MDS_linkGramDistanceMatrix.Rmd

---
title: "MDS: Link Squared Distance Matrix and Gram Matrix in Practice"
author: "Lieven Clement"
date: "statOmics, Ghent University (https://statomics.github.io)"
---

```{r, child="_setup.Rmd"}
```

```{r echo=FALSE}
library(tidyverse)
```

# Data

Part of the iris dataset

```{r}
X <- iris[1:5,1:4] %>% as.matrix
X
```

# Centering

\[
  \mathbf{H} = \mathbf{I} - \frac{1}{n} \mathbf{1}\mathbf{1}^T ,
\]
\[
 \mathbf{X}_c = \mathbf{H}\mathbf{X}
\]
\[
\mathbf{H}\mathbf{X}_c = \mathbf{X}_c
\]

```{r}
H <- diag(nrow(X)) - matrix(1/nrow(X),nrow=nrow(X),ncol=nrow(X))
Xc <- H%*%X
colMeans(Xc)
H%*%Xc-Xc
```

# Gram matrix

**Gram matrix**: $\mathbf{G}=\mathbf{X}\mathbf{X}^T$
Here we work on centered data so
$\mathbf{G}=\mathbf{X}_c\mathbf{X}_c^T$

```{r}
G <- Xc%*%t(Xc)
```

# Squared Distance matrix

\[
  \mathbf{D}_X = \mathbf{N} - 2\mathbf{X}\mathbf{X}^T + \mathbf{N}^T ,
\]

```{r}
N <- matrix(diag(G),nrow(Xc),nrow(Xc))
N
dist2 <- N-2*G+t(N)
dist2
dist(Xc)^2
```

# Link Gram Matrix and Squared Distance Matrix

\[\mathbf{G} =\mathbf{X}_c\mathbf{X}_c^T=-\frac{1}{2}\mathbf{H}\mathbf{D}_X\mathbf{H}\]

```{r}
G
-1/2*H%*%dist2%*%H
-1/2*H%*%dist2%*%H - G
```

# Show that N cancels out when multiplying with H

\[ \mathbf{H N H} = \mathbf{0}\]

\[ \mathbf{H N}^T \mathbf{H} = \mathbf{0}\]

```{r}
N%*%H
H%*%t(N)
```

```{r, child="_session-info.Rmd"}
```