## False nearest neighbors (FNN)

Purpose

Description

Tips

Macro Synopsis

Modules

Related Functions

References

### Purpose

Compute false nearest neighbours to find the proper embedding dimension for phase space reconstruction

### Description

The false nearest neigbours procedure is a method to obtain the optimum embedding
dimension for phase space reconstruction.
By checking the neighbourhood of points embedded in projection manifolds of increasing dimension, the algorithm eliminates 'false neighbours': This means that points apparently lying close together due to projection are separated in higher embedding dimensions.

A natural criterion for catching embedding errors is that the increase
in distance between two neighboured points is large when going from
dimension *d* to *d+1*. We state this criterion by designating as a
false nearest neighbour any neighbour for which the following is valid:

Here t and are the times corresponding to the neighbour and
the reference point, respectively.
denotes the distance in phase space with embedding dimension
*d*, and is the tolerance threshold.

However, this criterion by itself is not sufficient for determining a
proper embedding dimension. A problem turns out if a point is a
nearest neighbour of another without necessarily being close to it.
Therefore the number of false nearest neighbours will again
increase at higher dimensions. To handle this problem, a further criterion
is implemented: the loneliness criterion. It is represented by the
loneliness tolerance threshold.

The output produced by the function is the percentual amount of FNN
versus increasing embedding dimension and has a monotonic decreasing graph.
The optimum embedding dimension usually can be found near the crossing of
the 30 % threshold.

Parameters of "False nearest neighbors (FNN)" are:

- delay of coordinates
`tau` (number of samples), needed for phase space reconstruction
- maximum embedding dimension
`m` : The percentage of false nearest neighbours will be computed from embedding dimension 1 up to this maximum embedding dimension
- distance tolerance
`rtol` - This is a threshold for the embedding
criterion (see above). The choice of `rtol` 10 will usually identify the FNN clearly. Very high values of `rtol` will result in an underestimation of the number of FNN. Very low values for `rtol` will identify too many false nearest neighbours, in particular when the points on the attractor become sparse
- loneliness tolerance
`atol` - This is a threshold for the second
criterion (see above). A choice of `atol` 2 will be good in most
cases. Very high or low values of `atol` have the same effects as
described above for `rtol`

Note that since this function operates on a 'pure' time series, the scale and the shift of the given signal do not affect the result.

### Tips

The detected number of FNN is increased in case of a noisy signal.
This fact should be taken into account when using the FNN algorithm.

### Macro Synopsis

`y = FNN(x,tau,m,rtol,atol);`

signal x,y;

int tau, m;

float rtol, atol;

### Modules

Nonlinear

### Related Functions

Mutual information, Correlation integral,
Correlation dimension, Recurrence plot.

### References

Kennel et al. [27], Liebert et al. [28]