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Matrix algebracan then be usedto manipulatethis vector to computeboth the coefficient of determination and the standard error of the estimate illustratedin the following example as

ALTER TABLE employee ADD CONSTRAINT employee$salaryNonNegative CHECK (salary > 0)

P o l y n o m i oR e g r e s s i ow i t h M A T L A B l n ProblemSiotement Repeat Example141, usematrixoperations described this but as in section Solution First,enterthe datato be fit >> x - t0 12 3 4 5l'; > > y = 1 2 I 1 1 1 3 6 2 ' 7 2 4 A 9 6 1 1 1 ' ;

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The Boolean expression can t directly reference another table, but you can use a function to access the data in the table. Note that I said you can t see which other rows were modified, but the data is already in the table, so you can access it in the function s query. One example of using a function is limiting the cardinality of a value in a database (or the count of the times a value can appear). Using a unique constraint, you can

L\NEARIEASISQUARESAND NONI\NEAR RIGRESS\ON GENERAT Next, create the [Z] matnx:

Professional Development Programs Local school districts are major sponsors of the development of their staff Some of this kind of continuing education takes place on school time, with students being given a half or a full day off In such sessions, staff are introduced to new curricula; oriented to new federal, state, and local policies affecting instruction; and familiarized with technological developments Intermediate and cooperative school districts also are providers of inservice and continuing education for school personnel and sometimes for par-

>> Z = [ones(slze(x) ) x x^2]

easily limit the cardinality to 1, but if you want to allow a number different than 1, there isn t an easy way to do this declaratively. So we can use a function and a check constraint. Take the following table:

100 111 1,24 139

create table cardinalityLimit ( value varchar(10) )

7416 1 5

We can verify that LZf lzl resultsin the coefficientmatrix for the normal equations: >> z'*z 6 15 55 15 ss 225 55 225 979

If we want no more than two of the same values for the value column, you can build a simple function like this:

This is the sameresultwe obtainedwith summations Example 14I We can solvefor the in quadraticby implementingEq (14l0): coefficientsof the least-squares

>> a = \Z'*Z) \(Z'*y)

CREATE FUNCTION dbo.cardinalityLimit$countValue ( @value varchar(10) ) RETURNS INT AS BEGIN RETURN (SELECT count(*) FROM cardinalityLimit WHERE value = @value) END

2 47 B6 23593 18607 In order to compute 12 and s"7-, first compute the sum of the squares of the residuals: , > > S r = s u m( ( y - Z * a ) ^ 2 )

It counts the number of values in the table that match the @value parameter. In the following constraint, you ll check to see if the count is 2 or less:

3 't 466 Then r2 can be computed as > > 1 2 = I - S r / s u m( ( y - m e a n ( y ) ) ^ 2 ) 0998s and sr,7, can be computed as >> syx sqrt (Sr/ (lenqth (x) -lenqth (a) ) )

Our primary motivation for the foregoing has been to illustrate the unity among the three approaches and to show how they can all be expressed simply in the same matrix notation It also sets the stage for the next section where we will gain some insights into the preferred strategies for solving Eq (1a10) The matrix notation will also have relevance when we tum to nonlinear regression in Section 145

ALTER TABLE cardinalityLimit ADD CONSTRAINT cardinalityLimit$valueCardinality2 CHECK (dbo.cardinalityLimit$countValue(value) <= 2)

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