FINDING RELATIONS BETWEEN VARIABLES - unibo.itFINDING RELATIONS BETWEEN VARIABLES...

FINDING RELATIONS BETWEEN VARIABLES

Pearson’s Correlation

Pier Luigi Martelli - System and In Silico Biology. AA 2015-2016- University of Bologna

Relation between coupled variables

What couples of variables are in relation?Pier Luigi Martelli - System and In Silico Biology. AA 2015-2016- University of Bologna

Correlated variables


Uncorrelated variables


Covariance and Pearson’s Correlation index

Variable 1 Variable 2

Item1 x1(1) x2(1)

Item 2 x1(2) x2(2)

Item i x1(i) x2(i)

Item m x1(m) x2(m)

Mean M1 M2

m

i

m

i

ixm

M

ixm

M

1

22

1

11

)(1

)(1

m

i

m

i

MixMix

mxxcorr

MixMixm

xx

1 21

221121

22

1

1121

)()(

1

1),(

)()(1

1),cov(


Covariance and Pearson’s Correlation index

Variable 1 Variable 2

Item1 x1(1) x2(1)

Item 2 x1(2) x2(2)

Item i x1(i) x2(i)

Item m x1(m) x2(m)

Mean M1 M2

m

i

m

i

ixm

M

ixm

M

1

22

1

11

)(1

)(1

2

1

)()(1

1),cov( jjj

m

i

jjjj MixMixm

xx


Correlation

1,1),( 21 xxcorr


When is a correlation significant?

Given a correlation index:

A test variable can be computed under the null hypothesis that r=0

t is distributed as Student’s t test with n-2 degrees of freedom It assumes normality of x

m

i

MixMix

mxxr

1 21

221121

)()(

1

1),(


Graph showing the minimum value of Pearson's correlation coefficient that is significantly different from zero at the 0.05 level, for a given sample size.


Example: discovery of a misconduct

Repeatability test: 2 different experimentalist were asked to take the same solution and to perform 24 independent ELISA assays on a 6x4 plate.

They submitted to the assessor the following results out of the spectrophotometer, ordered following the well


S1 S2P1 0,481 0,496P2 0,485 0,501P3 0,479 0,495P4 0,506 0,522P5 0,467 0,48P6 0,474 0,491P7 0,469 0,48P8 0,475 0,489P9 0,514 0,52P10 0,52 0,524P11 0,526 0,531P12 0,494 0,509P13 0,535 0,54P14 0,524 0,526P15 0,481 0,492P16 0,502 0,509P17 0,479 0,484P18 0,491 0,495P19 0,503 0,515P20 0,472 0,481P21 0,481 0,486P22 0,503 0,512P23 0,448 0,454P24 0,519 0,526

The assessor suspectedthat the experimenter submitted two reads of the same plate

How to prove it?



0,44

0,45

0,46

0,47

0,48

0,49

0,5

0,51

0,52

0,53

0,54

0,55

0,44 0,46 0,48 0,5 0,52 0,54

S2

S1

R=0.978



R=0.978 n=24 t=22.05

Objection: the test is valid only when data are normally distributed and we cannot prove that.Any other idea?


Use the data set to generate random experiments

S1 S2P1 0,481 0,496P2 0,485 0,501P3 0,479 0,495P4 0,506 0,522P5 0,467 0,48P6 0,474 0,491P7 0,469 0,48P8 0,475 0,489P9 0,514 0,52P10 0,52 0,524P11 0,526 0,531P12 0,494 0,509P13 0,535 0,54P14 0,524 0,526P15 0,481 0,492P16 0,502 0,509P17 0,479 0,484P18 0,491 0,495P19 0,503 0,515P20 0,472 0,481P21 0,481 0,486P22 0,503 0,512P23 0,448 0,454P24 0,519 0,526

S1 Random(S2)P1 0,481 0,495P2 0,485 0,522P3 0,479 0,48P4 0,506 0,491P5 0,467 0,48P6 0,474 0,489P7 0,469 0,52P8 0,475 0,524P9 0,514 0,531P10 0,52 0,509P11 0,526 0,54P12 0,494 0,526P13 0,535 0,492P14 0,524 0,509P15 0,481 0,484P16 0,502 0,495P17 0,479 0,515P18 0,491 0,481P19 0,503 0,486P20 0,472 0,512P21 0,481 0,454P22 0,503 0,526P23 0,448 0,496P24 0,519 0,501

0,44

0,46

0,48

0,5

0,52

0,54

0,56

0,44 0,46 0,48 0,5 0,52 0,54

Rand

om (S2)

S1

R=0.25


Building a distribution by random resampling

Iterate the process of shuffling and computation of r many times (say 1000)

Compute a cumulative histogram counting the resamplings scoring with correlation ≥ r

0,00

200,00

400,00

600,00

800,00

1000,00

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90


Building a distribution by random resampling

The plot gives the probability (per thousand) of obtaining a given correlation with random pairings of the original data P-value independent on the assumptions on the data distribution

0,00

200,00

400,00

600,00

800,00

1000,00

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90

This is only an example plot: Compute by yourself the plot corresponding to the data available in misconduct.xls file


Bootstrapping

The term is often attributed to Rudolf Erich Raspe's story The Surprising Adventures of Baron Munchausen, where the main character pulls himself out of a swamp by his hair (specifically, his pigtail), but the Baron does not, in fact, pull himself out by his bootstraps


FINDING RELATIONS BETWEEN VARIABLES

Spearman’s Correlation

Pier Luigi Martelli - System and In Silico Biology. AA 2015-2016-University of Bologna

Pearson’s correlation index assumes linear dependence

R=0.816


Non parametric correlation: Spearman

Given a set of paired (xi,yi) sort separately the two variables, obtaining the ranks.

The Spearman’s correlation is the Pearson’s correlation of the ranked variables: [R(xi),R(yi)]


Under the null hypothesis (r=0)

Is distributed as a Student’s t test with n-2 degrees of freedom


Categorical data: Matthews correlation index

Secreted Non Secreted Total

With Signal peptide a b a + b

Without Signal Peptide c d c + d

total a + c b + d n

cdbdcaba

ad-bcMCC


EXTRACTING INFORMATION FROM HIGH DIMENSIONALDATA

Principal Component Analysis


High-dimensional descriptors

Many different descriptors can be adopted for characterizing a set of objects under investigation:

1) given a set of proteins, we can measure the residue composition (20 values), the dipeptide composition (400 values), length, average hydrophobicity..... of each sequence

2) Given a set of individuals we can measure dimensions, weight, haematic concentration of metabolites...

How can we extract a minimal set of descriptors without losing information on the variability of the set?


Data Reduction

Summarizing the p-dimensional description of nobjects by a smaller set of (k) derived (synthetic, composite) variables.


Data Reduction

“Residual” variation is information in A that is not retained in X

A good data reduction must balance between clarity of representation, ease of understanding

oversimplification: loss of important or relevant information.


Most important descriptors

When plotting these two dimensional data it is evident that the x-direction accounts for the largest part of the variance

Directions with largest variance better describe the data set


Most important descriptors

When variables are correlated the variable variance is not able to determine principal components


Example:2D

Markus Ringnér. What is principal component analysis?Nature Biotechnology 26, 303 - 304 (2008)

Suppose to measure the expression level of two genes (XBP1 and GATA3) in breast cancer samples expressing or not the Estrogen Receptor (ER+ in red, ER- in black)


Example:2D

The totalvariance isdecomposed intotwo orthogonalcomponents, PCA1 and PCA2


Example:2D

Analysis of the principal component can highlightimportant featuresin an unsupervisedway

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14 16 18 20

Variable X1

Va

ria

ble

X2

+

2D Example of PCA• variables X1 and X2 have positive covariance & each

has a similar variance.

67.61 V 24.62 V 42.32,1 C

35.81 X

91.42 X

www.plantbiology.siu.edu/PLB444/PCA.ppt

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8 10 12

Variable X1

Vari

ab

le X

2Configuration is Centered

• each variable is adjusted to a mean of zero (by subtracting the mean from each value).


-6

-4

-2

0

2

4

6

-8 -6 -4 -2 0 2 4 6 8 10 12

PC 1

PC

2Configuration is rotatedNew coordinates

PC1 PC2• PC 1 and PC 2 have zero covariance.• PC 1 has the highest possible variance (9.88)• PC 2 has a variance of 3.03


Covariance Matrix

Variable 1 Variable 2 Variable j Variable n

Item1 x1(1) x2(1) xj(1) xn(1)

Item 2 x1(2) x2(2) xj(2) xn(2)

Item i x1(i) x2(i) xj(i) xn(i)

Item m x1(m) x2(m) xj(m) xn(m)

2

21

2

21

22

2

212

1121

2

1

,cov,cov,cov

,cov,cov,cov

,cov,cov,cov

,cov,cov,cov

njnnn

njjjj

nj

nj

xxxxxx

xxxxxx

xxxxxx

xxxxxx

COV


Covariance and multidimensional normal distribution

M is a n-valued vector (means)

COV is a nxn symmetric matrix (covariance matrix), with determinant |COV|

MxCOVMx

COV

MxT

n

1

2

1

22

1exp

2

1),|(


Uncorrelated variables

The set of variable is uncorrelated if all the covariances (not the variances) are null:

that is if covariance matrix is in diagonal form.

iij

m

k

jjiiij MkxMkxm

COV 2

11

1


Eigenvector equation

COV is a square symmetrical matrix and can be reduced in diagonal form

by means of the eigenvector equation

it defines n real eigenvalues λi

and n real-m-valued orthonormal eigenvectors ui

ijiijVOC ~

0det

ICOV

COV

uu


Matrix transformation

The column normalized eigenvectors ui are orthogonal and define the n x n unitary matrix U

T

T

UU

IUU

1


22

11

21

21

2212

2111

uu

uu

UU

UUU


U defines an orthogonal rotation and/or reflection of the coordinate axes (preserving norms and angles)

xxxxxx

xx

TTT

~UU~~~

U~

T

T


cossin

sincosU

Try for example a counterclokwise rotation of 45° and transform the point (1,1)


Given the matrix A, the eigenvalues definethe diagonal matrix Λ and the eigenvectorsdefine the unitary matrix U so that:

Λ=UTAU

Prove with matrix:

12

21A



The diagonal matrix :

U then defines a coordinate rotation so that in the new system variables are not correlated

VOCUCOVU~

T



Coordinate transformation

2

1

2

1

2

1

2

1

x

x

y

y

y

y

x

x

TU

U

x1

x2

y2 y1



In particular the old coordinates of new axes, called LOADINGS, are the eigenvectors

2

1

2

1

2

1

2

1

x

x

y

y

y

y

x

x

TU

U

U11

x2

y1

22

12

21

11

1

0

0

1

U

U

U

U

U

U


U21U22

U12


Given any item x, represented by an m-valued vector

it can be expressed with the n ordered principal components, y, by using the basis U

Where

),..,( 21 m

T xxxx

n

i

ii uyx1

xuuxy T

ii

T

i



The eigenvalues measure the variances along the new coordinate axes.

Usually the percentage of the total variance accounted by any coordinate is reported

Coordinates are sorted from the highest to the lowest eigenvalue



Given a set of m items described with n variables:1) Compute the mean of each variable2) Subtract the mean to any measure3) Compute the covariance matrix4) Diagonalize the covariance matrix5) Sort the eigenvalues from the highest to the

lowest: the corresponding eigenvectors define the 1st,2nd....jth principal components

6) For each item i, the j-th component results from the scalar product of the i-th variable vector with the j-th eigenvector


Why “principal”?

Given a n-dimensional representation of mobserved samples and you want to representdata in a p-dimensional space, with p<n (e.g.,for representing them in 2- or 3-dimensional space). Which is the bestchoice, when only linear transformations areallowed?

To use the first p principal components.


Why “principal”? Minimum-error formulation

We have to search for a completeorthonormal m-dimensional basis V

and we have to use p-dimensions in thatcoordinate basis to approximate the points:

where ci are independent of k

i

n

i

i

T

k

n

i

ikik vvxvax

11

)(

n

pi

ii

p

i

ikik vcvbx11

~



The goal is to minimize the error

For any base V, the error is minimized when:

Then:

m

k

kk xxm

E1

2~1

i

T

ki

i

T

kkiki

vxc

vxab

n

pi

ii

T

kkkk vvxxxx1

~



The error is minimized, imposing thenormalization of each v (via Lagrange’smultipliers)

Then

choosing the lowesteigenvalues

n

pi

i

T

i

m

k

n

pi

i

T

ki

T

k vvvxvxn

E11 1

21 COV

i

T

ii

n

pi

i

T

iv vvvvMINIMIZEi

1

1

COV

iii vv

COV

n

pi

iE1

Covariance vs Correlation

using covariances among variables only makes sense if they are measured in the same units

even then, variables with high variances will dominate the principal components

these problems are generally avoided by standardizing each variable to unit variance and zero mean.


Correlation PCA

The correlation matrix can be chosen instead of the covariance. It gives a zero-mean, unit-covariance plot

m

i

ii MxMx

mxxcorr

1 21

121121

1

1),(

An ecological example• data from research on habitat definition in the endangered Baw Baw frog

• 16 environmental and structural variables measured at each of 124 sites

• correlation matrix used because variables have different units

Philoria frostiwww.plantbiology.siu.edu/PLB444/PCA.ppt

Axis Eigenvalue% of

VarianceCumulative % of Variance

1 5.855 36.60 36.60

2 3.420 21.38 57.97

3 1.122 7.01 64.98

4 1.116 6.97 71.95

5 0.982 6.14 78.09

6 0.725 4.53 82.62

7 0.563 3.52 86.14

8 0.529 3.31 89.45

9 0.476 2.98 92.42

10 0.375 2.35 94.77

Eigenvalues


Baw Baw Frog - PCA of 16 Habitat Variables

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

1 2 3 4 5 6 7 8 9 10

PC Axis Number

Eig

en

valu

e


Interpreting Eigenvectors

• correlations between variables and the principal axes are known as loadings

• each element of the eigenvectors represents the contribution of a given variable to a component

1 2 3

Altitude 0.3842 0.0659 -0.1177

pH -0.1159 0.1696 -0.5578

Cond -0.2729 -0.1200 0.3636

TempSurf 0.0538 -0.2800 0.2621

Relief -0.0765 0.3855 -0.1462

maxERht 0.0248 0.4879 0.2426

avERht 0.0599 0.4568 0.2497

%ER 0.0789 0.4223 0.2278

%VEG 0.3305 -0.2087 -0.0276

%LIT -0.3053 0.1226 0.1145

%LOG -0.3144 0.0402 -0.1067

%W -0.0886 -0.0654 -0.1171

H1Moss 0.1364 -0.1262 0.4761

DistSWH -0.3787 0.0101 0.0042

DistSW -0.3494 -0.1283 0.1166

DistMF 0.3899 0.0586 -0.0175


How many axes are needed?

• does the (k+1)th principal axis represent more variance than would be expected by chance?

• several tests and rules have been proposed

• a common “rule of thumb” when PCA is based on correlations is that axes with eigenvalues > 1 are worth interpreting


Example: Thermostability

The problem is to investigate the differences in Residue composition Codon composition Codon usage

Among thermophilic and mesophilicprokaryotes

Montanucci, Martelli, Fariselli, Casadio J Proteome Res, 2007


Example: Thermostability

The data set contains 116 fully sequenced genomes from prokaryotes

16 thermophilic species (11 archaea and 5 bacteria), with an OGT higher than 60 °C

100 mesophilic species (95 bacteria and 5 archaea) with an OGT lower than 45 °C.

7 quasi-mesophilic species (7 bacteria) with OGT between 45 and 60 °C


PCA on residue composition

Red:

Thermophilic

Green:

Intermediates

Blue:

Mesophilic


PCA on codon usage

Red:

Thermophilic

Green:

Intermediates

Blue:

Mesophilic


PCA on codon composition

Red:

Thermophilic

Green:

Intermediates

Blue:

Mesophilic


Components of 1st and 2nd PC (codon composition)

First two components expressed as a function of the codons


Relation between 2nd component and OGT


Pitfalls of PCA

A method such as PCA assumes that thereare linear relationships between thederived components and the originalvariables. This is apparent from the role ofthe covariance or correlation matrix. If therelationships are non linear a Pearsoncorrelation coefficient, which measures thestrength of linear relationships, wouldunderestimate the strength of the non-linear relationship.


EXTRACTING INFORMATION FROM HIGH DIMENSIONALDATA

Correspondence analysis


Pitfalls of PCA

PCA can be applied in a coherent way onlywhen continuous variables are taken intoaccount. Data are however available ascounts on categorical variable

contingency table


Contingency tables


Item 1 x11 x12 x1j x1n

Item 2 x21 x22 x2j x2n

Item i xi1 xi2 xij xin

Item m xm1 xm2 xmj xmn

Ex: Experiment = genomes, Variable = codon countExperiment = Text, Variable = number each character in the text


Text characters usage:is it distinctive of the authors

Yelland PM,The Mathematica JournalPier Luigi Martelli - System and In Silico Biology. AA 2015-

2016- University of Bologna

Margins


Item 1 x11 x12 x1j x1m r1

Item 2 x21 x22 x2j x2m r2

Item i xi1 xi2 xij xim ri

Item m xn1 xn2 xnj xnm rn

c1 c2 cj cm N

m

i

i

n

j

j

n

j

m

i

ij

m

i

ijj

n

j

iji

rcxN

xcxr

111 1

11


Correspondence matrix

:N

cm

j

jc

Mass of column j

:N

rm i

ir

Mass of row i

N

xm

ij

ij

Frequency value


Item 1 x11/N=m11 x12/N=m12 x1j/N=m1j x1n/N=m1m r1/N=mr:1

Item 2 x21/N=m21 x22/N=m22 x2j/N=m2j x2n/N=m2m r2/N=mr:2

Item i xi1/N=mi1 xi2/N=mi2 xij/N=mij xin/N=mim ri/N=mr:I

Item m xm1/N=mn1 xm2/N=mn2 xmj/N=mnj xmn/N=mnm rm/N=mr:m

c1/N=mc:1 c2/N=mc:2 cj/N=mc:j cn/N=mc:n 1


Are the variable dependent on the experiments?

If the variable j is independent of the experiment i

represent a baseline distribution for

We need a distance between distributions: it would be nice to evaluate the probability that is compatible with

:: jcirij mmm

:: jcir mm ijm

ijm :: jcir mm


Testing distributions

Is a frequency distribution of certain events observed in a sample consistent with a particular theoretical distribution?

Null hypothesis: the distributions do not differ

A simple example is the hypothesis that an ordinary six-sided die is "fair", i.e., all six outcomes are equally likely to occur.


Pearson’s chi-square test

The observed values (Oi) and the expected values (Ei) are compared over all the n classes (bins):

This test variable approximately follows a chi-square statistics: degrees of freedoms are: n-1-s, where s is the number of parameters describing the theoretical distribution (e.g. 2 for normal distribution, 1 for binomial distribution)


Chi-square distribution

Mean = kVariance = 2k


Critical values



Correspondence analysis can be viewed as a weighted PCA in which Euclidean distances are replaced by chi-squared distances that are more appropriate for count data.

Chi-square distances capture also NON-linear dependencies and can be applied to counts

As with PCA, derived variables are created from the original variables but this time they maximise the correspondence between row and column counts.


Are the variable dependent on the experiments?

If the variable j is independent of the experiment i

A good measure of distance between the actual matrix and the matrix computed with the independence hypothesis is

If the null hypothesis holds (independence), it follows a chi-square statistics (m-1)(n-1) DoF

1 1 ::

2

::2

m

i

n

j jcir

jcirij

mm

mmm

:: jcirij mmm


Text characters usage:is it distinctive of the authors

Yelland PM,The Mathematica Journal

0.01 value-P

5.4482

The hypothesis of independence between character usage and authors can be rejected.How can I recognize the works of different authors?


Normalizing rows: Row profiles


Item 1 m11/mr:1 m12/mr:1 m1j/mr:1 m1n/mr:1 1

Item 2 m21/mr:2 m22/mr:2 x2j/mr:2 m2n/mr:2 1

Item i mi1/mr:i mi2/mr:i mij/mr:i min/mr:i 1

Item m mm1/mr:m mm2/mr:m mmj/mr:m mmn/mr:m 1

Row centroid mc:1 mc:2 mc:j mc:n 1

:N

cm

j

jc

Mass of column jProfile of row i

:

:

:2

:1

irin

irij

iri

iri

i

mm

mm

mm

mm

R


Distance between row profiles

//

1 :

2

::

n

j jc

krkjirijik

r

m

mmmmd

Measures the distance between the rows iand k, upon renormalization of each row

Considering the row centroid z:

m

j jcir

jcirij

ir

m

j jc

jcirijiz

r

mm

mmm

mm

mmmd

1 ::

2

::

:1 :

2

:: 1

/


Chi square distance among authors

Yelland PM,The Mathematica JournalPier Luigi Martelli - System and In Silico Biology. AA 2015-


Column profiles

:N

xm i

ir

Mass of ROW iProfile of column j

:

:

:2

:1

jcnj

jcij

jcj

jcj

j

mm

mm

mm

mm

C

Variable 1

Variable 2 Variable j Variable n Column centroid

Item 1 m11/mc:1 m12/mc:2 m1j/mc:j m1m/mc:n mr:1

Item 2 m21/mc:1 m22/mc:2 x2j/mc:j m2m/mc:n mr:2

Item I mi1/mc:1 mi2/mc:2 mij/mc:j mim/mc:n mr:i

Item m mm1/mc:1 mm2/mc:2 mmj/mc:j mmn/mc:n mr:m

1 1 1 1 1


Distance between column profiles

//

1 :

2

::

m

i ir

kcikjcijjk

c

m

mmmmd

Measures the distance between the columns j and k, upon renormalization of each column

Considering the column centroid z:

m

i irjc

irjcij

jc

m

i ir

rjcijjz

c

mm

mmm

mm

mmmd

1 ::

2

::

:1 :

2

:: 1

/


Nomenclature: reminder

original count

correspondence values

mass of row i

mass of column j

: j

iji

ir mN

rm

ijx

N

xm

ij

ij

: i

ij

j

jc mN

cm


Nomenclature: reminder

profile of row i

profile of column j

:

:

:2

:1

irin

irij

iri

iri

i

mm

mm

mm

mm

R

:

:

:2

:1

jcmj

jcij

jcj

jcj

j

mm

mm

mm

mm

C


Mass centroid:row

We can see the system as a set of n (row-) masses (mr:i) distributed in the m-dimensional space in points: ri

The centroid is given by the column masses

:

:

2:

1:

:

1

:

:

:2

:1

1

:

nc

jc

c

c

ir

n

i

irim

irij

iri

iri

n

i

iir

m

m

m

m

m

mm

mm

mm

mm

Rm


Chi square as inertia with weighted distances

/

1 1 :

2

:

:

1 1 :

2

::

:

1 1 ::

2

::2

m

i

n

j jc

jc

i

j

ir

m

i

n

j jc

jcirij

ir

m

i

n

j jcir

jcirij

m

mRm

m

mmmm

mm

mmm


Standardized residuals

The chi-square residuals S measure the level of dependency of the individuals from the variables

Is it possible to find a linear transform of the individuals and a linear transform of variable so that: ?

If yes each new transformed individual would depend only on a single new transformed variable.

jcir

jcirij

ijmm

mmmS

::

::

ijiij dS ~



Searching a transformation that preserves the distances between rows or column and that allows the representation into low dimensional spaces with low loss of information.

Diagonalization of matrix S.

S is a matrix mxn


Single value decomposition

The matrix S (mxn) can be decomposed as

where U (mxm) and V (nxn) are base change matrices in the experiment (row) and variable (column) space, respectively

and Λ is a mxn diagonal matrix

TVΛUS

1 1 TTVVUU


Single value decomposition: solution 1

The decomposition can be reduced to the determination of eigen-vector and -values

and Λ2 is a nxn diagonal matrix

NB: STS is a sort of covariance matrix of the chi-square distances

T

nn

TT

TTTT

VVVUUV

VUVUSS

2

)()(



The diagonalization of STS gives n eigenvalues λ2, whose square roots are the eigenvalues of Λ sort them

The n column eigenvectors, sorted following the eigenvalues, define the matrix V (base change in column space)right singular vectors

VΛVSS

nn

TT 2



The decomposition can be reduced to t

and Λ2 is a mxm diagonal matrix

T

mm

TT

TTTT

UUUVVU

VUVUSS

2

))((



The diagonalization of SST gives m eigenvalues λ2, whose square roots are the eigenvalues of Λ sort them

The n column eigenvectors, sorted following the eigenvalues, define the matrix U (base change in row space) left singular vectors

UΛUSS

mm

TT 2


Are the two sets of eigenvaluesequal?

If λ2i is eigenvalue of SST with eigenvector ui

then λ2i is eigenvalue of STS with eigenvector

vi=STui

iii

iii

iii

vv

uu

uu

2T

T2TT

2T

SS

SSSS

SS


Single value decomposition

The number of non-null eigenvalues of matrices Λ2 is at most equal to min (m,n).

Λ is a mxn diagonal matrix containing the sortedsquare roots of eigenvaluesλ

n>m n<m

and U and V are the ortogonal change vector matricesPier Luigi Martelli - System and In Silico Biology. AA 2015-



Searching a transformation that preserves the distances between rows or column and that allows the representation into low dimensional spaces with low loss of information.

Diagonalization of matrix S.

ji

jiij

ijmm

mmmS

TVΛUS Pier Luigi Martelli - System and In Silico Biology. AA 2015-


Row Scores

kik

i

m

l

lkjl

i

ijik Um

Um

R

11

1

Represents new component k of row i. It conserves the distance from the centroid


Row components

The first 2 or 3 components are plotted, usually

111

1i

i

i Um

R

222

1i

i

i Um

R


2-D plot of the texts

Yelland PM,The Mathematica Journal

Proximity in the plot means low chi-square distance


This is the best approximation

As in the case of PCA, the CA find the low dimensional space for projecting the data that is allows the best approximated representation of data. In the case of PCA the euclidean distance among points is best approximated. In tha case of CA, it is the chi-square distance


Column Scores

kjk

j

n

l

lkjl

j

ijjk Vm

Vm

C

11

1

Represents new component k of column j. It conserves the distance from the centroid

m

i ij

ijij

j

jzmm

mmm

md

1

21


Column components

The first 2 or 3 components are plotted, usually

111

1j

j

j Vm

C

222

1 j

j

j Vm

C


Joint plot


Angles determine the type of dependency


The cosine of the angles determines the stregnth of the dependence


Example: codon usage in genes

Contingency table

AAA AAC AAT AAG ...........

gene1

gene2

.......


Row (gene) representation

Genes that are close each other, haveMinimum chi-square distance are coded with a similar codon distribution

High GC genes and Low GC genes separate along the first axis


Column (codon) representation

Rapid divergence of codon usage patterns within the rice genomeHuai-Chun Wang1and Donal A HickeyBMC Evolutionary Biology 2007, 7(Suppl 1):


Column (codon) representation

Column plot shows the separation of C/

G-ending codons and A/U-ending codons along the first axis. The separation of genes on the second axis appears to be largely due to frequency differences in C-ending and G-ending codons among the GC rich genes


Codon usage in prokaryotes

Use of Correspondence Analysis in Genome ExplorationFredj Tekaia


Codon usage in lehismania

Online Synonymous Codon Usage Analyses with the ade4 and seqinR packagesCharif, D., Thioulouse, J. Lobry, J.R., Perrière, G.,Pier Luigi Martelli - System and In Silico Biology. AA 2015-


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