---
title: "Preparing data and choosing sampler settings"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Preparing data and choosing sampler settings}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

Before calling `pgmm_rjmcmc()`, the analyst must specify the data orientation,
scaling, missing-value handling, latent-factor dimension, cluster range, and
starting covariance model.

## Matrix orientation

`bpgmm` expects a numeric matrix with variables in rows and observations in
columns. Many R data sets use the opposite convention, so transpose after
selecting numeric variables.

The package convention is

\[
X =
\begin{bmatrix}
x_{11} & \cdots & x_{1n} \\
\vdots &        & \vdots \\
x_{p1} & \cdots & x_{pn}
\end{bmatrix},
\]

where row \(j\) is variable \(j\) and column \(i\) is observation \(x_i\).

```{r}
library(bpgmm)

iris_numeric <- as.matrix(iris[, 1:4])
iris_labels <- as.integer(iris$Species)

dim(iris_numeric)

X <- t(iris_numeric)
dim(X)
```

Rows now correspond to variables and columns correspond to observations.

```{r}
rownames(X)
X[, 1:3]
```

## Check finite numeric inputs

The sampler requires finite numeric values. Handle missing values before
fitting. Common choices include complete-case filtering, domain-specific
imputation, or fitting the model to a subset of variables with reliable
measurements.

```{r}
all(is.finite(X))
storage.mode(X)
```

## Scale variables

Mixture models are sensitive to measurement scale. If variables are measured in
different units, standardizing each variable is usually a sensible default.
For each variable \(j\), the usual transformation is

\[
x_{ji}^{\mathrm{scaled}} =
\frac{x_{ji} - \bar{x}_{j\cdot}}{s_j},
\qquad
s_j^2 = \frac{1}{n - 1}\sum_{i=1}^n
(x_{ji} - \bar{x}_{j\cdot})^2 .
\]

```{r}
X_scaled <- t(scale(t(X)))

round(rowMeans(X_scaled), 6)
round(apply(X_scaled, 1, sd), 6)
```

The package does not scale internally because some scientific applications need
the original measurement scale. Scaling should be an explicit analysis choice.

`pgmm_rjmcmc()` also centers the data internally before sampling. On the centered
scale the cluster-mean prior mean is \(\bar{x} = 0\), consistent with the
augmented loading posterior in Supplement A.1. Sampled means in the output are
returned on the original scale.

## Hyperprior defaults

The default symmetric hyperpriors are `d_vec = s_vec = c(1, 1, 1)` with
`delta = 3` and `ggamma = 1`. These control the gamma hyperpriors on
\(\alpha_1\), \(\alpha_2\), and \(\beta\) described in the model-and-sampler
vignette.

## Choose `q_new`

`q_new` is the latent-factor dimension assigned to newly proposed clusters. It
controls the dimension of the factor-analyzer part of the covariance model.
In the paper's notation,

\[
\Lambda_k \in \mathbb{R}^{p \times q_k}, \qquad
y_{ki} \in \mathbb{R}^{q_k}.
\]

The package uses `q_new` as the \(q_k\) value for a newly created component.

Useful starting points:

- `q_new = 1` for very small examples or when covariance structure should be
  simple.
- `q_new = 2` or `3` for moderate-dimensional data.
- Larger values only when the number of variables and observations support the
  extra covariance flexibility.

The value should be smaller than the number of observed variables.

```{r}
p <- nrow(X_scaled)
q_new <- min(2, p - 1)
q_new
```

## Choose `m_range`

`m_range` is the allowed cluster-number range. A wide range gives RJMCMC more
freedom, but also increases the model space. Start with a scientifically
reasonable range, then assess sensitivity.

```{r}
m_init <- 3
m_range <- c(1, 5)
```

For data with a known reference label, such as `iris`, the reference partition
gives a simple check on the range. In unsupervised applications, use domain
knowledge and exploratory plots.

```{r, fig.width = 5.5, fig.height = 4.2, fig.alt = "Scatter plot of scaled iris variables colored by species."}
species_cols <- c("#0072B2", "#D55E00", "#009E73")
plot(
  X_scaled[1, ], X_scaled[2, ],
  col = species_cols[iris_labels],
  pch = 19,
  xlab = rownames(X_scaled)[1],
  ylab = rownames(X_scaled)[2],
  main = "Scaled iris data",
  asp = 1
)
legend(
  "topleft",
  legend = levels(iris$Species),
  col = species_cols,
  pch = 19,
  bty = "n"
)
```

## Choose the starting covariance model

The three-letter model labels describe whether loading matrices and noise
covariances are shared across clusters. `UUU` is flexible; `CCC` is more
constrained. A flexible starting model is often reasonable when using
`v_step = 1`, because the sampler can move across covariance structures.

The fitted covariance is always

\[
\Sigma_k = \Lambda_k\Lambda_k^\top + \Psi_k,
\]

but the label controls whether \(\Lambda_k\) and \(\Psi_k\) are shared and
whether \(\Psi_k\) is isotropic or diagonal.

```{r}
model_to_constraint("UUU")
constraint_to_model(c(1, 1, 1))
```

## A prepared call

The call below is not evaluated in the vignette because applied analyses should
use longer chains and repeated runs. It records how the prepared objects enter
the package interface.

```{r, eval = FALSE}
fit <- pgmm_rjmcmc(
  X = X_scaled,
  m_init = m_init,
  m_range = m_range,
  q_new = q_new,
  burn = 1000,
  niter = 5000,
  constraint = "UUU",
  m_step = 1,
  v_step = 1,
  split_combine = 1,
  verbose = FALSE
)

summarize_pgmm_rjmcmc(fit, true_cluster = iris_labels)
```

## Practical checklist

Before fitting:

- verify that `X` is numeric with variables in rows;
- remove or impute missing and non-finite values;
- decide whether variables should be scaled;
- choose a scientifically plausible `m_range`;
- choose `q_new` smaller than the number of variables;
- use multiple chains for applied analyses.
