Lecture 4: Principal components

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1 /3/6 Lecture 4: Prncpal components 3..6 Multvarate lnear regresson MLR s optmal for the estmaton data...but poor for handlng collnear data Covarance matrx s not nvertble (large condton number) Robustness problems Numercally large coeffcents x x

2 /3/6 Enhanced regresson model Data s mapped frst onto a subspace and only from there onto the output Data mapped from nput to output through a latent bass: Note lnearty: Y XF F X F F XF. he orgnal one-level model structure s stll vald 3 General modelng procedure General procedure to be followed when more sophstcated regresson methods are appled: 4

3 /3/6 Why the trouble? When the hdden bass has lower dmenson, some nformaton s lost n the mappng he goal s to defne the hdden bass so that the real nformaton s preserved, whle the nose s fltered out How to select the hdden subspace = bass? Expert nowledge s needed: What s the nature of nformaton and nose n that specfc case? Hypothess : data varance carres nformaton. hs s the assumpton underlyng Prncpal Component Analyss (PCA) 5 Varance maxmzaton he goal now s to select the new bass so that varance of the proected nput data s maxmzed gven the latent bass dmenson Assumpton: (co)varaton carres nformaton, whereas nose s equally dstrbuted to all varables Search for the frst new bass vector Varance of the proected data: Z X Z Z X X 6 3

4 /3/6 Varance maxmzaton Possble drectons: x θ θ θ 3 θ θ x θ 3 7 Varance maxmzaton he length of can be fxed and we defne a constrant so that = Constraned maxmzaton problem: f X X g Solved usng the lagrange multplers: dj d f g d d Lmt the vector length to one X X X X Soluton s an egenvector of the data covarance 8 4

5 /3/6 Egenvalue problem Assume now that n and he varance of the data proected to n Z Z X X n n => maxmum s reached when I n 9 Remanng varance he drecton of largest varance n X s found to be (.e. the most mportant egenvector of X X ) How about the remanng varance? Elmnate the largest varance drecton: * X X Z Now search for the drecton of largest varance n X that s orthogonal to : * * Z Z X X X Z X Z * 5

6 /3/6 Remanng varance Z Z X Z X Z X Z X Z X X X X X Z X Z => he largest remanng varance drecton s the second egenvector of X X,.e. and so on... Prncpal components he egenvectors of the data n covarance matrx are called prncpal components or loadngs he proected data ponts z,..., zn are called scores Covarance of the latent varables: Z X Z Z X X n => the latent varables are mutually uncorrelated (because ther covarance matrx s dagonal) 6

7 /3/6 Illustraton n D Orgnal bass Prncpal component bass x x x x x x 3 Varance of the scores he orgnal varance s preserved 4 7

8 /3/6 Example redstrbuted varance Data contans 7 varables and samples value - x x x3 x4 x5 x6 x # of sample 5 Example redstrbuted varance Wthout scalng orgnal data scores 3 4 varance varance 3 x x x3 x4 x5 x6 x7 varable z z z3 z4 z5 z6 z7 varable 6 8

9 /3/6 Example redstrbuted varance Wth scalng (.e. varances normalzed to ) orgnal data scores varance.5 varance 3 x x x3 x4 x5 x6 x7 varable z z z3 z4 z5 z6 z7 varable 7 Informaton vs. nose Informaton s assumed to be drectonal, whereas the nose s undrectonal (.e. equally present n all drectons) 8 9

10 /3/6 Prncpal component analyss (PCA) Dates bac to 9: Pearson, K: On Lnes and Planes of Closest Ft to Systems of Ponts n Space Also nown as the Karhunen-Loève decomposton Basc steps of PCA Preprocess the data Calculate the prncpal component drectons Calculate the scores and score varances Leave out the (hopefully) nsgnfcant prncpal components Analyze the rest of the loadngs and scores 9 How to utlze PCA results he loadngs descrbe the man dependences (covarances) between the orgnal varables he score data represents a low-dmensonal verson of the orgnal data Data clusterng Regresson Vsualzaton

11 /3/6 Relaton of PCA and SVD Data X, covarance R X X Egenvalue decomposton of the covarance: R Sngular value decomposton of the data: X Now: R X X n I n I n =>,, Z n Relaton of PCA and SVD If rows equal to samples: R X scores loadngs But on the other hand: If columns equal to samples: R X loadngs scores X X XX n Columns equal to varables = assumpton used n ths course Rows equal to varables

12 /3/6 PCA another vew PCA s a method of wrtng a matrx X of ran d as a sum of d matrces of ran : X Z Z Zdd n n n d terms 3 Nonlnear teratve partal least squares (NIPALS) An algorthm to teratvely calculate PCA (and PLS) Developed by Herman Wold H. Wold: Estmaton of prncpal components and related models by teratve least squares, n Multvarate Analyss (Ed., P.R. Krshnaah), Academc Press, NY, 966, pp he prncpal components are calculated one by one, startng from the man components Sometmes useful for large data sets 4

13 /3/6 NIPALS. Intalze the :th score Z X Z X. Calculate Z Z 3. Normalze to length : 4. Calculate Z X Random column of X X s proected on the score vector X s proected on the loadng vector 5. Return to step. untl has converged. Z 6. o calculate, deflate the data: X X Z and return to step. 5 Interpretaton of NIPALS Results equal the PCA obtaned from the egenproblem (sngular value decomposton): Loadngs Z X X X X X X X c Z Z c c Scores X Z XX Z Z Z c Z X XX Z cz 6 3

14 /3/6 Prncpal component regresson (PCR) model a response varable when there are a large number of predctor varables, and those predctors are hghly correlated or even collnear. construct new predctor varables, nown as components, as lnear combnatons of the orgnal predctor varables PCR creates components to explan the observed varablty n the predctor varables, wthout consderng the response varable at all. 7 Prncpal component regresson (PCR) Collnearty problem solved by frst proectng the data to the PCA subspace => score varables mutually uncorrelated => Z Z s well-behavng and nvertble Result: F F F Z Z Z Y X X X Y => PCR X X X Y X Y FPCR N X Y Compare to MLR: FMLR X X X Y X8 Y 4

15 /3/6 Based on covarances Agan, the PCR model can be calculated (f necessary) usng only the data covarance matrces: => Rxx X X PCR Rxy X Y xx F X X X Y R R xy 9 Example PCR Data: X, Y 5 Rxx = Rxy = Collnearty 3 5

16 /3/6 Example PCR PCA of X:.5 Latent varable varances PCA loadngs (N=3) varance.5 - x x x3 x4 x5.5 z z z3 z4 z5 varable - x x x3 x4 x5 3 - x x x3 x4 x5 3 Example PCR Covarance of the latent varables (wth N=3): Rzz = PCR mappng: Fpcr = theta*nv(z'*z)*z'*y; Fpcr =

17 /3/6 Example PCR Modelng result (for estmaton data) Yhat = X*Fpcr; y y 5 5 Predcted -5 Predcted Measured Measured 33 Sgnal vs. nose max N a a B A a C ad A B N C D n 34 7

18 /3/6 New problems Problems of MLR are solved Clever method to deal wth collnearty No essental nformaton s lost (hopefully)...but new ones emerge: Output s not taen care of n the constructon of the latent bass: he nput/output behavor can stll be very bad! 35 Summary Prncpal component analyss Covarance carres nformaton Optmal way to reduce the data dmenson Data descrbed as loadngs and scores Calculaton Egenvalues of the data covarance or Sngular value decomposton of the data or Iteratvely usng NIPALS Prncpal component regresson Clever way to deal wth collnear nput varables 36 8

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