get_elts_ab: The R implementation to get the elements necessary for...
In sqyu/genscore: Generalized Score Matching Estimators

Description Usage Arguments Details Value Examples

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

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

`hdx`	A matrix, h(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\'(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 μ=η=0) or not. Default to `TRUE`.
`profiled_if_noncenter`	A boolean, whether in the profiled setting (λ_η=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.

Computes the Γ matrix and the g vector for generalized score matching.

Here, Γ is block-diagonal, and in the non-profiled non-centered setting, the j-th block is composed of Γ_{KK,j}, Γ_{Kη,j} and its transpose, and finally Γ_{ηη,j}. In the centered case, only Γ_{KK,j} is computed. In the profiled non-centered case,

Γ_j=Γ_{KK,j}-Γ_{Kη,j}Γ_{ηη,j}^(-1)Γ_{Kη}'.

Similarly, in the non-profiled non-centered setting, g can be partitioned p parts, each with a p-vector g_{K,j} and a scalar g_{η,j}. In the centered setting, only g_{K,j} is needed. In the profiled non-centered case,

g_j=g_{K,j}-Γ_{Kη,j}Γ_{ηη,j}^(-1)g_{η,j}.

The formulae for the pieces above are

Γ_{KK,j}=1/n*∑_{i=1}^n h(Xij)*Xij^(2a-2)*Xi^a*(Xi^a)',

Γ_{Kη,j}=-1/n*∑_{i=1}^n h(Xij)*Xij^(a+b-2)*Xi^a,

Γ_{ηη,j}=1/n*∑_{i=1}^n h(Xij)*Xij^(2b-2),

g_{K,j}=1/n*∑_{i=1}^n (h'(Xij)*Xij^(a-1)+(a-1)*h(Xij)*Xij^(a-2))*Xi^a+a*h(Xij)*Xij^(2a-2)*e_{j,p},

g_{η,j}=1/n*∑_{i=1}^n -h'(Xij)*Xij^(b-1)-(b-1)*h(Xij)*Xij^(b-2)),

where 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 t1 and t2 defined as

t1=Γ_{ηη}^(-1)g_{η}, t2=Γ_{ηη}^(-1)Γ_{Kη}',

so that \hat{η}=t1-t2*vec(\hat{K}).

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 Γ after applying the multiplier.
`setting`	The setting. Same as input.
`g_K`	The g vector. In the non-profiled non-centered setting, this is the g sub-vector corresponding to K.
`Gamma_K`	The Γ matrix with no diagonal multiplier. In the non-profiled non-centered setting, this is the Γ sub-matrix corresponding to K.
`g_eta`	Returned in the non-profiled non-centered setting. The g sub-vector corresponding to η.
`Gamma_K_eta`	Returned in the non-profiled non-centered setting. The Γ sub-matrix corresponding to interaction between K and η.
`Gamma_eta`	Returned in the non-profiled non-centered setting. The Γ sub-matrix corresponding to η.
`t1,t2`	Returned in the profiled non-centered setting, where the η estimate can be retrieved from t1-t2\hat{K}* after appropriate resizing.

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)