Today we are resuming the discussion on model fitting and error estimation by Kleinstein and Hershberg.There are two ways by which we can select the appropriate model. The first is by observing trend lines which correspond to some well known mathematical formula. The second is on the observation of underlying physical processes which contribute towards the system. These physical interpretations contribute to model parameters.
In order to fit the data points, a model may use least squares of errors. the errors called residuals may be both positive or negative which result in inaccurate measure. Instead the squares of the errors can be minimized to give a better fit.
We used the least squares error minimization to fit the data points. Another way to do this is using Maximum likelihood estimation. This method asks: "Given my set of model parameters, what is the probability that this data set occurred ?" This translates as likelihood for the parameters given the data.
The chi-square error measure and maximum likelihood estimation have a relation between the two.For a Gaussian distribution, the probability of the data set coming from the model parameters involves minimizing the negative natural log of probability which is the chi-square function of weighted residuals. Furthermore, if the variance is uniform, then the chi-square function yields the sum of squared residuals.
Linear regression analysis strikes a relationship between a dependent variable and an independent variable. The best values of slope and intercept are found by taking partial derivatives of the error function with respect to them. Regression parameters are different from correlation coefficient. The latter describes the strength of association between two variables. It is therefore symmetric for the pair of variables unlike regression parameters.
If we are not inclined towards error estimation then we can attempt Bootstrap method. This method uses actual data sets with its N points to generate synthetic data with just as many points. The synthetic differs from the original with N being the fraction of the original points replaced with duplicated originals. Since the order of data points does not matter, the estimation can take place with actual measurement noise.
#codingexercise
We talked about finding closest elements to a given value across three arrays. We introduced the notion of projections to find non-overlapping regions In the same spirit we continue that projections can be initiated from any sentinel values.
In order to fit the data points, a model may use least squares of errors. the errors called residuals may be both positive or negative which result in inaccurate measure. Instead the squares of the errors can be minimized to give a better fit.
We used the least squares error minimization to fit the data points. Another way to do this is using Maximum likelihood estimation. This method asks: "Given my set of model parameters, what is the probability that this data set occurred ?" This translates as likelihood for the parameters given the data.
The chi-square error measure and maximum likelihood estimation have a relation between the two.For a Gaussian distribution, the probability of the data set coming from the model parameters involves minimizing the negative natural log of probability which is the chi-square function of weighted residuals. Furthermore, if the variance is uniform, then the chi-square function yields the sum of squared residuals.
Linear regression analysis strikes a relationship between a dependent variable and an independent variable. The best values of slope and intercept are found by taking partial derivatives of the error function with respect to them. Regression parameters are different from correlation coefficient. The latter describes the strength of association between two variables. It is therefore symmetric for the pair of variables unlike regression parameters.
If we are not inclined towards error estimation then we can attempt Bootstrap method. This method uses actual data sets with its N points to generate synthetic data with just as many points. The synthetic differs from the original with N being the fraction of the original points replaced with duplicated originals. Since the order of data points does not matter, the estimation can take place with actual measurement noise.
#codingexercise
We talked about finding closest elements to a given value across three arrays. We introduced the notion of projections to find non-overlapping regions In the same spirit we continue that projections can be initiated from any sentinel values.
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