LocalLinearRegressionHatMatrix

class skfda.misc.hat_matrix.LocalLinearRegressionHatMatrix(*, bandwidth=None, kernel=<function normal>)

Local linear regression method.

Creates the matrix \(\hat{H}\), used in the kernel smoothing and kernel regression algorithms, as explained below.

For kernel smoothing algorithm to estimate the smoothed value for \(t_i'\) the following error must be minimised

\[AWSE(a, b) = \sum_{j=1}^n \left[ \left(y_j - \left(a + b (t_j - t'_i) \right) \right)^2 K \left( \frac {|t_j - t'_i|}{h} \right) \right ]\]

which gives the following expression for each cell

\[\hat{H}_{i,j} = \frac{b_j(t_i')}{\sum_{k=1}^{n}b_k(t_i')}\]

\[b_j(t_i') = K\left(\frac{t_j - t_i'}{h}\right) S_{n,2}(t_i') - (t_j - t_i')S_{n,1}(t_i')\]

\[S_{n,k}(t_i') = \sum_{j=1}^{n}K\left(\frac{t_j-t_i'}{h}\right) (t_j-t_i')^k\]

where \(\{t_1, t_2, ..., t_n\}\) are points with known value and \(\{t_1', t_2', ..., t_m'\}\) are the points for which it is desired to estimate the smoothed value [1].

For kernel regression algorithm:

Given functional data, \(\{X_1, X_2, ..., X_n\}\) where each function is expressed in a orthonormal basis with \(J\) elements and scalar response \(Y = \{y_1, y_2, ..., y_n\}\).

It is desired to estimate the values \(\hat{Y} = \{\hat{y}_1, \hat{y}_2, ..., \hat{y}_m\}\) for the data \(\{X'_1, X'_2, ..., X'_m\}\) (expressed in the same basis).

For each \(X'_k\) the estimation \(\hat{y}_k\) is obtained by taking the value \(a_k\) from the vector \((a_k, b_{1k}, ..., b_{Jk})\) which minimizes the following expression

\[AWSE(a_k, b_{1k}, ..., b_{Jk}) = \sum_{i=1}^n \left(y_i - \left(a_k + \sum_{j=1}^J b_{jk} c_{ik}^j \right) \right)^2 K \left( \frac {d(X_i - X'_k)}{h} \right)\]

Where \(c_{ik}^j\) is the \(j\)-th coefficient in a truncated basis expansion of \(X_i - X'_k = \sum_{j=1}^J c_{ik}^j\) and \(d\) some functional distance [2]

For both cases, \(K(\cdot)\) is a kernel function and \(h\) the bandwidth.

Parameters:

• bandwidth (float | None) – Window width of the kernel (also called h).

• kernel (Callable[[NDArrayFloat], NDArrayFloat]) – Kernel function. By default a normal kernel.

References

[1]

Larry Wasserman. Nonparametric Regression, pages 61–123. Springer Texts in Statistics. Springer-Verlag, New York, 2006. URL: https://www.springer.com/gp/book/9780387251455 (visited on 2019-06-25).

[2]

Amparo Baíllo and Aurea Grané. Local linear regression for functional predictor and scalar response. Journal of Multivariate Analysis, 100(1):102–111, January 2009. URL: https://www.sciencedirect.com/science/article/pii/S0047259X08000973 (visited on 2023-04-03), doi:10.1016/j.jmva.2008.03.008.

Methods

get_metadata_routing()	Get metadata routing of this object.
get_params([deep])	Get parameters for this estimator.
set_params(**params)	Set the parameters of this estimator.

get_metadata_routing()

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:

routing – A MetadataRequest encapsulating routing information.

Return type:

MetadataRequest

get_params(deep=True)

Get parameters for this estimator.

Parameters:

deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

params – Parameter names mapped to their values.

Return type:

dict

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:

**params (dict) – Estimator parameters.

Returns:

self – Estimator instance.

Return type:

estimator instance

Examples using skfda.misc.hat_matrix.LocalLinearRegressionHatMatrix

Kernel Regression

Kernel Regression

Kernel Smoothing

Kernel Smoothing