# Consistent nonlinear estimate of the order using local polynomial regression.

### Description

A function to estimate the order of a time series using the nonparametric order selection method of Cheng and Tong (1992, 1994) as modified by Yao & Tong (1994; see also Fan, Yao & Tong 1996). The method uses leave-one-out cross-validation of the locally linear regression against lagged-abundances.

### Usage

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### Arguments

`x` |
A time series without missing values. |

`order` |
The candidate orders. The default is 1:5. |

`step` |
The time step for predicition. |

`deg` |
The degree of the local polynomial. |

`bandwidth` |
The candidate bandwidths to be considered. |

`cv` |
if TRUE leave-one-out crossvalidation will be performed. |

`echo` |
if TRUE a counter shows the progress |

### Details

The time series is normalized prior to cross-validation.

A Gaussian kernel is used for the locally linear regression.

The bandwidth is optimized using crossvalidation. If a single bandwidth is provided, no cross validation of bandwidth will be carried out. Highly nonlinear data will require more narrow bandwidths. If NA is returned it may be because the min bandwidth considered is too small relative to the density of data.

Missing values are NOT permitted.

If `deg`

is set to 0, the order is estimated on the basis
of the Nadaraya-Watson (locally constant) estimator of
the conditional expecation against lagged-abundances
(Cheng and Tong 1992, 1994). The function subsumes
the `nw.order`

of the previous S-plus nlt-library.

The function requires Loader's `locfit`

-library.

### Value

An object of class "ll.order" is returned consisting of the following components:

`grid` |
the grid of orders, bandwidths, and CV's. |

`grid$order` |
the orders. |

`grid$CV` |
the cross-validation score across the
grid of orders and bandwidths. (If |

`grid$GCV` |
the generalized cross-validation score. |

`grid$bandwidth` |
the bandwidths. |

`grid$df` |
the degrees of freedom of the fitted model. |

`order` |
the vector of orders considered. |

`deg` |
The degree of the local polynomial. |

### Author(s)

Ottar N. Bjornstad onb1@psu.edu

### References

Cheng, B. & Tong, H. (1992) On consistent nonparametric order determination and chaos. Journal of Royal Statistical Society B, 54, 427-449.

Cheng, B. & Tong, H. (1994) Orthogonal projection, embedding dimension and sample size in chaotic time series from a statistical perspective. Philosophical Transactions of the Royal Society London, A. , 348, 325-341.

Fan, J., Yao, Q., & Tong, H. (1996) Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika, 83, 189-206.

Yao, Q. & Tong, H. (1994) Quantifying the influence of initial values on non-linear prediction. Journal of Royal Statistical Society B, 56, 701-725.

Bjornstad, O.N., Sait, S.M., Stenseth, N.C., Thompson, D.J., & Begon, M. (2001) Coupling and the impact of specialised enemies on the dimensionality of prey dynamics. Nature, 409, 1001-1006.

Loader, C. (1999) Local Regression and Likelihood. Springer, New York.

### See Also

`summary.ll.order`

`plot.ll.order`

### Examples

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