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A Method of Trend Extraction Using Singular Spectrum Analysis
Theodore Alexandrov
TL;DR
Existing SSA component-selection methods do not establish which kinds of trend they can extract. The paper proposes a two-parameter SSA trend-extraction method with a selectable scale and threshold; in simulations with an exponential trend and white Gaussian noise, the optimal threshold selected the trend's first SVD component with high probability.
Problem
An existing SSA component-selection method does not establish which kinds of trend it can extract, limiting clarity about its applicability.
Source: §2.2 · p. 6Method
The method selects SSA trend components using a scale parameter and a method-specific threshold, with procedures for choosing the parameters.
Source: §1 · p. 2, §4 · p. 8, §5.2 · p. 18Results
For an exponential trend with white Gaussian noise, optimal C0 selection chose the trend's first SVD component with high probability, with type I error below 0.05 for σ ≤1.4.
Source: §5.1.2 · p. 15Takeaways & Limitations
The method provides a two-parameter, model-free way to specify trend scale and extract trends amid noise and oscillations.
Source: §1 · p. 2, §7 · p. 21Successful separation is limited to trend and residual classes that SSA can separate.
Source: §2.1 · p. 5
Abstract
from arXiv · showhide
The paper presents a new method of trend extraction in the framework of the Singular Spectrum Analysis (SSA) approach. This method is easy to use, does not need specification of models of time series and trend, allows to extract trend in the presence of noise and oscillations and has only two parameters (besides basic SSA parameter called window length). One parameter manages scale of the extracted trend and another is a method specific threshold value. We propose procedures for the choice of the parameters. The presented method is evaluated on a simulated time series with a polynomial trend and an oscillating component with unknown period and on the seasonally adjusted monthly data of unemployment level in Alaska for the period 1976/01-2006/09.
1 INTRODUCTION
The paper introduces an SSA-based method for trend extraction, addressing the lack of a precise, widely accepted definition of trend. It aims to extract trends without specifying time-series or trend models, including when noise and oscillations are present.
1 INTRODUCTION
- Trend is commonly described as a smooth additive component reflecting global change, but the paper notes that this definition lacks agreed precision.The questions of what smoothness means and what information trend contains remain unresolved among researchers and practitioners.Source: §1 · p. 1
- SSA represents a time series through eigenvalues and eigenvectors of a matrix constructed from it, in an approach likened to PCA for multivariate data.The paper notes that SSA has been used for trend and quasi-periodic component extraction, denoising, forecasting, and change-point detection.Source: §1 · p. 2
- The method was first proposed in and studied in detail in the author’s unpublished Ph.D. thesis.The paper’s outline covers parameter-choice procedures and applications to simulated data and Alaska unemployment data.Source: §1 · p. 2
- The proposed SSA method needs no time-series or trend model specification and has two method parameters, in addition to SSA’s window length.It allows users to specify the desired trend scale and is intended to extract trends amid noise and oscillations.Source: §1 · pp. 1–2
2 SINGULAR SPECTRUM ANALYSIS
SSA decomposes a time series and reconstructs a selected additive component; trend extraction therefore depends on choosing the window length and the SVD-component subgroup. The section describes conditions relevant to trend separation and limitations of existing selection methods.
2 SINGULAR SPECTRUM ANALYSIS
- Trend extraction in SSA depends on selecting the window length and the subgroup of SVD components used to reconstruct the time series.SSA embeds the series, identifies a subspace for the desired component, and reconstructs a series from that subspace; reconstruction uses the full series and is non-local.Source: §2 · pp. 3–4
- SSA’s SVD calculation has substantial computational complexity, although modern and parallel SVD algorithms can reduce this drawback.The paper also notes that an updating algorithm can be used when new data points arrive.Source: §2 · p. 4
2.1 Trend in SSA
- SSA requires no a priori deterministic or stochastic model, but successful trend extraction depends on trends and residuals being separable.The paper describes smooth, slowly varying trends represented by fewer than d SVD components, with a class including polynomial, exponential, and sinusoidal combinations.Source: §2.1 · p. 5
- Smooth, slowly varying trends are associated with SVD components whose EOFs are also smooth and slowly varying.This provides a basis for identifying trend components through their EOF properties.Source: §2.1 · p. 5
2.2 Existing methods of trend extraction in SSA
- Reconstructing a trend from the first SVD components can fail when the trend is small relative to residuals or has a complicated structure.A high-order polynomial may be characterized by many SVD components rather than only the first ones.Source: §2.2 · pp. 5–6
- Selecting trend components by monotonic EOFs can fail for non-monotonic trends and for noisy observations that corrupt EOFs.The paper notes that even a monotonic trend can yield non-monotonic EOFs; low-order polynomials can also produce non-monotonic EOFs.Source: §2.2 · p. 6
3 PROPOSED METHOD FOR TREND EXTRACTION
The method selects SSA singular-value-decomposition components according to their eigenvectors’ low-frequency content, using two parameters, ω0 and C0. The selected components are then reconstructed as the trend.
3 PROPOSED METHOD FOR TREND EXTRACTION
- The method defines a periodogram and cumulative contribution from frequencies in [0, ω0] to quantify low-frequency content.Its periodogram doubles values for frequencies in (0, 0.5), using a normalization different from standard spectral-analysis periodograms.Source: §3 · p. 7
- Components are selected using ω0 and C0 according to a criterion on their eigenvectors, which favors EOFs characterized mostly by low-frequency fluctuations.The passage does not state the criterion’s formula.Source: §3 · p. 7
- After selecting the trend components, the method reconstructs the trend using the SSA procedure described earlier.The paper discusses choosing ω0 and C0 in Sections 4 and 5, respectively.Source: §3 · p. 7
4 THE LOW-FREQUENCY BOUNDARY ω0
The low-frequency boundary ω0 sets the scale of the extracted trend and is chosen from the time-series periodogram to include trend frequencies while limiting oscillations. The section motivates this choice through periodogram behavior and gives a simulated example.
4 THE LOW-FREQUENCY BOUNDARY ω0
- Lowering ω0 produces a more slowly varying trend, and monthly data with possible 12-period seasonal oscillations suggests ω0 < 1/12, for example 0.075.The proposed choice balances capturing large trend-periodogram values near zero against including oscillations below the cutoff.Source: §4 · p. 8
- When trend, oscillation, and noise periodogram supports are nearly disjoint, the series periodogram is approximately the sum of their individual periodograms.This conclusion is presented for series whose trend has large low-frequency values, oscillations have periods shorter than 1/ω0, and noise spreads across frequencies.Source: §4 · p. 10
- The proposed ω0 selection uses the original series periodogram and its median as a reference for identifying frequencies beyond the trend’s frequency interval.The rationale models trend and oscillations as having nonoverlapping frequency supports; large noise-periodogram values outside the trend interval can make the selected boundary too large.Source: §4 · pp. 10–11
- For the same data sampled at m times the original resolution, use ω′0 = ω0/m to extract the same trend; after decimation by m, use ω′0 = mω0.The adjustment follows the paper’s definition of resolution from the spacing between measurement times.Source: §4 · p. 11
- The simulated example yields ω0 = 6/120 = 0.05; the series periodogram median is 0.0141, close to the noise periodogram median of 0.0126.Figure 1 presents the simulated series and component periodograms used in the example.Source: §4 · pp. 11–12, Figure 1 · p. 12
5 THE LOW-FREQUENCY CONTRIBUTION C0
The paper evaluates the low-frequency contribution threshold C0 in simulations of an exponential trend with multiplicative Gaussian noise, then proposes a heuristic for choosing C0. With an optimal C0, the method often selects the trend-associated first SVD component; the heuristic uses jumps in residual low-frequency contribution.
5.1 A simulation example: an exponential trend plus a Gaussian noise
- Simulations compared the method’s trend with the first-SVD reconstruction and true trend using weighted MSE across α=0, 0.01, 0.02 and noise levels 0.2–1.6, using 104 realizations of length-47 series and window length 24.Source: §5.1 · p. 13, §5.1.1 · p. 14
5.1.2 Simulation results
- For an exponential trend with white Gaussian noise, an optimal C0 yields an error below the ideal SSA trend’s own error, especially for σ ≤ 0.8; the method selects the trend-associated first SVD component with high probability.The estimated type I error is less than 0.05 for σ ≤ 1.4.Source: §5.1.2 · p. 15
- Figure 3 plots square roots of mean errors and optimal C0 values; the minimum errors are similar for α = 0 and α = 0.02 despite differing optimal C0 values.The α = 0 results are shown in gray; estimates for α = 0.01 are not reported.Source: §5.1.2 · pp. 14–15, Figure 3 · p. 16
- Choosing C0 too large can exclude the trend from reconstruction, while values below the optimum can retain accuracy because erroneously included noise components contribute little.Figure 4 plots error against C0 for α = 0 and 0.02 at σ = 0.8 and 1.4.Source: §5.1.2 · p. 16, Figure 4 · p. 17
5.2 Heuristic procedure for the choice of C0
- Based on Proposition 5.1, the heuristic expects trend-related reconstructed components to have large low-frequency contributions and associates maximal C0 values with jumps in RF,ω0(C0), using suggested ΔR of 0.05–0.1 and default ΔC of 0.01.Simulations and applications to different time series showed its ability to choose reasonable C0 values in many cases.Source: §5.2 · p. 18
6 EXAMPLES
The examples test trend extraction on a noisy simulated series and seasonally adjusted monthly Alaska unemployment data. In the simulation, the extracted trend approximates the original well and outperforms an ideal low-pass filter by mean square error.
Simulated example with polynomial trend
- The simulated series combines a polynomial trend, an exponentially modulated sine wave of unknown period, and white Gaussian noise.The series has length N = 300.Source: §6 · p. 19
- Mean square error was 0.79 for the extracted trend versus 3.14 for an ideal low-pass filter with cutoff frequency 0.02.The improvement was mostly due to better approximation at the first and last 50 time-series points.Source: §6 · p. 19, Figure 5 · p. 19
Trends of the unemployment level
- The Alaska example uses seasonally adjusted monthly unemployment data from 1976/01 to 2006/09, a setting described as having relatively little noise and abrupt changes.The data are identified as Bureau of Labor Statistics series LASST02000004.Source: §6 · p. 20
- The Alaska example extracts trends at ω0 = 0.01, 0.02, 0.05, 0.075, and 0.095; 0.075 is the default for monthly data, while the other values illustrate scale effects.The search for C0 used [0.5, 1] with steps ∆C = 0.01 and ∆R = 0.05.Source: §6 · p. 20, Figure 7 · p. 21
7 CONCLUSIONS
The method retains SSA's ability to extract trends without specifying time-series or trend models, including in noisy series with oscillations of unknown period. It requires selection of only two parameters.
7 CONCLUSIONS
- The method inherits SSA's ability to extract trends without specifying models of the time series or trend.It also handles noisy series containing oscillations of unknown period.Source: §7 · p. 21
- It is easy to use because it requires selection of only two parameters.Source: §7 · p. 21
- SSA extracts trends from noisy time series containing oscillations of unknown period.Source: §7 · p. 21