ESTIMATING THE REGRESSION LINE

read upc - 13 for javaUsing Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications.

tion, it turns out that if we let the conditional distribution of (a, {3) for each given value of a be a bivariate normal distribution and let the distribution of 1/a 2 be a gamma distribution, then this constitutes a conjugate prior distribution for the parameter vector (a, {3, a). By using a conjugate prior distribution we will be able to obtain explicit formulas for the expectations of a and {3 under the posterior distribution.

EAN-13 Supplement 5 printing with javagenerate, create ean13 none for java projects

Quantifying Prior Information. The decision to use a prior distribution from the family of conjugate distributions is not based on prior information but simply on convenience. Our prior information about the parameters enters the procedure when we select a particular conjugate distribution. For instance, we might quantify our prior information by specifying the expectations, standard deviations and correlation of a and {3 given a and the expectation and standard deviation of 1/a 2 If we are only interested in calculating estimates of a and {3, as in this section, it is not necessary to specify the expectation and standard deviation of 1/a 2 Rather than translate our prior information into a distribution for a and {3, it may be easier, and is equivalent, to translate it into a prior distribution for JL and {3, where JL = a + {3xm and xm is a "middle" x-value. Note that JL is the height of the regression line near the middle of the data. It would often be reasonable to regard our prior information about JL to be independent of our prior information about {3, the slope of the regression line. One case in which this assumption of independence is especially suitable is when our prior information comes from a previous least-squares analysis of similar data. In a normal simple linear regression model, if x m is the average x-value, then ilLS and ffiLS are independent.

EAN 13 decoder in javaUsing Barcode reader for Java Control to read, scan read, scan image in Java applications.

Bayes Estimates. Let JL = a + {3x; that is, take xm to be the average x-value in the current data set. Suppose that, based on previous information, we specify a conjugate prior distribution in which, conditional on a given value of a, the expectations of JL and {3 are elL and ef3 and their standard deviations are clLa and cf3a. Let the prior distributions of JL and {3 be independent. The Bayes estimates of a and {3 are given by the expectations of a and {3 under the posterior distribution:

Barcode barcode library in javausing barcode generator for java control to generate, create barcode image in java applications.

where

Bar Code barcode library on javaUsing Barcode decoder for Java Control to read, scan read, scan image in Java applications.

il - ffix C;;2 2)ef3 +( l:(X;-X)22)ffiLS C;;2+l:(X;-x) C;;2+l:(X;-x)

Paint ean 13 on .net c#using visual studio .net todraw ean13 on asp.net web,windows application

(7.3)

Attach ean 13 with .netgenerate, create ean-13 supplement 5 none on .net projects

ffi=(

Build ean-13 supplement 5 on .netusing barcode integrated for .net vs 2010 control to generate, create ean13 image in .net vs 2010 applications.

il = (

c; ~

Use pdf417 with javagenerate, create pdf417 2d barcode none with java projects

n ) elL

Bar Code barcode library in javause java barcode creation toaccess bar code with java

{c;2n+ n }

Java code-39 creationin javagenerate, create 39 barcode none in java projects

ffiLS

4-State Customer Barcode barcode library in javausing barcode implementation for java control to generate, create usps onecode solution barcode image in java applications.

is the least-squares estimate calculated from the current data.

Control code128 image in office wordusing office word tomake code 128 code set b on asp.net web,windows application

BA YESIAN REGRESSION

Control ucc ean 128 size with vb.netto print ean / ucc - 14 and uss-128 data, size, image with vb.net barcode sdk

Reasonableness of Formulas (7.3). These formulas show how past knowledge and current data are combined. The Bayes estimate of {3 is a weighted average of e(3' the prior expected value of {3 based on past knowledge, and ~LS' the least-squares estimate of {3 based on the current data. The weights are sensible in so far as more weight is put on the current data when there are more data, so that L:(x; - i)2 is larger, or when the prior knowledge is more imprecise, that is, c(3 is larger. The Bayes estimate of J.L is a similar weighted average of the prior expected value based on past knowledge and the least-squares estimate based on the current data. (Note that ji = ~ LSJ The Rainfall Data. Results are available from a previous regression analysis of annual rainfall in Seattle and Portland for the years 1950 through 1979. Let the subscript "0" identify these previous results. The numbers we need for our prior distribution are a o = 5.513, ~o = 0.8961, SD(a o) = 1.140u, SD(~o) = 0.02986u, and the correlation Corr(ao, ~o) = - 0.9871. (These numbers do not completely specify the prior distribution, but they are all we need for the purpose of estimating a and {3.) In order to use formulas (7.3), we let J.L = a + 35.14{3, where 35.14 is the average rainfall in Portland for the current data. Then ~o = 5.513 + (35.14)(0.8961) = 37.00 and SD(~o) = 0.1977u. The number 0.1977 is the square root of (1.140)2 + (35.14)2(0.02986)2 + 2(35.14X1.140XO.02986X -0.9871). To specify a prior distribution for the parameters it makes sense to let J.L have expectation 37.00 and standard deviation 0.1977u (conditional on u) and let {3 have expectation 0.8961 and standard deviation 0.02986u. In order to illustrate the use of (7.3) we suppose that the prior distributions of J.L and {3 are independent. Note, however, that the estimates ~o and ~o are not exactly independent because the average x-value in the previous data is not exactly 35.14. Next, from the current data, we calculate ji = 35.82, ~LS = 0.5063, and L(x; - i)2 = 497.2. Now we can use formulas (7.3) to obtain the Bayes estimates of a and {3. We calculate

Deploy bar code in .netgenerate, create bar code none in .net projects

(0.02986) -2/[ (0.02986) -2

RDLC Reports 2d matrix barcode encodingfor .netusing barcode integrating for rdlc report control to generate, create 2d matrix barcode image in rdlc report applications.

+ 497.2] = 0.6929

Barcode 3/9 maker on .netusing aspx topaint code 3/9 in asp.net web,windows application

~ = (0.6929)(0.8961) + (1 - 0.6929)(0.5063) = 0.7764

EAN13 barcode library in visual c#use vs .net gtin - 13 integrated tocreate gs1 - 13 on c#.net

(0.1977)-2/[(0.1977)-2

Microsoft Excel 2d barcode creatorwith microsoft excelgenerate, create 2d barcode none with office excel projects

+ 11] = 0.6993

= (0.6993)(37.00) + (1 - 0.6993)(35.82) = 36.65

a = 36.65 -

(0.7764)(35.14) = 9.363