---
title: "Introduction to ArchaeoChron"
author: "A. Philippe and M.-A. Vibet"
date: "2023-12-04"
output:
  markdown::html_format:
    options:
      toc: true
      number_sections: true
vignette: >
  %\VignetteIndexEntry{Introduction to ArchaeoChron}
  %\VignetteEngine{knitr::knitr}
  %\VignetteEncoding{UTF-8}
---




```r
## Load packages
library(ArchaeoChron)
library(ArchaeoPhases)
library(coda) # MCMC diagnostic
```

# Bayesian model for combining Gaussian dates

## Example of a sunspot

The sunspot data is constituted of several dates assumed to be contemporaneous of a single event. These dates do not need any calibration but their unit is in year before 2016.

## Simple model for combining Gaussian dates

Let's first use a simple Bayesian model.


```r
data(sunspot)

MCMC1 <- combination_Gauss(
  M = sunspot$Age[1:5],
  s = sunspot$Error[1:5],
  refYear = rep(2016, 5),
  studyPeriodMin = 900,
  studyPeriodMax = 1500,
  variable.names = c('theta')
)
#> [1] "Update period"
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```

The output of the function `combination_Gauss()` is a Markov chain of the posterior distribution of the Bayesian model.

### Convergence of the Markov chain

First, let's check the convergence of the Markov chain of each parameter.


```r
plot(MCMC1)
```

<div class="figure" style="text-align: center">
<img src="./figures/unnamed-chunk-2-1.png" alt="plot of chunk unnamed-chunk-2"  />
<p class="caption">plot of chunk unnamed-chunk-2</p>
</div>

```r
gelman.diag(MCMC1)
#> Potential scale reduction factors:
#> 
#>       Point est. Upper C.I.
#> theta          1          1
```

The Gelman diagnotic gives point estimates about 1, so convergence is reached.

### Statistics of the posterior distribution of the event.

Now, we can use the package **ArchaeoPhases** in order to describe the posterior distribution of the parameters.

Here we will focus on the posterior distribution of the date of interest, i.e. the parameter *theta*.


```r
MCMCSample1 <- cbind(MCMC1[[1]], MCMC1[[2]])
MCMCEvent1 <- as_events(MCMCSample1, calendar = CE())
plot(MCMCEvent1[, 1], level = 0.95)
```

<div class="figure" style="text-align: center">
<img src="./figures/unnamed-chunk-3-1.png" alt="plot of chunk unnamed-chunk-3"  />
<p class="caption">plot of chunk unnamed-chunk-3</p>
</div>

```r

(M1 <- summary(MCMCEvent1[, 1], level = 0.95))
#>        mad mean sd  min   q1 median   q3  max start  end
#> theta 1028 1028  3 1021 1027   1028 1029 1035  1025 1032
```

The date (mean posterior distribution) of the sunspot estimated using a simple Bayesian model that combines dates, is dates at 1028 years after Christ. This date is associated with a 95% confidence interval : [1025, 1032].

## Bayesian model for combining Gaussian dates and handling potential outliers

Let's use the function `combinationWithOutliers_Gauss()`.


```r
MCMC2 <- combinationWithOutliers_Gauss(
  M = sunspot$Age[1:5],
  s = sunspot$Error[1:5],
  refYear = rep(2016, 5),
  outliersIndivVariance = rep(1, 5),
  outliersBernouilliProba = rep(0.2, 5),
  studyPeriodMin = 800,
  studyPeriodMax = 1500,
  variable.names = c('theta')
)
#> [1] "Update period"
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plot(MCMC2)
```

<div class="figure" style="text-align: center">
<img src="./figures/unnamed-chunk-4-1.png" alt="plot of chunk unnamed-chunk-4"  />
<p class="caption">plot of chunk unnamed-chunk-4</p>
</div>

The output of the function `combinationWithOutliers_Gauss()` is a Markov chain of the posterior distribution of the Bayesian model.

Here we will focus on the posterior distribution of the date of interest i.e. the parameter 'theta*.


```r
MCMCSample2 <- cbind(MCMC2[[1]], MCMC2[[2]])
MCMCEvent2 <- as_events(MCMCSample2, calendar = CE())
plot(MCMCEvent2[, 1], level = 0.95)
```

<div class="figure" style="text-align: center">
<img src="./figures/unnamed-chunk-5-1.png" alt="plot of chunk unnamed-chunk-5"  />
<p class="caption">plot of chunk unnamed-chunk-5</p>
</div>

```r

(M2 <- summary(MCMCEvent2[, 1], level = 0.95))
#>        mad mean sd  min   q1 median   q3  max start  end
#> theta 1028 1028  3 1022 1027   1028 1029 1035  1025 1032
```

The date (mean posterior distribution) of the sunspot, estimated using a simple Bayesian model that combines dates and allows for outliers, is dates at 1028 years after Christ. This date is associated with a 95% confidence interval : [1025, 1032].


## Bayesian model for combining Gaussian dates with a random effect

Let's now use the function `combinationWithRandomEffect_Gauss()`.


```r
MCMC3 <- combinationWithRandomEffect_Gauss(
  M = sunspot$Age[1:5],
  s = sunspot$Error[1:5],
  refYear = rep(2016, 5),
  studyPeriodMin = 0,
  studyPeriodMax = 1500,
  variable.names = c('theta')
)
#> [1] "Update period"
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plot(MCMC3)
```

<div class="figure" style="text-align: center">
<img src="./figures/unnamed-chunk-6-1.png" alt="plot of chunk unnamed-chunk-6"  />
<p class="caption">plot of chunk unnamed-chunk-6</p>
</div>

The output of the function `combinationWithRandomEffect_Gauss()` is a Markov chain of the posterior distribution of the Bayesian model.

Here we will focus on the posterior distribution of the date of interest i.e. the parameter 'theta*.


```r
MCMCSample3 <- cbind(MCMC3[[1]], MCMC3[[2]])
MCMCEvent3 <- as_events(MCMCSample3, calendar = CE())
plot(MCMCEvent3[, 1], level = 0.95)
```

<div class="figure" style="text-align: center">
<img src="./figures/unnamed-chunk-7-1.png" alt="plot of chunk unnamed-chunk-7"  />
<p class="caption">plot of chunk unnamed-chunk-7</p>
</div>

```r

(M3 <- summary(MCMCEvent3[, 1], level = 0.95))
#>        mad mean sd min   q1 median   q3  max start  end
#> theta 1027 1026  5 998 1024   1027 1029 1052  1018 1033
```

The date (mean posterior distribution) of the sunspot, estimated using a simple Bayesian model that combines dates and allows for random effects, is dates at 1026 years after Christ. This date is associated with a 95% confidence interval : [1018, 1033].

## Event Model : Bayesian model for combining Gaussian dates with individual random effects

If we want to date that event, we can use the Event model for combining Gaussian dates. In that example, we will investigate the posterior distribution of the date of the event (called 'theta*) and the posterior distribution of the dates associated with this event.

### Generating the Markov chain of the posterior distribution

Finally, let's use the function `eventModel_Gauss()` and the first 10 dates of the dataset sunspot. The study period should be given in calendar year (BC/AD).


```r
MCMC4 <- eventModel_Gauss(
  M = sunspot$Age[1:5],
  s = sunspot$Error[1:5],
  refYear = rep(2016, 5),
  studyPeriodMin = 900,
  studyPeriodMax = 1500,
  variable.names = c('theta', 'mu')
)
#> [1] "Update period"
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```

The output of the `eventModel_Gauss()` is the Markov chain of the posterior distribution.

### Convergence of the Markov chain

First, let's check the convergence of the Markov chain of each parameter.


```r
plot(MCMC4)
```

<div class="figure" style="text-align: center">
<img src="./figures/unnamed-chunk-9-1.png" alt="plot of chunk unnamed-chunk-9"  />
<p class="caption">plot of chunk unnamed-chunk-9</p>
</div><div class="figure" style="text-align: center">
<img src="./figures/unnamed-chunk-9-2.png" alt="plot of chunk unnamed-chunk-9"  />
<p class="caption">plot of chunk unnamed-chunk-9</p>
</div>

```r
gelman.diag(MCMC4)
#> Potential scale reduction factors:
#> 
#>       Point est. Upper C.I.
#> mu[1]          1          1
#> mu[2]          1          1
#> mu[3]          1          1
#> mu[4]          1          1
#> mu[5]          1          1
#> theta          1          1
#> 
#> Multivariate psrf
#> 
#> 1
```

The Gelman diagnotic gives point estimates about 1, so convergence is reached.

### Statistics of the posterior distribution of the event.

Here we will focus on the posterior distribution of the event of interest (parameter 'theta*).


```r
MCMCSample4 <- cbind(MCMC4[[1]], MCMC4[[2]])
MCMCEvent4 <- as_events(MCMCSample4, calendar = CE())
plot(MCMCEvent4[, 6], level = 0.95)
```

<div class="figure" style="text-align: center">
<img src="./figures/unnamed-chunk-10-1.png" alt="plot of chunk unnamed-chunk-10"  />
<p class="caption">plot of chunk unnamed-chunk-10</p>
</div>

```r

(M4 <- summary(MCMCEvent4[, 6], level = 0.95))
#>        mad mean sd  min   q1 median   q3  max start  end
#> theta 1027 1026  5 1007 1024   1027 1029 1048  1018 1034
```

The date (mean posterior distribution) of the sunspot, estimated using a simple Bayesian model that combines dates and allows for individual random effects, is dates at 1026 years after Christ. This date is associated with a 95% confidence interval : [1018, 1034].
