Class Note for ECE 596A at UA
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This 17 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 12 views.
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Date Created: 02/06/15
ECE 596C Network Simulation Probability amp Statistics Basic concepts Independent events Randon1vanabh Cumulative distribution function CDF Probability density functionPDF Probability mass function Mean or expected value Vanance Coef cient of variation Ratio of standard deviation to mean Covanance CovXY E X EX Y EY EXY EX EY ECE 596C Network Simulation Basic concepts Correlation coef cient or correlation Ratio of covariance ofx and y to the product of the standard deviations ofx and y Mean and variance of sums Quantile The x value at which the CDF takes a value y is called yquantile or 100ypercentile Median 05quantile or 50percentile Mode The most likely value Normal distribution ECE 596C Network Simulation Summarizing data by a single number Indices of central tendencies Mean median and mode Some characteristics Mean exists and is unique Mode may not exist or may not be unique Outliers may drastically affect mean Mean has linearity property while mode and median do not Selecting among mean median and mode If data is categorical use mode lftotal is of interest use mean If distribution is skewed use median Otherwise use mean ECE 596C Network Simulation common misuse Of means Using means of signi cantly different values Using means without regard to skewness of distribution Simple way to measure skewness is the ratio of maximum to minimum Multiplying means to get the mean of product Taking mean of a ratio with different bases ECE 596C Network Simulation Geometric mean Geometric mean of x1 x2 xn nth root ofthe product ofxl x2 xn Used when product of observed quantities is of interest Example Performance improvement of individual layers are known Compute average improvement per layer Other examples Protocol Performance Layer Improvement 7 l8 6 13 5 l l 4 8 3 10 2 28 l 5 Cache hitmiss ratios over several levels Percentage performance improvement between successive versions Average error rate per hop on a multihop path in a network ECE 596C Network Simulation Mean of ratios De ning mean of ratios Average b1 b2 bn Example CPU busy times measured over several intervals Compute the mean busy duration Approximates to arithmetic and harmonic means Ifthe denominator is constant arithmetic mean Ifthe numerator is constant harmonic mean ECE 596C Network Simulation a1 a2 ana1a2an b1b2bn cl 9 Maggi CPU Busy 1 45 1 45 1 45 1 45 1 00 20 Su m 200 Mean 40 Mean of ratios f numerator and denominator follows multiplicative property ai c bi c may be estimated by geometric mean of aibi Example Program COde Size Ratio Before After Benchmarking a program optimizer P1 119 89 075 P2 158 134 085 P3 142 121 085 P4 8612 7579 088 P5 7133 7062 099 P6 184 112 061 P7 2908 2879 099 P8 433 307 071 Geometric Mean 082 ECE 596C Network Simulation Summarizing variability Indices of dispersion Range minimum and maximum of values observed Variance or standard deviation 10 and 90percentiles Semiinterquartile range Mean absolute deviation 1 n S l 2 E 2 amp evarlance 3 n 1 z 1xz 31 Use sample standard deviation instead of variance Even better use ratio of standard deviation to mean removes unit of measurement ECE 596C Network Simulation Summarizing variability Quartiles 25 50 75percentiles 01 25percentile 02 50percentile Q3 75percentile Semilnterquartile Range SIQR SIQR Q3 Q1 075 025 2 2 Mean absolute deviation 1 77 Mean Absolute Deviation Z 5 n i1 How are these measures affected by outliers ECE 596C Network Simulation 10 Selecting index of dispersion ls distribution bounded Ifyes use range ls distribution unimodal symmetrical Ifyes use coef cient of variance Otherwise use percentiles or SIQR ECE 596C Network Simulation 11 Con dence interval for the mean Population vs sample Not possible to obtain population mean from nite size samples Provide probabilistic guarantees as Probability01 g u S 02 1 oz Con dence interval Con dence coef cient Using Central Limit Theorem lfx1 x2 xn are independent and come from same population then sample mean and for large samples is approximated by normal distribution Standard deviation of the sample mean Standard error ECE 596C Network Simulation 12 Con dence interval for the mean Con dence interval computed as s s 5 Zl a2W75 Zl az Zla2 1 aZquantile ofa unit normal variate Above applies to samples of greater than 30 For smaller samples use tvariate with n1 degrees of freedom ECE 596C Network Simulation 13 Testing for zero mean Check if the measured value is signi cantly different from 0 Mean Same procedure may be applied to other values ECE 596C Network Simulation 14 Comparing two alternatives Paired vs unpaired observations Paired observations have onetoone correspondence on every sample Testing similarity between paired observations is straightforward Comparing two unpaired samples ttest 1Compute sample means 2 Compute sample standard deviations 3 Compute difference of sample means 4 Compute the standard deviation of the mean of difference 5 Compute the effective number ofthe degrees of freedom 6 Compute the con dence interval for the mean difference 7 Observe the presence of 0 in the desired con dence interval ECE 596C Network Simulation 15 Approximate visual test Simpler method is to compare two unpaired samples Perform the ttest only in the third scenario Mean Mean Mean ECE 596C Network Simulation 16 Other considerations What con dence level to use Hypothesis testing vs con dence intervals Onesided con dence intervals Sample size to use for determining mean with a desired con dence interval ECE 596C Network Simulation 17
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