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 <= 122/1082 = .0123. • 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 For a random variable X >= 0, Derivation: E[x]=, where fx (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, c2 is constant and (Y-E[Y])2 continues to increase. But, how do fy|Y-E[Y]| and fY (Y-E[Y])2relate? (|Y-E[Y]|)2 = (Y-E[Y])2 As long as Y – E[Y] is >= 1, then u2 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. 287. • 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 • http://en.wikipedia.org/wiki/Image:Poisson_distribution_PMF.png
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, 1997. • Lathi Text: • Modern Digital and Analog Communication Systems, 3rd ed., B. P. Lathi. New York, Oxford: Oxford University Press, 1998. • Entropy-Based Decision Trees: • ID3: P. E. Utgoff, "Incremental induction of decision trees.," Machine Learning, vol. 4, pp. 161-186, 1989. • C4.5: J. R. Quinlan, C4.5: Programs for machine learning, 1st ed. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., 1993. • 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, 1997. • Other Decision Tree Methods: • CART: L. Breiman, J. H. Friedman, R. A. Olshen, C. J. Stone, Classification and Regression Trees. Belmont, CA: Wadsworth. 1984. • 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 2008. • Poisson Distribution Example: • http://en.wikipedia.org/wiki/Image:Poisson_distribution_PMF.png