init time-varying live

day2/live_code_examples/live_3_timevarying.R

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+library(mgcv)
+library(dplyr)
+library(marginaleffects)
+library(gratia)
+library(ggplot2); theme_set(theme_bw())
+
+#### Time-varying coefficients in mgcv ####
+
+# First a function to simulate from a squared exponential Gaussian Process
+# N: number of timepoints to simulate
+# c: constant (average coefficient value)
+# alpha: amplitude of variation
+# rho: length scale (how 'wiggly' should the time-varying effect be?)
+sim_gp = function(N, c, alpha, rho){
+  Sigma <- alpha ^ 2 *
+           exp(-0.5 * ((outer(1:N, 1:N, "-") / rho) ^ 2)) +
+           diag(1e-9, N)
+ c + mgcv::rmvn(1,
+                mu = rep(0, N),
+                V = Sigma)
+}
+
+# Simulate a time-varying coefficient that has a mean value of 0.5
+# (i.e. the effect tends to be positive, but can change smoothly over time)
+set.seed(3)
+N <- 150
+beta_t <- sim_gp(alpha = 0.75,
+               rho = 15,
+               c = 0.5,
+               N = N)
+
+# Plot the coefficient
+plot(beta_t, type = 'l', lwd = 3,
+     bty = 'l', xlab = 'Time',
+     ylab = 'Coefficient',
+     col = 'darkred')
+
+# Now simulate the covariate itself
+x <- rnorm(N, 5, sd = 1)
+plot(x, type = 'l', lwd = 3,
+     bty = 'l', xlab = 'Time',
+     ylab = 'Covariate (x)',
+     col = 'darkred')
+
+# Simulate the linear predictor, which is a simple linear model:
+# mu = beta_0 + x_t * beta_t
+beta_0 <- 3
+mu <- beta_0 + x * beta_t
+
+# And simulate some Gaussian noise
+y <- rnorm(N, mean = mu, sd = 0.5)
+plot(y, type = 'l', lwd = 3,
+     bty = 'l', xlab = 'Time',
+     ylab = 'Outcome (y)',
+     col = 'darkred')
+
+# Fit a model that assumes the coefficient is static
+dat <- data.frame(y, x, time = 1:N)
+mod0 <- gam(y ~ x + time, data = dat, method = 'REML')
+summary(mod0)
+plot_predictions(mod0, condition = 'x', points = 0.5)
+plot_predictions(mod0, by = 'time', points = 0.5)
+avg_slopes(mod0, variables = 'x')
+appraise(mod0, n_simulate = 100, method = 'simulate')
+
+# Now a model that allows the coefficient to vary over time;
+# this makes use of the 'by' argument in the s() function. The idea is
+# that we fit a smooth function of 'time' that gets multiplied with the covariate
+# x, effectively allowing us to estimate arbitrarily-nonlinear effects that change
+# over time
+mod1 <- gam(y ~ s(time, by = x, k = 50),
+            data = dat, method = 'REML')
+summary(mod1)
+draw(mod1)
+plot_predictions(mod1, by = c('time'), points = 0.5)
+
+summary_round = function(x){
+  round(fivenum(x, na.rm = TRUE), 4)
+}
+plot_predictions(mod1, by = c('time', 'x'),
+                 newdata = datagrid(x = summary_round,
+                                    time = 1:N),
+                 points = 0.5)
+
+# Easier to visualise the time-varying effect if we fix the x value
+mean_round = function(x){
+  round(mean(x, na.rm = TRUE), 4)
+}
+plot_predictions(mod1, by = c('time', 'x'),
+                 newdata = datagrid(x = mean_round,
+                                    time = 1:N))
+
+avg_slopes(mod1, variables = 'x')
+appraise(mod1, n_simulate = 100, method = 'simulate')
+
+# How would this smooth extrapolate?
+plot_predictions(mod1, by = c('time', 'x'),
+                 newdata = datagrid(x = mean_round,
+                                    time = 1:(N + 20)))
+
+# Not really very realistic; what we want is a proper time series
+# model for the coefficient. A quick way to do this in mgcv is to use
+# a Random Walk basis
+library(MRFtools) # devtools::install_github("eric-pedersen/MRFtools")
+rw_penalty <- mrf_penalty(object = 1:(N + 20),
+                          type = 'linear')
+dat$time_factor <- factor(1:N, levels = as.character(1:(N + 20)))
+dat$time_factor
+
+mod2 <- gam(y ~ s(time_factor, by = x,
+                  bs = "mrf",
+                  k = 25,
+                  xt = list(penalty = rw_penalty)),
+            data = dat,
+            drop.unused.levels = FALSE,
+            method = "REML")
+summary(mod2)
+draw(mod2)
+plot_predictions(mod2, by = c('time_factor', 'x'),
+                 newdata = datagrid(x = mean_round,
+                                    time_factor = 1:(N + 20)))
+
+# Because 'time' is a factor here, it is a bit harder to visualise. It is probably
+# easier to resort to our own plots
+newdat <- datagrid(model = mod2,
+                   x = mean_round,
+                   time_factor = 1:(N + 20)) %>%
+  # Add a continuous version of 'time' for plotting
+  dplyr::mutate(time = 1:(N + 20))
+preds <- predict(mod2,
+                 newdata = newdat,
+                 type = 'response', se = TRUE)
+newdat$pred <- preds$fit
+newdat$upper <- preds$fit + 1.96*preds$se.fit
+newdat$lower <- preds$fit - 1.96*preds$se.fit
+ggplot(newdat, aes(x = time, y = pred)) +
+  geom_line(linewidth = 1, alpha = 0.6) +
+  geom_ribbon(aes(ymin = lower, ymax = upper),
+              alpha = 0.3, col = NA) +
+  theme_classic() +
+  theme(legend.position = 'none')
+
+# These extrapolations are much more realistic!