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 = 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 =
Test if data comes from a normal distribution
>> [h,p,stats]=chi2gof(data,’edges’,[-inf,325:10:405,inf]);
>> h = 0
>> p =
>> chi2stat =
>> 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; ; ; ; ; ; ];
>> y = [0.75; 0.77; 0.73; 0.81; 0.80; 0.76; 0.72; 0.74; 0.71];
Experiment 1 2 3 4 5 6 7 8 9
Temperature
300
275
250
Pressure
1.0
0.75
1.25
Vapor Composition
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 = (bias) (T) (P) >> rstool(x,y,'interaction') >> beta2 = (bias) (T) (P) (T*P)
>> rstool(x,y,'quadratic') >> beta3 = (bias) (T) (P) (T*P) (T*T) (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 =

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 =

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'