Visualization 1 Applied Multivariate Statistics Spring 2012 Goals - - PowerPoint PPT Presentation

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Visualization 1 Applied Multivariate Statistics Spring 2012 Goals - - PowerPoint PPT Presentation

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Visualization 1 Applied Multivariate Statistics Spring 2012 Goals Covariance, Correlation (true / sample version) Test for zero correlation: Fishers z -Transformation Scatterplot / Scatterplotmatrix Covariance matrix /

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Visualization 1

Applied Multivariate Statistics – Spring 2012

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Goals

  • Covariance, Correlation (true / sample version)
  • Test for zero correlation: Fisher’s z-Transformation
  • Scatterplot / Scatterplotmatrix
  • Covariance matrix / Correlation matrix
  • Multivariate Normal Distribution
  • Mahalanobis distance

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  • Appl. Multivariate Statistics - Spring 2012
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Visualization in 1d

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Normaldistribution in 1d: Most common model choice

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'¹;¾2(x) =

1 p 2¼¾2 exp(¡1 2 ¢ (x¡¹)2 ¾2

)

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'¹;¾2(x) =

1 p 2¼¾2 exp(¡1 2 ¢ (x¡¹)2 ¾2

)

Normaldistribution in 1d: Most common model choice

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Squared Mahalanobis Distance =

  • Sq. Distance from mean in

standard deviations

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Two variables: Covariance and Correlation

  • Covariance:
  • Correlation:
  • Sample covariance:
  • Sample correlation:
  • Correlation is invariant to changes in units,

covariance is not (e.g. kilo/gram, meter/kilometer, etc.)

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Cov(X;Y ) = E[(X ¡ E[X])(Y ¡ E[Y ])] 2 [¡1;1] Corr(X; Y ) = Cov(X;Y )

¾X¾Y

2 [¡1; 1] d Cov(x; y) =

1 n¡1

Pn

i=1(xi ¡ x)(yi ¡ y)

rxy = d Cor(x; y) = c

Cov(x;y) ^ ¾x^ ¾y

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Scatterplot: Correlation is scale invariant

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Intuition and pitfalls for correlation Correlation = LINEAR relation

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Test for zero correlation: Fisher’s z-Test

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  • Appl. Multivariate Statistics - Spring 2012
  • X, Y (bivariate) normal distributed with true correlation ½
  • Take n samples
  • Compute sample correlation r

Compute Compute

  • For large n:
  • Use cor.test() in R to test and get confidence intervals

z = 1

2 log

¡1+r

1¡r

¢ » = 1

2 log

¡ 1+½

1¡½

¢ pn ¡ 1(z ¡ ») » N(0;1)

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Many dimensions: Scatterplot matrix

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Covariance matrix / correlation matrix: Table of pairwise values

  • True covariance matrix:
  • True correlation matrix:
  • Sample covariance matrix:

Diagonal: Variances

  • Sample correlation matrix:

Diagonal: 1

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  • Appl. Multivariate Statistics - Spring 2012

§ij = Cov(Xi;Xj) Cij = Cor(Xi;Xj) Sij = d Cov(xi; xj) Rij = d Cor(xi;xj)

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Multivariate Normal Distribution: Most common model choice

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  • Appl. Multivariate Statistics - Spring 2012

f(x;¹; §) =

1

p

2¼j§j exp

¡ ¡ 1

2 ¢ (x ¡ ¹)T§¡1(x ¡ ¹)

¢

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Multivariate Normal Distribution: Funny facts

If X1, …, Xp multivariate normal, then

  • every linear combination Y = a1 X1 + … + ap Xp

is normally distributed

  • every projection on a subspace is multivariate normally

distributed If margins are normally distributed, then it is NOT GUARANTEED that the underlying distribution is multivariate normal (i.e., “multivariate” is stronger than just normal margins)

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Multivariate Normal Distribution: Two examples 1000 random samples

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  • Appl. Multivariate Statistics - Spring 2012

§ = µ 10 3 3 2 ¶ ¹ = µ 5 10 ¶ ; § = µ 1 1 ¶ ¹ = µ ¶ ;

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Multivariate Normal Distribution: Most common model choice

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  • Appl. Multivariate Statistics - Spring 2012

f(x;¹; §) =

1

p

2¼j§j exp

¡ ¡ 1

2 ¢ (x ¡ ¹)T§¡1(x ¡ ¹)

¢

  • Sq. Mahalanobis Distance MD2(x)

=

  • Sq. distance from mean in

standard deviations IN DIRECTION OF X

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Mahalanobis distance: Example

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  • Appl. Multivariate Statistics - Spring 2012

§ = µ 25 1 ¶ ¹ = µ ¶ ;

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Mahalanobis distance: Example

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§ = µ 25 1 ¶ ¹ = µ ¶ ;

(20,0)

MD = 4

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Mahalanobis distance: Example

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§ = µ 25 1 ¶ ¹ = µ ¶ ;

(0,10)

MD = 10

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Mahalanobis distance: Example

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§ = µ 25 1 ¶ ¹ = µ ¶ ;

(10, 7)

MD = 7.3

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Concepts to know

  • Covariance, Correlation (true / sample version)
  • Test for zero correlation: Fisher’s z-Transformation
  • Scatterplot / Scatterplotmatrix
  • Covariance matrix / Correlation matrix
  • Multivariate Normal Distribution
  • Mahalanobis distance

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R commands to know

  • read.csv, head, str, dim
  • colMeans, cov, cor
  • mvrnorm, t, solve, %*%

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