How do you minimize the value of the error or uncertainty when fitting sample data?
Statisticians use least square fitting to ensure that they have optimized their sample data in the fitted model. When they do so, they are able to minimize the value of the statistical error/uncertainty. The least square fitting model can either be linear or non-linear.
The statistical error or uncertainty is described by the following equation;
Where,
ε is the statistical error or uncertainty
y is the dependent variable from the data set
f is the fitted function to the sample data set
n is the number of observations in the sample data set
Procedure
Follow these steps to come up with the least square estimators of the parameters in the fitted linear model:
- Find partial derivates of the fitted function f with respect to the parameters.
- Now set each of this derivates equal to zero.You will end up with a system of linear equations which can simply be represented in matrix form as:
Ax=b
Where A is an mx n matrix.
From the dimension of the matrix A we see that we have m simultaneous equations and n variables.
- Apply linear algebra techniques to solve this system of linear equations.
- The results obtained are the least square estimators of the parameters.
In a non-linear model, start with an approximate value and iterate over it to find the least square estimator of the parameter.
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