Constructing singular-spectrum forecast baselines

Treat the observed series as three parts

An observed market series can be treated as the combination of a slow trend, an oscillation whose period and amplitude may change, and a residual random component. Spectral analysis is a data-adaptive decomposition of that ordered series into trend, oscillatory, and residual parts taken from the singular structure of a trajectory matrix rather than from a fixed trigonometric basis.

Singular spectrum analysis forms its components from the observed series itself. It does not require a preliminary stabilization step or a fixed sine-cosine basis.

Embed consecutive segments

The construction begins by embedding consecutive segments of length m into a trajectory matrix so that repeatable structure in the original series is inherited by those segments. Window length is the spatial dimension m that sets how long each embedded segment is and how narrow the implied reconstruction filter becomes.

Choose series length, window length, and component count

The free parameters are series length N, window length m, and retained-component count L. Increasing m narrows the implied reconstruction filter.

For unknown periods a common choice is m equal to half the sample length. When a period is known, N is made divisible by that period and m is set to match it.

Rank the leading components

Leading components may be kept when their eigenvalues exceed one, or by reading breaks on a descending eigenvalue chart. Component count is the number L of leading singular components retained for reconstruction or forecast after that numerical or graphical cutoff.

In a 240-minute EUR/USD illustration with N equal to 50 and m equal to 25, the first singular value was taken as the trend. Slope changes were used to cap L near 23 to 25, and a working set of 3 to 10 components was described as often adequate.

Reconstruct with diagonal averaging

After grouping selected principal components, diagonal averaging reconstructs a series that can isolate trend, recover periodic structure, or reduce the residual. That step converts a grouped low-rank trajectory matrix back into a single reconstructed series.

Record the first projected point

Because the reconstructed path is recomputed as new bars arrive, a historical overlay of that path is not a valid out-of-sample reading. A forecasting variant instead records the first projected point of each bar. That forecasting moving SSA line can be placed next to a same-length exponential moving average that only filters.

Exponential smoothing is a recursively weighted average of ordered prices used as that same-length, filtration-only counterpart. A moving average is the same-length filtration of ordered prices used as a non-forecasting baseline against which a reconstructed or projected spectral smoother can be compared.

When a linear generating model is too restrictive

The linear form of the construction assumes a linear generating model. Kernel methods were introduced when that assumption was treated as too restrictive for market series.