The distance measure d is usually defined (although alternative definitions exist) as d(x,y) = sum( (xi-yi)^2 / (xi+yi) ) / 2 . It is often used in computer vision to compute distances between some bag-of-visual-word representations of images.
The name of the distance is derived from Pearson‘s chi squared test statistic X2(x,y) = sum( (xi-yi)^2 / xi) for comparing discrete probability distributions (i.e histograms). However, unlike the test statistic, d(x,y) is symmetric wrt. x and y, which is often useful in practice, e.g., when you want to construct a kernel out of the histogram distances.
Chi-Square Distance
Consider a frequency table with n rows and p columns, it is possible to calculate row profiles and column profiles. Let us then plot the n or p points from each profile. We can define the distances between these points. The Euclidean distance between the components of the profiles, on which a weighting is defined (each term has a weight that is the inverse of its frequency), is called the chi-square distance. The name of the distance is derived from the fact that the mathematical expression defining the distance is identical to that encountered in the elaboration of the chi square goodness of fit test.
MATHEMATICAL ASPECTS
|
|
| f i. | is the sum of the components of the ith row; |
| f .j | is the sum of the components of the jth column; |
|
|
is the ith row profile for j = 1,2,...,p. |
|
|
DOMAINS AND LIMITATIONS
The chi-square distance incorporates a weight that is inversely proportional to the total of each row (or column), which increases the importance of small deviations in the rows (or columns) which have a small sum with respect to those with more important sum package.
The chi-square distance has the property of distributional equivalence, meaning that it ensures that the distances between rows and columns are invariant when two columns (or two rows) with identical profiles are aggregated.
EXAMPLES
Consider a contingency table charting how satisfied employees working for three different businesses are. Let us establish a distance table using the chi-square distance.
Values for the studied variable X can fall into one of three categories:
- X 1: high satisfaction;
- X 2: medium satisfaction;
- X 3: low satisfaction.
The observations collected from samples of individuals from the three businesses are given below:
|
Business 1 |
Business 2 |
Business 3 |
Total |
|
|---|---|---|---|---|
|
X 1 |
20 |
?55 |
30 |
105 |
|
X 2 |
18 |
?40 |
15 |
?73 |
|
X 3 |
12 |
??5 |
?5 |
?22 |
|
Total |
50 |
100 |
50 |
200 |
The relative frequency table is obtained by dividing all of the elements of the table by 200, the total number of observations:
|
Business 1 |
Business 2 |
Business 3 |
Total |
|
|---|---|---|---|---|
|
X 1 |
0.1 |
0.275 |
0.15 |
0.525 |
|
X 2 |
0.09 |
0.2 |
0.075 |
0.365 |
|
X 3 |
0.06 |
0.025 |
0.025 |
0.11 |
|
Total |
0.25 |
0.5 |
0.25 |
1 |
We can calculate the difference in employee satisfaction between the the 3 enterprises. The column profile matrix is given below:
|
Business 1 |
Business 2 |
Business 3 |
Total |
|
|---|---|---|---|---|
|
X 1 |
0.4? |
0.55 |
0.6 |
1.55 |
|
X 2 |
0.36 |
0.4? |
0.3 |
1.06 |
|
X 3 |
0.24 |
0.05 |
0.1 |
0.39 |
|
Total |
1?? |
1?? |
1? |
3?? |
|
|
|
Business 1 |
Business 2 |
Business 3 |
|
|---|---|---|---|
|
Business 1 |
0 |
0.613 |
0.514 |
|
Business 2 |
0.613 |
0 |
0.234 |
|
Business 3 |
0.514 |
0.234 |
0 |
We can also calculate the distances between the rows, in other words the difference in employee satisfaction; to do this we need the line profile table:
|
Business 1 |
Business 2 |
Business 3 |
Total |
|
|---|---|---|---|---|
|
X 1 |
0.19? |
0.524 |
0.286 |
1 |
|
X 2 |
0.246 |
0.548 |
0.206 |
1 |
|
X 3 |
0.546 |
0.227 |
0.227 |
1 |
|
Total |
0.982 |
1.299 |
0.719 |
3 |
|
|
We can calculate d(1,3) and d(2,3) in a similar way. The differences between the degrees of employee satisfaction are finally summarized in the following distance table:
|
X 1 |
X 2 |
X 3 |
|
|---|---|---|---|
|
X 1 |
0 |
0.198 |
0.835 |
|
X 2 |
0.198 |
0 |
0.754 |
|
X 3 |
0.835 |
0.754 |
0 |
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