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LWU HRF Basis for Taylor Expansion

hrf_basis_lwu.Rd

Constructs the basis set for the Lag-Width-Undershoot (LWU) HRF model, intended for Taylor expansion-based fitting. The basis consists of the LWU HRF evaluated at a given expansion point theta0, and its partial derivatives with respect to its parameters (tau, sigma, rho).

Usage

hrf_basis_lwu(theta0, t, normalize_primary = "none")

Arguments

theta0

A numeric vector of length 3 specifying the expansion point c(tau0, sigma0, rho0) for the LWU parameters.

t

A numeric vector of time points (in seconds) at which to evaluate the basis.

normalize_primary

Character string, one of "none" or "height". If "height", the primary HRF column (h0(t)) is normalized to have a peak absolute value of 1. For Taylor expansion fitting as described in Fit_LRU.md, this should typically be "none" as the scaling is absorbed by the beta coefficient. Default is "none".

Value

A numeric matrix of dimension length(t) x 4. Columns are: \itemize \item\codeh0: LWU HRF evaluated at theta0, \eqnh(t; \tau_0, \sigma_0, \rho_0) \item\coded_tau: Partial derivative \eqn\partial h / \partial \tau evaluated at theta0 \item\coded_sigma: Partial derivative \eqn\partial h / \partial \sigma evaluated at theta0 \item\coded_rho: Partial derivative \eqn\partial h / \partial \rho evaluated at theta0

See also

hrf_lwu, grad

Other hrf_functions: hrf_bspline(), hrf_gamma(), hrf_gaussian(), hrf_inv_logit(), hrf_lwu(), hrf_mexhat(), hrf_sine(), hrf_spmg1(), hrf_time()

Examples

t_points <- seq(0, 30, by = 0.5)
theta0_default <- c(tau = 6, sigma = 1, rho = 0.35)

# Generate the basis set
lwu_basis <- hrf_basis_lwu(theta0_default, t_points)
dim(lwu_basis) # Should be length(t_points) x 4
#> [1] 61  4
head(lwu_basis)
#>                 h0         d_tau       d_sigma         d_rho
#> [1,] -1.289099e-06  3.984647e-06 -2.390788e-05 -3.726653e-06
#> [2,] -5.656107e-06  1.587675e-05 -8.732212e-05 -1.693161e-05
#> [3,] -2.069270e-05  4.813840e-05 -2.406920e-04 -6.976958e-05
#> [4,] -5.119678e-05  5.142627e-05 -2.314182e-04 -2.607488e-04
#> [5,]  2.612342e-05 -6.168367e-04  2.467347e-03 -8.838263e-04
#> [6,]  1.236518e-03 -5.613114e-03  1.964590e-02 -2.717065e-03

# Plot the basis functions
matplot(t_points, lwu_basis, type = "l", lty = 1,
        main = "LWU HRF Basis Functions", ylab = "Value", xlab = "Time (s)")
legend("topright", colnames(lwu_basis), col = 1:4, lty = 1, cex = 0.8)


# Example with primary HRF normalization (not typical for Taylor fitting step)
lwu_basis_norm_h0 <- hrf_basis_lwu(theta0_default, t_points, normalize_primary = "height")
plot(t_points, lwu_basis_norm_h0[,1], type="l", main="Normalized h0 in Basis")

max(abs(lwu_basis_norm_h0[,1])) # Should be 1
#> [1] 1