The Generalized Least Squares Weighting (GLSW) algorithm, proposed by Martens et al. (2003), is a technique used to mitigate the effects of external interferences in datasets. It constructs a filter to remove these interferences, allowing for more accurate data analysis and processing.
Arguments
- x1
A numeric matrix, data frame or tibble representing the first set of data.
- x2
A numeric matrix, data frame or tibble representing the second set of data.
- alpha
A numeric value specifying the weighting parameter. Typical values range from 1 to 0.0001. Default is 0.01.
Details
The algorithm works by first calculating a covariance matrix from the differences between two spectral datasets that should ideally be similar. These differences are considered to be the interferences or clutter present in the data. For example, if two sets of measurements have been taken under similar conditions, the differences between them could be attributed to external factors such as sensor noise, environmental conditions, or other sources of interference. Once the covariance matrix is calculated, GLSW applies a filtering matrix to down-weight the contributions of the identified interferences or clutter. This filtering matrix is constructed using a regularization parameter, denoted as alpha (\(\alpha\)).
The value of \(\alpha\) determines how strongly the algorithm down-weights the clutter components in the data. In cases where the interferences are well-characterized and distinct from the desired signal, a small \(\alpha\) value may be appropriate to achieve effective clutter removal. However, if the interferences are more subtle or intertwined with the desired signal, a larger \(\alpha\) value may be preferred to avoid over-suppression of the signal itself.
Let \(\textbf{X}\) be a data matrix. The GLSW filter is applied as follows:
$$\textbf{X}_{new} = \textbf{X} \cdot \textbf{G}$$
where, \(\textbf{X}_{new}\) is the filtered matrix and \(\textbf{G}\) the filtering matrix.