library(brms)
library(loo)
theme_set(theme_default())

CHAINS <- 4
SEED <- 1234
set.seed(SEED)

## Introduction

One of the most common goals of a time series analysis is to use the observed series to inform predictions for future observations. We will refer to this task of predicting a sequence of $$M$$ future observations as $$M$$-step-ahead prediction ($$M$$-SAP). Fortunately, once we have fit a model and can sample from the posterior predictive distribution, it is straightforward to generate predictions as far into the future as we want. It is also straightforward to evaluate the $$M$$-SAP performance of a time series model by comparing the predictions to the observed sequence of $$M$$ future data points once they become available.

Unfortunately, we are often in the position of having to use a model to inform decisions before we can collect the future observations required for assessing the predictive performance. If we have many competing models we may also need to first decide which of the models (or which combination of the models) we should rely on for predictions. In these situations the best we can do is to use methods for approximating the expected predictive performance of our models using only the observations of the time series we already have.

If there were no time dependence in the data or if the focus is to assess the non-time-dependent part of the model, we could use methods like leave-one-out cross-validation (LOO-CV). For a data set with $$N$$ observations, we refit the model $$N$$ times, each time leaving out one of the $$N$$ observations and assessing how well the model predicts the left-out observation. LOO-CV is very expensive computationally in most realistic settings, but the Pareto smoothed importance sampling (PSIS, Vehtari et al, 2017ab) algorithm provided by the loo package allows for approximating exact LOO-CV with PSIS-LOO-CV. PSIS-LOO-CV requires only a single fit of the full model and comes with diagnostics for assessing the validity of the approximation.

With a time series we can do something similar to LOO-CV but, except in a few cases, it does not make sense to leave out observations one at a time because then we are allowing information from the future to influence predictions of the past (i.e., times $$t + 1, t+2, \ldots$$ should not be used to predict for time $$t$$). To apply the idea of cross-validation to the $$M$$-SAP case, instead of leave-one-out cross-validation we need some form of leave-future-out cross-validation (LFO-CV). As we will demonstrate in this case study, LFO-CV does not refer to one particular prediction task but rather to various possible cross-validation approaches that all involve some form of prediction for new time series data. Like exact LOO-CV, exact LFO-CV requires refitting the model many times to different subsets of the data, which is computationally very costly for most nontrivial examples, in particular for Bayesian analyses where refitting the model means estimating a new posterior distribution rather than a point estimate.

Although PSIS-LOO-CV provides an efficient approximation to exact LOO-CV, until now there has not been an analogous approximation to exact LFO-CV that drastically reduces the computational burden while also providing informative diagnostics about the quality of the approximation. In this case study we present PSIS-LFO-CV, an algorithm that typically only requires refitting the time-series model a small number times and will make LFO-CV tractable for many more realistic applications than previously possible.

## $$M$$-step-ahead predictions

Assume we have a time series of observations $$\mathbf{y} = (y_1, y_2, \ldots, y_N)$$ and let $$L$$ be the minimum number of observations from the series that we will require before making predictions for future data. Depending on the application and how informative the data is, it may not be possible to make reasonable predictions for $$y_{i}$$ based on $$(y_1, \dots, y_{i-1})$$ until $$i$$ is large enough so that we can learn enough about the time series to predict future observations. Setting $$L=10$$, for example, means that we will only assess predictive performance starting with observation $$y_{11}$$, so that we always have at least 10 previous observations to condition on.

In order to assess $$M$$-SAP performance we would like to compute the predictive densities

$p({\mathbf{y}_{{i<M}}} \,|\, {\mathbf{y}_{{<i}}}) = p(y_i, \ldots, y_{i + M - 1} \,|\, y_{1},...,y_{i-1})$

for each $$i \in \{L + 1, \ldots, N - M + 1\}$$. The quantities $$p({\mathbf{y}_{{i<M}}} \,|\, {\mathbf{y}_{{<i}}})$$ can be computed with the help of the posterior distribution $$p(\theta \,|\, {\mathbf{y}_{{<i}}})$$ of the parameters $$\theta$$ conditional on only the first $$i-1$$ observations of the time-series:

$p({\mathbf{y}_{{i<M}}} \,| \, {\mathbf{y}_{{<i}}}) = \int p({\mathbf{y}_{{i<M}}} \,| \, {\mathbf{y}_{{<i}}}, \theta) \, p(\theta\,|\,{\mathbf{y}_{{<i}}}) \,d\theta.$

Having obtained $$S$$ draws $$(\theta_{<i}^{(1)}, \ldots, \theta_{<i}^{(S)})$$ from the posterior distribution $$p(\theta\,|\,{\mathbf{y}_{{<i}}})$$, we can estimate $$p({\mathbf{y}_{{i<M}}} | {\mathbf{y}_{{<i}}})$$ as

$p({\mathbf{y}_{{i<M}}} \,|\, {\mathbf{y}_{{<i}}}) \approx \sum_{s=1}^S p({\mathbf{y}_{{i<M}}} \,|\, {\mathbf{y}_{{<i}}}, \theta_{<i}^{(s)}).$

In the following, we consider factorizable models in which the response values are conditionally independent given the parameters and the likelihood can be written in the familiar form

$p(\mathbf{y} \,|\, \theta) = \prod_{i=1}^N p(y_i \,|\, \theta).$

In this case, $$p({\mathbf{y}_{{i<M}}} \,|\, {\mathbf{y}_{{<i}}}, \theta_{<i})$$ reduces to $p({\mathbf{y}_{{i<M}}} \,|\, \theta_{<i}) = \prod_{j = i}^{i + M -1} p(y_j \,|\, \theta_{<i}),$ due to the assumption of conditional independence between $${\mathbf{y}_{{i<M}}}$$ and $${\mathbf{y}_{{<i}}}$$ given $$\theta_{<i}$$.

Non-factorizable models, which do not make this assumption, are discussed in a separate vignette Leave-one-out cross-validation for non-factorizable models.

## Approximate $$M$$-SAP using importance-sampling

Unfortunately, the math above makes use of the posterior distributions from many different fits of the model to different subsets of the data. That is, to obtain the predictive density $$p({\mathbf{y}_{{i<M}}} \,|\, {\mathbf{y}_{{<i}}})$$ requires fitting a model to only the first $$i-1$$ data points, and we will need to do this for every value of $$i$$ under consideration (all $$i \in \{L + 1, \ldots, N - M + 1\}$$).

To reduce the number of models that need to be fit for the purpose of obtaining each of the densities $$p({\mathbf{y}_{{i<M}}} \,|\, {\mathbf{y}_{{<i}}})$$, we propose the following algorithm. Starting with $$i = N - M + 1$$, we approximate each $$p({\mathbf{y}_{{i<M}}} \,|\, {\mathbf{y}_{{<i}}})$$ using Pareto smoothed importance sampling (PSIS, Vehtari et al, 2017ab):

$p({\mathbf{y}_{{i<M}}} \,|\, {\mathbf{y}_{{<i}}}) \approx \frac{ \sum_{s=1}^S w_i^{(s)}\, p({\mathbf{y}_{{i<M}}} \,|\, \theta^{(s)})}{ \sum_{s=1}^S w_i^{(s)}},$

where $$w_i^{(s)}$$ are importance weights and $$\theta^{(s)}$$ are draws from the posterior distribution based on all observations. To obtain $$w_i^{(s)}$$, we first compute the raw importance ratios

$r_i^{(s)} \propto \frac{1}{\prod_{j = i}^N p(y_j \,|\, \,\theta^{(s)})},$

and then stabilize them using PSIS.

We then decrease $$i$$ by $$1$$ (i.e., we move backwards in time) and repeat the process. At some observation $$i$$, the variability of importance ratios $$r_i^{(s)}$$ will become too large and importance sampling fails. We will refer to this particular value of $$i$$ as $$i^\star_1$$. To identify the value of $$i^\star_1$$ we check for which value of $$i$$ does the estimated shape parameter $$k$$ of the generalized Pareto distribution first cross $$0.7$$ (Vehtari et al, 2017ab). Only then do we refit the model using only observations before $$i^\star_1$$ and then restart the process. In some cases we may only need to refit once and in other cases we will find a value $$i^\star_2$$ that requires a second refitting, maybe an $$i^\star_3$$ that requires a third refitting, and so on. We repeat the refitting as few times as is required (only if $$k > 0.7$$) until we arrive at $$i = L+1$$. Recall that $$L$$ is the minimum number of observations we have deemed acceptable for making predictions (setting $$L=0$$ means predictions of all observations should be computed).

## Autoregressive models

Autoregressive (AR) models are some of the most commonly used time-series models. An AR(p) model —an autoregressive model of order $$p$$— can be defined as

$y_i = \eta_i + \sum_{k = 1}^p \varphi_k y_{i - k} + \varepsilon_i,$

where $$\eta_i$$ is the linear predictor for the $$i$$th observation, $$\phi_k$$ are the autoregressive parameters and $$\varepsilon_i$$ are pairwise independent errors, which are usually assumed to be normally distributed with equal variance $$\sigma^2$$. The model implies a recursive formula that allows for computing the right-hand side of the above equation for observation $$i$$ based on the values of the equations for previous observations.

## Case Study: Annual measurements of the level of Lake Huron

To illustrate the application of PSIS-LFO-CV for estimating expected $$M$$-SAP performance, we will fit a model for 98 annual measurements of the water level (in feet) of Lake Huron from the years 1875–1972. This data set is found in the datasets R package, which is installed automatically with R.

Before fitting a model, we will first put the data into a data frame and then look at the time series

N <- length(LakeHuron)
df <- data.frame(
y = as.numeric(LakeHuron),
year = as.numeric(time(LakeHuron)),
time = 1:N
)

# save plot labels to reuse them
plot_labs <- labs(
y = "Water Level (ft)",
x = "Year",
title = "Water Level in Lake Huron (1875-1972)"
)

ggplot(df, aes(x = year, y = y)) +
geom_point(size = 1) +
plot_labs

The above plot shows rather strong autocorrelation of the time-series as well as some trend towards lower levels for later points in time.

We can specify an AR(10) model for these data using the brms package as follows:

control <- list(adapt_delta = 0.95)
fit <- brm(
y ~ 1,
data = df,
autocor = cor_ar(~time, p = 10),
prior = prior(normal(0, 0.5), class = "ar"),
control = control,
seed = SEED,
chains = CHAINS
)

The model implied predictions along with the observed values can be plotted, which reveals a rather good fit to the data.

preds <- posterior_predict(fit)
preds <- cbind(
Estimate = colMeans(preds),
Q5 = apply(preds, 2, quantile, probs = 0.05),
Q95 = apply(preds, 2, quantile, probs = 0.95)
)

ggplot(cbind(df, preds), aes(x = year, y = Estimate)) +
geom_smooth(aes(ymin = Q5, ymax = Q95), stat = "identity", size = 0.5) +
geom_point(aes(y = y)) +
labs(subtitle = "Mean (blue) and 90% predictive intervals (gray) vs. observed data (black)") +
plot_labs

To allow for reasonable predictions of future values, we will require at least $$L = 25$$ historical observations (15 years) to make predictions.

L <- 25

We first perform approximate leave-one-out cross-validation (LOO-CV) for the purpose of later comparison with exact and approximate LFO-CV for the 1-SAP case.

loo_cv <- loo(log_lik(fit)[, (L + 1):N])
print(loo_cv)

Computed from 4000 by 73 log-likelihood matrix

Estimate   SE
elpd_loo    -86.4  6.4
p_loo         8.4  1.5
looic       172.7 12.7
------
Monte Carlo SE of elpd_loo is 0.1.

All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.

## 1-step-ahead predictions leaving out all future values

The most basic version of $$M$$-SAP is 1-SAP, in which we predict only one step ahead. In this case, $${\mathbf{y}_{{i<M}}}$$ simplifies to $$y_{i}$$ and the LFO-CV algorithm becomes considerably simpler than for larger values of $$M$$.

Before we compute approximate LFO-CV using PSIS we will first compute exact LFO-CV for the 1-SAP case so we can use it as a benchmark later. The initial step for the exact computation is to calculate the log-predictive densities by refitting the model many times:

loglik_exact <- matrix(nrow = nsamples(fit), ncol = N)
for (i in N:max(L + 1, 2)) {
fit_i <- update(fit, newdata = df[-(i:N), ], recompile = FALSE)
loglik_exact[, i] <- log_lik(fit_i, newdata = df[1:i, ])[, i]
}

Then we compute the exact expected log predictive density (ELPD):

# some helper functions we'll use throughout

# more stable than log(sum(exp(x)))
log_sum_exp <- function(x) {
max_x <- max(x)
max_x + log(sum(exp(x - max_x)))
}

# more stable than log(mean(exp(x)))
log_mean_exp <- function(x) {
log_sum_exp(x) - log(length(x))
}

# compute log of raw importance ratios
# sums over observations *not* over posterior samples
sum_log_ratios <- function(ll, ids = NULL) {
if (!is.null(ids)) ll <- ll[, ids, drop = FALSE]
- rowSums(ll)
}

# for printing comparisons later
rbind_print <- function(...) {
round(rbind(...), digits = 2)
}
exact_elpds_1sap <- apply(loglik_exact, 2, log_mean_exp)
exact_elpd_1sap <- c(ELPD = sum(exact_elpds_1sap[-(1:L)]))

rbind_print(
"LOO" = loo_cv\$estimates["elpd_loo", "Estimate"],
"LFO" = exact_elpd_1sap
)
      ELPD
LOO -86.36
LFO -90.16

We see that the ELPD from LFO-CV for 1-step-ahead predictions is lower than the ELPD estimate from LOO-CV, which should be expected since LOO-CV is making use of more of the time series. That is, since the LFO-CV approach only uses observations from before the left-out data point but LOO-CV uses all data points other than the left-out observation, we should expect to see the larger ELPD from LOO-CV.

For illustrative purposes, we will first compute approximate 1-SAP without any refitting, even when the Pareto $$k$$ estimate is too large. This will of course be a poor approximation to exact 1-SAP, in particular if the Pareto $$k$$ estimates increase rather quickly beyond the threshold up to which PSIS tends to produce stable results ($$k < 0.7$$).

approx_elpds_1sap_no_refit <- rep(NA, N)
loglik <- log_lik(fit)
ids <- N:(L + 1)
ks <- NULL

for (i in ids) {
logratio <- sum_log_ratios(loglik, i:N)
psis_part <- suppressWarnings(psis(logratio))
ks <- c(ks, pareto_k_values(psis_part))
lw_i <- weights(psis_part, normalize = TRUE)[, 1]
approx_elpds_1sap_no_refit[i] <- log_sum_exp(lw_i + loglik[, i])
}

If we plot the Pareto $$k$$ estimates we can see that they do start to increase quickly and that once we work backwards to about halfway through the data points we get $$k > 0.7$$. This is indicated by the vertical line in the plot below, which corresponds to the point $$i^\star_1$$ discussed in Section Approximate $$M$$-SAP using importance-sampling above.

plot_ks <- function(ks, ids, thres = 0.7) {
dat_ks <- data.frame(ks = ks, ids = ids)
ggplot(dat_ks, aes(x = ids, y = ks)) +
geom_point(aes(color = ks > thres), shape = 3, show.legend = FALSE) +
geom_hline(yintercept = thres, linetype = 2, color = "red2") +
scale_color_manual(values = c("cornflowerblue", "darkblue")) +
labs(x = "Data point", y = "Pareto k") +
ylim(-0.5, 1.5)
}
plot_ks(ks, ids)

Consequently, it is not surprising that the resulting approximate ELPD for 1-SAP is far away from the exact ELPD we computed above:

approx_elpd_1sap_no_refit <-
c(ELPD = sum(approx_elpds_1sap_no_refit, na.rm = TRUE))
rbind_print(
"approx LFO (no refit)" = approx_elpd_1sap_no_refit,
"exact LFO" = exact_elpd_1sap
)
                        ELPD
approx LFO (no refit) -87.68
exact LFO             -90.16

Next, we compute approximate 1-SAP with refit at observations where the Pareto $$k$$ estimate exceeds the threshold of $$0.65$$.

k_thres <- 0.7

The code becomes a little bit more involved to handle the refitting procedure. Note that we can compute exact 1-SAP at the refitting points, which comes with no additional computational costs since we had to refit the model anyway.

loglik <- matrix(nrow = nsamples(fit), ncol = N)
approx_elpds_1sap <- rep(NA, N)

fit_part <- fit
ids <- N:(L + 1)
i_refit <- N
refits <- NULL
ks <- NULL

for (i in ids) {
loglik[, i] <- log_lik(fit_part)[, i]
logratio <- sum_log_ratios(loglik, i:i_refit)
psis_part <- suppressWarnings(psis(logratio))
k <- pareto_k_values(psis_part)
ks <- c(ks, k)
if (k > k_thres) {
# refit the model based on the first i-1 observations
i_refit <- i
refits <- c(refits, i)
fit_part <- update(fit_part, newdata = df[1:(i - 1), ], recompile = FALSE)
loglik[, i] <- log_lik(fit_part, newdata = df[1:i, ])[, i]
approx_elpds_1sap[i] <- log_mean_exp(loglik[, i])
} else {
lw_i <- weights(psis_part, normalize = TRUE)[, 1]
approx_elpds_1sap[i] <- log_sum_exp(lw_i + loglik[, i])
}
} 

We see that the final Pareto-$$k$$-estimates are mostly well below the threshold of 0.7 and that we only needed to refit the model 20 times at observations 86, 78, 71, 68, 64, 61, 60, 59, 58, 57, 56, 54, 53, 49, 48, 41, 40, 28, 27, 26.

plot_ks(ks, ids)

The approximate 1-SAP ELPD is remarkably similar to the exact 1-SAP ELPD computed above, which indicates our algorithm to compute approximate 1-SAP worked well for the present data and model.

approx_elpd_1sap <- sum(approx_elpds_1sap, na.rm = TRUE)
rbind_print(
"approx LFO (no refits)" = approx_elpd_1sap_no_refit,
"approx LFO (with refits)" = approx_elpd_1sap,
"exact LFO" = exact_elpd_1sap
)
                           ELPD
approx LFO (no refits)   -87.68
approx LFO (with refits) -89.22
exact LFO                -90.16

Plotting exact against approximate predictions, we see that no approximation value deviates far from its exact counterpart, providing further evidence for the good quality of our approximation.

dat_elpd <- data.frame(
approx_elpd = approx_elpds_1sap,
exact_elpd = exact_elpds_1sap
)

ggplot(dat_elpd, aes(x = approx_elpd, y = exact_elpd)) +
geom_abline(color = "gray30") +
geom_point(size = 2) +
labs(x = "Approximate ELPDs", y = "Exact ELPDs")

No observation shows a difference between the exact and approximate ELPD calculations of more than 0.71 and the average difference is 0.044, resulting in a close approximation.

## $$M$$-step-ahead predictions leaving out all future values

To illustrate the application of $$M$$-SAP for $$M > 1$$, we compute next exact and approximate LFO-CV for the 4-SAP case.

### Exact $$M$$-step-ahead predictions

The necessary steps are the same as for 1-SAP with the exception that the log-density values of interest are now the sums of the log predictive densities of four consequtive observations. Further, the stability of the PSIS approximation actually stays the same for all $$M$$ as it only depends on the number of observations we leave out, not on the number of observations we predict.

M <- 4
loglikm <- matrix(nrow = nsamples(fit), ncol = N)
for (i in (N - M + 1):max(L + 1, 2)) {
fit_i <- update(fit, newdata = df[-(i:N), ], recompile = FALSE)
ll <- log_lik(fit_i, newdata = df[1:(i + M - 1), ])
loglikm[, i] <- rowSums(ll[, i:(i + M - 1)])
}
exact_elpds_4sap <- apply(loglikm, 2, log_mean_exp)
(exact_elpd_4sap <- c(ELPD = sum(exact_elpds_4sap, na.rm = TRUE)))
     ELPD
-348.2316 

### Approximate $$M$$-step-ahead predictions

Computing the approximate PSIS-LFO-CV for the 4-SAP case is a little bit more involved than the approximate version for the 1-SAP case, although the underlying principles remain the same.

loglikm <- loglik <- matrix(nrow = nsamples(fit), ncol = N)
approx_elpds_4sap <- rep(NA, N)
fit_part <- fit
ids <- (N - M + 1):(L + 1)
i_refit <- N - M + 1
refits <- NULL
ks <- NULL

loglik[, (N - M + 2):N] <- log_lik(fit_part)[, (N - M + 2):N]

for (i in ids) {
ll <- log_lik(fit_part, newdata = df[1:(i + M - 1), ])
loglikm[, i] <- rowSums(ll[, i:(i + M - 1)])
loglik[, i] <- ll[, i]
logratio <- sum_log_ratios(loglik, i:i_refit)
psis_part <- suppressWarnings(psis(logratio))
k <- pareto_k_values(psis_part)
ks <- c(ks, k)
if (k > k_thres) {
# refit the model based on the first i-1 observations
i_refit <- i
refits <- c(refits, i)
fit_part <- update(
fit_part, newdata = df[1:(i - 1), ],
recompile = FALSE
)
ll <- log_lik(fit_part, newdata = df[1:(i + M - 1), ])
loglik[, i] <- ll[, i]
loglikm[, i] <- rowSums(ll[, i:(i + M - 1)])
approx_elpds_4sap[i] <- log_mean_exp(loglikm[, i])
} else {
lw_i <- weights(psis_part, normalize = TRUE)[, 1]
approx_elpds_4sap[i] <- log_sum_exp(lw_i + loglikm[, i])
}
} 

Again, we see that the final Pareto-$$k$$-estimates are mostly well below the threshold and that we only needed to refit the model 21 times at observations 84, 75, 68, 64, 63, 62, 61, 60, 59, 58, 57, 56, 53, 47, 45, 39, 38, 31, 30, 27, 26.

plot_ks(ks, ids)

The approximate ELPD computed for the 4-SAP case is not as close to its exact counterpart as in the 1-SAP case. Likely, we need to lower the threshold for the Pareto-k-values from 0.7 to 0.65 or even 0.6 in order to achieve better results.

approx_elpd_4sap <- sum(approx_elpds_4sap, na.rm = TRUE)
rbind_print(
"Approx LFO" = approx_elpd_4sap,
"Exact LFO" = exact_elpd_4sap
)
              ELPD
Approx LFO -356.87
Exact LFO  -348.23

Plotting exact against approximate pointwise predictions confirms that, for a few specific data points, the approximate predictions underestimate the exact predictions.

dat_elpd_4sap <- data.frame(
approx_elpd = approx_elpds_4sap,
exact_elpd = exact_elpds_4sap
)

ggplot(dat_elpd_4sap, aes(x = approx_elpd, y = exact_elpd)) +
geom_abline(color = "gray30") +
geom_point(size = 2) +
labs(x = "Approximate ELPDs", y = "Exact ELPDs")

## 1-step-ahead predictions leaving out blocks of future values

Depending on the particular time-series model and data, the Pareto $$k$$ estimates may exceed 0.7 rather quickly (i.e., after only few observations) and so many refits may be required even when carrying out the PSIS approximation to LFO-CV. In this case, another option is to exclude only the block of $$B$$ future values that directly follow the observations to be predicted while retaining all of the more distant values $${\mathbf{y}_{{i>B}}} = (y_{i + B}, \ldots, y_N)$$. This will usually result in lower Pareto $$k$$ estimates and thus less refitting.

This block-$$M$$-SAP version closely resembles the basic $$M$$-SAP only if values in the distant future, $${\mathbf{y}_{{>B}}}$$, contain little information about the current observations being predicted. Whether this assumption is justified will depend on the data and model.

We will demonstrate the LFO-CV approach for the block-$$M$$-SAP case using block-1-SAP.

### Exact LFO-CV for block-1-SAP

The differences between the code for LFO-CV for 1-SAP and the code for block-1-SAP are very small. We now just need to define a block size $$B$$ of left-out observations, which we set to $$B = 20$$ for the present case study.

B <- 20
loglik <- matrix(nrow = nsamples(fit), ncol = N)
for (i in N:(L + 1)) {
to <- min(i + B - 1, N)
fit_i <- update(fit, newdata = df[-(i:to), ], recompile = FALSE)
loglik[, i] <- log_lik(fit_i, newdata = df[1:i, ])[, i]
}

Comparing the exact block-1-SAP to the exact 1-SAP reveals no substantial difference for this particular data set and model, but this may very well be different for other data sets and models.

exact_elpds_1sap_block <- apply(loglik, 2, log_mean_exp)
exact_elpd_1sap_block <- c(ELPD = sum(exact_elpds_1sap_block, na.rm = TRUE))
rbind_print(
"Exact LFO-CV (1-SAP)" = exact_elpd_1sap,
"Exact LFO-CV (block-1-SAP)" = exact_elpd_1sap_block
)
                             ELPD
Exact LFO-CV (1-SAP)       -90.16
Exact LFO-CV (block-1-SAP) -90.22

If exact block-1-SAP and exact 1-SAP were different, it would not necessarily imply that observations depend on distant future values, but could instead be due to the fact that the model parameters were estimated more precisely in the case of block-1-SAP since more data are available, in particular for predicting observations early in the time series. If this is the case we would expect the differences between the pointwise ELPDs for the two cases to be larger for earlier observations. To some degree, this is also what we see when plotting the differences for the present example, although these differences happen to cancel each other out in this case:

dat_elpd_diff <- data.frame(
elpd_diff = exact_elpds_1sap_block - exact_elpds_1sap,
i = seq_along(exact_elpds_1sap_block)
)

ggplot(dat_elpd_diff[(L + 1):N, ], aes(x = i, y = elpd_diff)) +
geom_point(size = 2) +
labs(x = "Data Point", y = "ELPD diff")

We compute approximate PSIS-LFO-CV for block-1-SAP as follows.

loglik <- matrix(nrow = nsamples(fit), ncol = N)
approx_elpds_1sap_block <- rep(NA, N)
fit_part <- fit
ids <- N:(L + 1)
i_refit <- N
refits <- NULL
ks <- NULL

for (i in ids) {
loglik[, i] <- log_lik(fit_part)[, i]
to <- min(i + B - 1, i_refit)
logratio <- sum_log_ratios(loglik, i:to)
psis_part <- suppressWarnings(psis(logratio))
k <- pareto_k_values(psis_part)
ks <- c(ks, k)
if (k > k_thres) {
# refit the model without a block of B observations
i_refit <- i
refits <- c(refits, i)
fit_part <- update(fit_part, newdata = df[-(i:to), ], recompile = FALSE)
loglik[, i] <- log_lik(fit_part, newdata = df[1:i, ])[, i]
approx_elpds_1sap_block[i] <- log_mean_exp(loglik[, i])
} else {
lw_i <- weights(psis_part, normalize = TRUE)[, 1]
approx_elpds_1sap_block[i] <- log_sum_exp(lw_i + loglik[, i])
}
}

Here we needed 6 refit(s), which is remarkable given that we have a time-series of 98 observations and are predicting for 98 - L = 73 of them.

plot_ks(ks, ids)

Consequently, it is not surprising that PSIS-LFO-CV for block-1-SAP matches the exact version closely.

approx_elpd_1sap_block <- c(ELPD = sum(approx_elpds_1sap_block, na.rm = TRUE))
rbind_print(
"Exact LFO-CV (block-1-SAP)" = exact_elpd_1sap_block,
"Approx LFO-CV (block-1-SAP)" = approx_elpd_1sap_block
)
                              ELPD
Exact LFO-CV (block-1-SAP)  -90.22
Approx LFO-CV (block-1-SAP) -89.34

This is also visible in the plot of approximate against exact predictions.

dat_elpd_block <- data.frame(
approx_elpd = approx_elpds_1sap_block,
exact_elpd = exact_elpds_1sap_block
)

ggplot(dat_elpd_block, aes(x = approx_elpd, y = exact_elpd)) +
geom_abline(color = "gray30") +
geom_point(size = 2) +
labs(x = "Approximate ELPDs", y = "Exact ELPDs")

When increasing $$M$$ in block-M-SAP, we have to make sure to simultaneously increase the block size $$B$$, as well, in order to keep the assumptions of block-M-SAP satisfied. This also implies that the stability of the PSIS approximation will decrease as we increase $$M$$ in block-M-SAP because we have to leave out more observations.

## Conclusion

In this case study we have shown how to do carry out exact and approximate leave-future-out cross-validation for $$M$$-step-ahead prediction tasks. For the data and model used in our example the PSIS-LFO-CV algorithm provides reasonably stable and accurate results depite not requiring us to refit the model nearly as many times. For many other examples we expect similar performance, but the results will always vary from case to case depending on the particulars of the data and the time series model.

## References

Vehtari A., Gelman A., & Gabry J. (2017a). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27(5), 1413–1432. doi:10.1007/s11222-016-9696-4. Online. arXiv preprint arXiv:1507.04544.

Vehtari A., Gelman A., & Gabry J. (2017b). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.

## Appendix

### Appendix: Session information

sessionInfo()
R version 3.5.0 (2018-04-23)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 16299)

Matrix products: default

locale:
[1] LC_COLLATE=German_Germany.1252  LC_CTYPE=German_Germany.1252
[3] LC_MONETARY=German_Germany.1252 LC_NUMERIC=C
[5] LC_TIME=German_Germany.1252

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base

other attached packages:
[1] loo_2.0.0     brms_2.4.3    ggplot2_3.0.0 Rcpp_0.12.18

loaded via a namespace (and not attached):
[4] threejs_0.3.1        shiny_1.1.0          assertthat_0.2.0
[7] stats4_3.5.0         yaml_2.1.19          pillar_1.3.0
[10] backports_1.1.2      lattice_0.20-35      glue_1.2.0
[13] digest_0.6.15        promises_1.0.1       colorspace_1.3-2
[16] htmltools_0.3.6      httpuv_1.4.4.2       Matrix_1.2-14
[19] plyr_1.8.4           dygraphs_1.1.1.6     pkgconfig_2.0.1
[22] rstan_2.17.3         purrr_0.2.5          xtable_1.8-2
[25] mvtnorm_1.0-8        scales_0.5.0         later_0.7.3
[28] tibble_1.4.2         bayesplot_1.6.0      DT_0.4
[31] withr_2.1.2          shinyjs_1.0          lazyeval_0.2.1
[34] magrittr_1.5         crayon_1.3.4         mime_0.5
[37] evaluate_0.10.1      nlme_3.1-137         xts_0.11-0
[40] colourpicker_1.0     rsconnect_0.8.8      tools_3.5.0
[43] matrixStats_0.54.0   stringr_1.3.1        munsell_0.5.0
[46] bindrcpp_0.2.2       compiler_3.5.0       rlang_0.2.1
[49] grid_3.5.0           ggridges_0.5.0       rstudioapi_0.7
[52] htmlwidgets_1.2      crosstalk_1.0.0      igraph_1.2.1
[55] miniUI_0.1.1.1       base64enc_0.1-3      labeling_0.3
[58] rmarkdown_1.10       gtable_0.2.0         codetools_0.2-15
[61] inline_0.3.15        abind_1.4-5          markdown_0.8
[64] reshape2_1.4.3       R6_2.2.2             gridExtra_2.3
[67] rstantools_1.5.0     zoo_1.8-3            knitr_1.20
[70] bridgesampling_0.5-2 dplyr_0.7.6          bindr_0.1.1
[73] shinystan_2.5.0      shinythemes_1.1.1    rprojroot_1.3-2
[76] stringi_1.2.3        parallel_3.5.0       tidyselect_0.2.4
[79] coda_0.19-1