Constructing maximum-entropy spectra for dominant-cycle forecasts

Two jobs in one construction

The method is constructed to serve two jobs at once: estimate a data spectrum and generate adaptive short-horizon forecasts that stay consistent with the dominant cycles found as spectral peaks.

A maximum-entropy spectrum is a spectrum built from linear-prediction coefficients chosen to minimize forecast error while adding as little extra structure as possible to the observed series. The dominant cycle is the strongest low-frequency spectral peaks retained as the signal that later forecasts and filter lengths are required to follow.

A linear predictor as the shared engine

Construction assumes the current observation equals a linear combination of M past values plus a residual. The coefficients are chosen to minimize squared prediction error and, in the usual algorithms, the sum of forward and backward residuals.

After the coefficients are fit, the spectrum is computed from those coefficients. Multi-step forecasts are formed recursively by feeding earlier predictions back into the same linear predictor.

Trend channels around the point forecast

A trend channel is the upper and lower bands around a recursive point forecast. The bands are built by adding and subtracting twice the square root of the n-step forecast-error variance around the point forecast, described as a 95 percent interval. Preprocessing used to quiet trading noise adds a further variance term.

Model order as the main design choice

Model order is the number of lagged coefficients in the predictor. The number of spectral peaks is about half the coefficient count, so the choice controls both resolution and the risk of spurious peaks.

Too small an order misses cycles. Too large an order can add peaks that are not in the observations.

Scoring candidate orders

Orders from 1 up to the smaller of half the sample length or 25 are scored with a final prediction error. That rule rescales mean-square one-step error by sample length and coefficient count so a constructor can compare candidate models, using FPE(M)=(N+M)*P(M)/(N-M).

Theory picks the final-prediction-error minimum. Practice may keep a local minimum that preserves the intended signal spectrum.

Which peaks a forecast may inherit

For forecasting, only the lowest-frequency peaks near the dominant cycle are treated as signal. Those peaks are the ones within 15 dB, and the lowest-frequency peak is treated as the prevailing trend. The remaining peaks are treated as interference.

Fourier analysis as a construction check

Fourier analysis is treated as a prerequisite comparison tool because the maximum-entropy spectrum is described as more flexible in adaptation and display, while still being checked against Fourier peak locations on the same series.

Fourier analysis is a classical spectrum estimate used as a construction check on whether maximum-entropy peaks agree in frequency, and more loosely in relative strength, with an independent transform of the same series.

Lowering order and averaging samples

On a 57-week series, final-prediction-error local minima appeared at orders 6 and 16. The order-16 spectrum resolved two significant low-frequency peaks plus noise peaks above 7 cycles per year, and those frequencies generally matched a Fourier spectrum of the same data.

Two-point averaging reduced the series to 28 observations with final-prediction-error minima at orders 2 and 8. The order-8 averaged spectrum kept the dominant-cycle shape with fewer noise peaks and was selected for forecasting while still spanning 16 weeks of lags.