# Estimates spectral entropy of a time series Estimates **spectral entropy** from a univariate (or multivariate) normalized spectral density. ```r spectral_entropy( series = NULL, spectrum.control = list(), entropy.control = list(), mvspectrum.output = NULL, ... ) ``` ## Arguments - `series`: univariate time series of length $T$. In the rare case that users want to call this for a multivariate time `series`, note that the estimated spectrum is in general **not** normalized for the computation. Only if the original data is whitened, then it is normalized. - `spectrum.control`: list; control settings for spectrum estimation. See `complete_spectrum_control` for details. - `entropy.control`: list; control settings for entropy estimation. See `complete_entropy_control` for details. - `mvspectrum.output`: optional; one can directly provide an estimate of the spectrum of `series`. Usually the output of `mvspectrum`. - `...`: additional arguments passed to `mvspectrum`. ## Returns 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 by the (smoothed) periodogram (see `mvspectrum`); thus using discrete, and not continuous entropy. ## Examples ```r set.seed(1) eps <- rnorm(100) spectral_entropy(eps) phi.v <- seq(-0.95, 0.95, by = 0.1) kMethods <- c("mvspec", "pgram") SE <- matrix(NA, ncol = length(kMethods), nrow = length(phi.v)) for (ii in seq_along(phi.v)) { xx.tmp <- arima.sim(n = 200, list(ar = phi.v[ii])) for (mm in seq_along(kMethods)) { SE[ii, mm] <- spectral_entropy(xx.tmp, spectrum.control = list(method = kMethods[mm])) } } matplot(phi.v, SE, type = "l", col = seq_along(kMethods)) legend("bottom", kMethods, lty = seq_along(kMethods), col = seq_along(kMethods)) # AR vs MA SE.arma <- matrix(NA, ncol = 2, nrow = length(phi.v)) SE.arma[, 1] <- SE[, 2] for (ii in seq_along(phi.v)){ yy.temp <- arima.sim(n = 1000, list(ma = phi.v[ii])) SE.arma[ii, 2] <- spectral_entropy(yy.temp, spectrum.control = list(method = "mvspec")) } matplot(phi.v, SE.arma, type = "l", col = 1:2, xlab = "parameter (phi or theta)", ylab = "Spectral entropy") abline(v = 0, col = "blue", lty = 3) legend("bottom", c("AR(1)", "MA(1)"), lty = 1:2, col = 1:2) ``` ## References Jerry D. Gibson and Jaewoo Jung (2006). The Interpretation of Spectral Entropy Based Upon Rate DistortionFunctions . IEEE International Symposium on Information Theory, pp. 277-281. L. L. Campbell, Minimum coefficient rate for stationary random processes , Information and Control, vol. 3, no. 4, pp. 360 - 371, 1960. ## See Also `Omega`, `discrete_entropy`

spectral_entropy function