Stat 20: A guide to estimating regression parameters

Transcription

1 Stat 0: A guide to estimatig regressio parameters B. M. Bolstad, bolstad@stat.berkeley.edu November 4, 003 The goal of this documet is to outlie the steps that you should go through to estimate regressio parameters i this class. This mai text should be used i coectio with the flow diagram which gives you a decisio guide for the process of estimatig regressio coefficiets. 1 Writig the regressio model i the geeral form 1.1 Theory We ca write the geeral regressio model i the form µ (x i ) = β 0 g 0 (x i ) + β 1 g 1 (x i ) + + β p g p (x i ) (1) the β 0,..., β p are called the regressio parameters. They are ukow ad we will estimate them as part of the regressio process. The g 0 (x i ),..., g p (x i ) are called the basis fuctios. They represet geeral fuctios of the x data. I practice we will have a regressio model ad will be able to idetify the basis fuctios by lookig at the model. 1. Some examples Cosider x = x. The followig table shows some regressio models ad idetifies the basis fuctios Regressio Model Basis fuctios µ(x) = β 0 + β 1 x g 0 (x) = 1, g 1 (x) = x µ(x) = β 0 + β 1 x + β x g 0 (x) = 1, g 1 (x) = x, g (x) = x 1 µ(x) = β 0 + β 1 x x 1 µ(x) = β 0 + β 1 + β x si(x ) g 0 (x) = 1, g 1 (x) = 1, g x (x) = x g 0 (x) = 1, g 1 (x) = 1, g x (x) = si(x ) Rather tha just a sigle x value lets cosider x = (x 1, x, x 3 ). The followig table shows some regressio models ad idetifies the basis fuctios. Regressio Model Basis fuctios µ(x) = β 0 + β 1 x 1 g 0 (x) = 1, g 1 (x) = x 1 µ(x) = β 0 + β 1 x 1 + β x + β 3 x 3 g 0 (x) = 1, g 1 (x) = x 1, g (x) = x,, g 3 (x) = x 3 µ(x) = β 0 + β 1 x 1 + β x 1 x + β 3 cos(x 3 ) g 0 (x) = 1, g 1 (x) = x 1, g (x) = x 1 x,, g 3 (x) = cos(x 3 ) µ(x) = β 0 + β 1 x 1 + β x 1 x + β 3 x 1 x x 3 g 0 (x) = 1, g 1 (x) = x 1, g (x) = x 1 x,, g 3 (x) = x 1 x x 3 1

2 Checkig that the basis is orthogoal.1 Theory Give ay two basis fuctio g j (x), g k (x) we say that they are orthogoal if ad oly if g j (x i ) g k (x i ) = 0 () We say a set of basis fuctios is orthogoal if g j (x), g k (x) are orthogoal for all possible j,k with j k.. Some examples Cosider the basis fuctio 1, x. To check whether the basis is orthogoal you eed to check whether (1)(x i) = x i = 0 for your data set. For istace suppose that x = (1, 0, 1,, 0,, 3, 0, 3) the x i = = 0 so the basis 1 ad x is orthogoal for our data. If our basis fuctios were 1 ad x the we would eed to check that (1)(x i ) = x i = 0 assumig we had the same data the x i = = 8 which is ot equal to 0 so the fuctios 1 ad x are ot a orthogoal basis. Now for a more complicated example. Cosider the followig data x 1 x If our basis fuctios are 1, x 1, x the to check that the basis is orthogoal we check that (1)(x i1) = x i1 = = 0, (1)x i = x i = = 0 ad fially x i1x i = = 0. So the basis is orthogoal. Usig the same data suppose istead that the basis fuctios are 1, x 1, x 4. the to check if the basis is orthogoal we check that (1)(x i1) = x i1 = = 0, (1)(x i 4) = ( 1 4)+( 1 4)+( 1 4)+(0 4)+(0 4)+(0 4)+(1 4)+(1 4)+(1 4) = = 36 ad so 1 ad x are ot orthogoal therefore the basis is ot orthogoal.

3 3 Estimatig the parameters of a regressio model: The Least squares method 3.1 Theory The goal of the least squares method is to choose the values of β 0, β 1,..., β p which miimize the error sum of squares (SSE) (Y i β 0 g 0 (x i ) β 1 g 1 (x i ) β p g p (x i )) by differetiatig with respect to each of the β 0, β 1,..., β p givig us a system of p + 1 equatios each of which we set equal to zero. This set of p + 1 equatios is called the ormal equatios. Note that i geeral you ca show by differetiatig the SSE above with respect to each of β 0, β 1,..., β p that the geeral form for the ormal equatios is give by ˆβ 0 g j (x i ) g 0 (x i ) + + ˆβ p g j (x i ) g p (x i ) = g j (x i ) Y i for j = 0,..., p The least squares estimates ˆβ 0, ˆβ 1,..., ˆβ p are the foud by solvig the system of equatios. 3. Some examples The least squares method was used to derive formula for the regressio model µ(x) = β 0 + β 1 x, see the otes from Nov 10. Cosider the regressio model µ(x) = β 0 + β 1 x 1 + β x. The sum of squared errors for this model will be (y i β 0 β 1 x i1 β x i ) we wat the values of β 0, β 1 ad β which miimize this sum of squares. Call these values ˆβ 0, ˆβ 1 ad ˆβ. To fid these we must differetiate the sum of square errors with respect to β 0 ad with respect to β 1 ad with respect to β. This gives three equatios which we set equal to zero (these are called the ormal equatios) ad the solve for the ˆβ 0, ˆβ 1 ad ˆβ. So differetiatig the sum of squared error with respect to β 0 ad settig equal to zero we get (y i ˆβ 0 ˆβ 1 x i1 ˆβ ) x i ( 1) = 0 (we put the hat o each β because we set the equatio equal to 0) after simplifyig a little we get the first ormal equatio y i ˆβ 0 ˆβ 1 x i1 ˆβ x i = 0 Differetiatig the sum of squared error with respect to β 1 ad settig equal to zero we get (y i ˆβ 0 ˆβ 1 x i1 ˆβ ) x i ( x i1 ) = 0 3

4 after simplifyig a little we get the secod ormal equatio x i1 y i ˆβ 0 x i1 ˆβ 1 x i1 ˆβ x i1 x i = 0 Fially differetiatig with respect to β ad settig equal to zero we get (y i ˆβ 0 ˆβ 1 x i1 ˆβ ) x i ( x i ) = 0 after simplifyig a little we get the third ormal equatio x i y i ˆβ 0 x i ˆβ 1 x i1 x i ˆβ x i = 0 The three equatios ca the be solved for ˆβ 0, ˆβ 1 ad ˆβ. Rather tha just solvig these equatios ad gettig geeral formula s for this model we should istead evaluate the summary statistics usig the data, sice this will make maipulatig the equatios much easier. Suppose that we have the followig data: y x 1 x Ad so we see that = 9, y i = 184, x i1 = 0, x i = 1, x i1y i = 18, x iy i = 48, x i1x i = 0, x i1 = 6 ad x i = 7. Thus, the ormal equatios become ˆβ 0 ˆβ = ˆβ 1 = 0 48 ˆβ 0 7 ˆβ = 0 Now we just solve the three equatios for ˆβ 0, ˆβ1 ad ˆβ. The secod equatio gives ˆβ 1 = 18/6 = 3. From the first equatio we get ˆβ = ˆβ 0. Substitutig this ito the third equatio we get ad so we get 48 ˆβ 0 7(184 9 ˆβ 0 ) = ˆβ 0 = 0 ad so ˆβ 0 = 140/6 = 0 ad ˆβ = 184 9(0) = 4. 4

5 3.3 The special case of the model µ (x) = β 0 + β 1 x For the special of the simple liear regressio model µ (x) = β 0 + β 1 x we showed (see 10th Nov), usig the least squares approach that for this particular model we could estimate ˆβ 0 ad ˆβ 1 usig ˆβ 0 = ȳ ˆβ 1 x (3) ad ˆβ 1 = x iy i ȳ x x i x Alteratively, we showed o Nov 1 that formula give by your textbook ˆβ 1 = r s y s x (4) could be used i place of the formula above (ad that the two agreed completely). Note that i this case r is the correlatio, s y is the sample stadard deviatio of the y data ad s x is the sample stadard deviatio of the x data. 4 Estimatig the parameters of the regressio model if the basis is orthogoal 4.1 Theory If all the basis fuctios g 0 (x),..., g p (x) are orthogoal, the we ca use the followig formula to estimate the regressio parameters β 0,..., β p ˆβ j = g j (x i ) y i g j (x i) for j = 0,..., p 4. Some examples Suppose that we have used our data ad via the methods discussed i sectio have show that the basis 1, x, x 5 is orthogoal for our data. The i this case whe we fit the regressio model µ (x) = β 0 + β 1 x + β (x 5) we get the followig estimates parameter estimates: ˆβ 0 = (1)y i (1) = y i = ȳ ˆβ 1 = x iy i x i ˆβ = x i y i (x i ) Cosider a differet dataset, where we show that 1, x 1, x ad x 1 x is a orthogoal basis usig the methods of sectio. The whe it comes to fit the regressio model µ (x 1, x ) = β 0 +β 1 x 1 +β x +β 3 x 1 x we would get the followig parameter estimates ˆβ 0 = (1)y i (1) = y i = ȳ 5

6 ˆβ 1 = x i1y i x i1 ˆβ = x iy i x i ˆβ 3 = x i1x i y i (x i1x i ) 6