Casdagli test of nonlinearity via locally linear forecasts

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`x` |
time series |

`m, d, steps` |
embedding dimension, time delay, forecasting steps |

`series` |
time series name (optional) |

`n.ahead` |
n. of steps ahead to forecast |

`eps.min, eps.max` |
min and max neighbourhood size |

`neps` |
number of neighbourhood levels along which iterate |

`eps` |
neighbourhood size |

`onvoid` |
what to do in case of an isolated point: stop or enlarge neighbourhood size by an r% |

`r` |
if an isolated point is found, enlarge neighbourhood window by r% |

`trace` |
tracing level: 0, 1 or more than 1 for |

`llar`

does the Casdagli test of non-linearity. Given the embedding state-space (of dimension `m`

and time delay `d`

) obtained from time series `series`

, for a sequence of distance values `eps`

, the relative error made by forecasting time series values with a linear autoregressive model estimated on points closer than `eps`

is computed.
If minimum error is reached at relatively small length scales, a global linear model may be inappropriate (using current embedding parameters).
This was suggested by Casdagli(1991) as a test for non-linearity.

`llar.predict`

tries to extend the given time series by
`n.ahead`

points by iteratively
fitting locally (in the embedding space of dimension m and time delay d)
a linear model. If the spatial neighbourhood window is too small, your
time series last point would be probably isolated. You can ask to
automatically enlarge the window `eps`

by a factor of r%
sequentially, until enough neighbours are found for fitting the linear
model.

`llar.fitted`

gives out-of-sample fitted values from locally linear
models.

`llar`

gives an object of class 'llar'. I.e., a list of components:

`RMSE` |
vector of relative errors |

`eps` |
vector of neighbourhood sizes (in the same order of RMSE) |

`frac` |
vector of fractions of the time series used for RMSE computation |

`avfound` |
vector of average number of neighbours for each point in the time series
which can be plotted using the |

Function `llar.forecast`

gives the vector of n steps ahead locally linear iterated
forecasts.

Function `llar.fitted`

gives out-of-sample fitted values from locally linear
models.

For long time series, this can be slow, especially for relatively big neighbourhood sizes.

The C implementation was re-adapted from that in the TISEAN package ("ll-ar" routine, see references). However, here the euclidean norm is used, in place of the max-norm.

Antonio, Fabio Di Narzo

M. Casdagli, Chaos and deterministic versus stochastic nonlinear modelling, J. Roy. Stat. Soc. 54, 303 (1991)

Hegger, R., Kantz, H., Schreiber, T., Practical implementation of nonlinear time series methods: The TISEAN package; CHAOS 9, 413-435 (1999)

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