Nonlinear regression

Transcription

1 oliear regressio

2 How to aalyse data?

3 How to aalyse data? Plot!

4 How to aalyse data? Plot! Huma brai is oe the most powerfull computatioall tools Works differetly tha a computer

5 What if data have o liear correlatio?

6 1. Liearizatio trasform oliear problem ito liear Example y Be Ax logy logb + Ax Y b + ax

7 Few word about R I the case of liear regressio the r coeff. Idicates the rate of liear depedecy betwe data. However there is more geeral approach

8 Few word about R S r y i f x i 2 Error of the model S t y i y 2 Discrepacy betwee data ad sigle estimate (mea)

9 Few word about R y y i s y S t 1

10 Few word about R S r y i f x i 2 Error of the model S t y i y 2 Discrepacy betwee data ad sigle estimate (mea)

11 Few word about R sy x S r 2 Stadard error of the estimate Spread aroud the lie

12 Few word about R sy x S r 2 Stadard error of the estimate Spread aroud the lie

13 Few word about R r 2 S t S r S t Error reductio due to describig data i terms of a model (straight lie) Scale depedet - ormalizatio

14 y 0.5 x + 3, r Ascombe example

15 2. Polyomial f x a 0 + a 1 x + a 2 x 2 + a 3 x 3 Same approach as i the case of liear regressio least squares

16 2. Polyomial f x a 0 + a 1 x + a 2 x 2 + a 3 x 3 e i y i f x i y i a 0 a 1 x i a 2 x i 2 e i y i f x i y i 0 a x i

17 2. Polyomial f x a 0 + a 1 x + a 2 x 2 + a 3 x 3 e i 2 y i a x i 2 0 SSE a 0, a 1, e i 2 y i a x i 2 0

18 How to adust a ad b so SSE is the smallest? SSE(a, b) y i ax i b 2 How to calculate miimum of the SSE(a,b) fuctio? SSE a, b a 0 SSE a, b b 0

19 How to adust a ad b so SSE is the smallest? 2 SSE a 0, a 1, e i 2 y i a x i 0 SSE a 0, a 1,.. y i a x i 0 2 y i a x i y i a x i y i a x i 0 SSE a 0, a 1,.. y i a x i 0 2 y i a x i y i a x i 0 x i 2 x i y i a x i 0 SSE a 0, a 1,.. a 2 a 2 y i a x i 0 2 a 2 y i a x i y i a x i 0 x i x i y i a x i 0

20 How to adust a ad b so SSE is the smallest? 2 SSE a 0, a 1, e i 2 y i a x i 0 SSE a 0, a 1,.. a k a k y i a x i 0 2 a k y i a x i y i a x i 0 x i k 2 k x i y i a x i 0 SSE a 0, a 1,.. a k 2 x k i y i 0 a x i

21 How to adust a ad b so SSE is the smallest? We obtai set of +1 liear equatios SSE a 0, a 1,.. a k 2 x k i y i 0 a x i 0 x i k y i a x i 0 0 x i k y i 0 a x i +k 0 0 a x i +k x i k y i

22 How to adust a ad b so SSE is the smallest? We obtai set of +1 liear equatios SSE a 0, a 1,.. a k 2x i k y i 0 a x i 0 x i k y i a x i 0 0 x i k y i 0 a x i +k 0 0 a x i +k x i k y i

23 How to adust a ad b so SSE is the smallest? Row 1, k 0 0 a x i +k x i k y i a 0 + a 1 x i + a 2 x i a x i y i Row 2, k 1 a 0 x i + a 1 x i 2 + a 2 x i a x i +1 x i y i Row +1, k a 0 x i + a 1 x i +1 + a 2 x i a x i + x i y i

24 How solve it? Liear equatio - matrix Row 1, k 0 x i x i y i x i x i 2 x i x i x i x i 2 a 0 a 1 a x i y i x i y i

25 oliear regressio 2 ormal equatios

26 Liear regressio differet approach y 1 ax 1 + b y 2 ax 2 + b y ax + b y 1 y 2 y x 1 x 2 x a b y Az

27 Liear regressio differet approach y 1 y 2 y x 1 x 2 x a b y Az This system caot be solved. System is over coditioed to may equatios. A is ot a square matrix caot be iverted. Solutio? Let s make it squared matrix!

28 Liear regressio differet approach Solutio? Let s make it squared matrix! A T A C square matrix x 1 x 2 x x 1 x 2 x x i x i x i

29 Liear regressio differet approach Solutio? Let s make it square matrix! y Az y Az A T A T y A T Az x 1 x 2 x y 1 y 2 y x i y i y i

30 Example X W (kg) Y L (m) y ax b ly bx + la A Y Ax + B Y l l 1. 1 l l l 1. 4

31 Example X W (kg) Y L (m) y ax b Y Ax + B

32 Example X W (kg) Y L (m) y ax b Y Ax + B A B A B y x

33 Summig up Liear regressio problem ca be formulated as: Havig set of approximatio of every y i as liear fuctio of x i y i f x i + e i where f x ax + b we look for such parameters a, b such that SSE is the smallest. Solutio for this is: where z AA T 1 A T y A x 1 x 2 x z a b y y 1 y 2 y

34 Example Fit fuctio f x, a 0, a 1 a 0 1 e a 1x to Partial derivatives off SSE with respect to a 0 ad a 1 are: SSE y i a 0 1 e a 1x i 1 e a 1x i 0 We obtai set of oliear equatios SSE a 0, a 1 Oe way is to solve them o error cotrol. y i a 0 1 e a 1x SSE 2 y i a 0 1 e a 1x i a 0 xe a 1x i 0 Aother Gauss-ewto iterative techique.

35 Example oliear regressio is characterized by the fact that the predictio equatio depeds oliearly o oe or more ukow parameters. Some oliear regressio problems ca be liearized by a suitable trasformatio of the model formulatio. However, use of a oliear trasformatio (that is, liearizatio) requires cautio. The iflueces of the data values will chage, as will the error structure of the model ad the iterpretatio of ay iferetial results. These may ot be desired effects. O the other had, depedig o what the largest source of error is, a oliear trasformatio may distribute your errors i a ormal fashio, so the choice to perform a oliear trasformatio must be iformed by modelig cosideratios. The accuracy obtaiable with liearizatio trasformatios is ot as good as that obtaiable with the iterative techiques applied directly to the oliear problem. However, eve whe very high accuracy is required, liearizatio trasformatios are of value i providig good iitial estimates for the iterative techiques used with oliear regressio. (Ref: W. G. Dotso, Jr., Liearizatio Trasformatios for Least Squares Problems, Mathematics Magazie, Vol. 39, o. 3 (May, 1966).

36 Iterative techique Gauss-ewto method Problem: Havig set of data x i, y i ad fuctio f(x, a 0, a 1, a 2, ) fit: y i f x i, a 0, a 1, a 2, + e i for every i 1,, So the sum of radom errors e i squared is the smallest. Vector otatio: y f x, a + e

37 Iterative techique Gauss-ewto method To illustrate the process we use the case where there are two parameters a 0 ad a 1. The the trucated Taylor expressio that defies the terms of the vectors used for values of the model fuctio f i the steps of the iterative process have the form: f x i +1 f x i + f x i Δa 0 + f x i Δa 1 for every i 1,, We dot travel i x rather i a we look for better estimatio of a. The values of Δa 0 ad Δa 1 are to be determied from the least squares computatio at each step ad represet the icremets added to the latest estimates of the parameters to geerate the ext parameter estimates. This expressio is said to liearize the origial model with respect to the parameters.

38 How to solve truly o-liear problem y i f x i + e i f x i +1 f x i + f x i Δa 0 + f x i Δa 1 y i f x i + f x i Δa 0 + f x i Δa 1 + e i f x 1 f x 1 y 1 f x 1 y 2 f(x 2 ) y f x f x 2 f x f x 2 f x Δa 0 Δa 1 + e 1 e 2 e

39 How to solve truly o-liear problem f x 1 f x 1 y 1 f x 1 y 2 f(x 2 ) y f x f x 2 f x f x 2 f x Δa 0 Δa 1 + e 1 e 2 e ow we drop the radom error terms to obtai a over determied system to which we apply least square computatioal strategy to determie the ormal system. f x 1 f x 1 y 1 f x 1 y 2 f(x 2 ) y f x f x 2 f x f x 2 f x Δa 0 Δa 1

40 How to solve truly o-liear problem f x 1 f x 1 y 1 f x 1 y 2 f(x 2 ) y f x f x 2 f x f x 2 f x Δa 0 Δa 1 f x 1 f x 1 b y 1 f x 1 y 2 f(x 2 ) y f x J f x 2 f x f x 2 f x z Δa 0 Δa 1

41 How to solve truly o-liear problem f x 1 f x 1 y 1 f x 1 y 2 f(x 2 ) y f x f x 2 f x f x 2 f x Δa 0 Δa 1 b Jz b Jz J T J T b J T Jz J T

42 How to solve truly o-liear problem f x 1 f x 1 y 1 f x 1 y 2 f(x 2 ) y f x f x 2 f x f x 2 f x Δa 0 Δa 1 J T b J T Jz J T z J T J 1 J T b Δa 0 Δa 1

43 How to solve truly o-liear problem The etries of vector z Δa 0 Δa 1 are used to update the values of the parameters: a 0,+1 a 0, + Δa 0 a 1,+1 a 1, + Δa 1 f x 1 f x 1 y 1 f x 1 y 2 f(x 2 ) y f x f x 2 f x f x 2 f x Δa 0 Δa 1

44 How to solve truly o-liear problem The iterative procedure cotiues ad we test for covergece usig a approximate relative error test a 0,+1 a 0, a 0,+1 < tolerace a 1,+1 a 1, a 1,+1 < tolerace

45 Example Fit fuctio f x, a 0, a 1 a 0 1 e a 1x to x y

46 Example Fit fuctio f x, a 0, a 1 a 0 1 e a 1x to We use as iitial guesses a ad a Partial derivatives with respect to a 0 ad a 1 are: f 1 e a 1x f a 0 xe a 1x ext we evaluate etries of matrix J: f x 1 f x 1 J f x 2 f x f x 2 f x

47 Example The we compute etries of b matrix: b y 1 f x 1 y 2 f(x 2 ) y f x Usig ormal system equatio z J T J 1 J T b So ew set parameters a 0 ad a 1 are: z Δa 0 Δa a 0 a

48 Problems 1. It may coverge slowly. 2. It may oscillate ad cotiually chage directio. 3. It may ot coverge.

49 More about ormal Distributio How ca we measure the similarity of give distributio to (0,1)? Media, mode Stadard ormal curve 0.5 Mea x xi 0 Media Mea Mode x Std. deviatio σ i μ Z distributio μ 0 σ 1 Skewess s1 Kurtosis k 1 x μ x μ 4 1 x μ x μ

50 More about ormal Distributio How ca we measure the similarity of give distributio to (0,1)? Media, mode Mea x xi 0 Media Mea Mode x Std. deviatio σ i μ 2 2 Skewess s1 Kurtosis k 1 x μ x μ 4 1 x μ x μ