Predicting with (CO)VLMC

VLMC case

Mixvlmc implements one-step-ahead prediction. This is done in a straightforward way, apart for the initial values. Indeed, given a time series \((x_i)_{1\leq i\leq n}\) and a VLMC of order \(d\), the context of \(x_j\) for \(j>d\) can be determined from the \(d\) past values \[ x_{j-d}, x_{j-d+1}, \ldots, x_{j-1}. \] In the VLMC case, this context is associated to a conditional probability distribution for \(X_j\). This distribution can be reported directly as the “prediction” for \(x_j\), or a direct prediction can be made using \[ \arg\max_{x\in S}\mathbb{P}(X_j=x|X_{j-1}=x_{j-1},\ldots,X_{j-d}=x_{j-d}), \] where \(S\) is the state space of the VLMC. Notice than this can be used to predict \(x_{n+1}\) which is unknown, making actual predictions.

COVLMC case

The case of COVLMC is almost identical. The only difference comes from the covariate influence. One-step-ahead predictions can be made only if the time series of covariates is provided.

Once the context of \(x_j\) has been computed as in the case of a VLMC, we obtain from the COVLMC the conditional distribution \[ \mathbb{P}(X_j|X_{j-1}=x_{j-1},\ldots,X_{j-d}=x_{j-d}, Y_{j-1}=y_{j-1}, \ldots, Y_{j-h}=y_{j-h}), \] where \(h\) is the order of covariate dependency for this context.

The distribution can be reported or a prediction can be made using the mode of the distribution, exactly as in the VLMC case.

Initial values

The only difficulty comes in both cases from the initial values \(x_1\) to \(x_d\) for which no proper context can be determined. This issue appears in numerous situations when using (CO)VLMC models, in particular for likelihood calculation (see vignette("likelihood")) and for sampling (see vignette("sampling")).

We use the notion of extended context described in detail in the likelihood vignette. The key idea is to include in the (CO)VLMC additional contexts (and thus conditional distributions) to model the first few observations.

VLMC

Let us consider for example the sun spot time series, sunspot.year, seen as a binary time series, with high activity associated to a number of sun spots larger than the median number.

sun_activity <- as.factor(ifelse(sunspot.year >= median(sunspot.year), "high", "low"))

We adjust automatically an optimal VLMC as follows:

sun_model_tune <- tune_vlmc(sun_activity)
sun_model_tune
#> VLMC context tree on high, low 
#>  cutoff: 2.306 (quantile: 0.03175)
#>  Number of contexts: 9 
#>  Maximum context length: 5 
#>  Selected by BIC (236.262) with likelihood function "truncated" (-98.83247)

Predictions are obtained as follows:

sun_model <- as_vlmc(sun_model_tune)
sun_model_predictions <- predict(sun_model, sun_activity)

Notice that contrarily to many implementations of stats::predict(), we do not support predictions without new data, but the original time series can of course be used as the “new data”.

Predictions are relatively correct, as shown by the confusion matrix:

table(sun_model_predictions[-length(sun_model_predictions)], sun_activity)
#>       sun_activity
#>        high low
#>   high  119  27
#>   low    27 116

Notice that we remove the last prediction to be able to perform the comparison. We could have asked for it to be excluded directly by passing final_pred=FALSE to the predict.vlmc() function.

The BIC used by default in tune_vlmc may be too conservative for prediction or sampling oriented use of the final model. With the AIC, we obtain a more complex model as follows:

sun_model_tune_aic <- tune_vlmc(sun_activity, criterion = "AIC")
sun_model_tune_aic
#> VLMC context tree on high, low 
#>  cutoff: 2.306 (quantile: 0.03175)
#>  Number of contexts: 9 
#>  Maximum context length: 5 
#>  Selected by AIC (203.711) with likelihood function "truncated" (-98.83247)

As expected it has a slightly better confusion matrix on its estimation data set:

table(
  predict(as_vlmc(sun_model_tune_aic), sun_activity, final_pred = FALSE),
  sun_activity
)
#>       sun_activity
#>        high low
#>   high  119  27
#>   low    27 116

Despite the use of penalized likelihood, we may experiment some form of overfitting and a split sample approach could be used to study it. For instance, one can use AIC to build a VLMC on the first half of the sun spots time series and evaluate the quality of the predictions on the second half, as follows:

first_half <- 1:(length(sun_activity) %/% 2)
sun_model_tune_aic_half <- tune_vlmc(sun_activity[first_half], criterion = "AIC")
sun_model <- as_vlmc(sun_model_tune_aic_half)
table(
  predict(sun_model, sun_activity[-first_half], final_pred = FALSE),
  sun_activity[-first_half]
)
#>       
#>        high low
#>   high   67  13
#>   low    14  51

The predictions remain of a rather good quality.

COVLMC

To illustrate COVLMC predictions, we use the French CAC index provided in EuStockMarkets:

CAC_raw <- as.data.frame(EuStockMarkets)$CAC

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")
)

We use the other indexes of EuStockMarkets as covariates. As previously, we select a model with the AIC criterion:

CAC_covariates <- as.data.frame(EuStockMarkets)[c("DAX", "SMI", "FTSE")][-1, ]
CAC_covlmc <- tune_covlmc(CAC_dts, CAC_covariates, criterion = "AIC")
CAC_comodel <- as_covlmc(CAC_covlmc)

Finally, we obtain predictions using predict.covlmc():

CAC_pred <- predict(CAC_comodel, CAC_dts, CAC_covariates, final_pred = FALSE)

In this case, the predictions are of rather poor quality:

table(CAC_pred, CAC_dts)
#>         CAC_dts
#> CAC_pred Down Stay  Up
#>     Down   40   46  36
#>     Stay  453  625 467
#>     Up     36   65  91

This can be easily explained by the large ambiguity of the conditional distributions as can be observed by focusing on them rather than on predictions:

CAC_probs <- predict(CAC_comodel, CAC_dts, CAC_covariates, final_pred = FALSE, type = "probs")
CAC_probs[1:10, ]
#>            Down      Stay        Up
#>  [1,] 0.2841765 0.3961249 0.3196986
#>  [2,] 0.2878788 0.4073084 0.3048128
#>  [3,] 0.2878788 0.4073084 0.3048128
#>  [4,] 0.2878788 0.4073084 0.3048128
#>  [5,] 0.2878788 0.4073084 0.3048128
#>  [6,] 0.2878788 0.4073084 0.3048128
#>  [7,] 0.2878788 0.4073084 0.3048128
#>  [8,] 0.2878788 0.4073084 0.3048128
#>  [9,] 0.4248040 0.3124036 0.2627924
#> [10,] 0.3747649 0.4891093 0.1361259

The Shannon entropies of those conditional distributions are generally relatively close to the maximum in this setting, i.e. 1.09861, as shown on this graphical illustration

entropies <- data.frame(entropy = apply(CAC_probs, 1, \(x) -sum(x * log(x))))
ggplot(entropies, aes(x = entropy)) +
  geom_density() +
  geom_rug(alpha = 0.1) +
  geom_vline(xintercept = -log(1 / 3), col = 2)

This illustrate the fact that (CO)VLMC models are not predictive models but rather generative models. They may perform well on predictive tasks if the conditional distributions are peaked enough, which corresponds in a way to simple predictive cases.

For example, the case of the sun spots studied above for the VLMC has better predictive performances because of the presence of strongly peaked distributions (some of them are even deterministic), as shown below:

sun_probs <- predict(as_vlmc(sun_model_tune_aic), sun_activity,
  final_pred = FALSE,
  type = "probs"
)
sun_entropies <- data.frame(entropy = apply(
  sun_probs, 1,
  \(x) -sum(x * log(x), na.rm = TRUE)
))
ggplot(sun_entropies, aes(x = entropy)) +
  geom_histogram(bins = 50) +
  geom_rug(alpha = 0.5) +
  geom_vline(xintercept = -log(1 / 2), col = 2)

This while (CO)VLMC based predictions can be interesting, one will generaly obtained more insights about the original time series using simulations, as detailed in vignette("sampling").

VLMC

For instance for the sun spots complex AIC model, we obtain

sun_metrics <- metrics(as_vlmc(sun_model_tune_aic))
sun_metrics
#> VLMC context tree on high, low 
#>  cutoff: 2.306 (quantile: 0.03175)
#>  Number of contexts: 9 
#>  Maximum context length: 5 
#>  Confusion matrix: 
#>        high low 
#>   high 119  27  
#>   low  27   116 
#>  Accuracy: 0.8131 
#>  AUC: 0.9146

The ROC curve is computed by metrics() (for state space of size 2) and can be plotted simply using:

plot(sun_metrics$roc)

COVLMC

Similarly, the CAC model gives:

CAC_metrics <- metrics(CAC_comodel)
CAC_metrics
#> VLMC with covariate context tree on Down, Stay, Up 
#>  cutoff in quantile scale: 0.01046
#>  Number of contexts: 5 
#>  Maximum context length: 2 
#>  Confusion matrix: 
#>        Down Stay Up  
#>   Down 40   46   36  
#>   Stay 453  625  467 
#>   Up   36   65   91  
#>  Accuracy: 0.4067 
#>  AUC: 0.6451

When the state space contains three or more states, we report the generalized AUC proposed by Hand and Till in A Simple Generalisation of the Area Under the ROC Curve for Multiple Class Classification Problems.