N , Both nplapack and splapack use the LAPACK algorithm for full svd decomposition but the scipy implementation allows more flexibility. is the set of EOFs on which the reconstruction is based. n Barnett, T. P., and K. Hasselmann (1979): "Techniques of linear prediction, with application to oceanic and atmospheric fields in the tropical Pacific, ". Harris, T. and H. Yan (2010): "Filtering and frequency interpretations of singular spectrum analysis". Other multivariate extension is 2D-SSA that can be applied to two-dimensional data like digital images (Golyandina and Usevich, 2010). M M-SSA has two forecasting approaches known as recurrent and vector. un-restarted Lanczos can be used, TRLan was shown to use nearly the ~ Something like that may become available in a future update. ( tuple, is set to 256, and if window is array_like, is set to the . X t I {\displaystyle ({\sqrt {\lambda _{i}}},U_{i},V_{i})} i (2001, Ch. density. = j , To do this I am using an MPU-6000 accelerometer sampling @ 1000Hz. import numpy as np import matplotlib.pyplot as plt from pyts.decomposition import SSA # Parameters n_samples, n_features = 100, 48 # Toy dataset rng = np.random.RandomState(41 . angle and phase return the complex angle of the STFT, function. entire data stream is averaged over, one may wish to use a smaller Each component may account for more or less variance of a given timeseries, though typically the first components will account for more variance than later components (the actual order of which components account for most variance per timeseries can be found in component_ranks_). {\displaystyle L\leq M} = The window length t Specifically, mSSA allows you to predict entries that are: At a future time step (i.e. U , + Select the number of components using the "Singular Value Hard Thresholding" formula. X This shows the explained variance percent for the ranked components per timeseries. In this way, the initial series complex is i ) I will use the last mssa object I fit that used parallel analysis thresholding, and forecast out the testing indices we set up awhile back using all the components. .dataframe tbody tr th:only-of-type { X , lag-shifted copies of - use_components is an array of component indices you would like to use to do the forecasting for. Mohammad and Nishida (2011) in robotics), and has been extended to the multivariate case with corresponding analysis of detection delay and false positive rate. X {\displaystyle K} X Number of points to overlap between segments. 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 . = { decreasing magnitude sampled at 10 kHz. If array-like, each element must be array-like and contain the indices U A K x p Download all examples in Python source code: auto_examples_python.zip. Likely the result of most interest to you will be the reconstructed components available in the .component_ attribute. window_size. . 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. The decomposition is meaningful if each reconstructed You signed in with another tab or window. } An example of the implementation of this code can be found in Singular Spectrum Analysis Example.ipynb. t MSSA (Multivariate Singular Spectrum Analysis) is a Python package for multivariate time-series analysis that provides a range of tools for decomposing and forecasting complex time-series data. The latter have of to the segment times. A sinusoid with frequency smaller than 0.5 produces two approximately equal eigenvalues and two sine-wave eigenvectors with the same frequencies and 1 L {\displaystyle M} Oppenheim, Alan V., Ronald W. Schafer, John R. Buck I i Parameters: xarray_like. is measured in V and fs is measured in Hz. {\displaystyle {\textbf {E}}_{k}} N U : An oscillatory mode is characterized by a pair of Add a description, image, and links to the Singular Spectrum Analysis. Better automated grouping of components (e.g. I will update the list of references/credits at another time. d eigenvectors = x Signals such as time series can be seen as a sum of different signals such 0 topic, visit your repo's landing page and select "manage topics.". forecasting); Missing/corrupted by noise (i.e. , This estimator supports two algorithms: a fast randomized SVD solver, and a naive algorithm that uses ARPACK as an eigensolver on (X * X.T) or (X.T * X), whichever is more efficient. The percent variance explained corresponding to these rank-ordered components can be found in the matrix component_ranks_explained_variance_. } 2 , This example illustrates the , 1 Target values (None for unsupervised transformations). Often M-SSA is applied to a few leading PCs of the spatial data, with ESPRIT for high-resolution frequency estimation. Email: damien.delforge@uclouvain.be Todo. { Singular spectrum analysis (SSA) and the maximum entropy method (MEM) have been combined to predict a variety of phenomena in meteorology, oceanography and climate dynamics (Ghil et al., 2002, and references therein). For each elementary matrix, reverse the hankelization back into a timeseries vector using a process called "diagonal averaging". of spatial channels much greater than the number The components you get out of MSSA are additive in the sense that they will add back up to the original series. x Options are M imputation) This repository is the implementation of . implicit QR procedure. 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". {\displaystyle L\!\times \!K} . One decomposition algorithm is Singular Spectrum Analysis. N , b k (Penland et al., 1991; Keppenne and Ghil, 1993). d + determines the resolution of the method: larger values of ) and by , This constructs multiple "noise" matrices of the same shape as your input matrix, and will select the number of components for your real data who's eigenvalues exceed a user-specified percentile of the noise eigenvalues (typically 95th percentile is used). Python implementation of Monte Carlo Singular Spectrum Analysis for univariate time series. {\displaystyle {\it {X(t)}}} {\displaystyle \lambda _{1}\geq \ldots \geq \lambda _{L}\geq 0} It is for this reason that the default window is a Tukey window with 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. Groth, A. and M. Ghil (2011): "Multivariate singular spectrum analysis and the road to phase synchronization". k 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. M of the grouped decomposition is hankelized and then the obtained Hankel matrix is transformed into a new series of length On symmetric (or Hermitian) problems, when the basis vectors The guide explains the following steps of an SSA analysis. k M (Eds.) d x ) algorithm is Singular Spectrum Analysis. Components that have correlations with many other components are often deemed noise components and discarded. X . } , numbers L L L If detrend is a and the grouped SVD expansion of I am not a code optimization expert, so any contributions that help with this are more than welcome! S The window length Prior to singular value decomposition, the MSSA algorithm goes through these steps: The idea of the trajectory matrix is to convert a timeseries vector into a matrix of incremental "windows" over the timeseries. equivalent to the output of stft with no padding or boundary Commun Stat Simul Comput 32, 319352. 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. The steps are almost similar to those of a singular spectrum analysis. Singular Spectrum Analysis for time series forecasting in Python. N . Each matrix ) i 1 i into L }. C {\displaystyle d=L} length of the window. Two trajectory matrices can be organized as either vertical (VMSSA) or horizontal (HMSSA) as was recently introduced in Hassani and Mahmoudvand (2013), and it was shown that these constructions lead to better forecasts. Each column represents a window of width 5 moving across the timeseries one at a time until the end of the timeseries is reached. Alternatively, a closed matrix formulation of the algorithm for the simultaneous rotation of the EOFs by iterative SVD decompositions has been proposed (Portes and Aguirre, 2016). This is a gold standard strategy for selecting number of components in PCA and has been adapted here to for the special trajectory matrix format of MSSA. This is a (rank, P) matrix, with rank being the number of components and P the number of timeseries. This methodology unies all these versions of SSA into a very powerful tool of time series analysis and forecasting. provide more refined decomposition into elementary components and therefore better separability. to obtain spectral information on the time series, assumed to be stationary in the weak sense. Caterpillar-SSA emphasizes the concept of separability, a concept that leads, for example, to specific recommendations concerning the choice of SSA parameters. ( SSA proceeds by diagonalizing the 1 I This provides the basis for SSA recurrent and vector forecasting algorithms (Golyandina et al., 2001, Ch.2). the SVD step in Basic SSA gives See help(type(self)) for accurate signature. decide what Ritz values to save during restarting. Python Singular Spectrum Analysis using various embedding methods and SVD algorithms. | 2014. {\displaystyle \mathbf {X} } < X N MSSA component matrix (or "reconstructions"), Left singular vectors and singular values, Explained variance scores and percent variance explained for components. Thomakos, D. (2010) "Median Unbiased Optimal Smoothing and Trend. {\displaystyle N'\times M} Length of the FFT used, if a zero padded FFT is desired. Finally, the extended RCs are used in the SSA reconstruction process to produce the forecast values. = = Download : Download high-res image (535KB) Download : Download full-size image; Fig. Recurrent forecasting function. Now we can instantiate the MSSA object and fit to the training data. SingularSpectrumAnalysis (window_size=4, groups=None) [source] . Groth, A. and M. Ghil (2015): "Monte Carlo Singular Spectrum Analysis (SSA) revisited: Detecting oscillator clusters in multivariate datasets". = for a typical real-life series) and Nekrutkin, V. (2010) "Perturbation expansions of signal subspaces for long signals". I've chosen to leave off 48 months, or 4 years of wine sales data, to serve as my holdout test set for validation. x The sum of all the components for a timeseries (without reducing number of components) will perfectly reconstruct the original timeseries. nonstationary signals frequency content over time. 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. of each series into the block trajectory matrix in the multivariate case. t Colebrook (1978) applied a form of . {\displaystyle \mathbf {X} _{I}} X = System of series can be forecasted analogously to SSA recurrent and vector algorithms (Golyandina and Stepanov, 2005). U k = (1997): de Carvalho, M., Rodrigues, P. C. and Rua, A. U 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. {\displaystyle M_{t}} The MSSA components. L k You signed in with another tab or window. > detrended segment. The matrix is shape (n_components, P), where P is the number of timeseries. Specifies how to detrend each segment. Output. have been demonstrated to be very effective. Continue exploring. Grouping by looking at the w-correlation can be more of an art than a science. 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 . : N x of the lag-covariance matrix - reconstruction of the time series. For a univariate time series, the SSA gap filling procedure utilizes temporal correlations to fill in the missing points. I L K 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. ; {\displaystyle {\textbf {C}}_{X}} We can use the list I made above to set the new groups for timeseries 0. Comments (20) Run. {\displaystyle M} the size of each word). SSA can be effectively used as a non-parametric method of time series monitoring and change detection. i in a vector space of dimension X You specify timepoints out and optionally which timeseries and with which components to forecast with. of temporal lags, thus limiting the temporal and spectral information. If you do not center the data then the first component is just going to end up being the offset that centers the reconstruction anyway. {\displaystyle k^{*}=S} They are fine to occur in the testing set, we will just have fewer values in some cases to calculate our performance metric. 1 {\displaystyle U_{1},\ldots ,U_{d}} You can see that the trajectory matrix has 5 rows (window length 5). L = {\displaystyle {\widetilde {\mathbb {X} }}^{(k)}=({\widetilde {x}}_{1}^{(k)},\ldots ,{\widetilde {x}}_{N}^{(k)})} {\displaystyle X_{i}=(x_{i},\ldots ,x_{i+L-1})^{\mathrm {T} }\;\quad (1\leq i\leq K)} TRLan and ARPACK are theoretically equivalent. Continuation of the series by the LRR is similar to forward linear prediction in signal processing. implementors -- it is of no consequence to the end users. {\displaystyle d=\mathop {\mathrm {rank} } \mathbf {X} =\max\{i,\ {\mbox{such that}}\ \lambda _{i}>0\}} visualizes the different subseries extracted. {\displaystyle \{X_{l}(t):l=1,\dots ,L;t=1,\dots ,N\}} : A tag already exists with the provided branch name. ( months in a year), use a multiple of this periodicity when selecting a window size. 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 \mathbf {X} } N {\displaystyle L>d} 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). 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. Colebrook, J. M., (1978): "Continuous plankton records: Zooplankton and environment, Northeast Atlantic and North Sea,". history Version 1 of 1. A comprehensive causality test based on MSSA has recently introduced for causality measurement. , {\displaystyle U_{t}} ~ axis=-1). Decomposing time series into several time series can M It combines elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. i 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. Set general Parameters M = 30; % window length of SSA N = 200; % length of generated time series T = 22; % period length of sine function stdnoise = 0.1; % noise-to-signal ratio such that . Keywords: singular spectrum analysis, time series, image processing, analysis, forecasting, decomposition, R package. However, Groth and Ghil (2015) have demonstrated possible negative effects of this variance compression on the detection rate of weak signals when the number {\displaystyle \mathbb {X} } If detrend is False, no detrending is Having specified the data vector to be analyzed (here `data', the SOI time-series) and the sampling interval, the principal SSA options to be specified are the Window . 1 ] m {\displaystyle \mathbf {S} } The application of SSA in this bivariate framework produces a smoothed series of the common root component. This method is thoroughly described in SSA as a model-free tool of this article. -trajectory matrix), which are provided by the SVD step of SSA. If n_split=1, X_new {\displaystyle \mathbf {X} _{I_{k}}} X Length of each segment. For example, timepoint 1 will only appear once in the trajectory matrix, while others in the middle of a timeseries can appear up to window-size L times. eigenvalue problems, TRLan usually performed better because of the new There is no reason not to just deal with it prior to the decomposition. j [1] Singular Spectrum Analysis for Time Series | Nina Golyandina | Springer. Defaults to None. k i c ( p 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. U {\displaystyle \{1,\ldots ,d\}} of These authors provided an extension and a more robust application of the idea of reconstructing dynamics from a single time series based on the embedding theorem. Author: Damien Delforge. {\displaystyle |i-j|} Let us choose ( 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 max possible to update each component of a nested object. Vectors 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. {\displaystyle N\rightarrow \infty } , The same goes with sparpack and skarpack, as skarpack is just a wrapper to sparpack with fewer arguments allowed. analysis In time series analysis, singular spectrum analysis (SSA) is a nonparametric spectral estimation method. = Defaults to V Lomb-Scargle periodogram for unevenly sampled data. 139.7s. D d U The entire time series or parts of it that correspond to trends, oscillatory modes or noise can be reconstructed by using linear combinations of the PCs and EOFs, which provide the reconstructed components (RCs) a {\displaystyle L} Halko, et al., 2009, A randomized algorithm for the decomposition of matrices If for some series Left upper panel shows an observed time series of a relevant adaptation parameter. , Summary functions and printouts with relevant statistics on fits/decomposition/forecasts. Hence different modifications of SSA have been proposed and different methodologies of SSA are used in practical applications such as trend extraction, periodicity detection, seasonal adjustment, smoothing, noise reduction (Golyandina, et al, 2001). The set_component_groups function alternatively takes a dictionary of timeseries index keys and values of lists of lists for group indices if you want to set multiple timeseries groupings at once (or just prefer to do it with a dictionary).