# manhattan: Manhattan and related distances In kylebittinger/abdiv: Alpha and Beta Diversity Measures

## Description

The Manhattan or city block distance is the sum of absolute differences between the elements of two vectors. The mean character difference is a closely related measure.

## Usage

 ```1 2 3 4 5``` ```manhattan(x, y) mean_character_difference(x, y) modified_mean_character_difference(x, y) ```

## Arguments

 `x, y` Numeric vectors

## Details

For vectors `x` and `y`, the Manhattan distance is given by

d(x, y) = ∑_i |x_i - y_i|.

Relation of `manhattan()` to other definitions:

• Equivalent to R's built-in `dist()` function with `method = "manhattan"`.

• Equivalent to `vegdist()` with `method = "manhattan"`.

• Equivalent to the `cityblock()` function in `scipy.spatial.distance`.

• Equivalent to the `manhattan` calculator in Mothur.

• Equivalent to D_7 in Legendre & Legendre.

• Whittaker's index of association (D_9 in Legendre & Legendre) is the Manhattan distance computed after transforming to proportions and dividing by 2.

The mean character difference is the Manhattan distance divided by the length of the vectors. It was proposed by Cain and Harrison in 1958. Relation of `mean_character_difference()` to other definitions:

• Equivalent to D_8 in Legendre & Legendre.

• For binary data, equivalent to 1 - S_1 in Legendre & Legendre, where S_1 is the simple matching coefficient.

The modified mean character difference is the Manhattan distance divided by the number elements where either `x` or `y` (or both) are nonzero. Relation of `modified_mean_character_difference()` to other definitions:

• Equivalent to D_{19} in Legendre & Legendre.

• Equivalent to `vegdist()` with `method = "altGower"`.

• For binary data, it is equivalent to the Jaccard distance.

## Value

The distance between `x` and `y`. The modified mean character difference is undefined if all elements in `x` and `y` are zero, in which case we return `NaN`.

## References

Cain AJ, Harrison GA. An analysis of the taxonomist's judgment of affinity. Proceedings of the Zoological Society of London 1958;131:85-98.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```x <- c(15, 6, 4, 0, 3, 0) y <- c(10, 2, 0, 1, 1, 0) manhattan(x, y) # Whittaker's index of association manhattan(x / sum(x), y / sum(y)) / 2 mean_character_difference(x, y) # Simple matching coefficient for presence/absence data # Should be 2 / 6 mean_character_difference(x > 0, y > 0) modified_mean_character_difference(x, y) # Jaccard distance for presence/absence data modified_mean_character_difference(x > 0, y > 0) jaccard(x, y) ```

kylebittinger/abdiv documentation built on Jan. 31, 2020, 3:13 p.m.