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https://github.com/baggepinnen/singularspectrumanalysis.jl

A package for performing Singular Spectrum Analysis (SSA) and time-series decomposition

https://github.com/baggepinnen/singularspectrumanalysis.jl

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Keywords

detrending esprit forecast hankel-matrix singular-spectrum-analysis time-series time-series-analysis time-series-decomposition time-series-forecasting trend-detection

Keywords from Contributors

mathematics mixed-model hybrid-differential-equation high-performance nonlinear-dynamics computer-algebra nonlinear-programming julialang dynamical-systems bindings
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Repository

A package for performing Singular Spectrum Analysis (SSA) and time-series decomposition

Basic Info
  • Host: GitHub
  • Owner: baggepinnen
  • License: other
  • Language: Julia
  • Default Branch: master
  • Homepage:
  • Size: 94.7 KB
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  • Stars: 65
  • Watchers: 5
  • Forks: 5
  • Open Issues: 4
  • Releases: 11
Topics
detrending esprit forecast hankel-matrix singular-spectrum-analysis time-series time-series-analysis time-series-decomposition time-series-forecasting trend-detection
Created about 9 years ago · Last pushed over 2 years ago
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README.md

SingularSpectrumAnalysis

CI codecov

A package for performing Singular Spectrum Analysis (SSA) https://en.wikipedia.org/wiki/Singularspectrumanalysis

Simple Usage

The example below creates a simulated signal that has two strong seasonal components. The main entry function is analyze(y,L) that returns the trend and seasonal components. y is the signal to decompose and L is a window length to use for the internal embedding. ```julia using SingularSpectrumAnalysis, Plots

generate some data

L = 20 # Window length K = 100 N = KL; # number of datapoints t = 1:N; # Time vector T = 20; # period of main oscillation y = sin.(2pi/Tt); # Signal y .+= (0.5sin.(2pi/T4t)).^2 # Add another frequency e = 0.1randn(N); # Add some noise yn = y+e;

plot(ys)

yt, ys = analyze(yn, L, robust=true) # trend and seasonal components plot(yt, lab="Trend") plot!(ys, lab="Season") `` Therobust` keyword makes the analysis robust against large, sparse outliers, at the expense of longer computational time.

ESPRIT

  • esprit(x, L, r; fs=1, robust=false) Estimates r (positive) frequencies present in signal x using a lag-correlation matrix of size L.

Advanced usage

Internally a Hankel matrix is formed and the SVD of this is calculated. The singular values of the SVD can be plotted to manually determine which singular value belongs to the trend, and which pairs belong to seasonal components (these are always pairs). julia USV = hsvd(yn,L,robust=false) # Perform svd on the trajectory matrix, robust uses a robust version of svd, resistant to outliers plot(USV, cumulative=false) # Plot normalized singular values window

```julia seasonalgroupings = [1:2, 4:5] # Determine pairs of singular values corresponding to seasonal components trendi = 3 # If some singular value lacks a buddy, this is a trend component

trendi, seasonalgroupings = autogroup(USV) # This uses a heuristic

pairplot(USV,seasonalgroupings) # plot phase plots for all seasonal components yrt, yrs = reconstruct(USV, trendi, seasonal_groupings) # Reconstruct the underlying signal without noise, based on all identified components with significant singular values yr = sum([yrt yrs],dims = 2) # Form full reconstruction plot([y ys yr], lab=["y" "ys" "ys" "yr"]) ```

Forecasting

We provide the function fit_trend(yt, order) to fit an n:th order polynomial to the trend: julia yt, ys = analyze(yn, L) A,x = fit_trend(yt, 1) This returns the regressor matrix A and the polynomial coefficients x. This fit can be used to forecast the trend. To forecast the seasonal components, we make use of the package ControlSystemIdentification.jl to fit AR(na) models. We create a simulated signal to test with: julia using Random Random.seed!(0) L = 20 K = 10 N = K*L; t = 1:N; T = 20; y = sin.(2pi/T*t); # Add seasons y .+= (0.5sin.(2pi/T*4*t)).^2 # Add seasons y .+= LinRange(0,1,N) # Add trend e = 0.1randn(N); yn = y+e; # Add noise Next, we use SSA to find the trend and the seasonal components julia yt, ys = analyze(yn, L) # trend and seasons using ControlSystemIdentification pd = PredictionData(yt, ys, trend_order=1, ar_order=2) yth = trend(pd) ysh = seasons(pd) Next, we visualize the trends and seasonal components estimated by both SSA and AR models. julia plot(ys, layout=size(ys,2), lab="SSA", title="Estimated seasonal components") plot!(ysh, lab="AR") window

julia plot(yt, lab="SSA", title="Estimated trend") plot!(yth, lab="Polyfit") window

julia yr = yt+sum(ys, dims=2) plot(yn, lab="Measured", title="Full reconstructions") plot!(yr, lab="SSA") plot!(+(yth, ysh...), subplot=1, lab="AR", l=(:dash,)) window

To perform n-step prediction, use the function pred: julia pd = pred(pd,2) # Predict two steps yth = trend(pd) ysh = seasons(pd) The example above is implemented in forecast.jl.

Missing data / outliers

See the keyword argument robust. The robust estimation is handled by TotalLeastSquares.jl which performs a robust PCA of the Hankel matrix. This factorization handles large but sparse outliers very well. To indicate that a value is missing, you can set it to some large value that is very far from the other values and it will be identified as an outlier by the robust factorization. To obtain the inferred values for the missing data, call the low-level function directly julia X = hankel(y,L) # Form trajectory matrix X̂, E = rpca(X) ŷ = unhankel(X̂) Where ŷ is a clean version of the signal. The sparse matrix E contains the estimated noise values. See also function lowrankfilter which packages this procedure.

See further documentation and examples here.

Advanced low-level usage

See the implementation of functions hsvd and reconstruct

Reading

See http://www.jds-online.com/files/JDS-396.pdf for an easy-to-read introduction to SSA

Owner

  • Name: Fredrik Bagge Carlson
  • Login: baggepinnen
  • Kind: user
  • Location: Lund, Sweden

Control systems, system identification, signal processing and machine learning

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  • Total dependent packages: 1
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  • Total versions: 10
juliahub.com: SingularSpectrumAnalysis

A package for performing Singular Spectrum Analysis (SSA) and time-series decomposition

  • Versions: 10
  • Dependent Packages: 1
  • Dependent Repositories: 0
  • Downloads: 1 Total
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Average: 16.8%
Forks count: 21.7%
Dependent packages count: 23.0%
Last synced: 6 months ago

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