---
title: "Temporal Disaggregation of Time Series"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Introduction}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  fig.width  = 7.1,
  fig.height = 3.5,
  comment = "#>"
)

```

This is a short introduction to tempdisagg.
Our article on **temporal disaggregation of time series** (\doi{10.32614/RJ-2013-028}) in the R-Journal describes the package and the theory of temporal disaggregation in more detail.

For citation, please see `citation("tempdisagg")`.

## Disaggregating annual data to quarterly

Suppose we have an annual series and want to create quarterly values that sum
up to the annual values. Let us explore the annual sales of the pharmaceutical
and chemical industry in Switzerland, from which we want to create a quarterly
series.

```{r}
library(tempdisagg)
data(swisspharma)
plot(sales.a)
```

## No indicator

The most simple method is `denton-cholette` without an indicator. It
performs a simple interpolation that meets the temporal additivity constraint.
In R, this can be done the following way:

```{r}
m1 <- td(sales.a ~ 1, to = "quarterly", method = "denton-cholette")
```

`td()` produces an object of class `"td"`. The formula, `'sales.a ~ 1'`, indicates
that our low frequency variable will be disaggregated with a constant. The
resulting quarterly values of sales can be extracted with the `predict()`
function:

```r
predict(m1)
```

As there is no additional information on quarterly movements, the resulting
series is very smooth:

```{r}
plot(predict(m1))
```

## Using additional information

While this purely mathematical approach is easy to perform and does not need
any other data series, the economic value of the resulting series may be
limited. There might be a related quarterly series that follows a similar
movement than sales. For example, we may use quarterly exports of
pharmaceutical and chemical products:

```{r}
plot(exports.q)
m2 <- td(sales.a ~ 0 + exports.q, method = "denton-cholette")
```

Because we cannot use more than one indicator with the `denton-cholette` or
`denton` method, the intercept must be specified as missing in the formula
(`~ 0`). Contrary to the first example, the `to` argument is redundant, because
the destination frequency can be interfered from the time series properties of
`exports.q`.

The resulting model leads to a much more interesting series:

```{r}
plot(predict(m2))
```

As the indicator series is longer than the annual series, there is an
extrapolation period, in which the quarterly sales are forecasted.

With an indicator, the `denton-cholette` method simply transfers the movement
of the indicator to the resulting series. Even if in fact there were no
correlation between the two series, there would be a strong similarity between
the indicator and the resulting series.

## Regression based disaggregation

In contrast, regression based methods transfer the movement only if the
indicator series and the resulting series are actually correlated on the
annual level. For example, a Chow-Lin regression of the same problem as above
can be performed the following way:

```{r}
m3 <- td(sales.a ~ exports.q)
```

As `chow-lin-maxlog` is the default method, it does not need to be specified.
Like with the corresponding `lm` method, `summary()` produces an overview of the
regression:

```{r}
summary(m3)
```

There is indeed a strong correlation between exports and sales, as it has been
assumed in the example above. The coefficient of `exports.q` is highly
significant, and the very high adjusted R-squared points to a strong
relationship between the two variables. The coefficients are the result of a
GLS regression between the annual series.

The estimation of the AR1 parameter, *rho*, was estimated to be negative; in
order to avoid the undesirable side-effects of a negative *rho*, it has been
truncated to 0. This feature can be turned off:

```{r}
td(sales.a ~ exports.q, truncated.rho = -1)
```

Again, we can extract the resulting quarterly series of sales:

```{r}
plot(predict(m3))
```

Like all regression based methods, `chow-lin-maxlog` can also be used with
more than one indicator series (`imports.q` is only significant on a 10% level
in the following example, it probably will not help to produce a more accurate
temporal disaggregation):

```{r}
m4 <- td(formula = sales.a ~ exports.q + imports.q)
summary(m4)
```

## Comparison of methods

In our example, we actually know the true data on quarterly sales, so we can
compare the artificial values to the true values:

```{r}
plot(sales.q)
lines(predict(m2), col = "blue") # Denton-Cholette
lines(predict(m3), col = "red") # Chow-Lin
```

With an indicator series, both the Denton method and Chow-Lin produce a series
that is close to the true series. This is, of course, due to fact that in this
example, exports are a good indicator for sales. If the indicator is less
close to the series of interest, the resulting series will be less close to
the true series.