Spectral entropy of a time series

Computes spectral entropy from a univariate normalized spectral density, estimated using an AR model.

Usage

entropy(x)

Arguments

x

a univariate time series

Value

A non-negative real value for the spectral entropy \(H_s(x_t)\).

Details

The spectral entropy equals the Shannon entropy of the spectral density \(f_x(\lambda)\) of a stationary process \(x_t\): $$ H_s(x_t) = - \int_{-\pi}^{\pi} f_x(\lambda) \log f_x(\lambda) d \lambda, $$ where the density is normalized such that \(\int_{-\pi}^{\pi} f_x(\lambda) d \lambda = 1\). An estimate of \(f(\lambda)\) can be obtained using spec.ar with the burg method.

References

Jerry D. Gibson and Jaewoo Jung (2006). “The Interpretation of Spectral Entropy Based Upon Rate Distortion Functions”. IEEE International Symposium on Information Theory, pp. 277-281.

Goerg, G. M. (2013). “Forecastable Component Analysis”. Proceedings of the 30th International Conference on Machine Learning (PMLR) 28 (2): 64-72, 2013. Available at https://proceedings.mlr.press/v28/goerg13.html.

See also

spec.ar

Author

Rob J Hyndman

Examples

entropy(rnorm(1000))
#> entropy 
#>       1 
entropy(lynx)
#>   entropy 
#> 0.7331515 
entropy(sin(1:20))
#>     entropy 
#> 0.003481715