Random Variables & Entropy: Extension and Examples Brooks Zurn EE 270 / STAT 270 FALL 2007

Overview Density Functions and Random Variables Distribution Types Entropy

Density Functions PDF vs. CDF – PDF shows probability of each size bin – CDF shows cumulative probability for all sizes up to and including current bin – This data shows the normalized, relative size of a rodent as seen from an overhead camera for 8 behaviors

Markov & Chebyshev Inequalities What’s the point? Setting a maximum limit on probability This limits the search space for a solution – When looking for a needle in a haystack, it helps to have a smaller haystack. Can use limit to determine the necessary sample size

Markov & Chebyshev Inequalities Example: Mean height of a child in a kindergarten class is 3’6”. (Leon-Garcia text, p. 137 – see end of presentation) – Using Markov’s inequality, the probability of a child being taller than 9 feet is <= 42/108 =.389.  there will be fewer than 39 students over 9 feet tall in a class of 100 students.  Also, there will be NO LESS THAN 41 students who are under 9’ tall. -Using Chebyshev’s inequality (and assuming the variance = 1 foot) the probability of a child being taller than 9 feet is <= 12 2 /108 2 =  there will be no more than 2 students taller than 9’ in a class of 100 students. (this is also consistent with Markov’s Inequality).  Also, there will be NO LESS THAN 98 students under 9’ tall. This gives us a basic idea of how many student heights we need to measure to rule out the possibility that we have a 9’ tall student… SAMPLE SIZE!!

Markov’s Inequality Derivation: For a random variable X >= 0, E[x]=, where f x (x)=P[x-e/2£X£x+e/2]/e Assuming this also holds for X = a, because this is a continuous integral.

Markov’s Inequality Therefore for c > 0, the number of values of x > c is infinite, therefore the value of c will stay constant while x continues to increase.

Markov’s Inequality References: Lefebvre text.

Chebyshev’s Inequality Derivation (INCOMPLETE):

Chebyshev’s Inequality As before, c 2 is constant and (Y-E[Y]) 2 continues to increase. But, how do f y |Y-E[Y]| and f Y (Y-E[Y]) 2 relate? (|Y-E[Y]|) 2 = (Y-E[Y]) 2 As long as Y – E[Y] is >= 1, then u 2 will be > u and the inequality holds, as per Markov’s Inequality. Note: this is not a rigorous proof, and cases for which Y – E[Y] < 1 are not discussed. Reference: Lefebvre text.

Note These both involve the Central Limit Theorem, which is derived in the Leon-Garcia text on p Central Limit Theorem states that the CDF of a normalized sequence of n random variables approaches the CDF of a Gaussian random variable. (p. 280)

Overview Entropy – What is it? – Used in…

Entropy What is it? – According to Jorge Cham (PhD Comics),

Entropy “Measure of uncertainty in a random experiment” Reference: Leon-Garcia Text Used in information theory – Message transmission (for example, Lathi text p. 682) – Decision Tree ‘Gain Criterion’ Leon-Garcia text p. 167 ID3, C4.5, ITI, etc. by J. Ross Quinlan and Paul Utgoff Note: NOT same as the Gini index used as a splitting criterion by the CART tree method (Breiman et al, 1984).

Entropy ID3 Decision Tree: Expected Information for a Binary Tree where the entropy I is E(A) is the average information needed to classify A. ITI (Incremental Tree Inducer): - Based on ID3 and its successor, C4.5. -Uses a gain ratio metric to improve function for certain cases

Entropy ITI Decision Tree for Rodent Behaviors – ITI is an extension of ID3 Reference: ‘Rodent Data’ paper.

Distribution Types Continuous Random Variables – Normal (or Gaussian) Distribution – Uniform Distribution – Exponential Distribution – Rayleigh Random Variable Discrete (‘counting’) Random Variables – Binomial Distribution – Bernoulli and Geometric Distributions – Poisson Distribution

Poisson Distribution Number of events occurring in one time unit, time between events is exponentially distributed with mean 1/a. Gives a method for modeling completely random, independent events that occur after a random interval of time. (Leon-Garcia p. 106) Poisson Dist. can model a sequence of Bernoulli trials (Leon-Garcia p. 109) – Bernoulli gives the probability of a single coin toss. and References: Kao text, Leon-Garcia text.

Poisson Distribution

References Lefebvre Text: – Applied Stochastic Processes, Mario Lefebvre. New York, NY: Springer., 2003 Kao Text: – An Introduction to Stochastic Processes, Edward P. C. Kao. Belmont, CA, USA: Duxbury Press at Wadsworth Publishing Company, Lathi Text: – Modern Digital and Analog Communication Systems, 3 rd ed., B. P. Lathi. New York, Oxford: Oxford University Press, Entropy-Based Decision Trees: – ID3: P. E. Utgoff, "Incremental induction of decision trees.," Machine Learning, vol. 4, pp , – C4.5: J. R. Quinlan, C4.5: Programs for machine learning, 1st ed. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., – ITI: P. E. Utgoff, N. C. Berkman, and J. A. Clouse, "Decision tree induction based on efficient tree restructuring.," Machine Learning, vol. 29, pp. 5-44, Other Decision Tree Methods: – CART: L. Breiman, J. H. Friedman, R. A. Olshen, C. J. Stone, Classification and Regression Trees. Belmont, CA: Wadsworth Rodent Data: – J. Brooks Zurn, Xianhua Jiang, Yuichi Motai. Video-Based Tracking and Incremental Learning Applied to Rodent Behavioral Activity under Near-Infrared Illumination. To appear: IEEE Transactions on Instrumentation and Measurement, December 2007 or February Poisson Distribution Example: –

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