Data Analysis II (Data Mining Techniques) WS 07/08. Contents


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1 Contents 0. Introduction 1. Instance Based Learning 2. Naïve Bayes 3. Clustering 4. Decision Trees 5. Association Rules 6. Support Vector Machines Folie 1
2 Short Introduction Assumption: quantities of interest governed by probability distributions Optimal decisions by reasoning about these probabilities together with observed data Features: each observed training example increases or decreases the probability that a hypothesis is correct classify new instances by combining predictions of multiple hypotheses weighted by their probabilities difficulty: initial knowledge of probabilities required (or estimation based on background knowledge or assumption) Intro to Naive Bayes: Folie 2
3 Thomas Bayes (c April 17,1761) was a British mathematician and Presbyterian minister, known for having formulated a special case of Bayes' theorem, which was published posthumously. Biography Thomas Bayes was born in London. He is known to have published two works in his lifetime: Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures (1731), and An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst (published anonymously in 1736), in which he defended the logical foundation of Isaac Newton's calculus against the criticism of George Berkeley, Author of The Analyst. It is speculated that Bayes was elected as a Fellow of the Royal Society in 1742 on the strength of the Introduction to the Doctrine of Fluxions, as he is not known to have published any other mathematical works during his lifetime. Bayes died in Tunbridge Wells, Kent. He is interred in Bunhill Fields Cemetery in London where many Nonconformists are buried. Folie 3
4 2.1. Bayes Theorem revisted Task: determine best hypothesis for a space H giving the training data D Best Hypothesis: most probable hypothesis, given data and initial knowledge about prior probability of hypothesis from H Bayes Theorem: the method to calculate the probability of a hypothesis P(h) initial probability that hypothesis h holds P(D) prior probability that D is observed P(D h) conditional probability that D is observed given h holds P(h D) probability that h holds given D is observed See for in depth a/c of Bayes prob Folie 4
5 2.1. Bayes Theorem revisted Can read theorem 2 ways: simply counting cardinality of sets and applying definition of conditional probablilities Learning rule where we learn about the state of out world from fuzzy evidence ( p(h e) < 1!!) corollary can get hold of p(e h) under laboratory conditions (preparing h) or from old examples, where p(h) may have been different Only discriminating evidence counts ( p(e h1) >> p(e h2) ) Incremental learning p(h) p(h e1) p(h e1&e2) See for in depth a/c of Bayes prob Folie 5
6 2.1. Bayes Theorem revisted Given: a set of hypothesis what is the most probable one? determine best hypothesis for a space H giving the training data D maximum a posteriory (MAP) hypothesis h MAP = argmax P(h D) h H = argmax P(D) can be dropped constant h H independent of h = argmax P(D h) P(h) h H Assumption P(h i ) = P(h j ) for all h i, h j H maximum likelihood (ML) hypothesis h ML = argmax P(D h) h H Folie 6
7 2.1. Bayes Theorem revisted Example (1) Shall I take an umbrella? A piori knowledge: Without watching the forecast p(rain) =.30 rainy days were forecast correctly in 80% of those days Sunny days were forecast correctly in 60% of those days P(rain) = 0.30 P(~rain= sunny ) = 0.70 P(forecRain rain ) = 0.80 P(forecSunny rain) = 0.20 P(forecSunny sunny) = 0.60 P(forecRain sunny) = 0.40 Question: Should I take an umbrella even upon SUNNY forecast Using equation for h MAP : P(rain forecsunny) = P(forecSunny rain) * P(rain) / P(forecSunny) =.20 *.30 / (.20 * *.70) = 12.5 % Further treatment through utility functions ODDS[forecastSunny] = P(rain forecsunny) / P(sunny forecsunny) = = P(rain)/P(sunny) * P(forecastSunny rain)/p(forecastsunny sunny) = aprioriodds * conditionalodds =.30 /.70 *.20 /.60 = 1 / 7 Folie 7
8 Folie 8
9 2.1. Bayes Theorem revisted Example (1) Shall I take an umbrella? (II) Notes: (a) ODDS are easily calculated & have an intuitive interpretation (b) A fully fledged system will add a utility calculation ( OK, chances are 1:7 against rain, but the umbrella is lightweigt and my jacket will suffer from the rain ) (c) A priori plays an important part Folie 9
10 2.1. Bayes Theorem revisted Example (2) : Medical Diagnosis Problem Has a patient cancer or not? A piori knowledge: a lab test is an imperfect indicator for the cancer: correct positve result (+) in 98% in case patient has cancer, correct negative result () in 97% in case patient has not 0.008% of all people have this kind of cancer P(cancer) = P( cancer) = P(+ cancer ) = 0.98 P( cancer) = 0.02 P(+ cancer) = 0.03 P( cancer) = 0.97 Question: Should a new patient with + test be diagnosed with cancer or not? Using equation for h MAP : P(+ cancer) P(cancer) = P(+ cancer) P( cancer) = P(cancer +) =.0078 / ( ) = 2.55 % h MAP =( cancer) Folie 10
11 2.2. Basics: Naive Bayes in Operation Folie 11
12 The core ingredients Anzahl von outlook outlook Anzahl von outlook Temp play? overcas t rainy sunny Gesamtergebni s play? cold high mild Gesamtergebnis no no yes yes Gesamtergebnis Gesamtergebnis Anzahl von outlook Windy Anzahl von outlook Humid play? false true Gesamtergebnis play? high normal Gesamtergebnis no no yes yes Gesamtergebnis Gesamtergebnis Folie 12
13 2.2. Basics: Naive Bayes in Operation Shall we expect a play on a hot sunny humid calm day and a priori is somewhat like playing?? P(play hshc) n*2/9*2/9*3/9*6/9 *.60 P(NOplay hshc) n*2/5*3/5*4/5*2/5 *.40 P(play hshc) n* 72/6561 *.60 P(NOplay hshc) n*48/625 *.40 P(play hshc) 18% P(NOplay hshc) 82 % Folie 13
14 2.2. Basics: What is so naive about Naive Bayes? need Either p (class =i x) where x is a vector of variables or through Bayes`Theorem p ( x class) Problem even with categorical variables: Combinatorical explosion Exact treatment of the game example requires joint probabilities such as p(sunny hot Humid nowind play ) there are 3 *3*2*2*2 = 72 conditional probabilites to estimate in such a trivial example If we had 5 influence variables with 4 classes each and 3 outcomes to estimate 4*4*4*4*4*3 = 3072 cond. Probs Without additional assumptions on causation we cannot get an estimate from reasonably sized databases Folie 14
15 2.2. Basics BUT let us approximate (as we have already done passim) p ( x class) = p (X 1 = x 1 & X 2 = x 2 &... X n = x n class) _ p (X 1 = x 1 class) * p (X 2 = x 2 class)*... * p (X n = x n class) = j p (X j = x j class) in the example need p (outlook= sunny//overcast//rainy class) 6 Probabilities p (temperature class) 6 Probabilities p (humidity class ) 4 Probabilities p (windy class) 4 Probabilities Probabilities If we were to add a 5th variable: Boss s Dress (shorts business) 4 add l probs scales well Folie 15
16 2.2. Basics What price to pay??? independence assumption is very often badly violated Φ suppose that we had a weather radar picture on of the official forecast with accuracy p(seedepressiononradar=ra rain) =.85 p(seeclearskiesonradar=ra+ sunny)=.70 Naïve Bayes was ODDS[forecastSunny] = P(rain forecsunny) / P(sunny forecsunny) = = P(rain)/P(sunny) * P(forecastSunny rain)/p(forecastsunny sunny) = aprioriodds * conditionalodds =.30 /.70 *.20 /.60 = 1 / 7 And now is ODDS[forecastSunny,Ra+] = P(rain forecsunny) / P(sunny forecsunny) = = P(rain)/P(sunny) * P(forecastSunny rain)/p(forecastsunny sunny) *P(Ra+ rain)/p(ra+ sunny) = aprioriodds * conditionaloddsforecast * conditionaloddsradar =.30 /.70 *.20 /.60 *.15/.70 = oldodds *.15/.70 = % chance of rain?!?? BUT: Radar is the most important ingredient in the official forcast in reality, we learn almost nothing from the Radar online learning factor for Radar should be MUCH closer to 1 than.15/.70 =.2 Folie 16
17 2.2. Basics What price to pay??? independence assumption is very often badly violated STILL: Works well if only used for classification need only : Class c = max arg c j p (X j = x j c) BUT j p (X j = x j c) is a poor estimate of actual probabilities If we have highly correlated variables Rough & ready approach: delete highly correlated vars Filter approach Index see below Folie 17
18 Voice Recognition recognize next phoneme p( ð ) = dictionary count or p( ð last was ə ) p(lip ð) p(waveform ð) Say Æthelwulf Folie 18
19 Voice recognition Problem With 2 Indicator Bayes rule would be p(ð lip1, waveform2) = N * p( lip1, waveform2 ð) * p(ð) p(ə lip1, waveform2) = N * p( lip1, waveform2 ə) * p(ə) BUT p(lip ð) * p(waveform ð) p(lip,waveform ð) So we are using a naive independency assumption Folie 19
20 2.2. Basics Demo: Iris and Naive Bayes Folie 20
21 2.3. Advantages easily understood & easily employed readily generalised to continuous data <student contribution> missing value treatment trivial in learning cases and class prediction integration of different data sources speed one pass through the data in basic model eager learner extremely compact representation of knowledge p ( X = x i Class = c) as results 20 numbers in playthegame example a "glimpse" of theory Folie 21
22 2.3. Advantages Diagrammtitel when to use: moderate to large training sets available attributes are conditionally independent given class p(x=x & Y=y C=c) = p(x c) * p( y c) x, y, c X Y C Example: X = tallness Y = weight C = race typical applications diagnosis text recognition on keywords CRM X C Y conditional independence <> uncond. Indep. Pygmäen 130 Tutsi Berber 120 Linear (Berber) Linear (Tutsi) 110 Linear (Pygmäen) moreexamples Folie 22
23 2.4. Explaining Away Evidence: a phenomenon of real life lassa fever Let LF be described by very clear symptoms P(fever LF) =.9 P(runnigNose LF) =.8 (a) A standard German physician & its patients P(fever NP: normalpop ) =.05 P (runnignose NP) =.10 p(lf) =.001 P(LF fever & runningnose) =N *.9 *.8 * % P(NP fever & runningnose) = N *.05 *.10 *.999 Strongly (almost proportionally) dependent on a priori Folie 23
24 2.4. Explaining Away Evidence: a phenomenon of real life lassa fever (II) Let LF be described by very clear symptoms. (b) During a (plain) cold epidemic P(fever cold) =.7 P(runningNose cold) =.8 P(COLD) =.30 P(LF fever & runningnose) =N *.9 *.8 * % P(CO fever & runningnose) =N *.7 *.8 * % P(NP fever & runningnose) = N *.05 *.10 *.699 2% Folie 24
25 2.4. Explaining Away Evidence: a phenomenon of real life lassa fever (III) Suppose we had another symptom that does not show up reliably but is seldom shown by plain cold and almost never with healthy people P(greenEars LF) =.50 P(greenEars CO) =.10 P(greenEars NP) =.01 P(LF fever & runningnose& Gn) =N *.9 *.8 *.50 * % P(CO fever & runningnose & Gn) =N *.7 *.8 *.10 * % P(NP fever & runningnose & Gn) = N *.05 *.10 *.01 * % Still LF goes UNDIAGNOSED during the epidemic NOT because of medical stupidity: 97.7% of the patients displaying all 3 symptoms mentioned ARE INDEED suffering from a cold!!! Folie 25
26 2.4. Explaining Away Evidence: a phenomenon of real life Fever Headache Arthralgias/Myalgias Retrosternal Pain Weakness Dizziness Sore throat/pharyngitis Cough Vomiting Abdominal Pain/Tenderness Diarrhea Conjunctivitis/Subconjunctival Hemorrhage Chills Deafness Lymphadenopgathy Bleeding Confusion Swollen Neck or Face Percent From From: Folie 26
27 2.5. Extensions 1. Laplace Correction p(x 1 =a Class=c) = (N(a & c) + 1) / (N(c) + m) m = Number of classes or more general: p(x 1 =a Class=c) = (N(a & c) + λ*p c apriori ) / (N(c) + λ) λ = smoothing parameter ( feign λ additional drawings with apriori class membership of p c apriori e.g. = 1 / m ) useful if Zero counts occur in learning data P (white raven) observed = 0 because <1 Albinos P(raven white) = 0 a white bird ex post is NEVER classified as raven even if all other traits (beak, size, voice ) fit perfectly! «NBEM models converge to much more consistent results when using this arithmetic smoothing procedure.» maybe also advisable with a prioris Folie 27
28 P(E if H) P(Hapriori) P(H apres E) Laplace: additive Verknüpfung der Elemente Folie 28
29 2.5. Extensions PierreSimon, Marquis de Laplace (March 23, 1749 March 5, 1827) was a French mathematician and astronomer who put the final capstone on mathematical astronomy by summarizing and extending the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) ( ). Quotations * What we know is not much. What we do not know is immense. * "It is therefore obvious that..." (frequently used in the Celestial Mechanics when he had proved something and mislaid the proof, or found it clumsy. Notorious as a signal for something true, but hard to prove.) It does appear that Laplace was not modest about his abilities and achievements, and he probably failed to recognise the effect of his attitude on his colleagues. Lexell visited the Académie des Sciences in Paris in and reported that Laplace let it be known widely that he considered himself the best mathematician in France. The effect on his colleagues would have been only mildly eased by the fact that Laplace was right! Folie 29
30 2.5. Extensions 2. Treatment of interdependencies and better probability estimates 2.1 delete highly correlated 2.2 wrapper 2.3 index / weighted sum 2.4 Tree Augmented Networks (TAN) super parent (E. Keogh / M. Pazzani 1999) tree stump ( Zhang/Ling 2001): Folie 30
31 2.5. Extensions 3. Probability Networks in Operation Example GENIE by Univ of Pittsburg: Link P(D A, B, C, E, F) P(F A, B, C, D, E) forward use: given all p(x Y) and a prioris for root nodes: calculate a posterioris backward use: given priors and some evidence: calculate new a posterioris Folie 31
32 bermudaonline.org/shorts.htm Folie 32
33 2.5. Extensions 3. Probability Networks in Operation Key Inputs and Outputs Stochastic Node p(ownstate=j {setofallstatesofincomingnodes}) J*I 1 *I 2 Matrix Further components Utililty calculationfor Node i Ui = f (StateIncomingNodeA,B, ) Decision Node Utility Calculations will be one per state_of_the_node weather mood clothing ambience happiness p(shivering rain,shorts)=.97 p(sweating hot,shorts)=.50 p(sweating hot,oilskin)=.99 U(depressed,shivering) = 0 U(enthusisatic,sweating) = 10 a prioris & a posterioris for weatherstates Network should update my beliefs Nst for clothing: I decide on state Interest is in a table of expected utilities Clothing Shorts Khakis Oilskin E(U happ ) Folie 33
34 We have added a weather mood arc This guy wearing shorts and Shivering was nevertheless enthusisatic What can we lean about the the weather But he tells us that he was enthusiatic because he made a lot of money at the Stock exchange (odr fell in love) No backpropagation of belief (genie uses the control value concept) Folie 34
35 Further Reading Mitchell, M.: Machine Learning. McGraw Hill, 1997, Chapter 6 Jensen, Finn V., An introduction to Bayesian networks, London, for very nice and gentle intro to more sophisticated methods for more indepth theory Domingos, Pazzani: Folie 35
36 Treatment continuous Vars Under the assumption of Normality in each class p( Temp (xε,x+ ε ) class) ~ ε * 1/σ c *exp( (xμ c ) / σ c )²) Using the independence assumption again, each variable has ist own Normal distribution no yes Standardabweichung (Stichprobe) von tempnum Mittelwert von tempnum Standardabweichung (Stichprobe) von tempnum Mittelwert von tempnum Gesamt: Standardabweichung (Stichprobe) von tempnum Gesamt: Mittelwert von tempnum Folie 36
37 Treatment continuous Vars Shall we expect a play on a 73.5 F + ε sunny humid calm day and a priori is somewhat like playing?? P(play hshc) n* ε * [ 1 / 6.16 *exp( ( ) / 6.16 )² ] *3/9*6/9 *.60 P(NOplay hshc) n* ε * [ 1 / 7.89 *exp( ( ) / 7.89 )² ] *3/5*4/5*2/5 *.40 P(play hshc) n* ε *.0215 P(NOplay hshc) n* ε *.0273 P(play hshc) 44% P(NOplay hshc) 56 % Note: the ε cancels on normalization (quelle fortune!) Folie 37
38 From WEKA Folie 38
39 Decisionanalytic decision support systems The principles of decisionanalytic decision support, implemented in GeNIe and SMILE canbeappliedin practical decision support systems (DSSs). In fact, Decision Systems Laboratory has developed and is currently developing several such probabilistic DSSs in which GeNIe plays the role of a developer's environment and SMILE plays the role of the reasoning engine. A decision support system based on SMILE can be equipped with a dedicated user interface. Probabilistic DSSs, applicable to problems involving classification, prediction, and diagnosis, are a new generation of systems that are capable of modeling any realworld decision problem using theoretically sound and practically invaluable methods of probability theory and decision theory. Based on graphical representation of the problem structure, these systems allow for combining expert opinions with frequency data, gather, manage, and process information to arrive at intelligent solutions. Probabilistic DSSs are based on a philosophically different principle than rulebased expert systems. While the latter attempt to model the reasoning of a human expert, the former use an axiomatic theory to perform computation. The soundness of probability theory provides a clear advantage over rulebased systems that usually represent uncertainty in an adhoc manner, such as using certainty factors, leading to underresponsiveness or overresponsiveness to evidence and possibly incorrect conclusions. Probabilistic DSSs are applicable in many domains, among others in medicine (e.g., diagnosis, therapy planning), banking (e.g., credit authorization, fraud detection), insurance (e.g., risk analysis, fraud detection), military (e.g., target detection and prioritization, battle damage assessment, campaign planning), engineering (e.g., process control, machine and process diagnosis), and business (e.g., strategic planning, risk analysis, market analysis). An example DSS developed using GeNIe and SMILE is the medical diagnostic system Hepar II (Onisko et al. 1999, 2000). The system aids physicians in diagnosis of liver disorders. The structure of the model, currently consisting of almost 100 variables, has been elicited from physician experts, while its numerical parameters have been learned from a database of patient cases. The Hepar II system is equipped with a simple dedicated user interface that allows for entering various observations such as symptoms and results of medical tests and displays the probability distribution over various possible disorders in the order of most to least likely. The system is currently used both as a diagnostic aid and a training tool. Folie 39
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