fdars (version 0.3.3)

ssa.fd: Singular Spectrum Analysis (SSA) for Time Series Decomposition

Description

Performs Singular Spectrum Analysis on functional data to decompose each curve into trend, seasonal (oscillatory), and noise components. SSA is a model-free, non-parametric technique based on singular value decomposition of the trajectory matrix.

Usage

ssa.fd(fdataobj, window.length = NULL, n.components = 10)

Value

A list of class "ssa_result" with components:

trend

fdata object containing reconstructed trend component

seasonal

fdata object containing reconstructed seasonal component

noise

fdata object containing noise/residual component

singular.values

Singular values from SVD (sorted descending)

contributions

Proportion of variance explained by each component

window.length

Window length used

n.components

Number of components extracted

detected.period

Auto-detected period (if any)

confidence

Confidence score for detected period

call

The function call

Arguments

fdataobj

An fdata object.

window.length

Embedding window length (L). If NULL, automatically determined as min(n/2, 50). Larger values capture longer-term patterns.

n.components

Number of SVD components to extract. Default: 10. More components allow finer decomposition but increase noise.

Details

The SSA algorithm consists of four stages:

1. Embedding: The time series is converted into a trajectory matrix by arranging lagged versions of the series as columns.

2. SVD Decomposition: Singular value decomposition of the trajectory matrix produces orthogonal components.

3. Grouping: Components are grouped into trend (slowly varying), seasonal (oscillatory), and noise. Auto-detection uses sign change analysis and autocorrelation.

4. Reconstruction: Diagonal averaging (Hankelization) converts grouped trajectory matrices back to time series.

SSA is particularly suited for:

• Short time series where spectral methods fail

• Noisy data with weak periodic signals

• Non-stationary data with changing trend

• Separating multiple periodicities

References

Golyandina, N., & Zhigljavsky, A. (2013). Singular Spectrum Analysis for Time Series. Springer.

Elsner, J. B., & Tsonis, A. A. (1996). Singular Spectrum Analysis: A New Tool in Time Series Analysis. Plenum Press.

See Also

stl.fd, decompose, estimate.period

Examples

# Signal with trend + seasonal + noise
t <- seq(0, 10, length.out = 200)
X <- matrix(0.05 * t + sin(2 * pi * t / 1.5) + rnorm(length(t), sd = 0.3), nrow = 1)
fd <- fdata(X, argvals = t)

# Perform SSA
result <- ssa.fd(fd)
print(result)

# Plot components
plot(result)

# Examine singular value spectrum (scree plot)
plot(result, type = "spectrum")