Description Usage Arguments Details Value Note Author(s) References See Also Examples

Returns the power spectrum of the Ornstein-Uhlenbeck state space (OUSS) process at a particular frequency. This is the asymptotic expected periodogram power for long regular time series.

1 2 3 | ```
ps_ouss_asymptotic(freq, power_o, sigma,
rho, lambda,
power_e, epsilon, time_step)
``` |

`freq` |
Single number or numeric vector. The frequency for which to the power spectrum is to be calculated. |

`power_o` |
Single non-negative number. Power spectrum at zero-frequency generated by the underlying OU process, when sampled at the given |

`sigma` |
Single number. Standard deviation of OU fluctuations around equilibrium. Either |

`rho` |
Single number between 0 (exclusive) and 1 (inclusive). Correlation of the OU process between two subsequent time points. Either |

`lambda` |
Single non-negative number. Resilience (or relaxation rate) of the OU process. This is also the inverse correlation time of the OU process. Either |

`power_e` |
Single non-negative number. Asymptotic power spectrum at large frequencies due to the random measurement errors. Setting this to zero corresponds to the classical OU process. Either |

`epsilon` |
Single number. Standard deviation of Gaussian measurement error. Setting this to zero corresponds to the classical OU process. Either |

`time_step` |
Positive number. The time step of the time series that was (or will be) used for periodogram generation. |

The OUSS parameters `power_o`

, `lambda`

and `power_e`

will typically be maximum-likelihood fitted values returned by `evaluate.pm`

. `time_step`

is also returned by `evaluate.pm`

and is inferred from the analysed time series. More generally, `power_o`

and `power_e`

are proportional to the OUSS parameters `sigma^2`

and `epsilon^2`

(see `generate_ouss`

), respectively, but the exact scaling depends on the normalization used for the periodogram.

Returns a numeric vector of the same size as `freq`

, containing the corresponding powers of the OUSS process.

This function is the asymptotic version of `ps_ouss`

in the limit where `series_size`

becomes very large. If you want to compare the expected periodogram to the periodogram of a short time series use `ps_ouss`

instead.

Stilianos Louca

Louca, S., Doebeli, M. (2015) Detecting cyclicity in ecological time series, Ecology 96: 1724–1732

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ```
# generate OUSS time series
times = seq(0,20,0.25);
signal = generate_ouss(times, mu=0, sigma=1, lambda=1, epsilon=0.5);
# calculate periodogram and fit OUSS model
report = evaluate.pm(times=times, signal=signal, startRadius=2);
# plot periodogram
plot(report$frequencies, report$periodogram,
type="l", ylab="power", xlab="frequency", main="periodogram & fitted OUSS power spectrum");
# plot OUSS power spectrum
lines(report$frequencies,
ps_ouss_asymptotic( freq=report$frequencies,
power_o=report$power_o,
lambda=report$lambda,
power_e=report$power_e,
time_step=report$time_step),
col="red");
``` |

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