Functional Data Analysis T Chapters 10,11,12 Markus Kuusisto

Topics 10 PCA of mixed data 11 Canonical Correlation Analysis 12 Functional linear models

PCA of mixed data Both: functional part and vector part (x i,y i ) Canadian temperature: Registeration process finds suitable shift. - Vector part is size of shift - Functional part is shifted curve

Canadian temperature

Canadian temperature (shifted)

Using PCA, vector part y i y i are nuisance parameters -> we ignore y i are marginal importance -> we ignore them when calculating PCA, but afterwards we investigate connections between PCA scores and y i y i are primary importance with functions x i -> we treat them as a hybrid data (x i,y i )

The PCA of hybrid data PCA weight function ( ,v) PCA score of particular observation:  i = x i (s)  (s) ds + y’ i v inner product: z i = (x i, y i )  z 1, z 2  = x 1 x 2 + y’ 1 y 2 To find leading principal component maximize sample variance of the  ( ,v), z i  when ||( ,v)|| = 1

Balance between functional and vector variation Measure units between functional and vector parts usually are not comparable  z 1, z 2  = x 1 x 2 + C 2 y’ 1 y 2 Choice of C 2 C 2 = |T |, where T is interval of function x i C 2 = |T | / M, where M is length of y C 2 = Var(x) / Var(y)

Incorporating smoothing Roughness of z = (x, y ) - D 2 z = (D 2 x, 0) - || D 2 z || 2 = || D 2 x || 2 Calculating like in chapter 9

Canonical Correlation Analysis CCA is a way of measuring the linear relationship between two multidimensional variables Ordinary correlation analysis is dependent on the coordinate system in wich variables are described CCA finds the coordinate system where the correlation is maximized

Definition of CCA Consider the linear combination x = x T w x y = y T w y Function to be maximized is The maximum of  with respect to w x and w y is maximum canonical correlation The number of solutions are limited to the smallest dimensionality of x and y

Car marks example

CCA of car marks Correlation r 1 = r 2 = w x1 = w x2 = Price Value w y1 = w y2 = Economy Service Design Sport Safety Easy h

(x T w x2, y T w y2 )

Predicting by CCA

Learning w x corresponds output (x) w y corresponds 52 previous datapoints (y) Learning - Finding maximum canonical correlation and its weights w x, w y - Linear line fitting Predicting output x is done by projecting y = y T w y to fitted line.

Predicting recursively next 50 data points

Functional canonical correlation analysis Function to be maximized subject to constraints It is possible allways to find perfect correlation Maximization does not produce a meaningfult result

Unsmoothed canonical variate weight function that attain perfect correlation. A standard condition for classical CCA n > p + q + 1, - n is number of samples - p is length of x i and q is lenght of y i In functional case p and q are infinite, no unique solution Overfitting

Smoothing Smoothing is essential Choice of can be done –subjectively –by leave one out cross validation, maximazing squared correlation. (11.3.3) ccorsq calculated as above but with the observation (X i,Y i ) omited

Smoothed canonical weight functions

Functional linear models Previous we have been exploring the variability of a functional variables Now we explore how much of variation is explained by other variables In calssical statistics we do that by linear regression and the general linear models. Now functional linear models

Precipitation example Preciptitation (= total rainfall) of particular area where i indexes the 35 weather stations Does the precipitation depend on temperature of that area Overfitting without smoothing

A Functional response and a functional independent variable How does a precipitation profile depend on the associated temperature profile ? Concurrent: Precipitation now depends only on the temperature now Annual: Precipitation now depend on the temperature of the whole year

Short-term feed-forward: For reasons of parsimony, precipitation now depends on the temperature over an interval back in time. Local influence: Precipitation now depends on the temperature over an interval back in time and the season (is it summer or winter ?)

Predicting derivatives Dynamic model: Model is designed to explain a derivative of some order –homogenous first order linear differential equation –- nonhomogenous temperature in the equation is called forcing function

References Book: Functional Data Analysis, J.O.Ramsay, B.W.Silverman – matlab toolbox for FDA –Classical Canonical Correlation Analysis –Method about solving blind source separation problem based on CCA –Matlab functions: cca.m and ccabss.m –Car marks example –You may get confused because results presented here differs from the site above. Reason is that in that site the first and second canonical correlations are changed places. –Data of example ”Predicting by CCA”