# Projected Cramér--von Mises test statistic for the goodness-of-fit test of functional linear models Computation of the Projected Cramér--von Mises (PCvM) test statistic and its associated $A_\bullet$ matrix. For a sample of functional covariates $X_1, \ldots, X_n$, the test statistic is computed from c("$\n$", "$x_{1, p}, \\ldots, x_{n, p}$"), the coefficients (scores) of the sample in a $p$-truncated basis expansion, such as Functional Principal Components (FPC). The PCvM statistic is defined as [REMOVE_ME]$$ \mathrm{PCvM}_{n,p,q} = c \cdot \mathrm{tr}(\hat{\mathbf{E}}_q'\mathbf{A}_\bullet \hat{\mathbf{E}}_q)PCvM_{n, p, q} =c * tr(\hat E_q' A_\bullet \hat E_q) [REMOVE_ME_2]$$ where [REMOVE_ME]$$ c = 2 \pi^{(p + q) / 2 - 1} / (q \Gamma(p / 2) \Gamma(q / 2) n^2), [REMOVE_ME_2]$$ $\hat E_q$ is the $n x q$ matrix of multivariate residuals, and $A_\bullet$ is a $n x n$ matrix whose $ij$-th element is $\sum_{r = 1}^n A_{ijr}$, for $A_{ijr}$ depending on c("$\n$", "$(x_{i, p}, x_{j, p}, x_{r, p})$"). Its exact expression can be seen in Escanciano (2006) and García-Portugués et al. (2021). ```r flm_stat(E, p, Adot_vec, constant = TRUE) Adot(X) ``` ## Arguments - `E`: the matrix of multivariate residuals, with dimension `c(n, q)`. A vector if $q = 1$. - `p`: dimension of the covariates space. Must be a positive integer. - `Adot_vec`: output from `Adot`. A vector of length `n * (n - 1) / 2 + 1`. This corresponds to the most expensive computation in the test statistic. - `constant`: whether to include the constant of the PCvM test statistic, $c$, in its computation. Defaults to `TRUE`. - `X`: a matrix of size `c(n, p)` containing the coefficients (scores) of the functional data in a `p`-truncated **orthonormal** basis expansion, such as FPC. Must not contain repeated rows. ## Returns * `flm_stat`: the value of the test statistic, a scalar. * `A_dot`: a vector of length `n * (n - 1) / 2 + 1`. The first entry contains the common diagonal element of $A_\bullet$. The remaining entries are the upper triangular matrix (excluding the diagonal) of $A_\bullet$, stacked by columns. ## Description Computation of the Projected Cramér--von Mises (PCvM) test statistic and its associated $A_\bullet$ matrix. For a sample of functional covariates $X_1, \ldots, X_n$, the test statistic is computed from c("$\n$", "$x_{1, p}, \\ldots, x_{n, p}$"), the coefficients (scores) of the sample in a $p$-truncated basis expansion, such as Functional Principal Components (FPC). The PCvM statistic is defined as $$ \mathrm{PCvM}_{n,p,q} = c \cdot \mathrm{tr}(\hat{\mathbf{E}}_q'\mathbf{A}_\bullet \hat{\mathbf{E}}_q)PCvM_{n, p, q} =c * tr(\hat E_q' A_\bullet \hat E_q) $$ where $$ c = 2 \pi^{(p + q) / 2 - 1} / (q \Gamma(p / 2) \Gamma(q / 2) n^2), $$ $\hat E_q$ is the $n x q$ matrix of multivariate residuals, and $A_\bullet$ is a $n x n$ matrix whose $ij$-th element is $\sum_{r = 1}^n A_{ijr}$, for $A_{ijr}$ depending on c("$\n$", "$(x_{i, p}, x_{j, p}, x_{r, p})$"). Its exact expression can be seen in Escanciano (2006) and García-Portugués et al. (2021). ## Details `Adot` assumes that `X` does not have repeated rows or otherwise `NaN`s will be present in the result. If `X` has repeated rows, `Adot` will throw a warning. The implementation of the PCvM test statistic for scalar response is addressed in García-Portugués et al. (2014), whereas García-Portugués et al. (2021) presents its multivariate extension and shows that $A_\bullet$ induces a weighted quadratic norm (if there are no repetitions in the sample). The PCvM statistic is rooted in the proposal by Escanciano (2006). Both `flm_stat` and `A_dot` are coded in C++. ## Examples ```r ## flm_stat # Generate data n <- 200 q <- 2 p <- 3 E <- matrix(rnorm(n * q), nrow = n, ncol = q) X_fdata <- r_ou(n = n, t = seq(0, 1, l = 101)) # Compute FPC X_fpc <- fpc(X_fdata) # Adot Adot_vec <- Adot(X = X_fpc[["scores"]]) # Check equality constant <- n^(-2) * 2 * pi^((p / 2) - 1) / gamma(p / 2) constant * .Fortran("pcvm_statistic", n = as.integer(n), Adot_vec = Adot_vec, residuals = E[, 2], statistic = 0)$statistic flm_stat(E = E[, 2, drop = FALSE], p = p, Adot_vec = Adot_vec, constant = FALSE) ## Adot # Generate data n <- 200 X_fdata <- r_ou(n = n, t = seq(0, 1, l = 101)) # Compute FPC X_fpc <- fpc(X_fdata) # Using inprod_fdata and Adot Adot_vec <- Adot(X = X_fpc[["scores"]]) # Check with fda.usc::Adot with adequate inprod head(drop(Adot_vec)) head(fda.usc::Adot(X_fdata)) # Obtention of the entire Adot matrix Ad <- diag(rep(Adot_vec[1], n)) Ad[upper.tri(Ad, diag = FALSE)] <- Adot_vec[-1] head(Ad <- t(Ad) + Ad - diag(diag(Ad))) # Positive definite eigen(Ad)$values # # Warning if X contains repeated observations # Adot(X = rbind(1:3, 1:3, 3:5)) # Comparison with .Fortran("adot", PACKAGE = "fda.usc") n <- as.integer(n) a <- as.double(rep(0, (n * (n - 1) / 2 + 1))) inprod <- X_fpc[["scores"]] %*% t(X_fpc[["scores"]]) inprod <- inprod[upper.tri(inprod, diag = TRUE)] X <- X_fpc[["scores"]] microbenchmark::microbenchmark( .Fortran("adot", n = n, inprod = inprod, Adot_vec = a, PACKAGE = "fda.usc"), Adot(X = X), times = 50, control = list(warmup = 10)) ``` ## References García-Portugués, E., Álvarez-Liébana, J., Álvarez-Pérez, G. and Gonzalez-Manteiga, W. (2021). A goodness-of-fit test for the functional linear model with functional response. **Scandinavian Journal of Statistics**, 48(2):502--528. tools:::Rd_expr_doi("10.1111/sjos.12486") Escanciano, J. C. (2006) A consistent diagnostic test for regression models using projections. **Econometric Theory**, 22(6):1030–-1051. tools:::Rd_expr_doi("10.1017/S0266466606060506") García-Portugués, E., González-Manteiga, W. and Febrero-Bande, M. (2014). A goodness-of-fit test for the functional linear model with scalar response. **Journal of Computational and Graphical Statistics**, 23(3):761--778. tools:::Rd_expr_doi("10.1080/10618600.2013.812519") ## Author(s) Eduardo García-Portugués.

flm_stat function