Function-on-scalar regression model, denote \(n\) as total number of observations, \(p\) the number of coefficients, \(K\) as the number of B-splines, \(T\) as total time points.
Arguments
- Y
Numeric \(n \times T\) matrix, response function.
- X
Numeric \(n \times p\) matrix, design matrix
- lambda
Regularization parameter \(\gamma\)
- time
Time domain, numerical length of \(T\)
- nsp
Integer or 'auto', number of B-splines \(K\); default is 'auto'
- ord
B-spline order, default is
4; must be \(\geq 3\)- alpha
Bridge parameter \(\alpha\), default is
0.5- W
A \(T \times T\) weight matrix or
NULL(identity matrix); default isNULL- init
Initial \(\gamma\); default is
NULL- max_iter
Number of outer iterations
- inner_iter
Number of \(ADMM\) iterations (inner steps)
- CI
Logical, whether to calculate theoretical confidence intervals
- ...
Ignored
Value
A spfda.model object (environment) with following elements:
- B
B-spline basis functions used
- error
Root Mean Square Error ('RMSE')
- CI
Whether confidence intervals are calculated
- gamma
B-spline coefficient \(\gamma_{p \times K}\)
- generate_splines
Function to generate B-splines given time points
- K
Number of B-spline basis functions
- knots
B-spline knots used to fit the model
- predict
Function to predict responses \(\beta(t)\) given new
Xand/or time points- raw
A list of raw variables
Details
This function implements "Functional Group Bridge for Simultaneous
Regression and Support Estimation" (https://arxiv.org/abs/2006.10163).
The model estimates functional coefficients \(\beta(t)\) under model
\[y(t) = X\beta(t) + \epsilon(t)\] with B-spline basis expansion
\[\beta(t) = \gamma B(t) + R(t), \] where \( R(t) \) is B-spline
approximation error. The objective function
\[
\left\| (Y-X\gamma B)W \right\|_{2}^{2} + \sum_{j,m}
\left\| \gamma_{j}^{T}\mathbf{1}(B^{t} > 0) \right\|_{1}^{\alpha}.
\]
The input response variable is a matrix. If \(y_{i}(t)\) are observed
at different time points, please interpolate (e.g.
kernel) before feeding in.
Examples
dat <- spfda_simulate()
x <- dat$X
y <- dat$Y
fit <- spfda(y, x, lambda = 5, CI = TRUE)
BIC(fit)
#> [1] 102.6161
plot(fit, col = c("orange", "dodgerblue3", "darkgreen"),
main = "Fitted with 95% CI", aty = c(0, 0.5, 1), atx = c(0,0.2,0.8,1))
matpoints(fit$time, t(dat$env$beta), type = 'l', col = 'black', lty = 2)
legend('topleft', c("Fitted", "Underlying"), lty = c(1,2))
print(fit)
#> Model: function-on-scalar with group-bridge penalty
#> Log-lik: -50042.01 (df=269)
#> E-BIC: 102.6161
#> RMSE : 1.069733
#> Parameters:
#> K: 50
#> alpha: 0.5
#> lambda: 5
coefficients(fit)
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] -7.012115e-07 1.692958e-06 3.035323e-06 3.522342e-06 3.216476e-06
#> [2,] 3.899492e-07 1.953288e-02 5.440272e-02 7.455603e-02 9.087678e-02
#> [3,] 4.824930e-06 3.062445e-06 1.234592e-06 2.454777e-07 5.725427e-09
#> [,6] [,7] [,8] [,9] [,10]
#> [1,] 2.137624e-06 5.671988e-07 -1.012244e-06 -2.128009e-06 -2.328751e-06
#> [2,] 1.302951e-01 1.853794e-01 2.228928e-01 2.366571e-01 2.564559e-01
#> [3,] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [,11] [,12] [,13] [,14] [,15]
#> [1,] -1.547007e-06 -5.688930e-07 -9.232223e-08 -4.274177e-10 0.000000e+00
#> [2,] 2.988815e-01 3.457207e-01 3.800646e-01 4.090581e-01 4.438675e-01
#> [3,] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 1.033472e-08
#> [,16] [,17] [,18] [,19] [,20] [,21]
#> [1,] 0.000000e+00 0.000000e+00 0.000000e+00 0.0000000000 0.000000000 0.00000000
#> [2,] 4.809348e-01 5.106553e-01 5.316664e-01 0.5489712600 0.567639934 0.59259491
#> [3,] 1.177189e-07 4.256611e-07 7.768799e-07 0.0001224084 0.005213564 0.02625419
#> [,22] [,23] [,24] [,25] [,26] [,27]
#> [1,] 0.00000000 0.0000000 9.403889e-08 7.523111e-07 2.347954e-06 3.841781e-06
#> [2,] 0.62422273 0.6528084 6.712570e-01 6.884208e-01 7.148511e-01 7.441919e-01
#> [3,] 0.07275711 0.1500462 2.551439e-01 3.541341e-01 4.125178e-01 4.486505e-01
#> [,28] [,29] [,30] [,31] [,32]
#> [1,] 3.678295e-06 2.045085e-06 4.497385e-07 -5.036751e-07 -1.427826e-06
#> [2,] 7.634550e-01 7.737690e-01 7.869391e-01 8.065791e-01 8.253142e-01
#> [3,] 5.023854e-01 5.808870e-01 6.661413e-01 7.422388e-01 7.968911e-01
#> [,33] [,34] [,35] [,36] [,37]
#> [1,] -2.618263e-06 -3.481290e-06 -3.484719e-06 -2.823682e-06 -1.888758e-06
#> [2,] 8.386304e-01 8.500369e-01 8.633195e-01 8.790275e-01 8.968078e-01
#> [3,] 8.271295e-01 8.502456e-01 8.823304e-01 9.237079e-01 9.691749e-01
#> [,38] [,39] [,40] [,41] [,42]
#> [1,] -1.171155e-06 -1.163954e-06 -1.713652e-06 -2.174629e-06 -2.113618e-06
#> [2,] 9.156166e-01 9.342083e-01 9.529468e-01 9.732888e-01 9.910018e-01
#> [3,] 1.001809e+00 1.001474e+00 9.785429e-01 9.662679e-01 9.743196e-01
#> [,43] [,44] [,45] [,46] [,47]
#> [1,] -1.390520e-06 -2.877211e-08 1.626933e-06 3.121352e-06 3.730909e-06
#> [2,] 9.944654e-01 9.808218e-01 9.678718e-01 9.725311e-01 9.969943e-01
#> [3,] 9.810109e-01 9.757735e-01 9.775317e-01 9.997103e-01 1.017775e+00
#> [,48] [,49] [,50] [,51] [,52]
#> [1,] 2.817584e-06 8.714609e-07 -1.138201e-06 -2.635401e-06 -3.358082e-06
#> [2,] 1.036748e+00 1.064897e+00 1.047697e+00 9.989148e-01 9.680815e-01
#> [3,] 1.004357e+00 9.812730e-01 9.891524e-01 1.020332e+00 1.030809e+00
#> [,53] [,54] [,55] [,56] [,57]
#> [1,] -3.416356e-06 -3.392399e-06 -3.640053e-06 -3.933152e-06 -3.977445e-06
#> [2,] 9.726943e-01 9.874546e-01 9.930983e-01 9.902115e-01 9.829315e-01
#> [3,] 1.011226e+00 9.983699e-01 1.011343e+00 1.022885e+00 1.010329e+00
#> [,58] [,59] [,60] [,61] [,62]
#> [1,] -3.520256e-06 -2.380238e-06 -8.264034e-07 6.696018e-07 1.623502e-06
#> [2,] 9.788561e-01 9.840580e-01 9.869441e-01 9.696805e-01 9.394833e-01
#> [3,] 9.944715e-01 1.001267e+00 1.014489e+00 1.001491e+00 9.667175e-01
#> [,63] [,64] [,65] [,66] [,67]
#> [1,] 1.554354e-06 4.599397e-07 -1.040031e-06 -2.339615e-06 -2.956360e-06
#> [2,] 9.223332e-01 9.233987e-01 9.202185e-01 8.993214e-01 8.714216e-01
#> [3,] 9.426183e-01 9.383351e-01 9.318091e-01 9.060340e-01 8.599339e-01
#> [,68] [,69] [,70] [,71] [,72]
#> [1,] -2.589401e-06 -1.597398e-06 -4.956303e-07 2.616530e-07 2.960383e-07
#> [2,] 8.480295e-01 8.307513e-01 8.181831e-01 8.049764e-01 7.846144e-01
#> [3,] 7.984197e-01 7.417279e-01 7.088987e-01 6.818144e-01 6.278567e-01
#> [,73] [,74] [,75] [,76] [,77]
#> [1,] -4.538372e-08 1.223854e-07 9.022284e-07 1.364667e-06 9.976339e-07
#> [2,] 7.595492e-01 7.388705e-01 7.212256e-01 6.915201e-01 6.453107e-01
#> [3,] 5.465497e-01 4.619456e-01 3.880247e-01 3.248358e-01 2.674511e-01
#> [,78] [,79] [,80] [,81] [,82]
#> [1,] 3.746302e-07 6.079649e-08 2.814652e-10 0.000000000 0.0000000000
#> [2,] 6.039739e-01 5.852963e-01 5.787304e-01 0.567730432 0.5437698866
#> [3,] 2.019148e-01 1.208014e-01 4.769353e-02 0.009482859 0.0002251713
#> [,83] [,84] [,85] [,86] [,87]
#> [1,] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [2,] 5.028267e-01 4.545120e-01 4.186110e-01 4.032374e-01 4.009719e-01
#> [3,] 5.381960e-06 6.450455e-06 8.047045e-06 1.003097e-05 1.103181e-05
#> [,88] [,89] [,90] [,91] [,92]
#> [1,] 0.000000e+00 0.000000e+00 2.049878e-07 1.639902e-06 5.114856e-06
#> [2,] 3.991240e-01 3.755551e-01 3.189491e-01 2.663931e-01 2.578203e-01
#> [3,] 1.023347e-05 8.240444e-06 5.869965e-06 4.021469e-06 3.451704e-06
#> [,93] [,94] [,95] [,96] [,97]
#> [1,] 8.337264e-06 7.889571e-06 4.322783e-06 1.195489e-06 1.049538e-07
#> [2,] 2.647675e-01 2.323313e-01 1.685371e-01 1.287731e-01 1.233559e-01
#> [3,] 3.710993e-06 3.848403e-06 3.500939e-06 2.763473e-06 1.871041e-06
#> [,98] [,99] [,100]
#> [1,] 0.000000e+00 0.000000e+00 0.000000e+00
#> [2,] 1.025808e-01 4.481035e-02 0.000000e+00
#> [3,] 1.219721e-06 -2.290948e-08 -5.455109e-06