euclidean {abdiv} | R Documentation |

These distance and diversity measures are mathematically similar to the Euclidean distance between two vectors.

euclidean(x, y) rms_distance(x, y) chord(x, y) hellinger(x, y) geodesic_metric(x, y)

`x, y` |
Numeric vectors |

For vectors `x`

and `y`

, the Euclidean distance is defined as

*d(x, y) = √{∑_i (x_i - y_i) ^ 2}.*

Relation of `euclidean()`

to other definitions:

Equivalent to R's built-in

`dist()`

function with`method = "euclidean"`

.Equivalent to

`vegdist()`

with`method = "euclidean"`

.Equivalent to the

`euclidean()`

function in`scipy.spatial.distance`

.Equivalent to the

`structeuclidean`

calculator in Mothur, to`speciesprofile`

if`x`

and`y`

are transformed to relative abundance, and to`memeuclidean`

if`x`

and`y`

are transformed to presence/absence.Equivalent to

*D_1*in Legendre & Legendre.Equivalent to the

*distance between species profiles*,*D_18*in Legendre & Legendre if`x`

and`y`

are transformed to relative abundance.

The *root-mean-square* distance or *average* distance is similar
to Euclidean distance. As the name implies, it is computed as the square
root of the mean of the squared differences between elements of `x`

and `y`

:

*d(x, y) = √{\frac{1}{n} ∑_i^n (x_i - y_i) ^ 2}.*

Relation of `rms_distance()`

to other definitions:

Equivalent to

*D_2*in Legendre & Legendre.

The *chord* distance is the Euclidean distance after scaling each
vector by its root sum of squares, *√{∑_i x_i^2}*. The chord
distance between any two vectors ranges from 0 to *sqrt(2)*.
Relation of `chord()`

to other definitions:

Equivalent to

*D_3*in Legendre & Legendre.

The *Hellinger* distance is equal to the chord distance computed after
a square-root transformation. Relation of `hellinger()`

to other
definitions:

Equivalent to

*D_17*in Legendre & Legendre.Equivalent to the

`hellinger`

calculator in Mothur.

The *geodesic metric* is a transformed version of the chord distance.

*d(x, y) = \textrm{arccos} ≤ft(1 - \frac{d_c^2(x, y)}{2} \right),*

where *d_c* is the chord distance. It gives the length of the arc on a
hypersphere between the vectors, if the vectors are normalized to unit
length. Relation of `geodesic_metric()`

to other definitions:

Equivalent to

*D_4*in Legendre & Legendre.

The distance between `x`

and `y`

. The chord distance,
Hellinger distance, and geodesic metric are not defined if all elements
of either vector are zero. We return `NaN`

in this case.

x <- c(15, 6, 4, 0, 3, 0) y <- c(10, 2, 0, 1, 1, 0) euclidean(x, y) # The "distance between species profiles" euclidean(x / sum(x), y / sum(y)) rms_distance(x, y) chord(x, y) hellinger(x, y) # Hellinger is chord distance after square root transform chord(sqrt(x), sqrt(y)) geodesic_metric(x, y) # No species in common with x v <- c(0, 0, 0, 5, 0, 5) chord(v, x) sqrt(2)

[Package *abdiv* version 0.2.0 Index]