Matlab Statistics

1
Matlab Statistics

• Statistics toolbox
• Probability distributions
• Hypothesis tests
• Linear regression and response surface modeling
• Statistical process control
• 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

• Continuous distributions for data analysis 21
  distributions
• Includes normal distribution
• Continuous distributions for statistics 6
  distributions
• Includes chi-square and students t distributions
• Discrete distributions 8 distributions
• Includes binomial and Poisson distributions
• Multivariable distributions 10 distributions
• Includes multivariable extension of normal
  distribution
• 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
Confidence Intervals Example

• Polymer molecular weight (scaled by 10-5)
• gtgt muhat,sigmahat,muci,sigmaci
  normfit(data,alpha)
• data vector or matrix of data
• alpha confidence level 1-alpha
• muhat estimated mean
• sigmahat estimated standard deviation
• muci confidence interval on the mean
• sigmaci confidence interval on the standard
  deviation
• gtgt x 1.25 1.36 1.22 1.19 1.33 1.12 1.27 1.27
  1.31 1.26
• gtgt muhat,sigmahat,muci,sigmaci
  normfit(x,0.05)
• muhat 1.2580
• sigmahat 0.0697
• muci 1.2081
• 1.3079
• sigmaci 0.0480
• 0.1273

6
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.

7
Mean Hypothesis Test Example

• gtgt h ttest(data,m,alpha,tail)
• data vector or matrix of data
• m expected mean
• alpha confidence level 1-alpha
• Tail left (left handed alternative), right
  (left 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 1.2
• gtgt h ttest(x,1.3,0.05,'left')
• h 1

8
Variance Hypothesis Test Example

• gtgt h vartest(data,v,alpha,tail)
• data vector or matrix of data
• v expected variance
• alpha confidence level 1-alpha
• Tail left (left handed alternative), right
  (left handed alternative) or both (two-sided
  alternative)
• h 1 (reject hypothesis) or 0 (accept
  hypothesis)
• Hypothesis s2 0.0049 and not a different
  variance
• gtgt h vartest(x,0.0049,0.05,'both')
• h 0
• gtgt h vartest(x,0.0049,0.9,'both')
• h 1

9
Goodness of Fit Example

• gtgt h chi2gof(data)
• data vector or matrix of data
• h 1 (reject hypothesis) or 0 (accept
  hypothesis)
• Default values alpha 0.05 nbins 10
• gtgt h chi2gof(x)
• Warning After pooling, some bins still have
  low expected counts. The chi-square
  approximation may not be accurate.
• gtgt x90x-0.2 x-0.15 x-0.1 x-0.05 x x0.05 x0.1
  x0.15 x0.2
• gtgt h chi2gof(x90)
• h 0

10
Linear Models

• Linear regression
• Multiple linear regression build linear models
  between a group of input variables (factors) and
  an output variable (response)
• Quadratic response surface models build
  response surface models between a group of input
  variables (factors) and an output variable
  (response)
• Stepwise regression determine the most
  significant terms (linear, interaction,
  quadratic) to include in a regression model
• Generalized linear models techniques for
  nonlinear models linear in the unknown parameters
• Robust and nonparametric methods techniques
  that are insensitive to data outliers (robust) or
  do not assume any underlying distribution
  (nonparametric)
• Analysis of variance determine whether data
  from several groups have a common mean

11
Linear Regression Example

• gtgt b,bint regress(y,x)
• x vector or matrix of input values
• y vector of output values
• b linear model slope and intercept
• bint 95 confidence limits on the slope and
  intercept
• Reaction rate data
• gtgt x 0.1 0.3 0.5 0.7 0.9 1.2 1.5 2.0
• gtgt x x ones(size(x))
• gtgt y 2.3 5.7 10.7 13.1 18.5 25.4 32.1 45.2
• gtgt b,bint regress(y,x)
• b 22.5315
• -1.1533
• bint 21.0040 24.0590
• -2.8038 0.4972

12
Response Surface Model Example

• gtgt 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
• 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

13
Response Surface Model Example cont.

• gtgt rstool(x,y,'linear')
• gtgt beta 0.7411 (bias)
• 0.0005 (T)
• -0.1333 (P)
• gtgt rstool(x,y,'interaction')
• gtgt beta2 0.3011 (bias)
• 0.0021 (T)
• 0.3067 (P)
• -0.0016 (TP)
• gtgt rstool(x,y,'quadratic')
• gtgt beta3 -2.4044 (bias)
• 0.0227 (T)
• 0.0933 (P)
• -0.0016 (TP)
• -0.0000 (TT)
• 0.1067 (PP)

14
Statistical Process Control

• Can produce a wide variety of quality control
  charts
• gtgt controlchart(data,param1,val1,param2,val2,...)
• data data matrix with each row a subgroup of
  measurements containing replicate observations
  taken at the same time and the rows in time
  order.
• param, val matching sets of adjustable
  parameters and their values
• charttype xbar (mean, default), s
  (standard deviation) or i (individual
  observation)
• nsigma number of sigma multiples from the
  center line (default 3)
• Many other options available

15
Control Chart Example

• gtgt load parts
• gtgt who
• Your variables are
• runout
• gtgt controlchart(runout)
• gtgt controlchart(runout,'chart','s')

16
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

17
Full Factorial Designs

• gtgt d fullfact(L1,,Lk)
• L1 number of levels for first factor
• Lk number of levels for last (kth) factor
• d design matrix
• gtgt d ff2n(k)
• k number of factors
• d design matrix for two levels
• gtgt 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

18
Fractional Factorial Designs

• gtgt d,conf fracfact(gen)
• gen generator string for the design
• d design matrix
• conf cell array that describes the confounding
  pattern
• gtgt 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
• conf
• 'Term' 'Generator' 'Confounding'
• 'X1' 'a' 'X1'
• 'X2' 'b' 'X2'
• 'X3' 'c' 'X3'

19
Fractional Factorial Designs cont.

• gtgt gens 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
• gtgt fracfactgen('a b c d e f g',4,4)
• ans
• 'a'
• 'b'
• 'c'
• 'd'
• 'bcd'
• 'acd'
• 'abd'

20
Central Composite Designs

• gtgt d ccdesign(nfactors,param1,val1,param2,val2,.
  ..)
• nfactors number of factors
• d design matrix
• param, val matching sets of adjustable
  parameters and their values
• fraction fraction of full factorial design for
  cube portion expressed as an exponent of ½ 0
  (full, default), 1 (½ design), 2 (¼ design)
• type inscribed, circumscribed (default),
  or faced
• center' number of center points m (force m
  center points), orthogonal (achieve orthogonal
  design, default), uniform (achieve uniform
  precision)

21
Central Composite Designs cont.

• gtgt d ccdesign(2)
• d
• -1.0000 -1.0000
• -1.0000 1.0000
• 1.0000 -1.0000
• 1.0000 1.0000
• -1.4142 0
• 1.4142 0
• 0 -1.4142
• 0 1.4142
• 0 0
• 0 0
• 0 0
• 0 0
• 0 0
• 0 0
• 0 0
• 0 0

• gtgt d ccdesign(5,'fraction',1,'type','inscribed',
  'center',3)
• d
• -0.5000 -0.5000 -0.5000 -0.5000
  0.5000
• -0.5000 -0.5000 -0.5000 0.5000
  -0.5000
• -0.5000 -0.5000 0.5000 -0.5000
  -0.5000
• -0.5000 -0.5000 0.5000 0.5000
  0.5000
• -0.5000 0.5000 -0.5000 -0.5000
  -0.5000
• -0.5000 0.5000 -0.5000 0.5000
  0.5000
• -0.5000 0.5000 0.5000 -0.5000
  0.5000
• -0.5000 0.5000 0.5000 0.5000
  -0.5000
• 0.5000 -0.5000 -0.5000 -0.5000
  -0.5000
• 0.5000 -0.5000 -0.5000 0.5000
  0.5000
• 0.5000 -0.5000 0.5000 -0.5000
  0.5000
• 0.5000 -0.5000 0.5000 0.5000
  -0.5000
• 0.5000 0.5000 -0.5000 -0.5000
  0.5000
• 0.5000 0.5000 -0.5000 0.5000
  -0.5000
• 0.5000 0.5000 0.5000 -0.5000
  -0.5000
• 0.5000 0.5000 0.5000 0.5000
  0.5000
• -1.0000 0 0
  0 0

22
Box-Behnken Designs

• gtgt d bbdesign(nfactors)
• nfactors number of factors
• d design matrix
• gtgt d bbdesign(3)
• d
• -1 -1 0
• -1 1 0
• 1 -1 0
• 1 1 0
• -1 0 -1
• -1 0 1
• 1 0 -1
• 1 0 1
• 0 -1 -1
• 0 -1 1
• 0 1 -1
• 0 1 1
• 0 0 0
• 0 0 0