local_fit {resemble}

Local fit functions

Description

These functions define the way in which each local fit/prediction is done within each iteration in the mbl function.

Usage

local_fit_pls(pls_c, modified = FALSE, max_iter = 100, tol = 1e-6)

local_fit_wapls(min_pls_c, max_pls_c, modified = FALSE,
                max_iter = 100, tol = 1e-6)

local_fit_gpr(noise_variance = 0.001)

Arguments

pls_c

an integer indicating the number of pls components to be used in the local regressions when the partial least squares (local_fit_pls) method is used.

modified

a logical indicating whether the modified version of the pls algorithm (Shenk and Westerhaus, 1991 and Westerhaus, 2014). Default is FALSE.

max_iter

an integer indicating the maximum number of iterations in case tol is not reached. Defaul is 100.

tol

a numeric value indicating the convergence for calculating the scores. Default is 1-e6.

min_pls_c

an integer indicating the minimum number of pls components to be used in the local regressions when the weighted average partial least squares (local_fit_wapls) method is used. See details.

max_pls_c

integer indicating the maximum number of pls components to be used in the local regressions when the weighted average partial least squares (local_fit_wapls) method is used. See details.

noise_variance

a numeric value indicating the variance of the noise for Gaussian process local regressions (local_fit_gpr). Default is 0.001.

Details

These functions are used to indicate how to fit the regression models within the mbl function.

There are three possible options for performing these regressions:

• Partial least squares (pls, local_fit_pls): It uses the orthogonal scores (non-linear iterative partial least squares, nipals) algorithm. The only parameter which needs to be optimized is the number of pls components.

• Weighted average pls (local_fit_wapls): This method was developed by Shenk et al. (1997) and it used as the regression method in the widely known LOCAL algorithm. It uses multiple models generated by multiple pls components (i.e. between a minimum and a maximum number of pls components). At each local partition the final predicted value is a ensemble (weighted average) of all the predicted values generated by the multiple pls models. The weight for each component is calculated as follows:

  \[w_{j} = \frac{1}{s_{1:j}\times g_{j}}\]

  where \(s_{1:j}\) is the root mean square of the spectral reconstruction error of the unknown (or target) observation(s) when a total of \(j\) pls components are used and \(g_{j}\) is the root mean square of the squared regression coefficients corresponding to the \(j\)th pls component (see Shenk et al., 1997 for more details).

• Gaussian process with dot product covariance (local_fit_gpr): Gaussian process regression is a probabilistic and non-parametric Bayesian method. It is commonly described as a collection of random variables which have a joint Gaussian distribution and it is characterized by both a mean and a covariance function (Rasmussen and Williams, 2006). The covariance function used in the implemented method is the dot product. The only parameter to be taken into account in this method is the noise. In this method, the process for predicting the response variable of a new sample (\(y_u\)) from its predictor variables (\(x_u\)) is carried out first by computing a prediction vector (\(A\)). It is derived from a reference/training observations congaing both a response vector (\(Y\)) and predictors (\(X\)) as follows:

  \[A = (X X^{T} + \sigma^2 I)^{-1} Y\]

  where \(\sigma^2\) denotes the variance of the noise and \(I\) the identity matrix (with dimensions equal to the number of observations in \(X\)). The prediction of \(y_{u}\) is then done as follows:

  \[\hat{y}_{u} = (x_{u}x_{u}^{T}) A\]

The modified argument in the pls methods (local_fit_pls() and local_fit_wapls()) is used to indicate if a modified version of the pls algorithm (modified pls or mpls) is to be used. The modified pls was proposed Shenk and Westerhaus (1991, see also Westerhaus, 2014) and it differs from the standard pls method in the way the weights of the predictors (used to compute the matrix of scores) are obtained. While pls uses the covariance between response(s) and predictors (and later their deflated versions corresponding at each pls component iteration) to obtain these weights, the modified pls uses the correlation as weights. The authors indicate that by using correlation, a larger potion of the response variable(s) can be explained.

Value

An object of class local_fit mirroring the input arguments.

Author(s)

Leonardo Ramirez-Lopez

References

Shenk, J. S., & Westerhaus, M. O. 1991. Populations structuring of near infrared spectra and modified partial least squares regression. Crop Science, 31(6), 1548-1555.

Shenk, J., Westerhaus, M., and Berzaghi, P. 1997. Investigation of a LOCAL calibration procedure for near infrared instruments. Journal of Near Infrared Spectroscopy, 5, 223-232.

Rasmussen, C.E., Williams, C.K. Gaussian Processes for Machine Learning. Massachusetts Institute of Technology: MIT-Press, 2006.

Westerhaus, M. 2014. Eastern Analytical Symposium Award for outstanding Wachievements in near infrared spectroscopy: my contributions to Wnear infrared spectroscopy. NIR news, 25(8), 16-20.

See Also

mbl

Examples

local_fit_wapls(min_pls_c = 3, max_pls_c = 12)