manhattan {abdiv} | R Documentation |

## Manhattan and related distances

### 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

```
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) = \sum_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

```
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)
```

*abdiv*version 0.2.0 Index]