After computing the x- and y-values we add Gaussian noise to the y-values in data(:,2) and sort the data. The y-values, stored in data(:,2), have a exponential relationship with the x-values, stored in data(:,1). To see the difference in the results we first create a synthetic data set. Logarithmizing the y-values violates this assumption: the errors then have a log-normal distribution and the regression therefore places less weight on the larger y-values. The classic linear regression method makes two assumptions about the data: (1) there is a linear relationship between x and y, (2) the unknown errors around the means have a normal (Gaussian) distribution, with a similar variance for all data points. Linear regression minimizes the deviations Δ y between the data points xy and the value y predicted by the best-fit line y= b 0+ b 1 x using a least-squares criterion. This means that x is the independent variable defined by the experimenter and regarded as being free of errors. Second, classical regression assumes that y responds to x and that the entire dispersion in the data set is contained within the y-value (see Section 4.3). First, if the relationship between the data is of the type y= a 0+ a 1∙ e x, then logarithmizing y-values in the form log( y) does not provide a complete linearization of the data the parameters a 0 and a 1 are ignored. Log-transforming the y-values has two important consequences that can influence the result. Here is a MATLAB example to show how to do it better. A common error in the regression analysis is that bivariate data with an exponential relationship are log-transformed, then a best-fit line is calculated using a classical linear regression using the least squares, before the result is backtransformed.
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