An introduction to FGLMtrunc

# An introduction to `FGLMtrunc`

## Introduction

`FGLMtrunc` is a package that fits truncated Functional Generalized Linear Models as described in Liu, Divani, and Petersen (2020). It implements methods for both functional linear and functional logistic regression models. The solution path is computed efficiently using active set algorithm with warm start. Optimal smoothing and truncation parameters (\(\lambda_s, \lambda_t\)) are chosen by Bayesian information criterion (BIC).

To install `FGLMtrunc` directly from CRAN, type in R console this command:

    install.packages("FGLMtrunc")

To load the `FGLMtrunc` package, type in R console:

    library(FGLMtrunc)

The function for fitting model is `fglm_trunc`, which have arguments to customize the fit. Below are details on some required arguments:

- `X.curves` is required for matrix of functional predictors.

- `Y` is required for response vector.

- Either `nbasis` or `knots` is needed to define the interior knots of B-spline.

Please use `?fglm_trunc` for more details on function arguments. We will demonstrate usages of other commonly used arguments by examples.

## Functional Linear Regression (`family="gaussian"`)

Functional linear regression model is the default choice of function `fglm_trunc` with argument `family="gaussian"`. For illustration, we use dataset `LinearExample`, which we created beforehand following Case I in simulation studies section from Liu et. al. (2020). This dataset contains \(n=200\) observations, and functional predictors are observed at \(p=101\) timepoints on \([0,1]\) interval. The true truncation point is \(\delta = 0.54\).

    data(LinearExample)
    Y_linear = LinearExample$Y
    Xcurves_linear = LinearExample$X.curves
    timeGrid = seq(0, 1, length.out = 101)
    plot(timeGrid, LinearExample$beta.true, type = 'l',
         main = 'True coefficient function', xlab = "t", ylab=expression(beta(t)))

### Fitting `FGLMtrunc` model for linear regression

We fit the model using 50 B-spline basis with default `degree=3` for cubic splines. Since argument `grid` is not specified, an equally spaced sequence of length \(p=101\) on \([0,1]\) interval (including boundaries) will automatically be used.

    fit = fglm_trunc(Y_linear, Xcurves_linear, nbasis = 50)

`fglm_trunc` also supports parallel computing to speed up the running time of tuning regularization parameters. Parallel backend must be registered before hand. Here is an example of using parallel with `doMC` backend (we cannot run the code here since it is not available for Windows) :

    library(doMC)
    registerDoMC(cores = 2)
    fit = fglm_trunc(Y_linear, Xcurves_linear, nbasis = 50, parallel = TRUE)

One can also manually provides `grid` or `knots` sequences (or both). If `knots` is specified, `nbasis` will be ignored.

    k <- 50 - 3 - 1 #Numbers of knots = nbasis - degree - 1
    knots_n <- seq(0, 1, length.out = k+2)[-c(1, k+2)] # Remove boundary knots
    fit2 = fglm_trunc(Y_linear, Xcurves_linear, grid = timeGrid, knots = knots_n)

`fit` and `fit2` fitted models will have the same results.

`fit` is an object of class `FGLMtrunc` that contains relevant estimation results. Please use `?fglm_trunc` for more details on function outputs. Function call and truncation point will be printed with `print` function:

    print(fit)
    #>
    #> Call:  fglm_trunc(Y = Y_linear, X.curves = Xcurves_linear, nbasis = 50)
    #>
    #>
    #> Optimal truncation point: 0.52

### Plotting with fitted `FGLMtrunc` model

We can visualize the estimates of functional parameter \(\beta\) directly with `plot`:

    plot(fit)

    #> NULL

The plot shows both smoothing and truncated estimates of \(\beta\). We can set argument `include_smooth=FALSE` to show only truncated estimate.

### Predicting with fitted `FGLMtrunc` model

Predict method for `FGLMtrunc` fits works similar to `predict.glm`. Type `"link"` is the default choice for `FGLMtrunc` object. For linear regression, both type `"link"` and `"response"` return fitted values. `newX.curves` is required for these predictions.

    predict(fit, newX.curves = Xcurves_linear[1:5,])
    #>            [,1]
    #> [1,]  2.1337413
    #> [2,]  1.2286605
    #> [3,]  1.8207552
    #> [4,] -0.2430198
    #> [5,]  0.5901628

To get truncated estimate of \(\beta\), we can use either `fit$beta.truncated` or `predict` function:

    predict(fit, type = "coefficients")
    #>   [1] -1.194642e+00 -1.233189e+00 -1.274041e+00 -1.321971e+00 -1.382684e+00
    #>   [6] -1.461724e+00 -1.560226e+00 -1.676778e+00 -1.806592e+00 -1.942359e+00
    #>  [11] -2.075460e+00 -2.195993e+00 -2.293871e+00 -2.358836e+00 -2.381285e+00
    #>  [16] -2.352969e+00 -2.266606e+00 -2.117256e+00 -1.901351e+00 -1.619372e+00
    #>  [21] -1.273650e+00 -8.713600e-01 -4.219470e-01  6.139018e-02  5.633530e-01
    #>  [26]  1.068248e+00  1.560120e+00  2.023704e+00  2.444542e+00  2.810437e+00
    #>  [31]  3.112588e+00  3.344368e+00  3.503214e+00  3.588157e+00  3.601536e+00
    #>  [36]  3.546803e+00  3.429575e+00  3.256279e+00  3.034685e+00  2.773190e+00
    #>  [41]  2.481013e+00  2.167832e+00  1.843081e+00  1.516008e+00  1.196723e+00
    #>  [46]  8.962979e-01  6.237543e-01  3.847519e-01  1.897238e-01  6.071026e-02
    #>  [51]  8.314897e-03  1.796018e-06  0.000000e+00  0.000000e+00  0.000000e+00
    #>  [56]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
    ....

## Functional Logistic Regression (`family="binomial"`)

For logistic regression, we use dataset `LogisticExample`, which is similar to `LinearExample`, but the response \(Y\) was generated as Bernoulli random variable.

    data(LogisticExample)
    Y_logistic = LogisticExample$Y
    Xcurves_logistic = LogisticExample$X.curves

### Fitting `FGLMtrunc` model for logistic regression

Similarly, we fit the model using 50 B-spline basis with default choice of cubic splines. We need to set `family="binomial"` for logistic regression. Printing and plotting are the same as before.

    fit4 = fglm_trunc(Y_logistic, Xcurves_logistic, family="binomial", nbasis = 50)

    print(fit4)
    #>
    #> Call:  fglm_trunc(Y = Y_logistic, X.curves = Xcurves_logistic, family = "binomial", nbasis = 50)
    #>
    #>
    #> Optimal truncation point: 0.58
    plot(fit4)

    #> NULL

### Predicting with fitted `FGLMtrunc` model for logistic regression

For functional logistic regression, each `type` option returns a different prediction:

- `type="link"` gives the linear predictors which are log-odds.

- `type="response"` gives the predicted probabilities.

- `type="coefficients"` gives truncated estimate of functional parameter \(\beta\) as before.

    logistic_link_pred = predict(fit4, newX.curves = Xcurves_logistic, type="link")
    plot(logistic_link_pred, ylab="log-odds")

    logistic_response_pred = predict(fit4, newX.curves = Xcurves_logistic, type="response")
    plot(logistic_response_pred, ylab="predicted probabilities")

## Functional Linear Regression with scalar predictors

### Fitting `FGLMtrunc` model

`FGLMtrunc` allows using scalar predictors together with functional predictors. First, we randomly generate observations for scalar predictors:

    scalar_coef <- c(1, -1, 0.5) # True coefficients for scalar predictors
    set.seed(1234)
    S <- cbind(matrix(rnorm(400), nrow=200), rbinom(200, 1, 0.5))  # Randomly generated observations for scalar predictors. Binary coded as 0 and 1.
    colnames(S) <- c("s1", "s2", "s3")

Next, we modify the response vector from `LinearExample` so that it takes into account scalar predictors:

    Y_scalar <- Y_linear + (S %*% scalar_coef)

Then we fit `FGLMtrunc` model with the matrix of scalar predictors `S`:

    fit_scalar = fglm_trunc(X.curves=Xcurves_linear, Y=Y_scalar, S=S, nbasis = 50)
    fit_scalar
    #>
    #> Call:  fglm_trunc(Y = Y_scalar, X.curves = Xcurves_linear, S = S, nbasis = 50)
    #>
    #>  Intercept      s1      s2     s3
    #>     1.0865  1.0937 -1.0411 0.4233
    #>
    #> Optimal truncation point: 0.49

Fitted coefficients for scalar predictors are close to the true values.

### Predicting with scalar predictors

To make prediction with fitted model using scalar predictors, we need to specified argument `newS`:

    predict(fit_scalar, newX.curves = Xcurves_linear[1:5,], newS=S[1:5,])
    #>            [,1]
    #> [1,]  0.3837103
    #> [2,]  1.2553910
    #> [3,]  3.2764746
    #> [4,] -3.4882131
    #> [5,]  0.7946992