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1 Polyomial Regressio: Problem: Give 1 pairs of data poits, y i, i 0,1,,, fid a polyomial P x x a x where such that the error fuctio E,,,a y i P i0 is miimized Note that the fuctio E,,,a expresses the sum of error squares of P x at all for i 0,1,, This method is also called the Least-Squares Method ad the obtaied solutio is called the least-squares solutio of this problem As a compariso, recall that the th degree iterpolatig polyomial we studied i Sectios 1 ad satisfies the coditios: P y i for i 0,1,, 1 Liear Regressio: 1adP 1 x x E, y i i0 Note that E, is a cocave up quadratic fuctio i ad SoE, attais its miimum value at the critical poit a 0,a 1 which is the solutio of the followig system of two equatios: Rewrite the equatios as follows: E E i0 y i 1 0 i0 y i 0 y 0 y 1 y 1 x 1 x 0 y 0 y 1 x 1 y x x 1 x x 0 x 1 x 0 which are equivalet to the followig equatios: y 0 y 1 y 1 x 1 x y 0 y 1 x 1 y x x 1 x x 0 x 1 x Express above system i matrix-vector otatio: 1 x 1 x y 0 y 1 y (*) x 1 x x 0 x 1 x y 0 y 1 x 1 y x ad the solutio is of the form: Note that (**) 1 x 1 x x 1 x x 1 x 1 y 0 y 1 y y 0 y 1 x 1 y x 1 1 x 1 x x 1 x x 1 x x 1 x 1 x 1 ad 1 1 x
2 y 0 y 0 y 1 y y 0 y 1 x 1 y x x 1 x y 1 y Defie 1 y 0 A 1 x 1, y y 1 ad a 1 x The the system of equatios i (*) is equivalet to the form: A T Aa A T y ad the solutio i (**) is equivalet to the form: (***) a A T A 1 A T y The miimum approximatio error for the liear regressio: E mi, y i i0 Steps to compute the vector a i (***): (1) Compute c 1, c y x i, b 1 y i ad b y i () Compute A T A 1 A T y 1 1 c c c 1 c 1 1 b 1 b Example Let fx x 1 Give 0,1,,,8, Fid P 1 x x Solutio: (1) c 1 i0 y i 1, i , c x i 0 9 7, i y i 0 0 i , P 1 x x E mi,
3 y y x 1, y P 1 x,,y i x Noliear Regressio: P x x a x,ade,,a Cosider P x x a x,ade,,a ad A T A x 1 x x 1 x 1 1 x 1 x 1 1 x x 1 A T y x 1 x x 1 x y 0 y 1 y y i y i x i y i A T A A T y or a a A T 1 A A T y I geeral, cosider P x x a x,ade,,a,,a
4 A T A x 1 x x 1 x x 1 x 1 1 x 1 x 1 x 1 1 x x x ad A T y x 1 x x 1 x y 0 y 1 y y i y i x i y i A T A A T y or A T A 1 A T y a a Example Let y i si, where MatLab program: lect ex1m 0 i, i 0,,10
5 clear format log xv0:pi/0:pi/; yvsi(xv); plot(xv,yv, r* ) hold o plus1legth(xv); a11plus1; a1sum(xv); asum(xv^); A1[a11 a1;a1 a]; b1sum(yv); bsum(xv*yv); b[b1;b]; aiv(a1)*b; Pa(1)a()*xv; err1sum((yv-p)^); disp( least square error ),err1 plot(xv,p, - ) asum(xv^); asum(xv^); A[A1 [a;a];a a a]; bsum(xv^*yv); b(,1)b; aiv(a)*b; Pa(1)a()*xva()*xv^; errsum((yv-p)^); disp( least square error ),err plot(xv,p, b ) hold off title( * ysi(x), 0x\pi/, - yp_1(x), yp_(x) ) I MatLab: lect ex1 least square error err least square error err E mi, E mi,,a * y=si(x), 0<=x<=π/, - y=p 1 (x), -- y=p (x) Trasformatio to Liear: a Let y i ab, i 0,1,, Values of a ad b ca be idetified as follows: (1) ly i la lb,thatis,ly is liear i x: ly x We ca compute ad by the liear regressio usig,ly i () Use obtaied ad to idetify a ad b as follows: a e, b e b Let y i a r, i 0,1,, Values of a ad r ca be idetified as follows: (1) ly i la rl thatis,ly is liear i lx: ly lx We ca compute ad by the liear regressio usig l,ly i () Use obtaied ad to idetify a ad b as follows:
6 a e, r a1 Example Let y ab x Suppose we ow: 0,, 1,,,1,, Idetify a ad b l ly l l1, x 1, A 1 1 1, A T A 1, A T ly l a e adb e Example Let y ax r Compute the liear regressio of the data: 1,,,,,8 Idetify a ad r l l1 1 l1 ly l,lx l, A 1 l, l8 l 1 l A T A, A T ly a e , r Example Oe of the followig data sets follows a expoetial law ad the other follows a power law Which is which? x y y MatLab program: lect exm:
7 clear format log xv[;;;;;;]; yv1[179;77;709;707;10999;110;19]; yv[11;198;19;097;81;19;18]; clf subplot(1),plot(xv,log(yv1)) title( log(yv1)a_0a_1 xv1 ) subplot(),plot(log(xv),log(yv1)); title( log(yv1)a_0a_1 log(xv1) ) subplot(),plot(xv,log(yv)) title( log(yv)a_0a_1 xv ) subplot(),plot(log(xv),log(yv)); title( log(yv)a_0a_1 log(xv) ) log(yv1)= + xv1 log(yv1)= + log(xv1) lect exm y 1 follows the power law ad log(yv)= + xv log(yv)= + log(xv) y follows the expoetial law Exercises: 1 Cosider the followig data relatig the amout of varish additive ad the resultig varish dryig time: Additive (grams) Dryig time (hours) a Fid the liear regressio: P 1 x x ad its approximatio error b Fid the quadratic regressio: P x x a x ad its approximatio error Let y i 1 cos 1 si where 0 i, i 0,1,,10 Fid the (1) liear ad () quadratic least squares approximatios ad their approximatio errors 7
8 Cosider the followig data: x y y a Oe of data sets follows a expoetial law ad the other follows a power law Which is which? b Fida,b or a,r Cosider the followig data: x y y a Oe of data sets follows a expoetial law ad the other follows a power law Which is which? b Fida,b or a,r 8
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