---
title: "Advanced lattice metrics"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Advanced lattice metrics}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

```{r setup}
library(fcaR)
```

## Introduction

Beyond the standard **support** (frequency), `fcaR` provides advanced metrics to analyze the quality and robustness of formal concepts, especially in fuzzy settings.

  * **Stability:** Measures the robustness of a concept against noise.
  * **Separation:** Measures the specificity of the objects covered by a concept.
  * **Fuzzy Density:** Measures the cohesion of the concept in the original data.

## Stability

Intensional stability is defined as the probability that the intent of a concept is preserved when a random subset of its extent is removed.

$$ \sigma(C) = \frac{|\{ A \subseteq \text{Ext}(C) \mid A' = \text{Int}(C) \}|}{2^{|\text{Ext}(C)|}} $$

Let's compute it for the `planets` dataset:

```{r}
fc <- FormalContext$new(planets)
fc$find_concepts()

stab <- fc$concepts$stability()
head(stab)
```

Concepts with stability close to 1 are very robust (likely real patterns), while those near 0 are unstable (likely artifacts).

## Separation

Separation measures how many objects are "unique" to a concept $C$, i.e., they are covered by $C$ but not by any of its direct subconcepts (descendants).

$$ \text{Sep}(C) = |\text{Ext}(C)| - |\bigcup_{K \prec C} \text{Ext}(K)| $$

```{r}
sep <- fc$concepts$separation()
head(sep)
```

A high separation indicates that the concept introduces a significant set of new objects into the hierarchy.

## Fuzzy density

In Fuzzy FCA, concepts are "rectangles" of high values in the matrix, but not necessarily all 1s. Density measures the average value of the relation $I$ within the concept.

$$ \rho(C) = \frac{\sum_{g \in \text{Ext}(C), m \in \text{Int}(C)} I(g, m)}{|\text{Ext}(C)| \cdot |\text{Int}(C)|} $$

```{r}
# For binary data, density is always 1 (or 0 for empty concepts)
# We need to pass the original matrix I
dens <- fc$concepts$density(I = fc$incidence())
head(dens)
```

## Putting it all together

We can combine these metrics to filter and analyze the lattice.

```{r}
df <- data.frame(
  Concept = 1:fc$concepts$size(),
  Support = fc$concepts$support(),
  Stability = stab,
  Separation = sep,
  Density = dens
)

# Concepts with high stability and separation
subset(df, Stability > 0.8 & Separation > 0)
```