get_elts_ab: The R implementation to get the elements necessary for calculations for general a and b.

Description

The R implementation to get the elements necessary for calculations for general $a$ and $b$.

Usage

get_elts_ab(
  hdx,
  hpdx,
  x,
  a,
  b,
  setting,
  centered = TRUE,
  profiled_if_noncenter = TRUE,
  scale = "",
  diagonal_multiplier = 1
)

Value

A list that contains the elements necessary for estimation.

n: The sample size.
p: The dimension.
centered: The centered setting or not. Same as input.
scale: The scaling method. Same as input.
diagonal_multiplier: The diagonal multiplier. Same as input.
diagonals_with_multiplier: A vector that contains the diagonal entries of $\boldsymbol{\Gamma}$ after applying the multiplier.
setting: The setting. Same as input.
g_K: The $\boldsymbol{g}$ vector. In the non-profiled non-centered setting, this is the $\boldsymbol{g}$ sub-vector corresponding to $\mathbf{K}$.
Gamma_K: The $\boldsymbol{\Gamma}$ matrix with no diagonal multiplier. In the non-profiled non-centered setting, this is the $\boldsymbol{\Gamma}$ sub-matrix corresponding to $\mathbf{K}$.
g_eta: Returned in the non-profiled non-centered setting. The $\boldsymbol{g}$ sub-vector corresponding to $\boldsymbol{\eta}$.
Gamma_K_eta: Returned in the non-profiled non-centered setting. The $\boldsymbol{\Gamma}$ sub-matrix corresponding to interaction between $\mathbf{K}$ and $\boldsymbol{\eta}$.
Gamma_eta: Returned in the non-profiled non-centered setting. The $\boldsymbol{\Gamma}$ sub-matrix corresponding to $\boldsymbol{\eta}$.
t1,t2: Returned in the profiled non-centered setting, where the $\boldsymbol{\eta}$ estimate can be retrieved from $\boldsymbol{t_1}-\boldsymbol{t_2}\hat{\mathbf{K}}$ after appropriate resizing.

Arguments

hdx: A matrix, $h(\mathbf{x})$ applied to the distance of x from the boundary of the domain, should be of the same dimension as x.
hpdx: A matrix, $h'(\mathbf{x})$ applied to the distance of x from the boundary of the domain, should be of the same dimension as x.
x: An n by p matrix, the data matrix, where n is the sample size and p the dimension.
a: A number, must be strictly larger than $b/2$.
b: A number, must be >= 0.
setting: A string that indicates the distribution type. Returned without being checked or used in the function body.
centered: A boolean, whether in the centered setting (assume $\boldsymbol{\mu}=\boldsymbol{\eta}=0$) or not. Default to TRUE.
profiled_if_noncenter: A boolean, whether in the profiled setting ($\lambda_{\boldsymbol{\eta}}=0$) if non-centered. Parameter ignored if centered=TRUE. Default to TRUE.
scale: A string indicating the scaling method. Returned without being checked or used in the function body. Default to "norm".
diagonal_multiplier: A number >= 1, the diagonal multiplier.

Details

Computes the $\boldsymbol{\Gamma}$ matrix and the $\boldsymbol{g}$ vector for generalized score matching.

Here, $\boldsymbol{\Gamma}$ is block-diagonal, and in the non-profiled non-centered setting, the $j$-th block is composed of $\boldsymbol{\Gamma}_{\mathbf{KK},j}$, $\boldsymbol{\Gamma}_{\mathbf{K}\boldsymbol{\eta},j}$ and its transpose, and finally $\boldsymbol{\Gamma}_{\boldsymbol{\eta\eta},j}$. In the centered case, only $\boldsymbol{\Gamma}_{\mathbf{KK},j}$ is computed. In the profiled non-centered case, $$\boldsymbol{\Gamma}_{j}\equiv\boldsymbol{\Gamma}_{\mathbf{KK},j}-\boldsymbol{\Gamma}_{\mathbf{K}\boldsymbol{\eta},j}\boldsymbol{\Gamma}_{\boldsymbol{\eta}\boldsymbol{\eta},j}^{-1}\boldsymbol{\Gamma}_{\mathbf{K}\boldsymbol{\eta}}^{\top}.$$

Similarly, in the non-profiled non-centered setting, $\boldsymbol{g}$ can be partitioned $p$ parts, each with a $p$-vector $\boldsymbol{g}_{\mathbf{K},j}$ and a scalar $g_{\boldsymbol{\eta},j}$. In the centered setting, only $\boldsymbol{g}_{\mathbf{K},j}$ is needed. In the profiled non-centered case, $$\boldsymbol{g}_j\equiv\boldsymbol{g}_{\mathbf{K},j}-\boldsymbol{\Gamma}_{\mathbf{K}\boldsymbol{\eta},j}\boldsymbol{\Gamma}_{\boldsymbol{\eta\eta},j}^{-1}g_{\boldsymbol{\eta},j}.$$

The formulae for the pieces above are $$\boldsymbol{\Gamma}_{\mathbf{KK},j}\equiv\frac{1}{n}\sum_{i=1}^nh\left(X_j^{(i)}\right){X_j^{(i)}}^{2a-2}{\boldsymbol{X}^{(i)}}^a{{\boldsymbol{X}^{(i)}}^a}^{\top},$$ $$\boldsymbol{\Gamma}_{\mathbf{K}\boldsymbol{\eta},j}\equiv-\frac{1}{n}\sum_{i=1}^nh\left(X_j^{(i)}\right){X_j^{(i)}}^{a+b-2}{\boldsymbol{X}^{(i)}}^a,$$ $$\boldsymbol{\Gamma}_{\boldsymbol{\eta\eta},j}\equiv\frac{1}{n}\sum_{i=1}^nh\left(X_j^{(i)}\right){X_j^{(i)}}^{2b-2},$$ $$\boldsymbol{g}_{\mathbf{K},j}\equiv\frac{1}{n}\sum_{i=1}^n\left(h'\left(X_j^{(i)}\right){X_j^{(i)}}^{a-1}+(a-1)h\left(X_j^{(i)}\right){X_j^{(i)}}^{a-2}\right){\boldsymbol{X}^{(i)}}^a+ah\left(X_j^{(i)}\right){X_j^{(i)}}^{2a-2}\boldsymbol{e}_{j,p},$$ $$\boldsymbol{g}_{\boldsymbol{\eta},j}\equiv\frac{1}{n}\sum_{i=1}^n-h'\left(X_j^{(i)}\right){X_j^{(i)}}^{b-1}-(b-1)h\left(X_j^{(i)}\right){X_j^{(i)}}^{b-2},$$ where $\boldsymbol{e}_{j,p}$ is the $p$-vector with 1 at the $j$-th position and 0 elsewhere.

In the profiled non-centered setting, the function also returns $t_1$ and $t_2$ defined as $$\boldsymbol{t}_1\equiv\boldsymbol{\Gamma}_{\boldsymbol{\eta\eta}}^{-1}\boldsymbol{g}_{\boldsymbol{\eta}},\quad\boldsymbol{t}_2\equiv\boldsymbol{\Gamma}_{\boldsymbol{\eta\eta}}^{-1}\boldsymbol{\Gamma}_{\mathbf{K}\boldsymbol{\eta}}^{\top},$$ so that $\hat{\boldsymbol{\eta}}=\boldsymbol{t}_1-\boldsymbol{t}_2\mathrm{vec}(\hat{\mathbf{K}}).$

Examples

Run this code

n <- 50
p <- 30
eta <- rep(0, p)
K <- diag(p)
domain <- make_domain("R+", p=p)
x <- gen(n, setting="ab_1/2_7/10", abs=FALSE, eta=eta, K=K, domain=domain, finite_infinity=100, 
       xinit=NULL, seed=2, burn_in=1000, thinning=100, verbose=FALSE)
h_hp <- get_h_hp("min_pow", 1.5, 3)
h_hp_dx <- h_of_dist(h_hp, x, domain) # h and h' applied to distance from x to boundary
elts <- get_elts_ab(h_hp_dx$hdx, h_hp_dx$hpdx, x, a=0.5, b=0.7, setting="ab_1/2_7/10",
            centered=TRUE, scale="norm", diag=1.5)
elts <- get_elts_ab(h_hp_dx$hdx, h_hp_dx$hpdx, x, a=0.5, b=0.7, setting="ab_1/2_7/10",
            centered=FALSE, profiled_if_noncenter=TRUE, scale="norm", diag=1.7)
elts <- get_elts_ab(h_hp_dx$hdx, h_hp_dx$hpdx, x, a=0.5, b=0.7, setting="ab_1/2_7/10",
            centered=FALSE, profiled_if_noncenter=FALSE, scale="norm", diag=1.9)

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