STA 6329 Matrix Algebra Application to Statistics

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STA 6329 Matrix Algebra Application to
Statistics
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The life-expectancy webhttp//www.life-expectancy
.longtolive.com

• 1. Gender 2. date of birth
• 3. Stress level 4. smoking frequency
• 5. Alcohol consumption 6. Times brush your teeth
• 7. Time floss your teeth 8. Drink ___ oz coffee
• 9. Average hours of exercise/week 10. of meat
  in diet
• 11. How long your father lived? 12 How long your
  mother lived?
• 13. Your blood pressure SBP/DBP

Result Mark Yang will die on 09/09/2035 053334
am If
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How do they know? From data
Suppose we have n death records with good
life-style information.
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Data analysis and model building
Goal 1 How to solve ? in YX??? It turns out
that if ? is normally distributed, it is
equivalent to solve XX?XY.
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A successful story genetic evaluation of farm
animals
Purpose Find the largest u (best individual with
milk production genes), because all the ? are
nonnegative. Ref Wiggans, ei al. Animal model
evaluation of Ayrshine milk yield with all
lactation, herd-sire interaction and groups based
on unknown parents, J. Dairy Sci., 771319-29
(1988). Data n119,541 cows, m5,762 sires, k
2 (constant, season).
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There may be multiple responses.

• Example High school activities A, B, and C will
  affect university grades of subjects D, E, F.
• SAS type input

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A simple way to describe a group of random
variables
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From covariance to correlation
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Early investigation in correlation Galtons
height inheritance
Francis Galton (1822-1911)
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Positively correlated random variables

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Presidents and Heights
The Economist, April 27, 2002
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The taller the better? Short women have more
children.
n 5,000 women born in Britain in March 1958

• The Economist, Aug. 17, 2002

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Dimension Reduction
Goal 2 Given ?, how to find a matrix A s.t.
A?A is diagonal with as many 0 as possible in
the diagonal.
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Real Data Microarray Analysis

• X-axis time (6 points) y-axis gene clustered
  (yeast)
• Data from DeRisi, JL, Iyer, VR, Brown, PO.
  "Exploring the metabolic and genetic control of
  gene expression on a genomic scale." Science.
  1997 Oct 24278(5338)680-6.

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DNA to mRNA to phenotype
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From gene to protein
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Idea Image
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Application Breast classification
Science, Sept. 2000
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Original dimension 6 time points

• Each dot represents a gene

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Canonical Correlation
Goal 3 How do we solve the equation
?12?22-1?21-??110?
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Real Data fMRI (functional magnetic resonance
imaging) on brain activities
Red color means High correlation.

• O. Friman, et al. Detection of neural activity
  in functional MRI using Canonical Correlation
  Analysis. Magnetic Resonance in Medicine Vol 45,
  Number 2, Pages 323-330, 2001.

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Findings

• fMRI response over 6 time points in 9 pixel
  cluster (m54)
• Stimulus pattern over time 6 frequency component
  (n6).
• To find Which pixel cluster is correlated with
  the stimulus?
• The top picture is by t-test, the middle is by
  F-test and the bottom one is by canonical
  correlation which has the most discovery.

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Markov Chains
Goal 4 Give P, how to study the properties of
Pn?
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Typical Mutation Analysis
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General Markov Chains
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Proofs
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Real Data

• Nei, M. and Kumar, S. (2000) Molecular Evolution
  and Phylogenetics. New York, NY Oxford
  University Press.

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Time Series Example
Again, Goal 4 Give P, how to study the
properties of Pn?
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Kalman filter and GPS (Global Position System)

• http//en.wikipedia.org/wiki/Kalman_filter
• http//images.google.com/images?gbv2hlensafea
  ctiveqsitewww.julien-cayzac.comkalmanfilter
  andGPS

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Basic Kalman filter equations to GPS
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Summary What do we wish to know from matrix
algebra viewpoint (no statistics)?
Goal 1 How to solve ? in XX?XY? Goal 2
Given ?, how to find a matrix A s.t. A?A is
diagonal with as many 0 as possible in the
diagonal. Goal 3 How do we solve the equation
?12?22-1?21-??110? Goal 4 Give P, how to
study the properties of Pn? Numerical solution
is good, but analytical solution is better.