---
title: "Decomposing Series"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Decomposing Series}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options:
  markdown:
    wrap: 80
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7,
  fig.height = 4.5,
  fig.align = "center",
  message = FALSE,
  warning = FALSE
)
```

```{r setup}
#| include: false
library(trendseries)
library(dplyr)
library(tidyr)
library(ggplot2)
```

# Decomposing Series

The trend extraction methods covered in the other vignettes return a single
smooth component. Often, though, an economic series is more naturally described
as the sum of three parts: a slow-moving **trend**, a repeating **seasonal**
pattern, and an irregular **remainder**. `decompose_series()` splits a series
into these three components and adds them as columns to the original data frame.

```{r libs}
#| eval: false
library(trendseries)
library(dplyr)
library(tidyr)
```

The theme below is used throughout the vignette for consistent styling.

```{r theme}
#| code-fold: true
library(ggplot2)

theme_series <- theme_minimal(paper = "#fefefe") +
  theme(
    legend.position = "bottom",
    panel.grid.minor = element_blank(),
    strip.background = element_rect(fill = "#2c3e50"),
    strip.text = element_text(color = "#fefefe"),
    axis.ticks.x = element_line(color = "gray40", linewidth = 0.5),
    axis.line.x = element_line(color = "gray40", linewidth = 0.5),
    axis.title.x = element_blank(),
    palette.colour.discrete = c(
      "#2c3e50",
      "#e74c3c",
      "#f39c12",
      "#1abc9c",
      "#9b59b6"
    )
  )
```

## Trend extraction vs decomposition

It is worth being clear about the difference between this function and
`augment_trends()`.

- `augment_trends()` returns **only the trend** (`trend_*` columns). The
  seasonal and irregular movements are simply smoothed away.
- `decompose_series()` returns **all three components**
  (`trend_*`, `seasonal_*`, and `remainder_*`), and they add back up exactly to
  the original series.

Because there is a seasonal component, `decompose_series()` requires seasonal
data: monthly (`frequency = 12`) or quarterly (`frequency = 4`). For annual
series there is nothing seasonal to isolate, so use `augment_trends()` instead.

## A first decomposition

Let's start with the `gdp_construction` dataset, a quarterly index of Brazilian
construction activity that ships with `trendseries`.

```{r gdp-plot}
ggplot(gdp_construction, aes(date, index)) +
  geom_line(lwd = 0.7) +
  scale_x_date(date_breaks = "2 years", date_labels = "%Y") +
  labs(
    title = "Brazilian construction activity",
    y = "Index"
  ) +
  theme_series
```

Passing the data to `decompose_series()` adds three new columns. The
frequency is detected automatically from the date column.

```{r gdp-decomp}
gdp_parts <- gdp_construction |>
  decompose_series(value_col = "index")

gdp_parts
```

The three components are named after the method (`stl` by default):
`trend_stl`, `seasonal_stl`, and `remainder_stl`.

Reshaping to long format makes it easy to plot the original series alongside its
three components.

```{r gdp-long}
gdp_long <- gdp_parts |>
  pivot_longer(
    cols = c(index, trend_stl, seasonal_stl, remainder_stl),
    names_to = "component",
    values_to = "value"
  ) |>
  mutate(
    component = factor(
      component,
      levels = c("index", "trend_stl", "seasonal_stl", "remainder_stl"),
      labels = c("Observed", "Trend", "Seasonal", "Remainder")
    )
  )
```

```{r gdp-facet}
#| code-fold: true
#| fig-height: 6
ggplot(gdp_long, aes(date, value)) +
  geom_line(aes(color = component), lwd = 0.7, show.legend = FALSE) +
  facet_wrap(vars(component), ncol = 1, scales = "free_y") +
  scale_x_date(date_breaks = "2 years", date_labels = "%Y") +
  labs(
    title = "STL decomposition of construction activity",
    subtitle = "Observed = Trend + Seasonal + Remainder",
    y = NULL
  ) +
  theme_series
```

## STL decomposition

The default method is **STL** (Seasonal-Trend decomposition via Loess),
implemented with `stats::stl()`. The seasonal component is estimated with a loess
smoother, the trend with an adaptive moving average, and the remainder is
whatever is left over. The defaults (`s.window = "periodic"`, `robust = FALSE`)
suit most economic series with a stable seasonal pattern.

Fine control is available through the `params` list. The two most useful options
are an evolving seasonal pattern and robust fitting.

```{r stl-params, eval = FALSE}
# Allow the seasonal pattern to evolve slowly over time, and downweight outliers
gdp_robust <- gdp_construction |>
  decompose_series(
    value_col = "index",
    params = list(s.window = 13, robust = TRUE)
  )
```

A `s.window` of `"periodic"` forces an identical seasonal shape in every year;
supplying a positive odd integer instead lets that shape drift gradually. Setting
`robust = TRUE` reduces the influence of one-off spikes on the trend and seasonal
estimates.

## Regression decomposition

The **regression** method (`methods = "regression"`) fits a single OLS model

$$
y_t = f(t) + s(t) + \epsilon_t
$$

where $f(t)$ is a polynomial in time and $s(t)$ is a set of period (month or
quarter) dummy variables. The trend is the constant plus the polynomial terms,
the seasonal component is the period effect (centred to mean zero), and the
remainder is the model residual.

Unlike STL, the regression trend is a smooth global polynomial, which can be a
better description of a series with a steady long-run direction. For the examples
that follow we use the `oil_derivatives` dataset, which records monthly
petroleum-derivatives production in Brazil; we restrict it to the 1995–2006 window
and work on the log scale.

```{r reg-decomp}
suboil <- oil_derivatives |>
  filter(between(date, as.Date("1995-06-01"), as.Date("2006-12-01"))) |>
  mutate(lprod = log(production))

ggplot(suboil, aes(date, lprod)) +
  geom_line(lwd = 0.7) +
  labs(
    title = "Petroleum derivatives production",
    y = "Thousand barrels per day (log scale)"
  ) +
  theme_series
```

The `trend` argument selects the polynomial form: `"linear"` (the default),
`"quadratic"`, or `"cubic"`. Orthogonal polynomials are used by default for numerical stability.

```{r}
suboil_reg <- suboil |>
  decompose_series(
    value_col = "lprod",
    methods = "regression",
    trend = "cubic"
  )

ggplot(suboil_reg, aes(date)) +
  geom_line(aes(y = lprod, color = "Original"), lwd = 0.7) +
  geom_line(aes(y = trend_regression, color = "Trend (reg.)"), lwd = 0.7) +
  labs(title = "Regression cubic trend", y = "log production", color = NULL) +
  theme_series
```

If the series is trend-stationary, `remainder_regression` — the series with the
trend and seasonality removed — is approximately stationary.

```{r}
ggplot(suboil_reg, aes(date, remainder_regression)) +
  geom_line(lwd = 0.7) +
  labs(title = "Regression remainder", y = NULL) +
  theme_series
```

It is instructive to compare the two decomposition methods directly. Both produce
fixed seasonal patterns, but STL follows short-term fluctuations more closely,
producing a "cleaner" remainder.

```{r compare-trends}
#| code-fold: true
comparison <- suboil |>
  decompose_series(value_col = "lprod", methods = "stl") |>
  decompose_series(value_col = "lprod", methods = "regression", trend = "cubic")

comparison_long <- comparison |>
  pivot_longer(
    cols = trend_stl:remainder_regression,
    names_pattern = "(.*)_(.*)",
    names_to = c("component", "method"),
    values_to = "trend"
  ) |>
  mutate(
    component = factor(component, levels = c("trend", "seasonal", "remainder"))
  )

ggplot(comparison_long, aes(date, trend)) +
  # Original series
  geom_line(
    data = suboil,
    aes(y = lprod),
    layout = c(1, 2),
    lwd = 0.5,
    alpha = 0.5
  ) +
  # Components and methods
  geom_line(aes(y = trend, color = component), lwd = 0.7) +
  facet_grid(vars(component), vars(method), scales = "free_y") +
  guides(color = "none") +
  labs(title = "STL vs regression decomposition", y = NULL) +
  theme_series
```

## Grouped decomposition

Like `augment_trends()`, `decompose_series()` accepts a `group_cols` argument to
decompose several series at once. The data must be in tidy format. Here we
use the `electricity` dataset, which records monthly electricity consumption for
three sectors (residential, commercial, and industrial).

```{r elec-decomp}
electricity_parts <- electricity |>
  mutate(lvalue = log(value)) |>
  decompose_series(value_col = "lvalue", group_cols = "name_series")

glimpse(electricity_parts)
```

Each group is decomposed independently, and the components are stacked back into
a single data frame.

```{r elec-plot}
#| code-fold: true
#| fig-height: 5
ggplot(electricity_parts, aes(date)) +
  geom_line(aes(y = seasonal_stl), color = "#2c3e50", lwd = 0.8) +
  facet_wrap(vars(name_series), ncol = 1, scales = "free_y") +
  scale_x_date(date_breaks = "3 years", date_labels = "%Y") +
  labs(
    title = "Electricity consumption by sector",
    subtitle = "STL seasonal component extracted per group (log scale)",
    y = "seasonal factor (log)"
  ) +
  theme_series
```

## Multiplicative seasonality

So far we have handled multiplicative seasonality by logging the series before decomposing (`lprod = log(production)`, `lvalue = log(value)`). Many
macroeconomic series behave this way: the seasonal variations grow with the level of
the series, so an additive decomposition of the raw data would leave a seasonal
pattern that widens over time.

Every `decompose_series()` method is additive by construction. Instead of logging
manually, you can pass `transform = "log"`, which logs the series, decomposes it
additively, and exponentiates the components back to the original scale.

```{r}
#| eval: false
decompose_series(
  oil_derivatives,
  value_col = "production",
  transform = "log"
)
```

On the log scale the additive identity holds; after exponentiating, the
components satisfy the *multiplicative* identity
`trend * seasonal * remainder = value`. Note that this requires strictly positive data.

## Other methods

### Classical decomposition

`methods = "classic"` is the textbook decomposition implemented by
`stats::decompose()`. The trend is a centred moving average of order equal to the
frequency, the seasonal component is the average detrended value for each period,
and the remainder is what is left. It is simple and fast, but shouldn't be used
in practice.

```{r}
#| eval: false
decompose_series(
  oil_derivatives,
  value_col = "production",
  methods = "classic",
  transform = "log"
)
```

### Basic structural model

`methods = "bsm"` fits a Basic Structural (state-space) Model with
`stats::StructTS()`: stochastic level, slope, and seasonal components estimated
by maximum likelihood and extracted with the Kalman smoother. Unlike the
moving-average methods, it returns trend and seasonal estimates for *every*
observation and lets both components evolve over
time. The trade-off is that it relies on numerical optimisation, which can
occasionally fail to converge on short or irregular series.

```{r}
#| eval: false
decompose_series(
  suboil,
  value_col = "lprod",
  methods = "bsm"
)
```

### X-13ARIMA-SEATS (`seasonal`)

`methods = "seats"` runs the U.S. Census Bureau's X-13ARIMA-SEATS program through
the `seasonal` package — the seasonal-adjustment procedure used by many
statistical agencies. `seas()` is called with its automatic defaults: ARIMA model
selection, log/level transformation, outlier detection, and calendar adjustment.
The SEATS trend-cycle and seasonally adjusted series are then mapped to a
trend/seasonal/remainder triple that reproduces the original series exactly.
Because X-13 selects its own transformation internally, there is normally no need
to set `transform = "log"` for this method.

```{r seats, eval = requireNamespace("seasonal", quietly = TRUE)}
# requires the 'seasonal' package
seas_oil <- decompose_series(
  suboil,
  value_col = "lprod",
  methods = "seats"
)

ggplot(seas_oil, aes(date)) +
  geom_line(aes(y = lprod, color = "Original"), lwd = 0.7) +
  geom_line(aes(y = trend_seats, color = "Trend (SEATS)"), lwd = 0.7) +
  labs(title = "SEATS trend-cycle", y = "log production", color = NULL) +
  theme_series
```

```{r seats-seasonal, eval = requireNamespace("seasonal", quietly = TRUE)}
ggplot(seas_oil, aes(date, seasonal_seats)) +
  geom_line(lwd = 0.7) +
  labs(title = "SEATS seasonal component", y = NULL) +
  theme_series
```

Often our interest is mainly in the series *without* its seasonal component. For
convenience, `deseason_series()` wraps `decompose_series()` and returns the
seasonally adjusted series directly.

```{r deseason, eval = requireNamespace("seasonal", quietly = TRUE)}
deseas_oil <- deseason_series(
  suboil,
  value_col = "lprod",
  methods = "seats",
  # set to TRUE to also return the trend, seasonal, and remainder components
  components = FALSE
)

ggplot(deseas_oil, aes(date, seasadj_seats)) +
  geom_line(lwd = 0.7) +
  labs(
    title = "SEATS seasonally adjusted series",
    y = "log production"
  ) +
  theme_series
```

This is equivalent to using `seasadj = TRUE` in `decompose_series()`.

```{r}
#| eval: false
decompose_series(
  suboil,
  value_col = "lprod",
  methods = "seats",
  seasadj = TRUE
)
```

## Summary

- `decompose_series()` splits a seasonal (monthly or quarterly) series into
  trend, seasonal, and remainder components that sum to the original series.
- There are five available methods: `"stl"` (default), `"regression"`,
  `"classic"`, `"bsm"`, and `"seats"`
- All methods are additive; use `transform = "log"` for multiplicative
  seasonality, where the components instead multiply to the original series.
- `deseason_series()` is a convenience wrapper that returns only the seasonally
  adjusted series.