See also slides for curve fitting, regression, colab link
Transclude of thoughts/university/twenty-four-twenty-five/sfwr-4ml3/ols_and_kls.py
curve fitting.
how do we fit a distribution of data over a curve?
Given a set of data points
- (or )
ols.
Ordinary Least Squares (OLS)
Let be the prediction of a model , is the error, minimize
In the case of 1-D ordinary least square, the problems equates find to minimize
Optimal solution
where , , ,
Hyperplane equation
where is the intercept (bias)
Homogenous hyperplane:
minimize
Thus we can find
Example:
With and $$ X^{’}{n \times 3} = \begin{pmatrix} x^{1}{1} & x^{1}{2} & 1 \ x^{2}{1} & x^{2}{2} & 1 \ x^{3}{1} & x^{3}_{2} & 1 \end{pmatrix}
With $$ W = \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} $$ and $$ W^{'} = \begin{pmatrix} w_1 \\ w_2 \\ w_0 \end{pmatrix} $$ thusX^{’} \times W = \begin{pmatrix} w_0 + \sum{w_i \times x_i^{1}} \ \vdots \ w_0 + \sum{w_i \times x_i^{n}} \end{pmatrix}