MDM Reference • Spectral representation
Spectral Representation of Series
The spectral view of a series reports how much of its variation is carried at each rate. It is the same series as before, turned so that the eye sees frequencies rather than instants, and it often exposes structure that the time view conceals.
The idea of a component
Any sufficiently well-behaved series may be written as a sum of sinusoids of various rates, each with its own amplitude and phase. The square of the amplitude attached to a rate measures the share of variation carried at that rate. This distribution of variation across rates is the spectrum, and it is the central object of the analysis.
Estimating the spectrum
From a finite record the spectrum is estimated rather than known. The raw estimate, formed by squaring the readings against each candidate rate, is noisy and its variance does not shrink with the length of the record. To tame the noise, neighbouring rates are averaged, and the resulting smoothed estimate is far more stable than the raw one, at the expense of a little resolution.
The window
A finite record is a slice of a longer series, and taking a slice leaks variation from one rate into its neighbours. The remedy is to taper the ends of the record before the estimate is formed, so that the slice does not end abruptly. The taper trades a small widening of each peak for a large reduction in the leakage, and it is a standard part of careful practice.
Peaks and their reading
- A sharp peak marks a periodic component present in the series.
- A broad hump marks variation spread over a band of rates rather than one.
- A slowly falling profile marks a series whose neighbouring readings are strongly alike.
These readings are the practical yield of the spectral view, and they guide the modelling that follows.
Harmonics and their meaning
A repeating form that is not a pure sinusoid throws variation into a base rate and its whole-number multiples, which together are called harmonics. The pattern of harmonic amplitudes is a fingerprint of the repeating form, and it allows one to infer shape from spectrum. The reader should note that harmonics are an artefact of the repetition, not separate causes.
Bridging to the time view
The spectrum and the time-domain account are two faces of one series, joined by a reversible transformation. Neither is more fundamental, and the choice between them is a choice of convenience. The cell that follows carries the reader back to the time view and to the joint behaviour of readings at separated instants.
Reading a spectrum with care
An estimated spectrum is never read as a bare picture but always against its own uncertainty. Two peaks of similar height may not be distinguishable, and a shallow trough may be no more than noise. The desk advises a reader to carry a sense of the uncertainty alongside the picture, so that a claim of a peak is not made on a shoulder of noise.
Bandwidth and resolution
The sharpness with which a spectrum separates two nearby rates depends on the length of the record and on the amount of smoothing used. A longer record gains finer separation, while more smoothing gains a steadier picture at the expense of that fineness. The two can be traded against each other, but neither can be had in full at once.