(

t

)

  

t



f

(

x

)

dx where f

(

x

)

 1 2 

²

e



(

x

2   

² )²

 

F

is called the cumulative distribution function

(cdf)

and 

f

the probability distrbution function and 

²

are respectively the

mean (pdf)

of

X

and the

variance

of the distribution

Moments of a distribution  The

k th



'

k

 moment

E

(

x k

)

is defined as     

x k f

(

x

)

dx

  The first moment is the

mean

The

k th

moment about the mean  is 

k



E

(

x

 

)

k

    

(

x

 

)

k f

(

x

)

dx

 The second moment about the mean is called the

variance

 ²

Kurtosis: a useful moments’ function   Kurtosis  4  

4 =



4 3



² 2

0 for a normal distribution so it is a measure of Normality

Observations   Observations x

i

are realizations random variable X of a The pdf of X can be visualized by a histogram : a graphics showing the frequency of observations in classes

Estimating moments  The Mean of X is estimated from a set of n observations

(x 1 , x 2 , ..x

n )

as 

x

 1

n i n

  1

x i

The variance is estimated by Var(X) =  2 

n

1  1

i n

  1

(

x i



x

)

2

The fundamental of statistics    Drawing conclusions about a population on the basis on a set of measurments or observations on a sample from that population Descriptive : get some conclusions based on some summary measures and graphics (Data Driven) Inferential : test hypotheses we have in mind befor collecting the data (Hypothesis driven).

What about having many variables?

   Let X=(X

1 , X 2 , ..X

p

) be a set of p variables What is the

marginal

distribution of each of the variables X

i

and what is their

joint

distribution If f(X

1 , X 2 , ..X

p

) is the joint pdf then the marginal pdf is

f

(

X i

)

 

f

(

x

1

,..,

x i

 1

,

x i

 1

...,

x p

)

dx

1

....

dx p

Independance  Variables are said to be independent if

f(X 1 , X 2 , ..X

p )= f(X 1 ) . f(X 2 )…. f(X p )

Covariance and correlation  Covariance is the joint first moment of two variables, that is

Cov(X,Y)=E(X-



X )(Y-



Y )=E(XY)-E(X)E(Y)

 Correlation: a standardized covariance 

(

X

,

Y

)



Cov

(

X

,

Y

)

Var

(

X

).

Var

(

Y

)

  is a number between -1 and +1

For example: a bivariate Normal  Two variables X and Y have a bivariate Normal if

f

(

x

,

y

)

 2  1  2 1 1   2

e

 1  1  2  

(

x

  1 

²

1

)²

 2 

(

x

  1

)(

 1  2

y

  2

)



(

y

  

²

2 2

)²

    is the correlation between X and Y

Uncorrelatedness and independence  If  =0 (Cov(X,Y)=0) we say that the variables are uncorrelated  Two uncorrelated variables are independent if and only if their joint distribution is bivariate Normal  Two independent variables are necessarily uncorrelated

Bivariate Normal

f

(

x

,

y

)  2  1  2 1 1   2

e

 1  1  2   (

x

  1  ² 1 )²  2  (

x

  1  )( 1 

y

2   2 )  (

y

  2  ² 2 )²    If 

=0

then

f

(

x

,

y

)  1 2  1 ²

e

 (

x

  1 )² ² 1    So f(x,y)=f(x).f(y) 1 2  2 ²

e

  (

y

  2 )² ² 2    the two variables are thus independent

Many variables  We can calculate the Covariance or correlation matrix of

(X 1 , X 2 , ..X

p )



C=Var(X)=

      

v

(

c(x x

1 1 )

c(x

1

,x

2

,x p c(x

1

,x

2

) ) v

(

x c(x

2 2

)

)

,x

........

........

c(x

2

p ) c(x

1

,x

..........

v ,x

(

p p x ) ) p

)         A square (pxp) and symmetric matrix

A Short Excursion into Matrix Algebra

What is a matrix?

Operations on matrices Transpose

Properties

Some important properties

Other particular operations

Eigenvalues and Eigenvectors

Singular value decomposition

Multivariate Data

Multivariate Data  Data for which each observation consists of values for more than one variables;  For example: each observation is a measure of the expression level of a gene i in a tissue j  Usually displayed as a data matrix

Biological profile data

The data matrix       

x

11

x x

21

n

1

x x

12 22

x n

2

....

x

....

x

1

p

....

x

2

p np

      

n

observations (rows) for

p

variables (columns) an nxp matrix

Contingency tables   When observations on two categorial variables are cross-classified.

Entries in each cell are the number of individuals with the correponding combination of variable values

Eyes colour Blue Medium Dark Light Fair 326 343 98 688 Red Hair colour Medium 38 84 241 909 48 116 403 584 Dark 110 412 681 188

Mutivariate data analysis

Exploratory Data Analysis   Data analysis that emphasizes the use of informal graphical procedures not based on prior assumptions about the structure of the data or on formal models for the data Data= smooth + rough is the underlying regularity or pattern in the data. The objective of EDA is to separate the smooth minimal use of formal mathematics or statistics methods where the from the smooth rough with

Reduce dimensionality without loosing much information

Overview on the techiques  Factor analysis  Principal components analysis  Correspondance analysis  Discriminant analysis  Cluster analysis

Factor analysis A procedure that postulates that the correlations between a set of p observed variables arise from the relationship of these variables to a small number k of underlying , unobservable,

latent variables

, usually known as common factors where

k