## Simplified Analysis in Principal Coordinates

### Description

performs a simplified analysis in principal coordinates, using an object of class `dist`.

### Usage

```pcoscaled(distmat, tol = 1e-07)
```

### Arguments

 `distmat` an object of class `dist` `tol` a tolerance threshold, an eigenvalue is considered as positive if it is larger than `-tol*lambda1` where `lambda1` is the largest eigenvalue

### Value

returns a data frame containing the Euclidean representation of the distance matrix with a total inertia equal to 1

Daniel Chessel

### References

Gower, J. C. (1966) Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika, 53, 325–338.

### Examples

```    a <- 1 / sqrt(3) - 0.2
w <- matrix(c(0,0.8,0.8,a,0.8,0,0.8,a,
0.8,0.8,0,a,a,a,a,0),4,4)
w <- as.dist(w)
w <- cailliez(w)
w
pcoscaled(w)
dist(pcoscaled(w)) # w
dist(pcoscaled(2 * w)) # the same
sum(pcoscaled(w)^2) # unity
```

### Worked out examples

```
> ### Name: pcoscaled
> ### Title: Simplified Analysis in Principal Coordinates
> ### Aliases: pcoscaled
> ### Keywords: array
>
> ### ** Examples
>
>     a <- 1 / sqrt(3) - 0.2
>     w <- matrix(c(0,0.8,0.8,a,0.8,0,0.8,a,
+         0.8,0.8,0,a,a,a,a,0),4,4)
>     w <- as.dist(w)
>     w <- cailliez(w)
>     w
1         2         3
2 1.0000000
3 1.0000000 1.0000000
4 0.5773503 0.5773503 0.5773503
>     pcoscaled(w)
C1 C2
1  1.1547005  0
2 -0.5773503 -1
3 -0.5773503  1
4  0.0000000  0
>     dist(pcoscaled(w)) # w
1        2        3
2 2.000000
3 2.000000 2.000000
4 1.154701 1.154701 1.154701
>     dist(pcoscaled(2 * w)) # the same
1        2        3
2 2.000000
3 2.000000 2.000000
4 1.154701 1.154701 1.154701
>     sum(pcoscaled(w)^2) # unity
[1] 4
>
>
>
>
```