---
title: "Automatic Selection and Ensembling"
output: 
    rmarkdown::html_vignette:
        css: custom.css
        code_folding: hide
        toc: yes
vignette: >
  %\VignetteIndexEntry{Automatic Selection and Ensembling}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup, warning=FALSE,message=FALSE}
library(knitr)
library(tsissm)
library(xts)
library(tsaux)
library(data.table)
library(zoo)
```

## Introduction

The `issm_modelspec()` function is the primary entry point for modeling in the 
`tsissm package`. It defines a model specification with support for automatic 
selection across multiple configurations. In addition to selecting the single 
best model (e.g., by AIC), it can optionally return the top N ranked models, 
enabling downstream model ensembling for filtering, forecasting and simulation.

This vignette demonstrates that functionality using U.S. Advanced Retail Sales 
data available in the package.

## Advanced Retail Sales

This dataset represents the monthly, non-seasonally adjusted advance estimate of 
retail trade sales, based on a sub-sample of firms from the larger Monthly Retail 
Trade Survey.

As shown in the plot below, the series exhibits two prominent structural breaks: 
one around 2008 during the Global Financial Crisis, and another during the 
COVID-19 pandemic. To address this, we use the `auto_regressors()` function from 
the [tsaux](https://CRAN.R-project.org/package=tsaux) package, which detects and encodes three types of anomalies: 
additive outliers, temporary changes, and structural breaks (see this blog [post](https://www.nopredict.com/blog/posts/2022-05-15-messy-data-and-anomaly-detection/) 
for details).

Modeling such anomalies is important. When ignored, they are often absorbed by 
other components of the model, leading to biased coefficient estimates and inflated forecast variance.

```{r, fig.width=6,fig.height=3}
data("us_retail_sales")
opar <- par(mfrow = c(1,1))
y <- as.xts(us_retail_sales)
par(mar = c(2,2,2,2))
plot(as.zoo(y), ylab = "Sales (Mil's S)", xlab = "", main = "Advance Retail Sales")
grid()
par(opar)
```


## Model Selection

For this demonstration, we reserve the last 38 months of data for forecasting 
evaluation and select the top 4 models for ensembling.


```{r}
train <- y["/2021"]
test <- y["2022/"]
lambda_pre_estimate <- box_cox(lambda = NA)$transform(train) |> attr("lambda")
xreg <- auto_regressors(train, frequency = 12, lambda = lambda_pre_estimate, sampling = "months", h = nrow(test), method = "sequential", 
                        check.rank = TRUE, discard.cval = 3.5, maxit.iloop = 10, maxit.oloop = 10, 
                        forc_dates = index(test))
spec <- issm_modelspec(train, auto = TRUE, slope = c(TRUE, FALSE), seasonal = TRUE, 
                             seasonal_harmonics = list(c(3,4,5)), xreg = xreg$xreg[index(train), ], 
                             seasonal_frequency = 12, ar = 0:2, ma = 0:2, lambda = lambda_pre_estimate, top_n = 4)
mod <- spec |> estimate()
```


We first inspect the top selected model:

```{r}
summary(mod$models[[1]])
```

This model uses 5 harmonics and an ARMA(2,1) structure. Most coefficients appear 
statistically significant. We can also summarize the top 4 selected models:

```{r}
mod$table |> kable()
```

All top models use 5 harmonics. The main differences lie in the ARMA specification,
and one model does not include a slope term. The pairwise correlation among model 
predictions is very high — expected, given the small structural differences:

```{r}
mod$correlation |> kable()
```



## Prediction and Ensembling

The object returned by `estimate()` when top_n > 1 is of class `tsissm.selection`. 
This class supports `tsfilter()`, `predict()`, and `simulate()` methods, which 
apply operations to each retained model, returning individual results. These can 
then be ensembled using the `tsensemble()` function.

Below, we generate forecasts from the top model and from all 4 models, 
and then ensemble them using equal weights:

```{r}
p_top <- mod$models[[1]] |> predict(h = nrow(test), seed = 200, nsim = 4000, newxreg = xreg$xreg[index(test),])
p_all <- mod |> predict(h = nrow(test), seed = 200, nsim = 4000, newxreg = xreg$xreg[index(test),])
p_ensemble <- p_all |> tsensemble(weights = rep(1/4,4))
```

We visualize the top model’s forecast and overlay the ensembled forecast distribution. 
The actual data is also shown for reference.

```{r, fig.width=6,fig.height=3}
opar <- par(mfrow = c(1,1))
par(mar = c(2,2,2,2))
p_top |> plot(n_original = 50, gradient_color =  "aliceblue", interval_type = 2, interval_color = "steelblue", interval_width = 1, median_width = 1.5, xlab = "")
p_ensemble$distribution |> plot(gradient_color =  "aliceblue", interval_type = 3, interval_color = "green", interval_width = 1, median_width = 1.5, 
                                median_type = 1, median_color = "green", add = TRUE)
lines(index(test), as.numeric(test), col = 2, lwd = 1.5)
legend("topleft", c("Historical","Actual (Forecast)", "Top (Forecast)", "Ensemble (Forecast)"), col = c("red","red","black","green"), lty = c(1,1.5,1.5,1.5), bty = "n")
par(opar)
```

Note: You can overlay multiple predictive distributions using the `add = TRUE` argument in `plot()` for
class `tsmodel.distribution`.

The following table compares performance metrics between the top model and the 
ensemble forecast. The ensemble consistently outperforms the top model across all metrics:

```{r}
tab <- rbind(tsmetrics(p_top, actual = test, alpha = 0.1),
             tsmetrics(p_ensemble, actual = test, alpha = 0.1))
rownames(tab) <- c("Top","Ensemble")
tab |> kable()
```


## Conclusion

The `tsissm` package provides a flexible and robust framework for modeling, 
forecasting, and simulating time series data using the innovations state space model. 
By supporting full model enumeration, automatic selection, and simulation-based 
prediction, it offers a rich toolkit for handling complex seasonal patterns 
and heteroscedasticity, and the availability of regressors allows the handling
of identified outliers and structural breaks.

This vignette demonstrated how to apply automatic model selection with support 
for retaining the top N models and performing ensemble forecasting. By leveraging 
both predictive distributions and weighted ensembling, users can obtain more 
stable and accurate forecasts — particularly valuable when models are structurally 
similar but individually uncertain.
