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...Hypotheses Evaluating Sample error, true error Confidence intervals for observed hypothesis error Estimators Binomial distribution, Normal distribution, Central Limit Theorem Paired t-tests Comparing Learning Methods CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 1 Problems Estimating Error 1. Bias: If S is training set, errorS(h) is optimistically biased bias E[errorS (h)] - errorD (h) For unbiased estimate, h and S must be chosen independently 2. Variance: Even with...
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Hypotheses Evaluating Sample error, true error Confidence intervals for observed hypothesis error Estimators Binomial distribution, Normal distribution, Central Limit Theorem Paired t-tests Comparing Learning Methods CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 1 Problems Estimating Error 1. Bias: If S is training set, errorS(h) is optimistically biased bias E[errorS (h)] - errorD (h) For unbiased estimate, h and S must be chosen independently 2. Variance: Even with unbiased S, errorS(h) may still vary from errorD(h) CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 2 Two Definitions of Error The true error of hypothesis h with respect to target function f and distribution D is the probability that h will misclassify an instance drawn at random according to D. errorD (h) Pr [ f ( x) h( x)] xD The sample error of h with respect to target function f and data sample S is the proportion of examples h misclassifies 1 errorS (h) ( f ( x) h( x) ) n xS where ( f ( x) h( x) ) is 1 if f ( x) h( x), and 0 otherwise How well does errorS(h) estimate errorD(h)? CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 3 Example Hypothesis h misclassifies 12 of 40 examples in S. 12 errorS (h) = = .30 40 What is errorD(h)? CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 4 Estimators Experiment: 1. Choose sample S of size n according to distribution D 2. Measure errorS(h) errorS(h) is a random variable (i.e., result of an experiment) errorS(h) is an unbiased estimator for errorD(h) Given observed errorS(h) what can we conclude about errorD(h)? CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 5 Confidence Intervals If S contains n examples, drawn independently of h and each other n 30 Then With approximately N% probability, errorD(h) lies in interval errorS (h)(1 - errorS (h)) errorS (h) z N n where N% : 50% 68% 80% 90% 95% 98% 99% z N : 0.67 1.00 1.28 1.64 1.96 2.33 2.53 CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 6 Confidence Intervals If S contains n examples, drawn independently of h and each other n 30 Then With approximately 95% probability, errorD(h) lies in interval errorS ( h)(1 - errorS (h)) errorS (h) 1.96 n CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 7 errorS(h) is a Random Variable Rerun experiment with different randomly drawn S (size n) Probability of observing r misclassified examples: 0.14 0.12 0.10 P(r) 0.08 0.06 0.04 0.02 0.00 0 5 10 15 20 r 25 30 35 40 Binomial distribution for n=40, p=0.3 P(r ) = CS 5751 Machine Learning n! errorD (h) r (1 - errorD (h)) n - r r!(n - r )! Chapter 5 Evaluating Hypotheses 8 Binomial Probability Distribution 0.14 0.12 0.10 P(r) 0.08 0.06 0.04 0.02 0.00 0 5 10 15 20 r 25 30 35 40 Binomial distribution for n=40, p=0.3 P(r ) = n! p r (1 - p ) n - r r!(n - r )! Probabilty P(r) of r heads in n coin flips, if p = Pr (heads) Expected, or mean value of X : E[X] iP (i ) = np i =0 n Variance of X : Var(X) E[( X - E[ X ]) 2 ] = np (1 - p ) Standard deviation of X : X E[( X - E[ X ]) 2 ] = np (1 - p ) CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 9 Normal Probability Distribution Normal distribution with mean 0, standard deviation 1 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 P(r ) = 1 2 2 e - - 1 ( x ) 2 2 The probability that X will fall into the interval (a,b) is given by b a p ( x)dx Expected, or mean value of X : E[X] = Variance of X : Var(X) = 2 Standard deviation of X : X = CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 10 Normal Distribution Approximates Binomial errors (h) follows a Binomial distribution, with mean errorS ( h ) = errorD (h) standard deviation errorS ( h ) errorD (h)(1 - errorD (h)) = n Approximate this by a Normal distribution with mean errorS ( h ) = errorD (h) standard deviation errorS ( h ) errorS (h)(1 - errorS (h)) n Chapter 5 Evaluating Hypotheses 11 CS 5751 Machine Learning Normal Probability Distribution 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -3 -2.5 -2 -1.5 -1 -0.5 0 1 0.5 1.5 2 2.5 3 80% of area (probability) lies in 1.28 N% of area (probability) lies in z N N% : 50% 68% 80% 90% 95% 98% 99% zN : 0.67 1.00 1.28 1.64 1.96 2.33 2.53 Chapter 5 Evaluating Hypotheses 12 CS 5751 Machine Learning Confidence Intervals, More Correctly If S contains n examples, drawn independently of h and each other n 30 Then With approximately 95% probability, errorS(h) lies in interval errorD (h)(1 - errorD (h)) errorD ( h) 1.96 n equivalently, errorD(h) lies in interval errorD (h)(1 - errorD (h)) errorS (h) 1.96 n which is approximately errorS (h)(1 - errorS (h)) errorS (h) 1.96 n CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 13 Calculating Confidence Intervals 1. Pick parameter p to estimate errorD(h) 2. Choose an estimator errorS(h) 3. Determine probability distribution that governs estimator errorS(h) governed by Binomial distribution, approximated n 30 by Normal when 4. Find interval (L,U) such that N% of probability mass falls in the interval Use table of zN values CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 14 Central Limit Theorem Consider a set of independent, identically distributed random variablesY ...Yn , all governed by an arbitrary probability distribution 1 with mean and finite variance 2 . Define the sample mean 1 n Y Yi n i =1 Central Limit Theorem. As n , the distribution governing Y 2 approaches a Normal distribution, with mean and variance . n CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 15 Difference Between Hypotheses Test h1 on sample S1 , test h2 on S 2 1. Pick parameter to estimate d errorD (h1 ) - errorD (h2 ) 2. Choose an estimator d errorS1 (h1 ) - errorS 2 (h2 ) 3. Determine probability distribution that governs estimator d errorS1 (h1 )(1 - errorS1 (h1 )) n1 + errorS 2 (h2 )(1 - errorS 2 (h2 )) n2 4. Find interval (L, U) such that N% of probability mass falls in the interval ^ z errorS1 (h1 )(1 - errorS1 (h1 )) + errorS 2 (h2 )(1 - errorS 2 (h2 )) d N n1 n2 CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 16 Paired t test to Compare hA,hB 1. Partition data into k disjoint test sets T1,T2 ,...,Tk of equal size, where this size is at least 30. 2. For i from 1 to k do i errorTi (hA ) - errorTi (hB ) 3. Return the value d, where 1 k i k i =1 N% confidence interval estimate for d : t N,k-1s k 1 s ( i - ) 2 k (k - 1) i =1 Note i approximately Normally distributed CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 17 Comparing Learning Algorithms LA and LB 1. Partition data D0 into k disjoint test sets T1,T2 ,...Tk of equal size, where this size is at least 30. 2. For i from 1 to k , do use Ti for the test set, and the remaining data for training set Si Si {D0 - Ti } hA LA(Si ) hB LB(Si ) i errorTi (hA ) - errorTi (hB ) 3. Return the value , where 1 k i k i =1 CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 18 Comparing Learning Algorithms LA and LB What we would like to estimate: ES D [errorD ( LA ( S )) - errorD ( LB ( S ))] where L(S) is the hypothesis output by learner L using training set S i.e., the expected difference in true error between hypotheses output by learners LA and LB, when trained using randomly selected training sets S drawn according to distribution D. But, given limited data D0, what is a good estimator? Could partition D0 into training set S and training set T0 and measure errorT0 ( LA ( S 0 )) - errorT0 ( LB ( S 0 )) even better, repeat this many times and average the results (next slide) CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 19 Comparing Learning Algorithms LA and LB Notice we would like to use the paired t test on to obtain a confidence interval But not really correct, because the training sets in this algorithm are not independent (they overlap!) More correct to view algorithm as producing an estimate of ES D0 [errorD ( LA ( S )) - errorD ( LB ( S ))] instead of but even this approximation is better than no comparison CS 5751 Machine Learning Chapter 5 Evaluating Hypotheses 20 ES D [errorD ( LA ( S )) - errorD ( LB ( S ))]
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El Paso CC >> SPOT >> 221 (Fall, 2009)
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El Paso CC >> SPOT >> 221 (Fall, 2009)
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El Paso CC >> SPOT >> 221 (Fall, 2009)
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El Paso CC >> SPOT >> 221 (Fall, 2009)
Density Problems = m V mass volume Acceptable units for each variable : g g kg or or 3 3 mL m cm m g or kg Density = V cm 3 or mL or m 3 1. A sample of wood has a mass of 5843 grams and has a volume of 7304 cm3. What is the woods density? 2. A sam...
El Paso CC >> SPOT >> 221 (Fall, 2009)
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El Paso CC >> SPOT >> 221 (Fall, 2009)
Name:_Date:_LecturePeriod:_ HomeworkAssignment4 1.) ThesymbolforisotopicUraniumis electronsareintheatom? 238 .Howmanyprotons,neutronsand 92U 2.) Theprincipleisotopeofironhas26protonsand30neutrons.Whatisthemassnumber ofthisisotope?Writethesymbolfor...
El Paso CC >> SPOT >> 221 (Fall, 2009)
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El Paso CC >> SPOT >> 221 (Fall, 2009)
Name: _ Date:_ Lecture Period:_ Homework Assignment 6 1.) Name the following: a. CO b. CO2 c. N2O d. NO e. NO2 f. N2O5 g. SO2 h. SO3 i. PCl5 j. PCl3 k. CCl4 2.) Give formulas for the following molecular compounds: a. bromine pentafluoride b....
El Paso CC >> SPOT >> 221 (Fall, 2009)
Name: _ Date:_ Homework 7 Lecture Period:_ 23 Na 16 O 12 C 1 H 22.9893 amu 15.9960amu 12.000 amu 1.0078 amu 1.) For the elements listed above, write in the number of protons, neutrons, and electrons for each 2.) Calculate the mass, in ...
El Paso CC >> SPOT >> 221 (Fall, 2009)
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