Step 04 · Spectral & Models
Autoregressive and Moving-Average Forms
Two recipes describe most records with memory. One carries the past forward, and the other mixes the recent shocks. Together they form the basic grammar of stochastic series.
The autoregressive recipe
Imagine that each value in the record is a fraction of its predecessor plus a fresh shock. If the fraction is small, the record forgets quickly and jumps about; if the fraction is large, the record remembers for a long time and drifts. Written in words, the value at one moment equals a coefficient times the previous value plus a shock drawn fresh at that moment. The coefficient, called the memory parameter, sits between minus one and one for a stationary record.
The moving-average recipe
Now imagine instead that the value is a blend of the recent shocks. The value at one moment equals a shock at that moment plus a weight times the shock at the previous moment, and perhaps weights on earlier shocks as well. The record now has memory of a finite kind: once a shock is older than the last weight, its influence is gone entirely. This is one of the cleanest possible pictures of fading influence.
| Feature | Autoregressive | Moving-average |
|---|---|---|
| memory source | previous values | recent shocks |
| memory span | unbounded but fading | finite and sharp |
| stationary when | coefficient inside unit interval | always |
| typical dense shape | spiked peak | flat plateau |
Combining the recipes
Many records need both ideas at once. A combined form carries the past forward and mixes the recent shocks, and the two mechanisms contribute to the shape of the autocorrelation and the spectral density. The combined form is flexible enough to describe a wide range of memory, and its parameters can be read off from the autocorrelation and the density together. Because the two portraits disagree about some features and agree about others, using both is far safer than using either alone.
Stability and its warning signs
A record built by carrying the past forward is stationary only while its memory parameter stays inside the unit interval. Let the parameter reach one and the record becomes a wandering without a settled level; let it pass one, and the record explodes. These boundaries are not mere technicalities. They mark a real change in behaviour, and a reader who wants to understand why a particular record resists summarising should look first at how close its memory parameter sits to the boundary.
Span and honesty
The number of past terms kept is called the span of the recipe. A longer span fits the observed record more closely, but it gains that closeness with parameters that may describe noise rather than structure. The compendium favours the shortest span that captures the visible behaviour, because a short recipe that explains the shape is more useful than a long one that merely traces it.
Where the language meets the record
These forms are not predictions and they are not prescriptions; they are descriptions of memory. Their value lies in compressing a long and complicated record into a small set of numbers whose meaning a reader can hold in mind. Used that way, they complete the spectral material of the previous paper and prepare the final question of the step: when may one long record stand in for the many the recipes imagine?
