# Locally weighted models `locw` and `locwlv` are generic working functions returning predictions of KNN locally weighted (LW) models. One specific (= local) model is fitted for each observation to predict, and a prediction is returned. See the wrapper `lwplsr` (KNN-LWPLSR) for an example of use. In KNN-LW models, the prediction is built from two sequential steps, therafter referred to as $weighting "1"$ and $weighting "2"$, respectively. For each new observation to predict, the two steps are as follow: - $Weighting "1"$. The $k$ nearest neighbors (in the training data set) are selected and the prediction model is fitted (in the next step) only on this neighborhood. It is equivalent to give a weight = 1 to the neighbors, and a weight = 0 to the other training observations, which corresponds to a binary weighting. - $Weighting "2"$. Each of the $k$ nearest neighbors eventually receives a weight (different from the usual $1/k$) before fitting the model. The weight depend from the dissimilarity (preliminary calculated) between the observation and the neighbor. This corresponds to a within-neighborhood weighting. The prediction model used in step $"2"$ has to be defined in a function specified in argument `fun`. If there are $m$ new observations to predict, a list of $m$ vectors defining the $m$ neighborhoods has to be provided (argument `listnn`). Each of the $m$ vectors contains the indexes of the nearest neighbors in the training set. The $m$ vectors are not necessary of same length, i.e. the neighborhood size can vary between observations to predict. If there is a weighting in step $"2"$, a list of $m$ vectors of weights have to be provided (argument `listw`). Then `locw` fits the model successively for each of the $m$ neighborhoods, and returns the corresponding $m$ predictions. Function `locwlv` is dedicated to prediction models based on latent variables (LVs) calculations, such as PLSR. It is much faster and recommended. latin1 ```r locw(Xtrain, Ytrain, X, listnn, listw = NULL, fun, verb = FALSE, ...) locwlv(Xtrain, Ytrain, X, listnn, listw = NULL, fun, nlv, verb = FALSE, ...) ``` ## Arguments - `Xtrain`: Training X-data ($n, p$). - `Ytrain`: Training Y-data ($n, q$). - `X`: New X-data ($m, p$) to predict. - `listnn`: A list of $m$ vectors defining weighting "1". Component $i$ of this list is a vector (of length between 1 and $n$) of indexes. These indexes define the training observations that are the nearest neighbors of new observation $i$. Typically, `listnn` can be built from `getknn`, but any other list of length $m$ can be provided. The $m$ vectors can have equal length (i.e. the $m$ neighborhoods are of equal size) or not (the number of neighbors varies between the observations to predict). - `listw`: A list of $m$ vectors defining weighting "2". Component $i$ of this list is a vector (that must have the same length as component $i$ of `listnn`) of the weights given to the nearest neighbors when the prediction model is fitted. Internally, weights are "normalized" to sum to 1 in each component. Default to `NULL` (weights are set to $1 / k$ where $k$is the size of the neihborhodd). - `fun`: A function corresponding to the prediction model to fit on the $m$ neighborhoods. - `nlv`: For `locwlv` : The number of LVs to calculate. - `verb`: Logical. If `TRUE`, fitting information are printed. - `...`: Optional arguments to pass in function `fun`. ## Returns - **pred**: matrix or list of matrices (if `nlv` is a vector), with predictions ## References Lesnoff M, Metz M, Roger J-M. Comparison of locally weighted PLS strategies for regression and discrimination on agronomic NIR data. Journal of Chemometrics. 2020;n/a(n/a):e3209. doi:10.1002/cem.3209. ## Examples ```r n <- 50 ; p <- 30 Xtrain <- matrix(rnorm(n * p), ncol = p, byrow = TRUE) ytrain <- rnorm(n) Ytrain <- cbind(ytrain, 100 * ytrain) m <- 4 Xtest <- matrix(rnorm(m * p), ncol = p, byrow = TRUE) ytest <- rnorm(m) Ytest <- cbind(ytest, 10 * ytest) k <- 5 z <- getknn(Xtrain, Xtest, k = k) listnn <- z$listnn listd <- z$listd listnn listd listw <- lapply(listd, wdist, h = 2) listw nlv <- 2 locw(Xtrain, Ytrain, Xtest, listnn = listnn, fun = plskern, nlv = nlv) locw(Xtrain, Ytrain, Xtest, listnn = listnn, listw = listw, fun = plskern, nlv = nlv) locwlv(Xtrain, Ytrain, Xtest, listnn = listnn, listw = listw, fun = plskern, nlv = nlv) locwlv(Xtrain, Ytrain, Xtest, listnn = listnn, listw = listw, fun = plskern, nlv = 0:nlv) ```

locw function