On CIFAR-10 we empirically demonstrate that as networks grow wider and deeper, untrained subnetworks perform just as well as the dense network with learned weights. The following plot shows both the OLS fitted line (black) and WLS fitted line (red) overlaid on the same scatterplot.We experiment on small and large scale datasets for image recognition, namely CIFAR-10 and Imagenet. Notice that the regression estimates have not changed much from the ordinary least squares method. The summary of this weighted least squares fit is as follows: Then we fit a weighted least squares regression model using the just-created weights. We therefore fit a simple linear regression model of the absolute residuals on the predictor and calculate weights as 1 over the squared fitted values from this model. A plot of the absolute residuals versus the predictor values is as follows: The weights we will use will be based on regressing the absolute residuals versus the predictor. We will turn to weighted least squares to address this possiblity. Below is the summary of the simple linear regression fit for this data:Ī plot of the residuals versus the predictor values indicates possible nonconstant variance since there is a very slight "megaphone" pattern: A scatterplot of the data is given below.įrom this scatterplot, a simple linear regression seems appropriate for explaining this relationship.įirst an ordinary least squares line is fit to this data. The response is the cost of the computer time ( Y) and the predictor is the total number of responses in completing a lesson ( X). The data below ( ca_learning_new.txt) was collected from a study of computer-assisted learning by n = 12 students. Use of weights will (legitimately) impact the widths of statistical intervals.Įxample 1: Computer-Assisted Learning Dataset.In designed experiments with large numbers of replicates, weights can be estimated directly from sample variances of the response variable at each combination of predictor variables.In some cases, the values of the weights may be based on theory or prior research.In cases where they differ substantially, the procedure can be iterated until estimated coefficients stabilize (often in no more than one or two iterations) this is called iteratively reweighted least squares. Weighted least squares estimates of the coefficients will usually be nearly the same as the "ordinary" unweighted estimates.The difficulty, in practice, is determining estimates of the error variances (or standard deviations).Some key points regarding weighted least squares are: We consider some examples of this approach in the next section. \[\begin^2\).Īfter using one of these methods to estimate the weights, \(w_i\), we then use these weights in estimating a weighted least squares regression model. The method of weighted least squares can be used when the ordinary least squares assumption of constant variance in the errors is violated (which is called heteroscedasticity). The method of ordinary least squares assumes that there is constant variance in the errors (which is called homoscedasticity). This approach uses the framework of generalized linear models, which we discuss in Lesson 12. For some applications we can explicitly model the variance as a function of the mean, E( Y).This leads to generalized least squares, in which various forms of nonconstant variance can be modeled. A generalization of weighted least squares is to allow the regression errors to be correlated with one another in addition to having different variances.This leads to weighted least squares, in which the data observations are given different weights when estimating the model – see below. Weight the variances so that they can be different for each set of predictor values.We explored this in more detail in Lesson 7. Apply a variance-stabilizing transformation to the response variable, for example a logarithmic transformation (or a square root transformation if a logarithmic transformation is "too strong" or a reciprocal transformation if a logarithmic transformation is "too weak").Some remedies for refining a model exhibiting excessive nonconstant variance includes the following: For example, if the residual variance increases with the fitted values, then prediction intervals will tend to be wider than they should be at low fitted values and narrower than they should be at high fitted values. Excessive nonconstant variance can create technical difficulties with a multiple linear regression model.
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