pylops.LinearRegression¶
- pylops.LinearRegression(taxis, dtype='float64')[source]¶
Linear regression.
Creates an operator that applies linear regression to a set of points. Values along the \(t\)-axis must be provided while initializing the operator. Intercept and gradient form the model vector to be provided in forward mode, while the values of the regression line curve shall be provided in adjoint mode.
- Parameters
- taxis
numpy.ndarray Elements along the \(t\)-axis.
- dtype
str, optional Type of elements in input array.
- taxis
- Raises
- TypeError
If
taxisis notnumpy.ndarray.
See also
RegressionPolynomial regression
Notes
The LinearRegression operator solves the following problem:
\[y_i = x_0 + x_1 t_i \qquad \forall i=0,1,\ldots,N-1\]We can express this problem in a matrix form
\[\mathbf{y}= \mathbf{A} \mathbf{x}\]where
\[\mathbf{y}= [y_0, y_1,\ldots,y_{N-1}]^T, \qquad \mathbf{x}= [x_0, x_1]^T\]and
\[\begin{split}\mathbf{A} = \begin{bmatrix} 1 & t_{0} \\ 1 & t_{1} \\ \vdots & \vdots \\ 1 & t_{N-1} \end{bmatrix}\end{split}\]Note that this is a particular case of the
pylops.Regressionoperator and it is in fact just a lazy call of that operator withorder=1.
