Measurements

Data Analysis Using the Method of Least Squares by John Wolberg

By John Wolberg

The most well-liked approach to facts research of quantitative experiments is the tactic of least squares. usually, even though, the total energy of the strategy is neglected and extremely few books care for this topic on the point that it merits. the aim of information research utilizing the strategy of Least Squares is to fill this hole and contain the kind of details required to aid scientists and engineers follow the tactic to difficulties of their specific fields of curiosity. additionally, graduate scholars in technological know-how and engineering doing paintings of experimental nature can make the most of this publication. fairly, either linear and non-linear least squares, using experimental blunders estimates for facts weighting, methods to incorporate earlier estimates, method for choosing and testing models, prediction research, and a few non-parametric equipment are mentioned.

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A question sometimes asked is: if we increase or decrease the weights how does this affect the results? For example, m for unit weighting what happens if we use a value of w other than 1? The answer is that it makes no difference. The values of the terms of the V vector will be proportional to w and -1 1 all the terms of the C matrix will also be proportional to w. , the product of C -1V) will be independent of w. A similar argument can be made for statistical weighting. For example, if all the values of σy are increased by a factor of 10, the values of wi will be decreased by a factor of 100.

3 Data Weighting 39 The question that must be answered is how do we relate wi to the σ 's associated with the i th data point? 1) i=n = ¦ w i (Yi − yi ) = ¦ w i (Yi − f (X i )) 2 i =1 2 i =1 We will see that the least squares solution is based upon adjusting the unknown values of the ak's that are included in the function f such that S is minimized. 2) For cases in which the uncertainties associated with the values of xi are negligible our objective should be that the residuals Ri are proportional to the values of σ yi .

5 the uncertainties in the model parameters were considered. If the only purpose of the experiment is to determine the parameters of the model, then only these uncertainties are of interest. However, there are many situations in which we are interested in using the model for making predictions. , the vector X). In this section attention is turned towards the uncertainties σf of these predictions. Typically, one assumes that the model is “correct” and thus the computed values of y are normally distributed about the true values.

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