A fancy interactive htmlwidgets of the perspective plot for Qindex model(s) using package plotly.
integrandSurface(
...,
newdata = data,
proj_Q_p = TRUE,
proj_S_p = TRUE,
proj_beta = TRUE,
n = 501L,
newid = seq_len(min(50L, .row_names_info(newdata, type = 2L))),
ylim = range(X, newX),
xcol = "dodgerblue",
ycol = "deeppink",
zcol = "darkolivegreen",
surface_col = c("white", "lightgreen")
)Function integrandSurface returns a pretty htmlwidgets created by R package plotly
to showcase the perspective plot of the estimated sign-adjusted integrand surface \(\hat{S}(p,q)\).
If a set of training/test subjects is selected (via parameter newid), then
the estimated sign-adjusted line integrand curve \(\hat{S}\big(p, Q_i(p)\big)\) of subject \(i\) is displayed on the surface \(\hat{S}(p,q)\);
the quantile curve \(Q_i(p)\)
is projected on the \((p,q)\)-plain of the 3-dimensional \((p,q,s)\) cube,
if proj_Q_p=TRUE (default);
the user-specified \(\tilde{p}\) is marked on the \((p,q)\)-plain of the 3D cube,
if proj_Q_p=TRUE (default);
\(\hat{S}\big(p, Q_i(p)\big)\)
is projected on the \((p,s)\)-plain of the 3-dimensional \((p,q,s)\) cube,
if one and only one Qindex model is provided in in
put argument ... and proj_S_p=TRUE (default);
the estimated linear functional coefficient \(\hat{\beta}(p)\) is shown on the \((p,s)\)-plain of the 3D cube,
if one and only one linear Qindex model is provided in input argument ... and proj_beta=TRUE (default).
one or more Qindex models based on a same training set.
data.frame, with at least the response \(y^{\text{new}}\) and the double matrix of functional predictor values \(X^{\text{new}}\) of the test set. The test functional predictor values \(X^{\text{new}}\) are tabulated on the same \(x\)-grid as the training functional predictor values \(X\). If missing, the training set will be used.
logical scalar, whether to show
the projection of \(\hat{S}\big(p, Q_i(p)\big)\)
(see sections Details and Value)
to the \((p,q)\)-plain, default TRUE
logical scalar, whether to show
the projection of \(\hat{S}\big(p, Q_i(p)\big)\) to the \((p,s)\)-plain, default TRUE
logical scalar, whether to show
\(\hat{\beta}(p)\) on the \((p,s)\)-plain when applicable, default TRUE
integer scalar, fineness of visualization,
default 501L. See parameter n.grid of function vis.gam.
integer scalar or vector,
row indices of newdata to be visualized.
Default 1:2, i.e., the first two test subjects.
Use newid = NULL to disable visualization of newdata.
length-2 double vector, range on \(y\)-axis. Default is the range of \(X\) and \(X^{\text{new}}\) combined.
character scalars, colors of the \(x\)-, \(y\)- and \(z\)-axes
length-2 character vector, color of the integrand surface(s), for lowest and highest \(z\) values
The quantile index (QI), $$\text{QI}=\displaystyle\int_0^1\beta(p)\cdot Q(p)\,dp$$ with a linear functional coefficient \(\beta(p)\) can be estimated by fitting a functional generalized linear model (FGLM, James, 2002) to exponential-family outcomes, or by fitting a linear functional Cox model (LFCM, Gellar et al., 2015) to survival outcomes. More flexible non-linear quantile index (nlQI) $$ \text{nlQI}=\displaystyle\int_0^1 F\big(p, Q(p)\big)\,dp $$ with a bivariate twice differentiable function \(F(\cdot,\cdot)\) can be estimated by fitting a functional generalized additive model (FGAM, McLean et al., 2014) to exponential-family outcomes, or by fitting an additive functional Cox model (AFCM, Cui et al., 2021) to survival outcomes.
The estimated integrand surface of quantile indices and non-linear quantile indices, defined on \(p\in[0,1]\) and \(q\in\text{range}\big(Q_i(p)\big)\) for all training subjects \(i=1,\cdots,n\), is $$ \hat{S}_0(p,q) = \begin{cases} \hat{\beta}(p)\cdot q & \text{for QI}\\ \hat{F}(p,q) & \text{for nlQI} \end{cases} $$
Ideally, we would wish that, in the training set, the estimated linear and/or non-linear quantile indices $$ \widehat{\text{QI}}_i = \displaystyle\int_0^1 \hat{S}_0\big(p, Q_i(p)\big)dp $$ be positively correlated with a more intuitive quantity, e.g., quantiles \(Q_i(\tilde{p})\) at a user-specified \(\tilde{p}\), for the interpretation of downstream analysis, Therefore, we define the sign-adjustment term $$ \hat{c} = \text{sign}\left(\text{corr}\left(Q_i(\tilde{p}), \widehat{\text{QI}}_i\right)\right),\quad i =1,\cdots,n $$ as the sign of the correlation between the estimated quantile index \(\widehat{\text{QI}}_i\) and the quantile \(Q_i(\tilde{p})\), for training subjects \(i=1,\cdots,n\).
The estimated sign-adjusted integrand surface is \(\hat{S}(p,q) = \hat{c} \cdot \hat{S}_0(p,q)\).
The estimated sign-adjusted quantile indices \(\int_0^1 \hat{S}\big(p, Q_i(p)\big)dp\) are positively correlated with subject-specific sample medians (default \(\tilde{p} = .5\)) in the training set.
James, G. M. (2002). Generalized Linear Models with Functional Predictors, tools:::Rd_expr_doi("10.1111/1467-9868.00342")
Gellar, J. E., et al. (2015). Cox regression models with functional covariates for survival data, tools:::Rd_expr_doi("10.1177/1471082X14565526")
Mathew W. M., et al. (2014) Functional Generalized Additive Models, tools:::Rd_expr_doi("10.1080/10618600.2012.729985")
Cui, E., et al. (2021). Additive Functional Cox Model, tools:::Rd_expr_doi("10.1080/10618600.2020.1853550")