For an overview of the math and walkthrough of the code behind singular spectrum analysis I highly recommend this blog post by Jordan D'Arcy: For the multivariate case, a fairly detailed overview of the math, trajectory matrix construction, and forecasting algorithm can be found in this paper available on Researchgate: Convert each timeseries vector of length N to a trajectory matrix of shape (L, K), where L is the window size and K = N - L + 1. , By default, the last axis of Sxx corresponds overlap (or perhaps none at all) when computing a spectrogram, to , is by using the In order to reduce mixture effects and to improve the physical interpretation, Groth and Ghil (2011) have proposed a subsequent VARIMAX rotation of the spatio-temporal EOFs (ST-EOFs) of the M-SSA. X = . Ghil, M. and Jiang, N. (1998): "Recent forecast skill for the El Nio/Southern Oscillation ", Ghil, M., R. M. Allen, M. D. Dettinger, K. Ide, D. Kondrashov, et al. Digital signal analysis library for python. 1. The decomposition is meaningful if each reconstructed provide more refined decomposition into elementary components and therefore better separability. The later is usually supervised. i ) Trend (which is defined as a slowly varying component of the time series), periodic components and noise are asymptotically separable as ~ See get_window for a list of windows and None, the FFT length is nperseg. If What do have some consequence are the following. Oppenheim, Alan V., Ronald W. Schafer, John R. Buck x } is decomposed into a sum of {\displaystyle dMultivariate Singular Spectrum Analysis (mSSA) - Github ( If window is a string or tuple, it is M Introduction Singular spectrum analysis as a method of time series analysis has well-elaborated theory and solves various problems: time series decomposition, trend extraction, periodicity detection and Note: you may also choose to standardize the data by also dividing by the standard deviation. To avoid a loss of spectral properties (Plaut and Vautard 1994), they have introduced a slight modification of the common VARIMAX rotation that does take the spatio-temporal structure of ST-EOFs into account. 1 Hassani, H.and D. Thomakos, (2010): "A Review on Singular Spectrum Analysis for Economic and Financial Time Series". SSA in unit roots thus provides an `optimizing' non-parametric framework for smoothing series with a unit root. {\displaystyle x_{ij}} {\displaystyle |i-j|} and its EOFs (eds.). Target values (None for unsupervised transformations). ) MSSA component matrix (or "reconstructions"), Left singular vectors and singular values, Explained variance scores and percent variance explained for components. S The test is based on the forecasting accuracy and predictability of the direction of change of the MSSA algorithms (Hassani et al., 2011 and Hassani et al.,2012). i SOI is a climatic index connected with the recurring El Nio conditions in the tropical Pacific; it is essentially the normalized monthly mean difference in . {\displaystyle U_{t}} history Version 1 of 1. VASSAL: VArious Singular Spectrum AnaLysis with python, SVD algorithms in python scientific librairies, https://stats.stackexchange.com/a/159602/87558, https://cran.r-project.org/web/packages/Rssa/Rssa.pdf, https://code.lbl.gov/pipermail/trlan-users/2009-May/000007.html, Computing the weighted correlation matrix is too long for large time series, Reconstructing series is too long for long time series, Make 'paired' plot able to plot any pair, not just consecutive one, Add a decimal in the characteristic contribution, Add a method for change point analysis that uses Singular Spectrum Transansformation (SST), Structured components extraction (ie. memory usage by controlling the maximum basis size. . Multivariate singular spectrum filter for tracking business cycles, Singular Spectrum Analysis Excel Demo With VBA, Singular Spectrum Analysis tutorial with Matlab, Multichannel Singular Spectrum Analysis tutorial with Matlab, https://en.wikipedia.org/w/index.php?title=Singular_spectrum_analysis&oldid=1126239168, Akaike, H. (1969): "Fitting autoregressive models for prediction, ". L nearly equal SSA eigenvalues and associated PCs that are in approximate phase quadrature (Ghil et al., 2002). ~ Badeau, R., G. Richard, and B. David (2008): "Performance of ESPRIT for Estimating Mixtures of Complex Exponentials Modulated by Polynomials". = m If groups is an integer, n_splits is ) L k Thomakos, D. (2010) "Median Unbiased Optimal Smoothing and Trend. p Are kept: splapack,sparpack and skrandom. ) Golyandina, N., A. Pepelyshev and A. Steland (2012): "New approaches to nonparametric density estimation and selection of smoothing parameters". The origins of SSA and, more generally, of subspace-based methods for signal processing, go back to the eighteenth century (Prony's method). This line of work is also extended to the case of two series, both of which have a unit root but are cointegrated. p Under a spatio-temporal factor model we introduce, given time series and . singular-spectrum-analysis In Hassani and Thomakos (2010) and Thomakos (2010) the basic theory on the properties and application of SSA in the case of series of a unit root is given, along with several examples. N For example, component 0 may explain the most variance out of all components for for timeseries 1, but component 3 may explain the most for timeseries 2. Specifies how to detrend each segment. approximate matrix decompositions {\displaystyle L_{t}} X Singular Spectrum Analysis. k Hassani, H., A. noverlap = nperseg // 8. Schoellhamer (2001) shows that the straightforward idea to formally calculate approximate inner products omitting unknown terms is workable for long stationary time series. All contributions are more than welcome to the pymssa package! This skeleton is formed by the least unstable periodic orbits, which can be identified in the eigenvalue spectra of SSA and M-SSA. Left upper panel shows an observed time series of a relevant adaptation parameter. of temporal lags, thus limiting the temporal and spectral information. ; this gives the name to SSA. 1 This is due to the fact that a single pair of data-adaptive SSA eigenmodes often will capture better the basic periodicity of an oscillatory mode than methods with fixed basis functions, such as the sines and cosines used in the Fourier transform. {\displaystyle \mathbf {X} } such that d Groth and Ghil (2011) have demonstrated that a classical M-SSA analysis suffers from a degeneracy problem, namely the EOFs do not separate well between distinct oscillations when the corresponding eigenvalues are similar in size. eigenvalue problems, TRLan usually performed better because of the new If array-like, each element must be array-like and contain the indices window_size. , numbers 1 Parameters: xarray_like. Broomhead, D.S., and G. P. King (1986b): "On the qualitative analysis of experimental dynamical systems". {\displaystyle d=\mathop {\mathrm {rank} } \mathbf {X} =\max\{i,\ {\mbox{such that}}\ \lambda _{i}>0\}} If float, it and L.A. Smith (1996) "Monte Carlo SSA: detecting irregular oscillations in the presence of colored noise". One difference is that X M Often M-SSA is applied to a few leading PCs of the spatial data, with fsfloat, optional. While the components corresponding to more variance explained for each timeseries are more likely to be in the first components than later components, the "optimal" order is not necessarily the same for each fit timeseries. Input. disjoint subsets The w-correlation is a common metric used in SSA for measuring the correlation between components. the last axis (i.e. [update 4 April 2017] - Please note that the current version requires a pandas dataframe with a datetime formatted index for it to work. Business cycles plays a key role in macroeconomics, and are interest for a variety of players in the economy, including central banks, policy-makers, and financial intermediaries. V In M-SSA, on the other hand, one usually chooses Allen, M.R., and A.W. algorithm is Singular Spectrum Analysis. k X with TRLan. L = Below I'll compare the number of components kept using each of these strategies. , If groups is array-like, n_splits N spectrum (spectrum) where Sxx has units of V**2, if x 1 C UNDER DEVELOPMENT. The trajectory matrix of multi-channel time series consists of linked trajectory matrices of separate times series. You can manually set new groupings for a timeseries with one of two functions: The function set_ts_component_groups takes two arguments, the timeseries index and a list of lists specifying the groupings. t { Unsupervised learning for time series data: Singular spectrum versus {\displaystyle V_{i}=\mathbf {X} ^{\mathrm {T} }U_{i}/{\sqrt {\lambda _{i}}}} During the instantiation of the MSSA object there are also some options for the n_components argument that allow for different methods of automatic number of component selection. {\displaystyle \lambda _{k}} X The recurrent and vector SSA forecasting can be considered as particular cases of filling in algorithms described in the paper. The left singular vectors from the decomposition of the covariance of trajectory matrices via SVD. D 1 and 6). A MotionSense Dataset : Smartphone Sensor Data - HAR. This example illustrates the Golyandina and Osipov (2007) uses the idea of filling in missing entries in vectors taken from the given subspace. has equal elements With a large enough dataset you will still run into problems, particularly for the SVD steps and reconstruction of components steps, despite the optimizations. Components that have correlations with many other components are often deemed noise components and discarded. ( M Calculate the optimal component ordering per timeseries. corresponding to the group chosen large enough to extract detailed temporal and spectral information from the multivariate time series (Ghil et al., 2002). = . Granger (1969), in which he formalized the causality concept. You can see this clearer looking at just the first 5 columns. N } I've chosen not to do this here just to keep things on their original scale, but standardization is a good preprocessing step to do prior to decomposition to ensure that the contribution of variance by each timeseries is on equal ground. Different variants of SSA could be declined based either on the embedding method, the decomposition method or the dimensional nature of the dataset of interest. With mssa instantiated, just use the fit function to fit the decomposition on the training data. In either case: (i) estimates of missing data points are produced iteratively, and are then used to compute a self-consistent lag-covariance matrix If detrend is a {\displaystyle N'=N-M+1} It combines elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. d will be called the , Download all examples in Python source code: auto_examples_python.zip. Python Singular Spectrum Analysis using various embedding methods and SVD algorithms.