Sampling from a VLMC
Introduction
Let us consider a simple example with an independent sample:
The optimal VLMC according to the BIC has non memory:
Simulating a sequence using the model is done via the standard
stats::simulate() function as follows:
Even if this is useless here because of the independence, we drop the
first 100 samples. In general simulate.vlmc() is used as
stats::simulate() and supports the standard parameters:
nsimfor the number of simulated values, here the length of the new time seriesseedto specify the random seed used during the simulation (the initial state of the random number generator is restored to its previous after simulating the values)
In addition, one can specify the initial values of the sequence via
the init parameter, for instance:
is guaranteed to start with \(0, 0\):
This provides an alternative to dropping the initial values of the simulated time series by setting those values to the one observed in the data set. Notice that this practice has no theoretical justification.
CAC time series
Let us consider the French CAC index provided in
EuStockMarkets:
We turn it into a discrete time series with three values:
- Stay if the value of the index on day t+1 is between 99.5% and 100.5% of the value on day t
- Up if the value increased by at least 0.5%
- Down if the value decreased by at least 0.5%
CAC_rel_evol <- diff(CAC_raw) / CAC_raw[-length(CAC_raw)]
CAC_dts <- factor(
ifelse(CAC_rel_evol >= 0.005, "Up",
ifelse(CAC_rel_evol <= -0.005, "Down", "Stay")
),
levels = c("Down", "Stay", "Up")
)Then we adjust a VLMC to the time series using the AIC criterion:
We use here the AIC to favour predictive performances and as a consequence the obtained model is quite complex with 30 contexts.
The original discrete time series does not exhibit long subsequences of constant values as shown in the following graphical representation.
CAC_rle <- rle(as.integer(CAC_dts))
CAC_rle_df <- data.frame(value = CAC_rle$values, length = CAC_rle$lengths)
ggplot(CAC_rle_df, aes(x = length)) +
geom_bar() +
labs(
title = "Distribution of the lengths of constant subsequences",
x = "Length",
y = "Count"
)
To study this aspect of the time series, a natural statistics is the probability of observing a constant subsequence of a length between 5 and 10 among all subsequences of length 10 that can be generated by the model. Notice that the patterns used in the statistics must be of fixed lengths to be covered by the consistency theorems mentioned above. We implement the statistics as follows:
long_sequence <- function(dts) {
dts_int <- as.integer(dts)
## RLE cannot be used directly as we need to account for overlapping
## patterns
dts_freq <- frollapply(dts_int, 10, \(x) max(rle(x)$length) >= 5)
mean(dts_freq, na.rm = TRUE)
}We generate 100 bootstrap samples, using a burning time proportional
to the depth of the model, in order to allow for a proper mixing to take
place. We use the seed parameter of
stats::simulate() to ensure reproducibility.
bootstrap_samples <- vector(100, mode = "list")
burning_time <- 50 * depth(CAC_model)
for (b in seq_along(bootstrap_samples)) {
bootstrap_samples[[b]] <- simulate(CAC_model,
burning_time + length(CAC_dts),
seed = b
)[-(1:burning_time)]
}Then we compute the statistics on the bootstrap samples.
The bootstrap distribution of this statistics is illustrated on the following figure in which the red vertical line represents the value of the statistics on the original sequence.
ggplot(mapping = aes(x = bootstrap_ls)) +
geom_density() +
geom_rug() +
labs(
title = "Bootstrap distribution of the probability of long constant subsequences",
x = "Probability"
) +
geom_vline(xintercept = long_sequence(CAC_dts), color = "red")
We build a discrete version by considering two states: the background
base active power below 0.4 kW and the active use of electric appliance
above this limit.