# Chi2Dist: chi2 distance between the rows of a rectangular matrix. In DistatisR: DiSTATIS Three Way Metric Multidimensional Scaling

## Description

Computesthe I*I matrix D which is the chi2 distance matrix between the rows of an I*J rectangular matrix X (with non-negative elements), and provides the I*1 m vector of mass (where the mass of a row is the sum of the entries of this row divided by the grand total of the matrix). When the distance matrix and the associated vector of masses are used as input to the function `mmds` the results will give the factor scores of the correspondence analysis of the matrix X. The `function` is used by the function `Chi2DistanceFromSort` that computes the chi2 distance for the results of a sorting task.

## Usage

 ```1 2``` ```Chi2Dist(X) ```

## Arguments

 `X` A rectangle matrix with non-negative elements

## Value

Sends back a list

 `\$Distance` the squared chi2 distance matrix computed the rows of matrix X. `masses` the vector of masses of the rows of of matrix X.

Herve Abdi

## References

The procedure and references are detailled in (Paper available from www.utdallas.edu/~herve): Abdi, H. (2007). Distance. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage. pp. 304–308.

And in:
Abdi, H., & Valentin, D. (2006). Mathematiques pour les Sciences Cognitives (Mathematics for Cognitive Sciences). Grenoble: PUG.

Abdi, H., & Williams, L.J. (2010). Principal component analysis. Wiley Interdisciplinary Reviews: Computational Statistics,2, 433–459.

`Chi2DistanceFromSort` `distatis` `mmds`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26``` ```# Here is a data matrix from Abdi & Williams (2012) # page 449, Table 15. Punctuation of 6 French authors Punctuation = matrix(c( 7836, 13112, 6026, 53655, 102383, 42413, 115615, 184541, 59226, 161926, 340479, 62754, 38177, 105101, 12670, 46371, 58367, 14299), ncol =3,byrow = TRUE) colnames(Punctuation) <-c('Period','Comma','Other') rownames(Punctuation) <-c('Rousseau','Chateaubriand', 'Hugo','Zola','Proust','Giroudoux') # 1. Get the Chi2 distance matrix # between the rows of Punctuation Dres <- Chi2Dist(Punctuation) # check that the mds of the Chi2 distance matrix # with CA-masses gives the CA factor scores for I # 2. Use function mmds from DistatisR # testmds <- mmds(Dres\$Distance,masses=Dres\$masses) # Print the MDS factor scores from mmds print('Factor Scores from mds') print(testmds\$FactorScores) print('It matches CA on X (see Abdi & Williams, 2010. Table 16, p. 449)') # Et voila! ```