Download Babbage's Calculating Engines by C. Babbage [first publ. 1889] PDF

By C. Babbage [first publ. 1889]

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Extra resources for Babbage's Calculating Engines

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Notice that the estimates of the regression coefficients closely match those that we have computed manually (it is not unusual to find minor differences between manual and computer software regression calculations because of round-off). In subsequent sections we will show how some of the other quantities in the output are obtained and how to interpret them. 2 Regression output from JMP. (a) Response y whole model, actual by predicted plot. (b) Residual by predicted plot. 3 PROPERTIES OF THE LEAST SQUARES ESTIMATORS AND ESTIMATION OF ???? 2 The method of least squares produces an unbiased estimator of the parameter β in the multiple linear regression model.

If these influential points are “bad” values, then they should be eliminated. On the other hand, there may be nothing wrong with these points, but if they control key model properties, we would like to know it, because it could affect the use of the model. In this subsection we describe and illustrate several useful measure of influence. Leverage Points The disposition of points in x-space is important in determining model properties. In particular remote observations potentially have disproportionate leverage on the parameter estimates, the predicted values, and the usual summary statistics.

This property may be easily demonstrated by finding the expected value of b as follows: E(b) = E[(X′ X)−1 X′ y] = E[(X′ X)−1 X′ (Xβ + ????)] = E[(X′ X)−1 X′ Xβ + (X′ X)−1 X′ ????] =β because E(????) = 0 and (X′ X)−1 X′ X = I. Thus b is an unbiased estimator of β. The variance property of b is expressed by the covariance matrix Cov(b) = E{[b − E(b)][b − E(b)]′ } The covariance matrix of b is a p × p symmetric matrix whose (j, j)th element is the variance of bj and whose (i, j)th element is the covariance between bi and bj .

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