LICORS (version 0.2.0)

estimate_state_adj_matrix: Estimate adjacency matrix for equivalent FLC distributions based on states

Description

This function estimates the adjacency matrix \(\mathbf{A}\) of all pairwise equivalent FLC distributions given the states \(s_1, \ldots, s_K\). See Details below.

Usage

estimate_state_adj_matrix(states = NULL, FLCs = NULL, pdfs.FLC = NULL, alpha = NULL, 
    distance = function(f, g) return(mean(abs(f - g))))

Arguments

states

vector of length \(N\) with entry \(i\) being the label \(k = 1, \ldots, K\) of PLC \(i\)

FLCs

\(N \times n_f\) matrix of FLCs (only necessary if distance= "KS")

pdfs.FLC

\(N \times K\) matrix of all \(K\) state-conditional FLC densities evaluated at each FLC \(\ell^{+}_i\), \(i=1, \ldots, N\) (only necessary if distance = function(f, g) return(...)).

alpha

significance level for testing. Default: alpha=NULL (this will return a p-value matrix if method == "KS")

distance

either a Kolmogorov-Smirnov test (distance = "KS") or a function metric (e.g. \(L_q\) distance). For a distance function, distance requires as input a function of \(f\) and \(g\) that returns one value.

Default: distance = function(f, g) return(mean(abs(f-g))) \(\rightarrow\) \(L_1\) distance.

Value

A \(K \times K\) adjacency matrix with a trimmed version of exp(-distance) or p-values. If alpha!=NULL then it returns the thresholded \(0/1\) matrix. However, here \(1\) stands for equivalent, i.e. not rejecting. The matrix is obtained by checking for pval>alpha (rather than the usual pval<alpha).

Details and user-defined distance function

The \((i,j)\)th element of the adjacency matrix is defined as $$ \mathbf{A}_{ij} = distance(P(X \mid s_i), P(X \mid s_j)) = distance(f, g), $$ where distance is either

a metric

in the function space of pdfs \(f\) and \(g\), or

a two sample test

for \(H_0: f=g\), e.g. a Kolmogorov-Smirnov test (distance="KS").

Again we use a functional programming approach and allow the user to specify any valid distance/similarity function distance = function(f, g) return(...).

If distance="KS" the adjacency matrix contains p-values of a Kolmogorov-Smirnov test or the thresholded versions (if alpha!=NULL) - see Return for details.

Otherwise distance is an R function that takes as an input two vectors f and g (e.g. the wKDE estimates for two states), and returns a non-negative, real number to estimate their distance. Default is the \(L_1\) distance distance = function(f, g) return(mean(abs(f-g))).

Examples

# NOT RUN {
WW <- matrix(runif(10000), ncol = 10)
WW <- normalize(WW)
temp_flcs <- cbind(rnorm(nrow(WW)))
temp_pdfs.FLC <- estimate_LC_pdfs(temp_flcs, WW)
AA_ks <- estimate_state_adj_matrix(states = weight_matrix2states(WW), FLCs = temp_flcs, 
    distance = "KS")
AA_L1 <- estimate_state_adj_matrix(pdfs.FLC = temp_pdfs.FLC)

par(mfrow = c(1, 2), mar = c(1, 1, 2, 1))
image2(AA_ks, zlim = c(0, 1), legend = FALSE, main = "Kolmogorov-Smirnov")
image2(AA_L1, legend = FALSE, main = "L1 distance")
# }