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Matlab: Statistics
1. Probability distributions
2. Hypothesis tests
3. Response surface modeling
4. Design of experiments
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
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
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
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.
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
1.25
1.36 1.22 1.19 1.33 1.12 1.27 1.27 1.31 1.26
x  1.258 s 2  0.0049

Hypothesis: m0 = 1.3 instead of m1 < m0
>> h = ttest(x,1.3,0.1,'left')
h=1
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
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
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
Response Surface Modeling

Develop linear and quadratic regression models from
data
y   0  1 x1   2 x2   3 x3  12 x1 x2  13 x1 x   23 x2 x3 
 

 
Bias
Main effects
Binary interactio n effects
11x12   22 x22   33 x32

Quadratic effects

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
Response Surface Model Example

VLE data – liquid composition held constant
Experiment
Temperature
Pressure
Vapor
Composition
1
300
1.0
0.75
2
275
1.0
0.77
3
250
1.0
0.73
4
300
0.75
0.81
5
275
0.75
0.80
6
250
0.75
0.76
7
300
1.25
0.72
>> 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];
8
275
1.25
0.74
9
250
1.25
0.71
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)
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
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
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
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
Fractional Factorial Designs cont.
>> gen = fracfactgen(model,K,res)
 model: string containing terms that must be
estimable in the design
K
 K: 2 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'