---
title: "Probabilistic Reconciliation via Conditioning with `bayesRecon`"
author: "Nicolò Rubattu, Giorgio Corani, Dario Azzimonti, Lorenzo Zambon"
date: "2023-11-26"
lang: "en"
output: html_vignette
bibliography: references.bib
cite:
- '@zambon2022efficient'
- '@zambon2023properties'
- '@corani2023probabilistic'
- '@corani2021probabilistic'
vignette: >
%\VignetteIndexEntry{Probabilistic Reconciliation via Conditioning with `bayesRecon`}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
---
```{r setup, include = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
eval=TRUE ### !!!! set to FALSE here to render only the text !!!!
)
set.seed(42)
```
```{r klippy, echo=FALSE, include=TRUE, eval=FALSE}
klippy::klippy(position = c('top', 'right'), tooltip_message = 'Copy', tooltip_success = 'Done', color="black")
```
# Introduction
This vignette shows how to perform *probabilistic reconciliation* with
the `bayesRecon` package. We provide three examples:
1. *Temporal hierarchy for a count time series*: we build a temporal hierarchy over a count time series, produce the base forecasts using `glarma` and reconcile them via Bottom-Up Importance Sampling (BUIS).
2. *Temporal hierarchy for a smooth time series*: we build a temporal hierarchy over a smooth time series, compute the base forecasts using `ets` and we reconcile them in closed form using Gaussian reconciliation. The covariance matrix is diagonal.
3. *Hierarchical of smooth time series*: this is an example of a cross-sectional hierarchy. We generate the base forecasts using `ets` and we reconcile them via Gaussian reconciliation.
The covariance matrix is full and estimated via shrinkage.
# Installation
The package, available at this
[CRAN page](https://cran.r-project.org/package=bayesRecon), can be installed and loaded with the usual commands:
```{r install, eval=FALSE}
install.packages('bayesRecon', dependencies = TRUE)
```
Load the package:
```{r load}
library(bayesRecon)
```
# Temporal hierarchy over a count time series
We select a monthly time series of counts from the *carparts* dataset, available from
the expsmooth package [@expsmooth_pkg].
The data set contains time series of sales of cars part from Jan. 1998 to Mar. 2002.
For this example we select time series #2655, which we make available as `bayesRecon::carparts_example`.
This time series has a skewed distribution of values.
```{r carpart-plot, dpi=300, out.width = "100%", fig.align='center', fig.cap="**Figure 1**: Carpart - monthly car part sales.", fig.dim = c(6, 3)}
layout(mat = matrix(c(1, 2), nrow = 1, ncol = 2), widths = c(2, 1))
plot(carparts_example, xlab = "Time", ylab = "Car part sales", main = NULL)
hist(carparts_example, xlab = "Car part sales", main = NULL)
```
<br><br>
We divide the time series into train and test; the test set contains the last 12 months.
```{r train-test}
train <- window(carparts_example, end = c(2001, 3))
test <- window(carparts_example, start = c(2001, 4))
```
<!-- In order to do hierarchical forecasting: -->
<!-- 1. we build a temporal hierarchy; -->
<!-- 2. we compute the *base forecasts*) at each temporal scale. -->
We build the temporal hierarchy using the `temporal aggregation` function.
We specify the aggregation levels using the
`agg_levels` argument; in this case they are
*2-Monthly*, *Quarterly*, *4-Monthly*, *Biannual*, and *Annual*.
``` {r temp-agg}
train.agg <- bayesRecon::temporal_aggregation(train, agg_levels = c(2, 3, 4, 6, 12))
levels <- c("Annual", "Biannual", "4-Monthly", "Quarterly", "2-Monthly", "Monthly")
names(train.agg) <- levels
```
The function returns a list of aggregated time series, ordered from the most aggregated (top of the hierarchy) to the most disagreggated (bottom of the hierarchy). We plot them below.
``` {r temp-agg-plot, dpi=300, fig.show="hold", out.width="100%", out.heigth="100%", fig.align='center', fig.cap="**Figure 2**: The aggregated time series of the temporal hierarchy.", fig.dim=c(6,3.5)}
par(mfrow = c(2, 3), mai = c(0.6, 0.6, 0.5, 0.5))
for (l in levels) {
plot(train.agg[[l]], xlab = "Time", ylab = "Car part sales", main = l)
}
```
<br><br>
We compute the *base forecasts* using the package [`glarma`](https://cran.r-project.org/package=glarma),
a package specific for forecasting count time series.
We forecast 12 steps ahead at the monthly level, 4 steps ahead at the quarterly level, etc. by iterating over the levels of the hierarchy,
At each level, we fit a `glarma` model
with Poisson predictive distribution and we forecast;
each forecast is constituted by 20000 integer samples.
Eventually we collect the samples of the 28 predictive distributions (one at the *Annual* level, two at the *Biannual* level, etc.) in a list.
The code below takes about 30 seconds on a standard computer.
``` {r hier-fore, cache=TRUE}
# install.packages("glarma", dependencies = TRUE)
#library(glarma)
fc.samples <- list()
D <- 20000
fc.count <- 1
# iterating over the temporal aggregation levels
for (l in seq_along(train.agg)) {
f.level <- frequency(train.agg[[l]])
print(paste("Forecasting at ", levels[l], "...", sep = ""))
# fit an independent model for each aggregation level
model <- glarma::glarma(
train.agg[[l]],
phiLags = if (f.level == 1) 1 else 1:(min(6, f.level - 1)),
thetaLags = if (f.level == 1) NULL else f.level,
X = cbind(intercept = rep(1, length(train.agg[[l]]))),
offset = cbind(intercept = rep(0, length(train.agg[[l]]))),
type = "Poi"
)
# forecast 1 year ahead
h <- f.level
tmp <- matrix(data = NA, nrow = h, ncol = D)
for (s in (1:D)) {
# each call to 'forecast.glarma' returns a simulation path
tmp[, s] <- glarma::forecast(
model,
n.ahead = h,
newdata = cbind(intercept = rep(1, h)),
newoffset = rep(0, h)
)$Y
}
# collect the forecasted samples
for (i in 1:h) {
fc.samples[[fc.count]] <- tmp[i, ]
fc.count <- fc.count + 1
}
}
```
Reconciliation requires the aggregation matrix $\mathbf{A}$, which we obtain using the function `get_reconc_matrices`.
It requires:
* the aggregation factors of the hierarchy, which in this example are $\{2, 3, 4, 6, 12\}$;
* the length of the forecasting horizon at the bottom level, which is 12 in this example.
``` {r aggregationMatrix}
recon.matrices <- bayesRecon::get_reconc_matrices(agg_levels = c(2, 3, 4, 6, 12), h = 12)
# Aggregation matrix
A <- recon.matrices$A
```
To reconcile using Bottom-Up Important Sampling (BUIS) we
we use the function `reconc_BUIS`, passing to it
the $\mathbf{A}$ matrix, the *base forecasts*, the type of the base forecasts (`in_type`="samples") and whether the samples are discrete or integer (`distr` = "discrete").
``` {r reconc}
recon.res <- bayesRecon::reconc_BUIS(
A,
base_forecasts = fc.samples,
in_type = "samples",
distr = "discrete",
seed = 42
)
```
Here we obtain samples from the reconciled forecast distribution.
```{r res}
reconciled_samples <- recon.res$reconciled_samples
dim(reconciled_samples)
```
We now compute the Mean Absolute Error (MAE) and the Continuous Ranked Probability Score (CRPS) for the bottom (i.e., *monthly*) time series. For computing CRPS, we use the package [`scoringRules`](https://cran.r-project.org/package=scoringRules).
```{r metrics}
# install.packages("scoringRules", dependencies = TRUE)
library(scoringRules)
ae.fc <- list()
ae.reconc <- list()
crps.fc <- list()
crps.reconc <- list()
for (h in 1:length(test)) {
y.hat_ <- median(fc.samples[[nrow(A) + h]])
y.reconc_ <- median(recon.res$bottom_reconciled_samples[, h])
# Compute Absolute Errors
ae.fc[[h]] <- abs(test[h] - y.hat_)
ae.reconc[[h]] <- abs(test[h] - y.reconc_)
# Compute Continuous Ranked Probability Score (CRPS)
crps.fc[[h]] <-
scoringRules::crps_sample(y = test[h], dat = fc.samples[[nrow(A) + h]])
crps.reconc[[h]] <-
scoringRules::crps_sample(y = test[h], dat = recon.res$bottom_reconciled_samples[, h])
}
mae.fc <- mean(unlist(ae.fc))
mae.reconc <- mean(unlist(ae.reconc))
crps.fc <- mean(unlist(crps.fc))
crps.reconc <- mean(unlist(crps.reconc))
metrics <- data.frame(
row.names = c("MAE", "CRPS"),
base.forecasts = c(mae.fc, crps.fc),
reconciled.forecasts = c(mae.reconc, crps.reconc)
)
metrics
```
# Temporal hierarchy over a smooth time series
In this second example, we select a smooth monthly time series (N1485) from the M3 forecasting competition [@makridakis2000m3]. The data set is available in
the Mcomp package [@mcomp_pkg] and we make available this time series as `bayesRecon::m3_example`.
```{r m3-plot, dpi=300, out.width = "100%", fig.align='center', fig.cap="**Figure 3**: M3 - N1485 time series.", fig.dim = c(6, 3)}
plot(M3_example$train, xlab = "Time", ylab = "y", main = "N1485")
```
<br>
We build the temporal hierarchy using the `temporal_aggregation` function.
Without specifying `agg_levels`, the function generates by default all the feasible aggregation: 2-Monthly, Quarterly, 4-Monthly, Biannual, and Annual.
```{r m3-agg}
train.agg <- bayesRecon::temporal_aggregation(M3_example$train)
levels <- c("Annual", "Biannual", "4-Monthly", "Quarterly", "2-Monthly", "Monthly")
names(train.agg) <- levels
```
We generate the base forecasts using `ets` from the [forecast](https://cran.r-project.org/package=forecast) package [@pkg_forecast].
At every level we predict 18 months ahead.
All the predictive distributions are Gaussian.
```{r m3-fore}
# install.packages("forecast", dependencies = TRUE)
library(forecast)
H <- length(M3_example$test)
H
fc <- list()
level.idx <- 1
fc.idx <- 1
for (level in train.agg) {
level.name <- names(train.agg)[level.idx]
# fit an ETS model for each temporal level
model <- ets(level)
# generate forecasts for each level within 18 months
h <- floor(H / (12 / frequency(level)))
print(paste("Forecasting at ", level.name, ", h=", h, "...", sep = ""))
level.fc <- forecast(model, h = h)
# save mean and sd of the gaussian predictive distribution
for (i in 1:h) {
fc[[fc.idx]] <- list(mean = level.fc$mean[[i]],
sd = (level.fc$upper[, "95%"][[i]] - level.fc$mean[[i]]) / qnorm(0.975))
fc.idx <- fc.idx + 1
}
level.idx <- level.idx + 1
}
```
Using the function `get_reconc_matrices`, we get matrix $\mathbf{A}$.
```{r m3-rmat, dpi=300, out.width = '70%', fig.align='center', fig.cap="**Figure 4**: M3 - The aggregation matrix A (red=1, yellow=0).", fig.dim = c(8, 8)}
rmat <- get_reconc_matrices(agg_levels = c(2, 3, 4, 6, 12), h = 18)
par(mai = c(1,1,0.5,0.5))
image(1:ncol(rmat$A), 1:nrow(rmat$A),
t(apply(t(rmat$A),1,rev)),
xaxt='n', yaxt='n', ylab = "", xlab = levels[6])
axis(1, at=1:ncol(rmat$A), label=1:ncol(rmat$A), las=1)
axis(2, at=c(23,22,19,15,9), label=levels[1:5], las=2)
```
<br>
The function `reconc_gaussian` implements the exact Gaussian reconciliation.
We also run `reconc_BUIS`, to check the consistency
between the two approaches.
```{r m3-reco}
recon.gauss <- bayesRecon::reconc_gaussian(
A = rmat$A,
base_forecasts.mu = sapply(fc, "[[", 1),
base_forecasts.Sigma = diag(sapply(fc, "[[", 2)) ^ 2
)
reconc.buis <- bayesRecon::reconc_BUIS(
A = rmat$A,
base_forecasts = fc,
in_type = "params",
distr = "gaussian",
num_samples = 20000,
seed = 42
)
# check that the algorithms return consistent results
round(rbind(
c(rmat$S %*% recon.gauss$bottom_reconciled_mean),
rowMeans(reconc.buis$reconciled_samples)
))
```
We now compare *base forecasts* and *reconciled forecasts*:
```{r m3-plotfore, dpi=300, out.width = "100%", fig.align='center', fig.cap="**Figure 5**: M3 - Base and reconciled forecasts. The black line shows the actual data (dashed in the test). The orange line is the mean of the base forecasts, the blu line is the reconciled mean. We also show the 95% prediction intervals.", fig.dim = c(6, 4)}
yhat.mu <- tail(sapply(fc, "[[", 1), 18)
yhat.sigma <- tail(sapply(fc, "[[", 2), 18)
yhat.hi95 <- qnorm(0.975, mean = yhat.mu, sd = yhat.sigma)
yhat.lo95 <- qnorm(0.025, mean = yhat.mu, sd = yhat.sigma)
yreconc.mu <- rowMeans(reconc.buis$bottom_reconciled_samples)
yreconc.hi95 <- apply(reconc.buis$bottom_reconciled_samples, 1,
function(x) quantile(x, 0.975))
yreconc.lo95 <- apply(reconc.buis$bottom_reconciled_samples, 1,
function(x) quantile(x, 0.025))
ylim_ <- c(min(M3_example$train, M3_example$test, yhat.lo95, yreconc.lo95) - 1,
max(M3_example$train, M3_example$test, yhat.hi95, yreconc.hi95) + 1)
plot(M3_example$train, xlim = c(1990, 1995.6), ylim = ylim_,
ylab = "y", main = "N1485 Forecasts")
lines(M3_example$test, lty = "dashed")
lines(ts(yhat.mu, start = start(M3_example$test), frequency = 12),
col = "coral", lwd = 2)
lines(ts(yreconc.mu, start = start(M3_example$test), frequency = 12),
col = "blue2", lwd = 2)
xtest <- time(M3_example$test)
polygon(c(xtest, rev(xtest)), c(yhat.mu, rev(yhat.hi95)),
col = "#FF7F5066", border = "#FF7F5066")
polygon(c(xtest, rev(xtest)), c(yhat.mu, rev(yhat.lo95)),
col = "#FF7F5066", border = "#FF7F5066")
polygon(c(xtest, rev(xtest)), c(yreconc.mu, rev(yreconc.hi95)),
col = "#0000EE4D", border = "#0000EE4D")
polygon(c(xtest, rev(xtest)), c(yreconc.mu, rev(yreconc.lo95)),
col = "#0000EE4D", border = "#0000EE4D")
```
# Gaussian reconciliation of a cross-sectional hierarchy
In this example, we consider the hierarchical time series *infantgts*, which is available
from the `hts` package [@hts_pkg]
We make it available also in our package as `bayesRecon::infantMortality`.
It contains counts of infant mortality (deaths) in Australia, disaggregated by state and sex (male and female).
<!-- State codes refer to: New South Wales (NSW), Victoria (VIC), Queensland (QLD), South Australia (SA), Western Australia (WA), Northern Territory (NT), Australian Capital Territory (ACT), and Tasmania (TAS). -->
We forecast one year ahead using `auto.arima` from the [`forecast` ](https://cran.r-project.org/package=forecast) package.
We collect the residuals, which we will later use to compute the covariance matrix.
```{r infants-forecasts}
# install.packages("forecast", dependencies = TRUE)
library(forecast)
fc <- list()
residuals <- matrix(NA,
nrow = length(infantMortality$total),
ncol = length(infantMortality))
fc.idx <- 1
for (s in infantMortality) {
s.name <- names(infantMortality)[fc.idx]
print(paste("Forecasting at ", s.name, "...", sep = ""))
# fit an auto.arima model and forecast with h=1
model <- auto.arima(s)
s.fc <- forecast(model, h = 1)
# save mean and sd of the gaussian predictive distribution
fc[[s.name]] <- c(s.fc$mean,
(s.fc$upper[, "95%"][[1]] - s.fc$mean) / qnorm(0.975))
residuals[, fc.idx] <- s.fc$residuals
fc.idx <- fc.idx + 1
}
```
Now we build the $\mathbf{A}$ matrix.
```{r infants-s, dpi=300, out.width = '70%', fig.align='center', fig.cap="**Figure 6**: Infants mortality - The aggregation matrix A (red=1, yellow=0).", fig.dim = c(8, 8)}
# we have 16 bottom time series, and 11 upper time series
A <- matrix(data = c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,
1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,
0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1), byrow=TRUE, ncol = 16)
# plot of A
par(mai = c(1.5,1,0.5,0.5))
image(1:ncol(A), 1:nrow(A),
t(apply(t(A),1,rev)),
xaxt='n', yaxt='n', ann=FALSE)
axis(1, at=1:ncol(A), label=names(infantMortality)[12:27], las=2)
axis(2, at=c(1:11), label=rev(names(infantMortality)[1:11]), las=2)
```
<br>
We use `bayesRecon::schaferStrimmer_cov` to estimate the covariance matrix of the residuals with shrinkage [@schafer2005shrinkage].
```{r infants reconc}
# means
mu <- sapply(fc, "[[", 1)
# Shrinkage covariance
shrink.res <- bayesRecon::schaferStrimmer_cov(residuals)
print(paste("The estimated shrinkage intensity is", round(shrink.res$lambda_star, 3)))
Sigma <- shrink.res$shrink_cov
```
We now perform Gaussian reconciliation:
```{r infants-recon}
recon.gauss <- bayesRecon::reconc_gaussian(A,
base_forecasts.mu = mu,
base_forecasts.Sigma = Sigma)
bottom_mu_reconc <- recon.gauss$bottom_reconciled_mean
bottom_Sigma_reconc <- recon.gauss$bottom_reconciled_covariance
# Obtain reconciled mu and Sigma for the upper variable
upper_mu_reconc <- A %*% bottom_mu_reconc
upper_Sigma_reconc <- A %*% bottom_Sigma_reconc %*% t(A)
upper_mu_reconc
```
# References
<div id="refs"></div>