimate the true accuracy of c∗,\nbecause Dval was used in selecting c∗⇒c∗is optimised towards Dval!\n\u0011 The “Optimiser’s Curse” ...\n21 / 56", "keywords": [ "the", "classi", ...
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) = true accuracy of c relative to the underlying distribution p)\n▶When you evaluate several classifiers, the estimates will have different\nrandom errors ϵ (some positive, some possibly negative)\n▶Even if the estimation process is unbiased (ϵ is symmetrically distributed\naround 0), picking the model with the highest...
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Model Selection\nEvaluation Measures\nCost-sensitive Learning\nLiterature\nModel Selection\nModel Selection: Choosing the ‘Right’ Classifier Complexity\nModel Selection – Strategy 2\nIterative Model Simplification (“Pruning”):\n1 Split D into 3 sets Dtrain, Dval, Dtest\n2 First learn a complex model c from training data ...
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Evaluation of Classifiers\nOverfitting & Model Selection\nEvaluation Measures\nCost-sensitive Learning\nLiterature\nModel Selection\nModel Selection: Choosing the ‘Right’ Classifier Complexity\nModel Selection – Strategy 2\nIterative Model Simplification (“Pruning”):\n1 Split D into 3 sets Dtrain, Dval, Dtest\n2 First lear...
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"text": "The General Setting\nEmpirical Evaluation of Classifiers\nOverfitting & Model Selection\nEvaluation Measures\nCost-sensitive Learning\nLiterature\nEmpirical Accuracy / Error / Cost\nEvaluation (1): Empirical Accuracy / Error\nEmpirical Accuracy measured on test data Dtest:\n▶Global measure of performance of a cl...
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\n=\nTP + FP = total # of p predictions\nRecall = proportion of p instances that are correctly recognised:\nRec =\nTP\nTP + FN = TP\nP\nPrecision = proportion of p predictions that are correct:\nPrec =\nTP\nTP + FP = TP\nPP\n31 / 56", "keywords": [ "evaluation", "recall", "precision", ...
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\nEvaluation Measures\nCost-sensitive Learning\nLiterature\nRecall & Precision\nEvaluation (3): Recall and Precision\nIf you need one single measure to compare binary classifiers:\nF1 Score = harmonic mean of recall and precision:\nF1 = 2 · Prec · Rec\nPrec + Rec =\n2TP\n2TP + FP + FN\n▶Strikes a compromise between reca...
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p examples\nN\n=\nTN + FP = total # of n examples\nTrue Positive Rate = proportion of positive instances predicted correctly:\nTPR =\nTP\nTP + FN = TP\nP\n(= Rec)\nFalse Positive Rate = proportion of ‘false alarms’:\nFPR =\nFP\nFP + TN = FP\nN\n35 / 56", "keywords": [ "roc", "and", "rate",...
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nEmpirical Evaluation of Classifiers\nOverfitting & Model Selection\nEvaluation Measures\nCost-sensitive Learning\nLiterature\nROC Space and ROC Analysis\nROC Space\n38 / 56", "keywords": [ "roc", "general", "evaluation", "space", "issues", "the", "setting", ...
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Model Selection\nEvaluation Measures\nCost-sensitive Learning\nLiterature\nROC Space and ROC Analysis\nWhich of these ...\n▶is the best one?\n▶and under what conditions?\n41 / 56", "keywords": [ "general", "the", "evaluation", "roc", "and", "issues", "settin...
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Note: this also includes the ‘all pos’ (1,1) and ‘all neg’ (0,0) corner classifiers!\n▶It is convex (if all classifiers are below the diagonal, the hull is the diagonal)\n▶It can be computed in O(n log n) time\n▶Any classifier below the hull is suboptimal under all possible conditions\nbecause:\n▶Any desired point on the ...
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slope of ‘iso-performance line’ for these conditions\n▶Find highest point(s) on ROC convex hull that intersect(s) a line with that\nslope\n45 / 56" }, { "id": "lecture-04-p046", "sourceId": "lecture-04", "lectureId": "lecture-04", "page": 46, "title": "Question:", "text": "Th...
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lecture-04", "lectureId": "lecture-04", "page": 48, "title": "⇒C2 is preferable", "text": "The General Setting\nEmpirical Evaluation of Classifiers\nOverfitting & Model Selection\nEvaluation Measures\nCost-sensitive Learning\nLiterature\nROC Space and ROC Analysis\nSelecting the Best Classifier wit...
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er that is optimised to\nthis cost scenario\n▶(in other words: a classifier that minimises the expected cost, rather than the\nerror rate)\nObservation:\n▶Many learning algorithms are designed to try to minimise the number of\nerrors (not some cost measure) on the training set\n⇒Try to force the learning algorithm to gi...
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(e.g., Bayes, Logistic Regression, Neural Networks)\nChanging the Objective Function:\n▶Change the function the learner is trying to minimise on the training data\n▶Possible with, e.g., neural networks, which are trained by minimising a given\nerror (‘loss’) function\n51 / 56" }, { "id": "lecture-04-p052"...
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the evaluation results were submitted to the NCTA in the form of. The General Setting\nEmpirical Evaluation of Classifiers\nOverfitting & Model Selection\nEvaluation Measures\nCost-sensitive Learning\nLiterature\nWhat to expect in the exam: A few typical questions ...\nConsider different Corona virus tests and think of t...
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be greater than the number\nof negative cases, and the costs of false positives and false negatives are the same?\n▶the expected class distribution is as in the confusion matrices, and a missed Corona\ncase is ten times as bad as a false positive result?\n▶it is expected that there will be no positive cases at all in A...
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following are excellent textbooks on various aspects of Machine Learning\nand Classification. I recommend them to anyone seriously interested in these\ntopics. However, you will not need to read the books in order to follow and\nunderstand this class.\nDuda, R., Hart, P. and Stork, D. (2001).\nPattern Classification (2nd...
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lecture-05-p002", "sourceId": "lecture-05", "lectureId": "lecture-05", "page": 2, "title": "2 / 49", "text": "Recap: Simple ‘Baseline’ Classifiers\nOther Popular Classifiers\nLiterature\nGoals of this Lecture\n▶Introduce some popular learning algorithms for classification\n▶Review their relat...
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/ 49", "keywords": [ "the", "classi", "baseline", "accuracy", "default", "your", "substantially", "better", "ers", "most", "frequent", "training" ], "retrievalText": "Common Machine Learning Algorithms. \u0...
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complex decision boundaries (low bias)\nDisadvantages / Problems:\n▶Slow at classification time (grows linearly with |D|)\n▶Does not produce an interpretable model\n▶Sensitive to irrelevant features (Curse of Dimensionality)\n⇒problematic in high-dimensional feature spaces\n6 / 49", "keywords": [ "classi",...
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ncool\nnormal\ntrue\nDon’t Play\n-\n7\novercast\ncool\nnormal\ntrue\nPlay\n+\n8\nsunny\nmild\nhigh\nfalse\nDon’t Play\n-\n9\nsunny\ncool\nnormal\nfalse\nPlay\n+\n10\nrain\nmild\nnormal\nfalse\nPlay\n+\n11\nsunny\nmild\nnormal\ntrue\nPlay\n+\n12\novercast\nmild\nhigh\ntrue\nPlay\n+\n13\novercast\nhot\nnormal\nfalse\nPla...
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/ 49", "keywords": [ "decision", "tree", "the", "given", "simple", "trees", "training", "set", "desirable", "properties", "classi", "ers" ], "retrievalText": "Common Machine Learning Algorithms. Desirable P...
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Newton, Preface to “Principia”a\n(cited after Li Vit´anyi, 2013)\nahttps://www.thelatinlibrary.com/newton.regulae.html\n10 / 49" }, { "id": "lecture-05-p011", "sourceId": "lecture-05", "lectureId": "lecture-05", "page": 11, "title": "maximises a local optimality criterion.", ...
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"make", "leaf", "label", "exit", "algorithm", "split" ], "retrievalText": "Common Machine Learning Algorithms. ⇒make current node a leaf, label with majority class in N, and EXIT. Recap: Simple ‘Baseline’ Classifiers\nOther Popular Classifiers\nLiterature\nA Classic: De...
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a Bernoulli trial as a function of success\nprobability, often called the binary entropy\nfunction. The entropy is maximised at 1 bit per trial\nwhen the two possible outcomes are equally\nprobable, as in an unbiased coin toss (or uniform\nbinary class distribution):\n14 / 49", "keywords": [ "distribution...
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:\nE(D) = −\nk\nX\ni=1\nN(ωi)\n|D|\nlog N(ωi)\n|D|\nwhere N(ωi) is the number of examples of class ωi in D\n15 / 49" }, { "id": "lecture-05-p016", "sourceId": "lecture-05", "lectureId": "lecture-05", "page": 16, "title": "|Di|", "text": "Recap: Simple ‘Baseline’ Classifiers\nO...
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n= .940 −6/14 · 1.00 −8/14 · .811\n= 0.048 [bits]\n17 / 49", "keywords": [ "bits", "log2", "the", "classi", "ers", "root", "play", "common", "machine", "algorithms", "recap", "simple" ], "retrievalText": "C...
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current node a leaf, label with class ω, and EXIT\n2 If no more features available for splitting in current node N\n⇒make current node a leaf, label with majority class in N, and EXIT\n3 Calculate IGDN (Ai) for all features Ai\n(i.e., try out all features for splitting)\n4 Select feature A = arg max\nAi IG(Ai) with max...
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p020", "sourceId": "lecture-05", "lectureId": "lecture-05", "page": 20, "title": "1", "text": "Recap: Simple ‘Baseline’ Classifiers\nOther Popular Classifiers\nLiterature\nA Classic: Decision Trees and the ID3 Algorithm\nNumeric Features\nDay\nOutlook\nTemp.\nHumidity\nWindy?\nClass\n1\nsunn...
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A:\n▶Sort examples in current node N according to A\n▶Try each mid-point xi between two values as possible split point\n▶Compute IG(A ≤xi) that would be produced by this split\n▶Select tA = arg maxxi IG(A ≤xi)\n▶This test competes against all other features A for the best split overall\nTemperature\n64\n65\n68\n69\n70\...
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Stop splitting a node further (even if it still contains examples of several\nclasses) if remaining features don’t seem to say much about the class any\nmore (to be determined by some appropriate statistical test)\n2. Post-Pruning:\n▶First construct (possibly complex) tree that is maximally consistent with (has\nminimu...
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′\ni = n · |Di|\n|D|\n24 / 49" }, { "id": "lecture-05-p025", "sourceId": "lecture-05", "lectureId": "lecture-05", "page": 25, "title": "|D|", "text": "Recap: Simple ‘Baseline’ Classifiers\nOther Popular Classifiers\nLiterature\nA Classic: Decision Trees and the ID3 Algorithm\nP...
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if there is no Ti that improves accuracy when removed: EXIT\n- otherwise: choose Ti with maximum improvement and replace with leaf\n4 Go to 3.\nQuestion:\n▶Why additional validation set Dval?\n▶Why not use original training set D for making pruning decisions?\n5see (Quinlan, 1987)\n26 / 49", "keywords": [ ...
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Recap: Simple ‘Baseline’ Classifiers\nOther Popular Classifiers\nLiterature\nRobust Learning via Ensemble Building: Random Forest\nRobust Learning via Ensembles: The Random Forest Classifier6\nBasic Idea:\n▶Instead of learning one decision tree, build N (a large number of) different\ntrees from the same data ( \u0011 a fo...
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c8b53cc83d4ef0bd4105d0736750f08111388cd9cceac93f13fe2507987a694b
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D, but introduce randomisation in\nnode splitting process: at each node, select and consider only a random\nsubset of the features as candidates for splitting.\nStatistical Analysis Shows:\n▶Feature randomisation is more effective in producing uncorrelated trees.\n29 / 49" }, { "id": "lecture-05-p030", ...
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3b45bfeaae4478926db99c792f44a079e10f0687d4c8a8d36ad865130733d2f0
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{ "id": "lecture-05-p032", "sourceId": "lecture-05", "lectureId": "lecture-05", "page": 32, "title": "robust against noise and have simple and efficient learning algorithms", "text": "Recap: Simple ‘Baseline’ Classifiers\nOther Popular Classifiers\nLiterature\nRobustness via Linear Clas...
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39095771f0915de5fefcada609d51a2b3dbaca5609ba3f58d17d94a924f09bd3
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space, with features a1 = x2\n1,\na2 = x2\n2, a3 =\n√\n2x1x2. The circular decision\nboundary in 2D becomes a linear boundary (a\nplane) in these three dimensions.\n7Figures from (Russell & Norvig, 2002)\n33 / 49", "keywords": [ "classi", "the", "boundary", "ers", "linear",...
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66c6a7ca6180fa22bc8f44a8f1e216db4dacca622fe7f3e74ff2722a27596cd1
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\nSupport Vector Machines (SVMs)8\nBasic Idea of SVMs (for binary classification tasks):\n▶Map data into a (very) high-dimensional space by a transform\nϕ : Rd1 →Rd2, x 7→ϕ(x) with d1 < d2\n▶Find the optimal linear decision boundary between the two classes in this\nhigh-dimensional space\n▶‘Optimal’ = the linear boundar...
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06b46bce02fe8d2c4f5f879004b2969f131868eb4920065c5e47fc8fd52055d0
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Classifiers\nLiterature\nRobustness via Linear Classification in High-dimensional Spaces: Support Vector Machines\nSimplified Scenario: Linear Separation in Original Space\n▶Given: Feature space X defined by a set of d features X = {X1, ..., Xd}\n▶Given: Training set D = {(xi, yi)} with xi ∈X and yi ∈Ω= {−1, +1}10\n▶Find: ...
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= −1\n▶Distance between these is the margin\n(of width\n2\n∥w∥)\n▶Max margin hyperplane lies halfway\nbetween these\n11Figure from https://en.wikipedia.org/wiki/Support_vector_machine\n37 / 49", "keywords": [ "classi", "ers", "linear", "can", "two", "the", "...
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10b167b91ec85728a65600f15f5f19b0b3d827316b7a2acecda658a2c84603c7
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nif\nyi = −1\nwhich can be summarised as:\nyi(w · xi + b) ≥1\nfor all xi ∈D\nResulting Optimisation Problem:\nFind w and b that minimise ∥w∥2\nsubject to the constraints yi(w · xi + b) ≥1, for i = 1, ..., n\n38 / 49" }, { "id": "lecture-05-p039", "sourceId": "lecture-05", "lectureId": "lecture...
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9070d918da11835373fd08e7f07edcd41a0caa5454154dd19433d2eff5bffb38
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do this explicitly: map training data D to ϕ(D) and solve above\noptimisation problem in Rd2\n▶Obtain linear classifier (with d2-dimensional weight vector w) in\nhigh-dimensional space\n▶To classify a new object x: map x into new space and apply\nhigh-dimensional classifier to it: ˆy(x) = sgn(w · ϕ(x) + b)\nProblem:\n▶Di...
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kernel", "vectors", "dot", "and", "function", "ers", "cation", "high-dimensional", "support", "for" ], "retrievalText": "Common Machine Learning Algorithms. n. Recap: Simple ‘Baseline’ Classifiers\nOther Popular Classifiers\nLiterature\nR...
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(xi) · ϕ(x)) + b\n!\nto\nClass Prediction in SVM\nˆy(x) = sgn\n n\nX\ni=1\nαiyiK(xi, x) + b\n!\n42 / 49" }, { "id": "lecture-05-p043", "sourceId": "lecture-05", "lectureId": "lecture-05", "page": 43, "title": "Figure : Illustration of the mapping ϕ:", "text": "Recap: Simple ‘...
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95e95ca0e30cc9924ea8c1c73b65f17966851852f4b820edad5e1a87fbce81f7
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Quadratic Kernel\nϕ(x)\n=\n{x2\nn, ..., x2\n1,\n√\n2xnxn−1, ...,\n√\n2xnx1, ...,\n√\n2xn−1xn−2, ...,\n√\n2xn−1x1, ...,\n√\n2x2x1,\n√\n2cxn, ...,\n√\n2cx1, c}\n44 / 49", "keywords": [ "kernel", "classi", "the", "quadratic", "simple", "ers", "mapping", ...
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4b82b5d796c15080b5d22a05be82bd2c08f8d223904ced9ba809a7b898df5956
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\n2 x1x3\n...\n√\n2c x1\n√\n2c x2\n...\nc\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n·\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\ny2\n1\ny2\n2\n...\n√\n2 y1y2\n√\n2 y1y3\n...\n√\n2c y1\n√\n2c y2\n...\nc\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n=\n(ϕ(x) · ...
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06d62ae7f6233f186371357d4b379068ceb25a9b0ef380ba88036aa39501749a
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n\n\n\n\n\n\n\n\n\n·\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\ny2\n1\ny2\n2\n...\n√\n2 y1y2\n√\n2 y1y3\n...\n√\n2c y1\n√\n2c y2\n...\nc\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n=\n(ϕ(x) · ϕ(y))\n▶with ϕ(x) = {x2\n1, x2\n2, ...,\n√\n2x1x2,\n√\n2x1x3, ...,\n√\n2cx1,\n...
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-p047", "sourceId": "lecture-05", "lectureId": "lecture-05", "page": 47, "title": "robustness", "text": "Recap: Simple ‘Baseline’ Classifiers\nOther Popular Classifiers\nLiterature\nRobustness via Linear Classification in High-dimensional Spaces: Support Vector Machines\nWhat you should remem...
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49", "keywords": [ "the", "classi", "set", "training", "new", "algorithms", "terms", "adding", "and", "machine", "may", "affected" ], "retrievalText": "Common Machine Learning Algorithms. learning algorithm...
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6aff5e74d91198ca140f8649f33cecd43948da9d0177a26f6c900d3dd3e3caa7
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).\nRandom Rorests. Machine Learning 45(1), 5-32.\nCortes, C. and Vapnik, V. (1995).\nSupport-Vector Networks. Machine Learning 20(3), 273-297.\nHyafil, L. and Rivest, R. (1976).\nConstructing Optimal Binary Decision Trees is NP-Complete. Information Processing\nLetters 5(1), 15-17.\nLi, M. and Vit´anyi, P. (2013).\nAn ...
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lectureId": "lecture-06", "page": 3, "title": "3 / 25", "text": "Feature Selection & Construction\nModel Selection and Parameter Optimisation\nEnsemble Methods\nPost-Processing of Predictions\nLiterature\nOutline\n1 Feature Selection & Construction\n2 Model Selection and Parameter Optimisation\n3 Ense...
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., based on knowledge/intuition about the task)\n▶Automatic selection of informative features via simple statistical tests\n▶Automatic search for optimal feature subset, governed by cross-validation\n5 / 25", "keywords": [ "the", "selection", "classi", "are", "fewer", ...
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"selection", "more", "one", "complex", "heuristic", "strategies", "start", "set", "the", "beam", "improving" ], "retrievalText": "Improving Learning Results. More Complex Heuristic Search Strategies:. Feature Selection & Constru...
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: "lecture-06", "lectureId": "lecture-06", "page": 9, "title": "(X −Y ), (X · Y ), (X/Y ), ...", "text": "Feature Selection & Construction\nModel Selection and Parameter Optimisation\nEnsemble Methods\nPost-Processing of Predictions\nLiterature\nFeature Construction\nPossible Operations on Featu...
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9ad04b5ca92307aa7c8757cf029b12634a79b54c64642bf1cdcbe27da7862a36
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▶This is what you have been doing for the past weeks ...\n10 / 25" }, { "id": "lecture-06-p011", "sourceId": "lecture-06", "lectureId": "lecture-06", "page": 11, "title": "1 Voting: combine different types of classifiers", "text": "Feature Selection & Construction\nModel Selec...
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e5fb9b3ec0c2b3d604f7b700039b96042c9411440154496a2d179b4cc15ca87a
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training\ndata D\n▶Classification of new object x: apply all N classifiers to x; result: N class\npredictions c1(x), ..., cN(x)\n▶Class predicted by meta-classifier is the class that gets the most votes:\nc(x) = arg max\nωj∈Ω\nN\nX\ni=1\n1 [ci(x) = ωj]\nVariant: Weighted Majority\n▶Weight individual classifiers’ votes by t...
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5d6aec3929965f80d406b711da38a936366fde5fd04da1b91a38d34713d49c1e
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\ndescribed only through the predictions of the base learners:\nyi = ⟨c1(xi), ..., cN(xi)⟩\nif base learners predict class labels only; or\nyi = ⟨Pc1(ω1), ..., Pc1(ωk); ...; PcN (ω1), ..., PcN (ωk)⟩\nif base learners ci predict a probability distribution Pci(Ω)\nClassification of new object x:\n▶Apply base classifiers to...
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8d9e68b823207db8fcaac348ffbd3f45b7df057cc5a6c1857974ff89addcb683
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to focus on weaknesses of ct\nBoosting: The General Algorithm Schema\nInitialise: Let D1 = D with uniform weights\nFor t = 1...N do:\n▶Learn classifier ct from weighted training seta Dt\n▶Compute ct’s error ϵt on its training set Dt\n▶Create data set Dt+1 by reweighting examples in Dt:\nincrease weight if example was mi...
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436c70925b0845a68b2c97b7ff50c7817d0fed7d369aa9d9e95b834a697090fd
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we may know something about the application task that is\nhard to integrate into the features or the learning process\n▶Example: Plausibility of class predictions in the context of other predictions\n▶May be easier to encode this in a post-processing algorithm\nConcrete example from a previous MLPC classification projec...
mlpc-exam
lib/data/slide-index.json
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over a set of Austrian TV\nshows: only 13 of the true segments are shorter than 8 sec; many long segments. Right: Histogram of\nlengths of predicted sequences of continuous “music” class labels; 2.105 are shorter than 8 sec.\n19 / 25" }, { "id": "lecture-06-p020", "sourceId": "lecture-06", "le...
mlpc-exam
lib/data/slide-index.json
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cc11ee8c195f1624238821c3c6836f5b0424d0bf3f1eb06cbab16ac69b75f485
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than some lower bound\nLmin\n⇒Change all the labels in this segment to no music\n21 / 25" }, { "id": "lecture-06-p022", "sourceId": "lecture-06", "lectureId": "lecture-06", "page": 22, "title": "22 / 25", "text": "Feature Selection & Construction\nModel Selection and Paramete...
mlpc-exam
lib/data/slide-index.json
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