singular spectrum analysis python singular spectrum analysis python
T . Accessed November 19, 2017. The attribute grouped_components_ generates component matrices that follow the specifications laid out in the component_groups_ dictionary. , 139.7s. i By default, the last axis of Sxx corresponds Decomposing time series into several time series can SSA can be an aid in the decomposition of time series into a sum of components, each having a meaningful interpretation. R be some integer called the window length and 2 are called temporal empirical orthogonal functions (EOFs). 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. T N What do have some consequence are the following. trend or seasonality), Finding structure with randomness: Stochastic algorithms for constructing Singular Spectrum Analysis. Each matrix There are some null values in the dataset, but they all happen to occur in the most recent 48 months, which we are leaving off for testing. False return a two-sided spectrum. The name "singular spectrum analysis" relates to the spectrum of eigenvalues in a singular value decomposition of a covariance matrix, and not directly to a frequency domain decomposition. {\displaystyle K=N-L+1} X You can see that the trajectory matrix has 5 rows (window length 5). , With larger datasets the steps can often take much longer, even with the numba optimizations in place. L "trajectory matrix" i We present a new method of trend extraction in the framework of the Singular Spectrum Analysis approach. 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. k King (1986a): "Extracting qualitative dynamics from experimental data". C nonstationary signals frequency content over time. Allen, M.R., and A.W. t As of the time of this writing, the MSSA class features: While the MSSA class provides the main interface to the MSSA algorithm and tools, you will notice if you look through the code that the majority of the actual functions and math are imported from the optimized submodule. # The first subseries consists of the trend of the original time series. Powered by, pyts.decomposition.SingularSpectrumAnalysis, # Author: Johann Faouzi , # We decompose the time series into three subseries, # Show the results for the first time series and its subseries. The w-correlation is a weighted correlation that takes into account the fact that some timepoints appear more frequently in the trajectory matrix than others. This data has 7 timeseries and 187 observations (some of which are null values equal to groups. Before I actually deploy this system on an aircraft, I have taped the sensor to a speaker and used a frequency generator to generate a 100Hz tone . eigenvalue problems, TRLan usually performed better because of the new be useful in order to keep the most important information. = / Danilov, D. and Zhigljavsky, A. This shows the explained variance percent for the ranked components per timeseries. Hassani, H., A. Soofi and A. Zhigljavsky (2011): "Predicting Daily Exchange Rate with Singular Spectrum Analysis". To my knowledge, skrandom has no equivalent. = They are fine to occur in the testing set, we will just have fewer values in some cases to calculate our performance metric. Defaults to None. . ~ of the lag-covariance matrix [update 4 April 2017] - Please note that the current version requires a pandas dataframe with a datetime formatted index for it to work. provide more refined decomposition into elementary components and therefore better separability. Discrete-Time Signal Processing, Prentice Hall, 1999. This example illustrates the decomposition of a time series into several subseries using this algorithm and visualizes the different subseries extracted. C Trend (which is defined as a slowly varying component of the time series), periodic components and noise are asymptotically separable as A Monte-Carlo test (Allen and Smith, 1996; Allen and Robertson, 1996; Groth and Ghil, 2015) can be applied to ascertain the statistical significance of the oscillatory pairs detected by SSA. ) X X Gallery generated by Sphinx-Gallery Scikit-learn compatibility; Plotting a time series Bozzo, E., R. Carniel and D. Fasino (2010): "Relationship between singular spectrum analysis and Fourier analysis: Theory and application to the monitoring of volcanic activity". Note, if using output that is not one sided, then use the following: Copyright 2008-2023, The SciPy community. In general, the A crucial result of the work of these authors is that SSA can robustly recover the "skeleton" of an attractor, including in the presence of noise. , gives the total variance of the original time series MSSA can be pretty slow and memory intensive, especially if you are dealing with large timeseries data. x SSA is a powerful tool for decomposition, reconstruction, and forecasting of climatic time series (Ghil et al., 2002 ; Plaut et al., 1995 ; Yiou et . and the grouped SVD expansion of After fitting the MSSA, there is an attribute labeled component_groups_ that is a dictionary with timeseries indices as keys and a list of component indices as values. , If float, it represents the percentage of the size of each time series and must be between 0 and 1. pymssa Uses numba-optimized Functions (where possible), A Brief Note on the Math and Algorithms in MSSA, component_ranks_ and component_ranks_explained_variance_, Assigning and accessing component groupings, Introducing SSA for Time Series Decomposition, Multivariate singular spectrum analysis: A general view and new vector forecasting approach, Uni- or Multi-variate decomposition of timeseries using Singular Spectrum Analysis. The two methods are both extensions of classical principal component analysis (PCA) but they differ in emphasis: EEOF analysis typically utilizes a number I X We can see that (using the default settings), parallel analysis is the most conservative, variance explained thresholding is the least conservative, and svht falls somewhere in between on the more conservative side. i Its roots lie in the classical Karhunen (1946)Love (1945, 1978) spectral decomposition of time series and random fields and in the Ma (1981)Takens (1981) embedding theorem. 1 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. This problem is a shortcoming of principal component analysis in general, not just of M-SSA in particular. X implementors -- it is of no consequence to the end users. , differ between the central part of the time series and the vicinity of its endpoints (Ghil et al., 2002). Here is a list of some things that come to mind that are not part of this package yet but that could definitely be added, if you are looking for ideas: This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. UNDER DEVELOPMENT. It combines elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing.Its roots lie in the classical Karhunen (1946)-Love (1945, 1978) spectral decomposition of time series and random fields and in the Ma . The research of Nina Golyandina from Russia was invaluable in aiding my understanding of this method. We can use the list I made above to set the new groups for timeseries 0. {\displaystyle N} {\displaystyle I_{1},\ldots ,I_{m}} Compute S = X X.T, where X is the vertically-stacked trajectory matrices. Caterpillar-SSA emphasizes the concept of separability, a concept that leads, for example, to specific recommendations concerning the choice of SSA parameters. One way to address this question was proposed by 1 ( j {\displaystyle {\textbf {A}}_{k}} If Set / {\displaystyle x_{1},\ldots ,x_{N}} Compatibility with python 3; Computing the weighted correlation matrix is too long for large time series Partition the set of indices In either case: (i) estimates of missing data points are produced iteratively, and are then used to compute a self-consistent lag-covariance matrix Desired window to use. In practice , (Golyandina et al., 2001, Ch.5). X Let the series be governed by the minimal LRR DFT-even by default. X {\displaystyle N} Since verbose=True you can see the steps taken in the fit procedure. L Multivariate Singular Spectrum Analysis: A General View and New Vector Forecasting Approach;. + Halko, et al., 2009, A randomized algorithm for the decomposition of matrices Selecting the `Singular Spectrum Analysis' button from the Analysis Tools menu on the main panel launches the following window (shows its state after pressing Get Default Values button, see below): Figure 11: SSA control panel. {\displaystyle U_{t}} see: https://stats.stackexchange.com/a/159602/87558, see: https://cran.r-project.org/web/packages/Rssa/Rssa.pdf, rSSA package uses either 'nutrlan', 'propack', 'full svd' with the dgesdd routine On more difficult There is a good chance that some timeseries components are highly correlated and should be "grouped" together (via adding together). np.linspace(0, window_size, groups + 1).astype('int64'). U For a project I am attempting to use an accelerometer to measure vibration in an RC aircraft and determine frequency from the result. . TRLan uses Ritz vectors while ARPACK uses the vectors produced by the {\displaystyle \mathbf {X} } , 1 x MSSA has many applications. s i and the sum of the eigenvalues, i.e., the trace of (Penland et al., 1991; Keppenne and Ghil, 1993). Perform the singular value decomposition (SVD) of the trajectory matrix Singular Spectrum Analysis (SSA) is a non-parametric and model free method for time series decomposition, reconstruction (and foracasting). M , To do this I am using an MPU-6000 accelerometer sampling @ 1000Hz. License. X If True, will return the parameters for this estimator and Desired window to use. {\displaystyle M_{t}} = {\displaystyle N'=N-M+1} Penland, C., Ghil, M., and Weickmann, K. M. (1991): "Adaptive filtering and maximum entropy spectra, with application to changes in atmospheric angular momentum,", Pietil, A., M. El-Segaier, R. Vigrio and E. Pesonen (2006) "Blind source separation of cardiac murmurs from heart recordings". ( t {\displaystyle {\sqrt {\lambda _{i}}}V_{i}=\mathbf {X} ^{\mathrm {T} }U_{i}} The discrepancies between these two approaches are attributable to the organization of the single trajectory matrix 1 , You can access the ranks of components for each timeseries from the component_ranks_ attribute. N An appropriate amount of overlap will depend on the choice of window Singular Spectrum Analysis for time series forecasting in Python. X Logs. ( noverlap = nperseg // 8. M The columns are types of wine, including: You can see the header of this dataset below. Zhigljavsky., K. Patterson and A. Soofi (2011): " A comprehensive causality test based on the singular spectrum analysis". 2nd step: Singular Value Decomposition (SVD). In order to further overcome the difficulties of the existing models in dealing with the nonstationary and nonlinear characteristics of high-frequency financial time series data, especially their weak generalization ability, this paper proposes an ensemble method based on data denoising methods, including the wavelet transform (WT) and singular spectrum analysis (SSA), and long-term short-term . Code. The same goes with sparpack and skarpack, as skarpack is just a wrapper to sparpack with fewer arguments allowed. ) Cross spectral density by Welchs method. for a typical real-life series) and t }. A Size of the sliding window (i.e. c Hence it is kept. This methodology became known in the rest of the world more recently (Danilov and Zhigljavsky, Eds., 1997; Golyandina et al., 2001; Zhigljavsky, Ed., 2010; Golyandina and Zhigljavsky, 2013; Golyandina et al., 2018). are the singular values and provide the singular spectrum of X Singular Spectrum Analysis for time series forecasting in Python, Digital signal analysis library for python. { {\displaystyle M} Defaults to None, but if window is str or L account for the partial variance in the Are you sure you want to create this branch? Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Automated options for selection of number of components: Calculation of the w-correlation matrix of component correlations. {\displaystyle \lambda _{1}\geq \ldots \geq \lambda _{L}\geq 0} N Ghil, Vautard and their colleagues (Vautard and Ghil, 1989; Ghil and Vautard, 1991; Vautard et al., 1992; Ghil et al., 2002) noticed the analogy between the trajectory matrix of Broomhead and King, on the one hand, and the KarhunenLoeve decomposition (Principal component analysis in the time domain), on the other. direction Hassani, H.and D. Thomakos, (2010): "A Review on Singular Spectrum Analysis for Economic and Financial Time Series". implicit QR procedure. Are kept: splapack,sparpack and skrandom. M I'm also going to center the data. Signals such as time series can be seen as a sum of different signals such is decomposed into a sum of , Thus, SSA can be used as a time-and-frequency domain method for time series analysis independently from attractor reconstruction and including cases in which the latter may fail. L Components that have correlations with many other components are often deemed noise components and discarded. i There is still a ton of things that can be added to make this a more comprehensive MSSA package; I've really only scratched the surface here. This is experimental and I'm not totally confident its doing what its supposed to yet, so I am going to leave it out of this demo for now. S i L L Comments (0) Run. L Axis along which the spectrogram is computed; the default is over N Singular Spectrum Analysis methods implementation in Python, Extracting social-economic signals from internet traffic data. Logs. L forecasting); Missing/corrupted by noise (i.e. {\displaystyle \mathbf {X} _{I_{j}}} t C {\displaystyle L_{t}} The general recommendation seems to be either: There are some other more advanced methods not yet implemented in this package, like "Sequential SSA", which decomposes the timeseries at different window lengths to pull out different frequencies of signal, then combines the results. r Normally you would want to select a smaller set of components, but since we have already used some stringent thresholding during the fitting procedure we can just use all the remaining components for the forecast and not have to worry as much about overfitting issues. history Version 1 of 1. Apparently this leads to higher forecasting accuracy and currently the recurrent formula is the one implemented. Python implementation of Monte Carlo Singular Spectrum Analysis for univariate time series. Vectorized forecasting formula. , depend only on the lag Let . Some other variants are not 1-dimensional such as: Multichannel SSA (multiple time series), 2d-SSA (arrays and images), nd-SSA (nd arrays). As of the time of this writing, I am not aware of any other implementation in python of multivariate SSA, though there are packages and implementations of univariate SSA. k In time series analysis, singular spectrum analysis (SSA) is a nonparametric spectral estimation method. Accordingly, we have four different forecasting algorithms that can be exploited in this version of MSSA (Hassani and Mahmoudvand, 2013). Separation of two time series components can be considered as extraction of one component in the presence of perturbation by the other component. Schoellhamer (2001) shows that the straightforward idea to formally calculate approximate inner products omitting unknown terms is workable for long stationary time series. The library includes such methods of the signal analysis, signal processing and signal parameter estimation as ARMA-based techniques; subspace-based techniques; matrix-pencil-based methods; singular-spectrum analysis (SSA); dynamic-mode decomposition (DMD); empirical mode decomposition; variational mod K the orthonormal system of the eigenvectors of the matrix ( L = M fsfloat, optional. The explained variance of the SVD components, Percent of explained variance for each component. a In the plot above, we can see that there are maybe 11 groups of components before the components start to have "messy" correlation with the others. } Trends can be extracted by grouping of eigentriples with slowly varying eigenvectors. A tag already exists with the provided branch name. k 2010). {\displaystyle \mathbf {X} } Defaults to True, but for analysis The EMH suggests that the information contained in the price series of an asset is reflected instantly, fully, and perpetually in the assets current price. {\displaystyle {\textbf {R}}_{K}} Issues. i (Guest Editor) (2010) "Special issue on theory and practice in singular spectrum analysis of time series". Vautard, R., Yiou, P., and M. Ghil (1992): "Singular-spectrum analysis: A toolkit for short, noisy chaotic signals", Weare, B. C., and J. N. Nasstrom (1982): "Examples of extended empirical orthogonal function analyses,". This is definitely not as comprehensive as the Rssa package (which I believe is maintained by the creator of SSA), but it implements the fundamental components of MSSA. i We can look at the w-correlation for the grouped components to validate that we have removed most of the correlation between components by grouping. A question that frequently arises in time series analysis is whether one economic variable can The w-correlation is a common metric used in SSA for measuring the correlation between components. i d On Multivariate Singular Spectrum Analysis and its Variants. Now we can instantiate the MSSA object and fit to the training data. = , Grouping by looking at the w-correlation can be more of an art than a science. length of the window. A sinusoid with frequency smaller than 0.5 produces two approximately equal eigenvalues and two sine-wave eigenvectors with the same frequencies and N (2016): "Matrix formulation and singular-value decomposition algorithm for structured varimax rotation in multivariate singular spectrum analysis". Power spectral density by Welchs method. | The function w_correlation will give back the w-correlation matrix between components for a provided (N, n_components) reconstruction matrix. The library includes such methods of the signal analysis, signal processing and signal parameter estimation as ARMA-based techniques; subspace-based techniques; matrix-pencil-based methods; singular-spectrum analysis (SSA); dynamic-mode decomposition (DMD); empirical mode decomposition; variational mod, A package for performing Singular Spectrum Analysis (SSA) and time-series decomposition. } d l 1 Length of each segment. MSSA component matrix (or "reconstructions"), Left singular vectors and singular values, Explained variance scores and percent variance explained for components. (2002). Trend extraction is an important task in applied time series analysis, in particular in economics and engineering. {\displaystyle U_{1},\ldots ,U_{d}} d Compute the largest k singular values/vectors for a sparse matrix. The values of the normalization factor In this subsection, we focus on phenomena that exhibit a significant oscillatory component: repetition increases understanding and hence confidence in a prediction method that is closely connected with such understanding. For a multivariate data set, gap filling by M-SSA takes advantage of both spatial and temporal correlations. 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. {\displaystyle L} {\displaystyle (i=1,\ldots ,d)} An example of the implementation of this code can be found in Singular Spectrum Analysis Example.ipynb. of length ( {\displaystyle \pi /2} U Fraedrich, K. (1986) "Estimating dimensions of weather and climate attractors". 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 {\displaystyle N\rightarrow \infty } U of X ] To do that, SSA performs the subspace tracking in the following way. Download : Download high-res image (535KB) Download : Download full-size image; Fig. E X and x L L It is for this reason that the default window is a Tukey window with i Easy interface to assign and retrieve component groupings. X Portes, L. L. and Aguirre, L. A. ) temporal principal components (PCs) {\displaystyle I=I_{1},\ldots ,I_{m}} First, the noise is filtered out by projecting the time series onto a subset of leading EOFs obtained by SSA; the selected subset should include statistically significant, oscillatory modes. One decomposition algorithm is Singular Spectrum Analysis. For this small dataset the steps are nearly instantaneous. L required parameters. Defaults to 1.0. X entire data stream is averaged over, one may wish to use a smaller = Robertson (1996): "Distinguishing modulated oscillations from coloured noise in multivariate datasets". Pull requests. N density. , ; and (ii) cross-validation is used to optimize the window width 1 The gap-filling version of SSA can be used to analyze data sets that are unevenly sampled or contain missing data (Kondrashov and Ghil, 2006; Kondrashov et al.
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