Omega {ForeCA} | R Documentation |
Estimate forecastability of a time series
Description
An estimator for the forecastability of a univariate time series
.
Currently it uses a discrete plug-in estimator given the empirical spectrum (periodogram).
Usage
Omega(
series = NULL,
spectrum.control = list(),
entropy.control = list(),
mvspectrum.output = NULL
)
Arguments
series |
a univariate time series; if it is multivariate, then
|
spectrum.control |
list; control settings for spectrum estimation.
See |
entropy.control |
list; control settings for entropy estimation.
See |
mvspectrum.output |
an object of class |
Details
The forecastability of a stationary process is defined as
(see References)
where is the normalized spectral density of
.
In particular
.
For white noise forecastability
; for a sum of sinusoids it equals
%.
However, empirically it reaches
only if the estimated spectrum has
exactly one peak at some
and
for all
.
In practice, a time series of length T
has Fourier frequencies
which represent a discrete
probability distribution. Hence entropy of
must be
normalized by
, not by
.
Also we can use several smoothing techniques to obtain a less variance estimate of
.
Value
A real-value between and
(%).
means not
forecastable (white noise);
means perfectly forecastable (a
sinusoid).
References
Goerg, G. M. (2013). “Forecastable Component Analysis”. Journal of Machine Learning Research (JMLR) W&CP 28 (2): 64-72, 2013. Available at http://jmlr.org/proceedings/papers/v28/goerg13.html.
See Also
spectral_entropy
, discrete_entropy
,
continuous_entropy
Examples
nn <- 100
eps <- rnorm(nn) # white noise has Omega() = 0 in theory
Omega(eps, spectrum.control = list(method = "pgram"))
# smoothing makes it closer to 0
Omega(eps, spectrum.control = list(method = "mvspec"))
xx <- sin(seq_len(nn) * pi / 10)
Omega(xx, spectrum.control = list(method = "pgram"))
Omega(xx, entropy.control = list(threshold = 1/40))
Omega(xx, spectrum.control = list(method = "mvspec"),
entropy.control = list(threshold = 1/20))
# an AR(1) with phi = 0.5
yy <- arima.sim(n = nn, model = list(ar = 0.5))
Omega(yy, spectrum.control = list(method = "mvspec"))
# an AR(1) with phi = 0.9 is more forecastable
yy <- arima.sim(n = nn, model = list(ar = 0.9))
Omega(yy, spectrum.control = list(method = "mvspec"))