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Statistical Methods for Optimization Algorithms

Statistical Methods for Optimization Algorithms. Robert Collier collierr@uoguelph.ca. A random variable is a variable. ...with a value that is the result of a measurement taken of a random process. Fundamental Concepts.

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Statistical Methods for Optimization Algorithms

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  1. Statistical MethodsforOptimization Algorithms Robert Collier collierr@uoguelph.ca

  2. A random variable is a variable... ...with a value that is the result of a measurement taken of a random process. Fundamental Concepts

  3. A probability distribution function is a function that describes the probability... ...that a random variable is assigned a specific value. Uniform Distribution Binomial Distribution Normal Distribution Fundamental Concepts A random variable is a variable... ...with a value that is the result of a measurement taken of a random process.

  4. A cumulative distribution function is a function that describes the probability... ...that a random variable is assigned a value less than or equal to a specific value. Fundamental Concepts A random variable is a variable... ...with a value that is the result of a measurement taken of a random process. A probability distribution function is a function that describes the probability... ...that a random variable is assigned a specific value. Uniform Distribution Binomial Distribution Normal Distribution

  5. The mean (μ) of a random variable is... ...the weighted average of the possible values that could be assigned. The variance (σ2) of a random variable is... ...the average squared differencebetween each observation and the mean... ...and the square root of the variance is known as the standard deviation (σ). Elementary Statistics

  6. It is very important to recognize that a sample data set... ...will often have a different mean, variance, and standard deviation... ...than that of the system from which the sample was taken. Example: Consider the sample {1, 3, 1, 2} of the outcome of a six-sided die... True Mean (μ) = 3.500000 Sample Mean (x) = 1.750000 True Variance (σ) = 2.916667 Sample Variance (s2) = 0.916667 Elementary Statistics The mean (μ) of a random variable is... ...the weighted average of the possible values that could be assigned. The variance (σ2) of a random variable is... ...the average squared differencebetween each observation and the mean... ...and the square root of the variance is known as the standard deviation (σ).

  7. The sum of many random variables... ...taken independently, from the same distribution... ...approaches the normal distribution. Central Limit Theorem

  8. If there is one die... If there are two dice... If there are three dice... 1.00 1.00 1.00 0.75 0.75 0.75 0.50 0.50 0.50 0.25 0.25 0.25 ...the distribution of the sumapproaches the normal distribution. 0.00 0.00 0.00 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 Central Limit Theorem The sum of many random variables... ...taken independently, from the same distribution... ...approaches the normal distribution. Example: Consider the probability distribution for a group of six-sided dice... This is a crucialproperty, because many statistical tests assume normality.

  9. There are functions that generate random values from a normal distribution... in Excel: NORMINV(RAND(), μ, σ) in Matlab: random('Normal', μ, σ, 1, 1) Sample Mean Distribution

  10. in Excel: NORMINV(RAND(), μ, σ) in Matlab: random('Normal', μ, σ, 1, 1) If one of these functions is used to generate a random sample... ...then the mean of the sampleis, itself, a random variable... ...meaning that it has a probability distribution function of its own. Sample Mean Distribution There are functions that generate random values from a normal distribution...

  11. in Excel: NORMINV(RAND(), μ, σ) in Matlab: random('Normal', μ, σ, 1, 1) If the randomsample is onlya single value... ...the sample mean has the same distribution as the source. Sample Mean Distribution There are functions that generate random values from a normal distribution... If one of these functions is used to generate a random sample... ...then the mean of the sampleis, itself, a random variable... ...meaning that it has a probability distribution function of its own.

  12. However, as the random sample size increases... ...the distribution of the sample mean seems to narrow. For sample size 5... For sample size 10... For sample size 100... Sample Mean Distribution

  13. With the sample size increasing, the standard deviation... ...must be smaller than that of the source distribution. The shape of this distribution approaches normal if the sample size is large... ...but if the sample size is smaller it will have a flatter peak and fatter tails... ...and it is referred to as the t-distribution. Sample Mean Distribution However, as the random sample size increases... ...the distribution of the sample mean seems to narrow. For sample size 5... For sample size 10... For sample size 100...

  14. By specifying an interval in under which some percent of the curve is located... ...the true mean lies within that same interval, with the same percent certainty. Confidence Intervals

  15. specified by parameter α, where: α = 1.000 –0.950 = 0.050 since, for this example, the sample size was 50, this point is at: sample mean + t0.05/2 (49) (sample mean + 2.01) 95.0 % of the area under the curve sample mean The need for confidence intervals can be easily demonstrated... Confidence Intervals By specifying an interval in under which some percent of the curve is located... ...the true mean lies within that same interval, with the same percent certainty.

  16. The remaining slides have been omitted from this preview. This graduate level course in genetic algorithms was developed and taught by Robert Collier, MSc, BSc. It was most recently offered at the University of Guelph during the summer 2011 semester.

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