Matlab: Statistics

1. Matlab: Statistics • Probability distributions • Hypothesis tests • Response surface modeling • Design of experiments

2. Statistics Toolbox Capabilities • Descriptive statistics • Statistical visualization • Probability distributions • Hypothesis tests • Linear models • Nonlinear models • Multivariate statistics • Statistical process control • Design of experiments • Hidden Markov models

3. Probability Distributions • 21 continuous distributions for data analysis • Includes normal distribution • 6 continuous distributions for statistics • Includes chi-square and t distributions • 8 discrete distributions • Includes binomial and Poisson distributions • Each distribution has functions for: • pdf — Probability density function • cdf — Cumulative distribution function • inv — Inverse cumulative distribution • functionsstat — Distribution statistics function • fit — Distribution fitting function • like — Negative log-likelihood function • rnd — Random number generator

4. Normal Distribution Functions • normpdf – probability distribution function • normcdf – cumulative distribution function • norminv – inverse cumulative distribution function • normstat – mean and variance • normfit – parameter estimates and confidence intervals for normally distributed data • normlike – negative log-likelihood for maximum likelihood estimation • normrnd – random numbers from normal distribution

5. Hypothesis Tests • 17 hypothesis tests available • chi2gof – chi-square goodness-of-fit test. Tests if a sample comes from a specified distribution, against the alternative that it does not come from that distribution. • ttest – one-sample or paired-sample t-test. Tests if a sample comes from a normal distribution with unknown variance and a specified mean, against the alternative that it does not have that mean. • vartest – one-sample chi-square variance test. Tests if a sample comes from a normal distribution with specified variance, against the alternative that it comes from a normal distribution with a different variance.

6. Mean Hypothesis Test Example >> h = ttest(data,m,alpha,tail) • data: vector or matrix of data • m: expected mean • alpha: significance level • Tail = ‘left’ (left handed alternative), ‘right’ (right handed alternative) or ‘both’ (two-sided alternative) • h = 1 (reject hypothesis) or 0 (accept hypothesis) • Measurements of polymer molecular weight • Hypothesis: m0 = 1.3 instead of m1 < m0 >> h = ttest(x,1.3,0.1,'left') h = 1

7. Variance Hypothesis Test Example >> h = vartest(data,v,alpha,tail) • data: vector or matrix of data • v: expected variance • alpha: significance level • Tail = ‘left’ (left handed alternative), ‘right’ (right handed alternative) or ‘both’ (two-sided alternative) • h = 1 (reject hypothesis) or 0 (accept hypothesis) • Hypothesis: s2 = 0.0049 and not a different variance >> h = vartest(x,0.0049,0.1,'both') h = 0

8. Goodness of Fit • Perform hypothesis test to determine if data comes from a normal distribution • Usage: [h,p,stats]=chi2gof(x,’edges’,edges) • x: data vector • edges: data divided into intervals with the specified edges • h = 1, reject hypothesis at 5% significance • h = 0, accept hypothesis at 5% significance • p: probability of observing the given statistic • stats: includes chi-square statistic and degrees of freedom

9. Goodness of Fit Example • Find maximum likelihood estimates for µ and σ of a normal distribution >> data=[320 … 360]; >> phat = mle(data) phat = 364.7 26.7 • Test if data comes from a normal distribution >> [h,p,stats]=chi2gof(data,’edges’,[-inf,325:10:405,inf]); >> h = 0 >> p = 0.8990 >> chi2stat = 2.8440 >> df = 7

10. Response Surface Modeling • Develop linear and quadratic regression models from data • Commonly termed response surface modeling • Usage: rstool(x,y,model) • x: vector or matrix of input values • y: vector or matrix of output values • model: ‘linear’ (constant and linear terms), ‘interaction’ (linear model plus interaction terms), ‘quadratic’ (interaction model plus quadratic terms), ‘pure quadratic’ (quadratic model minus interaction terms) • Creates graphical user interface for model analysis

11. Response Surface Model Example • VLE data – liquid composition held constant >> x = [300 1; 275 1; 250 1; 300 0.75; 275 0.75; 250 0.75; 300 1.25; 275 1.25; 250 1.25]; >> y = [0.75; 0.77; 0.73; 0.81; 0.80; 0.76; 0.72; 0.74; 0.71];

12. Response Surface Model Example cont. >> rstool(x,y,'linear') >> beta = 0.7411 (bias) 0.0005 (T) -0.1333 (P) >> rstool(x,y,'interaction') >> beta2 = 0.3011 (bias) 0.0021 (T) 0.3067 (P) -0.0016 (T*P) >> rstool(x,y,'quadratic') >> beta3 = -2.4044 (bias) 0.0227 (T) 0.0933 (P) -0.0016 (T*P) -0.0000 (T*T) 0.1067 (P*P)

13. Design of Experiments • Full factorial designs • Fractional factorial designs • Response surface designs • Central composite designs • Box-Behnken designs • D-optimal designs – minimize the volume of the confidence ellipsoid of the regression estimates of the linear model parameters

14. Full Factorial Designs >> d = fullfact(L1,…,Lk) • L1: number of levels for first factor • Lk: number of levels for last (kth) factor • d: design matrix >> d = ff2n(k) • k: number of factors • d: design matrix for two levels >> d = ff2n(3) d = 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1

15. Fractional Factorial Designs >> [d,conf] = fracfact(gen) • gen: generator string for the design • d: design matrix • conf: cell array that describes the confounding pattern >> [x,conf] = fracfact('a b c abc') x = -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 1

16. Fractional Factorial Designs cont. >> gen = fracfactgen(model,K,res) • model: string containing terms that must be estimable in the design • K: 2K total experiments in the design • res: resolution of the design • gen: generator string for use in fracfact >> gen = fracfactgen('a b c d e f g',4,4) gen = 'a' 'b' 'c' 'd' 'bcd' 'acd' 'abd'