---
title: "Iterative Proportional Fitting"
output: rmarkdown::html_vignette
author:
  - Gregor de Cillia
vignette: >
  %\VignetteIndexEntry{Iterative Proportional Fitting}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: lib.bib
---

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

This vignette explains the usage of the `ipf()` function, which has been used for calibrating the labour force survey of Austria for several years.
It is based on the Iterative Proportional Fitting algorithm and gives some flexibility about the details of the implementation. See [@mekogu2016] or `vignette("methodology")` for more details.

## Setup

We will assume the output of `demo.eusilc()` is our population.
From this population, a sample without replacement is drawn.
The sample covers 10 percent of the population.
We assign a weight of one for all observations of the population and a weight of ten for all observations of the sample.

```{r setup}
library(surveysd)
population <- demo.eusilc(1, prettyNames = TRUE)
population[, pWeight := 1]
pop_sample <- population[sample(1:.N, floor(.N*0.10)), ]
pop_sample[, pWeight := 10]
```

## One constraint, one variable

We will start with an example where we want to adapt the weights of `pop_sample` such that the weighted number of males and females matches the ones of `population`.
We can see that this is currently not the case.

```{r}
(gender_distribution <- xtabs(pWeight ~ gender, population))
xtabs(pWeight ~ gender, pop_sample)
```

Due to random sampling (rather than stratified sampling), there are differences between the gender distributions.
We can pass `gender_distribution` as a parameter to `ipf()` to obtain modified weights.

```{r}
pop_sample_c <- ipf(pop_sample, conP = list(gender_distribution), w = "pWeight")
```

The resulting dataset, `pop_sample_c` is similar to `pop_sample` but has an additional column with the adjusted weights.

```{r}
dim(pop_sample)
dim(pop_sample_c)
setdiff(names(pop_sample_c), names(pop_sample))
```

We can now calculate the weighted number of males and females according to this new weight.
This will result in a match for the constraints.

```{r}
xtabs(calibWeight ~ gender, pop_sample_c)
xtabs(pWeight ~ gender, population)
```

In this simple case, `ipf` just performs a post stratification step.
This means, that all males and all females have the same weight.

```{r, fig.align="center", out.width="100%"}
xtabs(~ calibWeight + gender, pop_sample_c)
```

```{r, include = FALSE}
overrepresented_gender <- pop_sample_c[calibWeight < 10, ][1, gender]
```

All `r overrepresented_gender`s have been weighted down (`calibWeight < 10`) to compensate for the overrepresentation in the sample.

## One constraint, two variables

Let's now assume that we want to put constraints on the number of males and females for each age group. 
The numbers from the original population can be obtained with `xtabs()`.

```{r}
(con_ga <- xtabs(pWeight ~ gender + age, population))
xtabs(pWeight ~ gender + age, pop_sample)
```

Again, we can see that those constraints are not met.
Supplying the contingency table `con_ga` to `ipf()` will again resolve this.

```{r}
pop_sample_c2 <- ipf(pop_sample, conP = list(con_ga), w = "pWeight")
xtabs(pWeight ~ gender + age, population)
xtabs(calibWeight ~ gender + age, pop_sample_c2)
```

## Two constraints

Now we assume that we know the number of persons living in each nuts2 region from registry data.

```{r}
registry_table <- xtabs(pWeight ~ region, population)
```

However, those registry data does not provide any information about age or `gender`.
Therefore, the two contingency tables (`con_ga` and `registry_table`) have to be specified independently.
This can be done by supplying a list to `conP`

```{r}
pop_sample_c2 <- ipf(pop_sample, conP = list(con_ga, registry_table), w = "pWeight")
xtabs(pWeight ~ gender + age, population)
xtabs(calibWeight ~ gender + age, pop_sample_c2)
xtabs(pWeight ~ region, population)
xtabs(calibWeight ~ region, pop_sample_c2)
```

this time, the constraints are not matched perfectly.
That is, because we provided more than one constraint.
therefore, the `ipf()` algorithm had to work iteratively.

## Household Constraints

If the dataset has a household structure, household constraints can be passed
via the parameter `conH`. If this parameter is used, it is also necessary to
supply `hid`, which defines the column names that contains household ids.

```{r}
(conH1 <- xtabs(pWeight ~ hsize + region, data = population[!duplicated(hid)]))
pop_sample_hh <- ipf(pop_sample, hid = "hid", conH = list(conH1), w = "pWeight",
                     bound = 10)
xtabs(calibWeight ~ hsize + region, data = pop_sample_hh[!duplicated(hid)])
```

## Tolerances

If `conP` or `conH` contain several contingency tables or if `conP` and `conH`
are used at the same time, the ipf algorithm will operate iteratively. This
means that the calibrated dataset will satisfy the constraints only
approximately. The default tolerances of the approximation can be overwritten
using the parameters `conP` and `conH`.

Lowering the tolerances will improve the match between the constraints and
the contingency tables according to the calibrated weights. However, lower
tolerances will also make it so more iterations are necessary until a
convergence is met. If the constraints are too small, ipf will return with
a warning that indicates that a convergence could not be reached.

```{r}
ipf(pop_sample, conP = list(con_ga, registry_table), w = "pWeight",
    verbose = TRUE, epsP = 0.01)
ipf(pop_sample, conP = list(con_ga, registry_table), w = "pWeight",
    verbose = TRUE, epsP = 0.0001)
```

We see that changing the tolerances from `0.01` (one percent) to `0.0001`
increases the number of required iterations.

## References