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From the Library of Wow! eBook in .NET Drawer data matrix barcodes in .NET From the Library of Wow! eBook

From the Library of Wow! eBook using barcode development for vs .net control to generate, create gs1 datamatrix barcode image in vs .net applications. Code 11 16 . Building Models Using ANOVA and Regression Response predicted by model at xi Residual y 5a 1bx Simple linear regression Our condition for the coef cients is that they minimize the sum of the squared error. The conditions for this are that the derivative of the squared error with respect to each model coef cient is 0, leading to the following requirements: 0 m n 2 0 m n 2 a a e ij = 0a a a (yij a bxi ) = 0 0a i=1 j=1 i=1 j=1 0 m n 2 0 m n 2 a a e ij = 0b a a (yij a bxi ) = 0 0b i=1 j=1 i=1 j=1 Using equations (16.15) and (16.

16), you can solve for a in terms of b:. m n m (16.15). (16.16). a a yij a= i=1 j=1 a xi (16.17). You can use eq uations (16.16) and (16.17) to solve for b:.

m n m i=1 n j=1 a a xiyij i=1 j=1 a xi a yij mn m 2 i=1 m 2 a xi i=1 a a xi b m (16.18). This is the pr ECC200 for .NET ocess to calculate the model coef cients for the case of simple linear regression. The result is that the coef cients a and b will place a straight line through the data set that.

From the Library of Wow! eBook Model Building Using Regression minimizes the sum of the square of the residuals. Before using the model, you have to determine whether or not the model is adequate. Is the Model Good You have to examine your model to see if it meets certain minimum criteria.

These are some of the main criteria: Is the model statistically signi cant This asks whether the model explains enough of the variability in the data to be useful. You can think of the data as being produced by two processes. The model is a mathematical description of some physical process that converts inputs to outputs.

The other process is the random process that produces Gaussian variations about each sample mean. You would like the contribution of the model to total variability to be signi cantly larger than the contribution of the noise. Recall that in 15 we discussed the F-distribution that is used to compare the variance of two processes.

Is the model a good t to the data A qualitative look at goodness of t would probably conclude that the data set of Figure 16.3 is a good t to the straight line passing through the samples, since the data samples all straddle the t line. Our eyes and brains are pretty good in this case.

What if the t were not so good You need an objective way to determine the goodness of t that s based on statistics. You can t a straight line to a couple of data points and a quadratic model to three data points, but is it a good idea In each of these cases you have no way to test the model to see if it s a good t. An important part of planning any experiment is to include enough data points to allow the estimation of error.

Is the model an ef cient one If you can build a model that is statistically signi cant, has all the important factors, and ts the data without over tting it, you are in good shape. You might be tempted to add higher-order terms, thinking that they will t the data better. However, building a model is a situation where less is more.

We discuss the concept of parsimonious models in the next chapter about designed experiments. In a nutshell, the idea is to make your model no more complex than is necessary..

A good t. From the Library of Wow! eBook 16 . Building Models Using ANOVA and Regression Statistical Si 2d Data Matrix barcode for .NET gni cance of the Model To answer the question of the goodness of the model, we use ANOVA. This is analogous to a single-factor ANOVA.

In the case of simple regression you are interested in the statistical signi cance of the model instead of a single factor. The approach described here applies to data sets where you have multiple observations at each level of xi. To start, express the total sum of the squares as the sum of two components:.

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