Generate diverse time series
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A MAR model is a mixture of \(k\) Gaussian ARIMA\((p,d,0)(P,D,0)_m\) processes of the form \[
(1-B)^{d_i}(1-B^{m_i})^{D_i} (1-\phi_i(B))(1-\Phi_i(B)) y_t = c_i + \sigma_{i,t}\epsilon_t
\] with probability \(\alpha_i\), where \(B\) is the backshift operator, \(m_i\) is the seasonal period, \(\epsilon_t\) is a N(0,1) variate, and \(\phi_i(B)\) and \(\Phi_i(B)\) are polynomials in \(B\) of order \(p_i\) and \(P_i\) respectively.
The function mar_model() generates a MAR model with randomly selected parameters. The orders are uniformly sampled such that \(p \in \{0,1,2,3\}\), \(d \in \{0,1,2\}\), \(P\in \{0,1,2\}\) and \(D \in\{0,1\}\) (with the restriction that \(d+D \le 2\)). The parameters \(\phi_{j,i}\) and \(\Phi_{j,i}\) are uniformly sampled from the stationary parameter space, while the \(\sigma_{i}\) values are uniformly sampled on \((1,5)\) and the mixture weights are uniformly sampled on \((0,1)\). The number of components is uniformly sampled on \(\{1,2,3,4,5\}\). If required, each of these parameters can be specified by the user, rather than randomly selected.
The resulting model object can be passed to generate() to return a tsibble of time series generated from the model. Alternatively, it can be passed to simulate() to return one time series using either the ts or msts class (depending on whether there is more than one seasonal period).
Suppose we want to generate a random MAR model, and then generate 9 quarterly time series from it, each of length 5 years.
qmar <- mar_model(seasonal_periods = 4)
qmar
#> Mixture AR model with 2 components:
#> ARIMA(0,2,0)(2,0,0)[4] with weight 0.43
#> ARIMA(1,2,0)(1,0,0)[4] with weight 0.57
This shows \(k=2\) components with weights 0.43, 0.57. Now we can generate time series from this model.
qmar %>%
generate(nseries = 9, length = 20) %>%
autoplot(value)
Each of these series comes from the same MAR model, but with different stochastic inputs. Although the two ARIMA models are seasonal, the seasonality is too weak to been in the plots.
Generate time series with targetted features
The functions generate_target() and simulate_target() can efficiently generate time series with targetted features. These use a genetic algorithm to tune the MAR parameters until the distance between the target feature vector and the feature vector of the synthetic time series is as small as possible. As before, the generate... function returns a tsibble while the simulate... function returns a ts or msts object.
Suppose we want to use generate a time series with the same level of trend and seasonality as the USAccDeaths data. First we create a function to measure the features we want to target. This time we will use simulate() rather than generate() so that the resulting time series has the same class as the USAccDeaths data.
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my_features <- function(y) {
c(stl_features(y)[c("trend", "seasonal_strength", "peak", "trough")])
}
y <- simulate_target(
length = length(USAccDeaths),
seasonal_periods = frequency(USAccDeaths),
feature_function = my_features, target = my_features(USAccDeaths)
)
# Make new series same scale and frequency as USAccDeaths
y <- ts(scale(y) * sd(USAccDeaths) + mean(USAccDeaths))
tsp(y) <- tsp(USAccDeaths)
cbind(USAccDeaths, y) %>% autoplot()
cbind(my_features(USAccDeaths), my_features(y))
#> [,1] [,2]
#> trend 0.8024570 0.9587332
#> seasonal_strength 0.9447945 0.8887976
#> peak 7.0000000 7.0000000
#> trough 2.0000000 2.0000000
Next we will demonstrate the generate_target() function with target features specified by the spectral entropy and the first two autocorrelation coefficients.
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my_features <- function(y) {
c(entropy(y), acf = acf(y, plot = FALSE)$acf[2:3, 1, 1])
}
df <- generate_target(
length = 60, feature_function = my_features, target = c(0.5, 0.9, 0.8)
)
df %>%
as_tibble() %>%
group_by(key) %>%
summarise(value = my_features(value),
feature=c("entropy","acf1", "acf2"),
.groups = "drop")
#> # A tibble: 30 x 3
#> key value feature
#> <chr> <dbl> <chr>
#> 1 Series 1 0.481 entropy
#> 2 Series 1 0.902 acf1
#> 3 Series 1 0.783 acf2
#> 4 Series 10 0.491 entropy
#> 5 Series 10 0.900 acf1
#> 6 Series 10 0.790 acf2
#> 7 Series 2 0.464 entropy
#> 8 Series 2 0.914 acf1
#> 9 Series 2 0.825 acf2
#> 10 Series 3 0.443 entropy
#> # ... with 20 more rows
ARIMA and ETS models
Just as mar_model() returns a MAR model, arima_model() and ets_model() will return ARIMA and ETS models. In all cases, elements will be selected randomly if the corresponding argument is omitted. For example,
mod <- arima_model(frequency = 4)
mod
#> Series:
#> ARIMA(2,1,1)(1,0,2)[4] with drift
#>
#> Coefficients:
#> ar1 ar2 ma1 sar1 sma1 sma2 drift
#> -0.0261 0.7914 0.3401 -0.8150 -0.4493 -0.5255 -2.8906
#>
#> sigma^2 = 23.17
This can then be passed to generate() or simulate() to obtain synthetic data from the model. The simulate methods for ARIMA and ETS models are actually from the forecast package rather than the gratis package.
