Statistics in MATLAB

1
Statistics in MATLAB

• COMM2M
• Harry R. Erwin, PhD
• University of Sunderland

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Resources

• http//www.mathworks.com/access/helpdesk/help/pdf_
  doc/stats/stats.pdf This can be found in the
  COMM2M Lectures folder as STATS.PDF.
• Higham and Higham, 2000, MATLAB Guide, SIAM.
• James E. Gentle, 2002, Elements of Computational
  Statistics, Springer.
• Wendy L. Martinez Angel R.  Martinez, 2002,
  Computational Statistics Handbook with MATLAB,
  Chapman Hall/CRC.
• Michael J. Crawley, 2005, Statistics An
  Introduction Using R, Wiley. Our Statistics Study
  Group is working through this.

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Doing Computational Statistics

• Usually you do computational statistics to
  explore the structure of data. The questions you
  might ask are rather open-ended. Your
  understanding is facilitated by a model.
• A model embodies what you currently know about
  the data. You can formulate it either as a
  data-generating process or a set of rules for
  processing the data.

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Statistical Models

• Often expressed as a set of equations relating
  data elements.
• Can include probability distributions for the
  elements. If this is the case, you have a
  stochastic model.
• The model should be free to evolve based on data
  mining.

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Common Stochastic Models

• Parameterized statistical distributions, such as
  the normal distribution, binomial distribution,
  or the chi-squared distribution.
• Sometimes more complicated, where you need to use
  simulation, resampling, and visualization to
  determine the parameters of the model.

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Structure-in-the-data

• Of most interest, for example
• Modes
• Gaps
• Clusters
• Symmetry
• Shape
• Deviations from normality

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Visualization

• Multiple views are necessary
• Be able to zoom in on the data as a few points
  can obscure the interesting structure.
• Scaling of the axes may be necessary, since our
  eyes are not perfect tools for detecting
  structure.
• Watch out for time-ordered or location-ordered
  data, particularly if time or location are not
  explicitly reported.

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Plots

• Use simple plots to start with.
• Watch for rounded datashown by horizontal strata
  in the data. That often signals other problems.

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Statistical Activities

• Data collection (ideally the statistician has a
  say on how they are collected)
• Description of a dataset
• Averages
• Spreads
• Extreme points
• Inference within a model or collection of models
• Model selection

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How to Do It

• Start by determining what sort of statistical
  analysis should you do. You need to know
• Which variable is the response variable?
• Which are the explanatory variables?
• What kind are the explanatory variables?
• What kind of response variable do you have?

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Basic Method of Analysis

• If all explanatory variables are continuous, plan
  on a regression analysis.
• If all explanatory variables are categorical,
  plan for an analysis of variance (ANOVA).
• If you have a mix, plan for an analysis of
  covariance (ANCOVA)

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Effect of Response Variable

• If the response variable is continuous, then plan
  on a normal regression, ANOVA, or ANCOVA.
• If the response variable is a proportion, do a
  logistic regression.
• If a count, you need a log linear model.
• If binary, you need a binary logistic analysis
• If time to event or time at death, you will be
  doing a survival analysis.

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Variation

• You want to understand how the response is
  dependent on variation in the explanatory
  variables, but you are also interested in lack of
  dependence.
• Design the simplest model that explains the data
  adequately.

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Significance

• You have to determine what the probability of a
  false alarm will bethat is, that you will think
  something is significant that really isnt.
• Typical values are 5, 1, and 0.1.
• Dont test every hypothesis. Some will be true by
  chance.

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Good and Bad Hypotheses

• There are vultures in the local park.
• There are no vultures in the local park.
• Which is testable?
• The null hypothesis is testable. You test it by
  taking measurements and showing that if the null
  hypothesis is true, the chance of those
  measurements is nearly zero.

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Experimental Design

• Replication
• Increases reliability, so be thorough. Usually
  the answer is 30.
• Randomization
• Reduces bias, so do it properly
• Almost never done properly
• Discuss

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Controls

• No controls, no conclusions.
• A control experiment is one where you dont apply
  the treatment or dont enable the part of your
  experiment that is supposed to produce the
  different outcome.

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Replication

• Must be independent
• Not part of a time series
• Not grouped together in space
• Of an appropriate spatial scale
• Covers the normal variation in initial conditons.

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Error Types
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Typical ? and ? values

• You usually want the probability of rejecting the
  null hypothesis (?) when it is true to be less
  than 5.
• You usually want the probability of accepting the
  null hypothesis (?) when it is false to be less
  than 20.
• The power of a test is 1- ?, or greater than 80
  in this case.
• Rule of Thumb the number of replicates to reject
  the null hypothesis with probability 80 is about
  8s2/d2, where s2 is the variance in the response
  and d is the size of the difference to be
  detected in a single sample.

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Inference

• Strong inference
• A clear hypothesis
• An acceptable test
• Weak inference
• Natural experiments
• Conclusions from natural experiments are
  hypotheses.

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How Long to Go On?

• To stop the experiment as soon as a pleasing
  result is obtained?
• To keep going until the theoretically correct
  result is obtained?
• Discuss.

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Statistics in MATLAB

• MATLAB has some useful statistical tools you can
  use to do all this (although most computational
  statistics is done using FORTRAN, SAS, R, or
  S-Plus).
• Supports the usual range of statistical tasks,
  including both analysis and visualization.
• Following is an overview of the capabilities of
  the MATLAB statistics toolbox.

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Statistics Capabilities

• Probability distributions
• Descriptive statistics
• Linear and non-linear models
• Hypothesis testing
• Multivariate statistics
• Plotting
• Statistical process control,
• Design of experiments, and
• Hidden Markov models.

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Random number generators

• There are functions in the Statistics Toolbox
  that return random output.
• These allow the user to observe probability
  distributions, evaluate statistical tests, and
  use resampling techniques.

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Probability distributions

• These are used to display possible probability
  distributions and create histograms.
• MATLAB provides the pdf, cdf, cdf-1, a random
  number generator, and mean and variance
  estimators for each distribution.

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Continuous Distributions Provided

• Beta
• Exponential
• Extreme value
• Gamma
• Lognormal
• Normal
• Rayleigh
• Uniform
• Weibull

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Continuous Statistical Distributions

• Chi-square
• Non-central Chi-square
• F
• Non-central F
• t
• Non-central t

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Discrete distributions

• Binomial
• Discrete uniform
• Geometric
• Hypergeometric
• Negative binomial
• Poisson

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Descriptive statistics

• mean
• median
• variance
• standard deviation
• Grouped data

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Linear and non-linear models

• ANOVA
• Covariance analysis (ANCOVA)
• Multiple linear regression
• Quadratic response surface models
• Stepwise regression
• GLM
• Robust and nonparametric methods
• Nonlinear least squares
• Regression and Classification Trees (CART)

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Hypothesis testing

• Null hypothesis
• Alternative hypotheses
• Significance level
• p-value
• Confidence intervals
• A number of tests are provided (this is a hard
  area)

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Multivariate statistics

• Principal components analysis
• Factor analysis
• MANOVA
• Cluster analysis
• Multidimensional scaling

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Plotting and Visualization

• Box plots
• Distribution plots
• Scatter plots

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Statistical process control

• Quality of manufactured goods
• Control charts
• Capability studies

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Design of experiments

• Full factorial designs
• Fractional factorial designs
• Response surface designs
• D-optimal designs

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Hidden Markov Models

• Concepts
• Markov chains
• Analysis of hidden Markov models (HMMs).

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Conclusions

• MATLAB provides a basic engineering toolkit for
  these statistical activities.
• Not as broad as R or S-plus, but compatible with
  data collected or generated by other toolkits.
• Supports all activities well.
• More specialized work (e.g., Bayesian analysis)
  requires either your own extensions or more
  specialized toolkits.