# integrationWeights: Functions to compute integration weights In FDboost: Boosting Functional Regression Models

## Description

Computes trapezoidal integration weights (Riemann sums) for a functional variable `X1` that has evaluation points `xind`.

## Usage

 ```1 2 3``` ```integrationWeights(X1, xind, id = NULL) integrationWeightsLeft(X1, xind, leftWeight = c("first", "mean", "zero")) ```

## Arguments

 `X1` for functional data that is observed on one common grid, a matrix containing the observations of the functional variable. For a functional variable that is observed on curve specific grids, a long vector. `xind` evaluation points (index) of functional variable `id` defaults to `NULL`. Only necessary for response in long format. In this case `id` specifies which curves belong together. `leftWeight` one of `c("mean", "first", "zero")`. With left Riemann sums different assumptions for the weight of the first observation are possible. The default is to use the mean over all integration weights, `"mean"`. Alternatively one can use the first integration weight, `"first"`, or use the distance to zero, `"zero"`.

## Details

The function `integrationWeights()` computes trapezoidal integration weights, that are symmetric. Per default those weights are used in the `bsignal`-base-learner. In the special case of evaluation points (`xind`) with equal distances, all integration weights are equal.

The function `integrationWeightsLeft()` computes weights, that take into account only the distance to the prior observation point. Thus one has to decide what to do with the first observation. The left weights are adequate for historical effects like in `bhist`.

`bsignal` and `bhist` for the base-learners.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34``` ```## Example for trapezoidal integration weights xind0 <- seq(0,1,l = 5) xind <- c(0, 0.1, 0.3, 0.7, 1) X1 <- matrix(xind^2, ncol = length(xind0), nrow = 2) # Regualar observation points integrationWeights(X1, xind0) # Irregular observation points integrationWeights(X1, xind) # with missing value X1[1,2] <- NA integrationWeights(X1, xind0) integrationWeights(X1, xind) ## Example for left integration weights xind0 <- seq(0,1,l = 5) xind <- c(0, 0.1, 0.3, 0.7, 1) X1 <- matrix(xind^2, ncol = length(xind0), nrow = 2) # Regular observation points integrationWeightsLeft(X1, xind0, leftWeight = "mean") integrationWeightsLeft(X1, xind0, leftWeight = "first") integrationWeightsLeft(X1, xind0, leftWeight = "zero") # Irregular observation points integrationWeightsLeft(X1, xind, leftWeight = "mean") integrationWeightsLeft(X1, xind, leftWeight = "first") integrationWeightsLeft(X1, xind, leftWeight = "zero") # obervation points that do not start with 0 xind2 <- xind + 0.5 integrationWeightsLeft(X1, xind2, leftWeight = "zero") ```