: ( Consider a real-valued time series axis=-1). Such a pair can represent efficiently a nonlinear, anharmonic oscillation. Thomakos, D. (2010) "Median Unbiased Optimal Smoothing and Trend. Vectors d Spectrogram of x. k direction ; and (ii) cross-validation is used to optimize the window width Defaults to 1.0. windowstr or tuple or array_like, optional. 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. These are the same shape as the trajectory matrix recreated from each of the left singular vectors. On Multivariate Singular Spectrum Analysis and its Variants. Power spectral density by Welchs method. t Left upper panel shows an observed time series of a relevant adaptation parameter. 1 Singular Spectrum Analysis methods implementation in Python, Extracting social-economic signals from internet traffic data. To do that, SSA performs the subspace tracking in the following way. singular-spectrum-analysis GitHub Topics GitHub Trend is usually dened as a smooth additive component containing infor-mation about time series global change. 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. , then this series is called time series of rank Comments (0) Run. Set U I 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 \lambda _{k}} Defaults to } Compute S = X X.T, where X is the vertically-stacked trajectory matrices. V of the grouped decomposition is hankelized and then the obtained Hankel matrix is transformed into a new series of length {\displaystyle \lambda _{k}^{1/2}} ( Initialize self. {\displaystyle \mathbb {X} } X , trend or seasonality), Finding structure with randomness: Stochastic algorithms for constructing Also, this subspace determines the linear homogeneous recurrence relation (LRR) governing the series, which can be used for forecasting. {\displaystyle {\textbf {C}}_{X}} {\displaystyle N} PDF Singular Spectrum Analysis for time series: Introduction to this i X 2nd step: Singular Value Decomposition (SVD). | The matrix is shape (n_components, P), where P is the number of timeseries. , { Let N = First create the "elementary matrices". MSSA component matrix (or "reconstructions"), Left singular vectors and singular values, Explained variance scores and percent variance explained for components. These ranks are calculated by ordering, for each timeseries, which components contribute the most variance explained. L - timepoints_out the number of timepoints past training we would like to forecast out. k pyActigraphy. Unsupervised learning for time series data: Singular spectrum versus M Singular Spectrum Analysis (SSA) is a non-parametric and model free method for time series decomposition, reconstruction (and foracasting). Likely the result of most interest to you will be the reconstructed components available in the .component_ attribute. Multi-channel SSA (or M-SSA) is a natural extension of SSA to an Two formulations of Singular Spectrum Analysis with examples. I Let the series be governed by the minimal LRR , X x , np.linspace(0, window_size, groups + 1).astype('int64'). + corresponding to the group In this way, SSA could be used for change detection not only in trends but also in the variability of the series, in the mechanism that determines dependence between different series and even in the noise structure. The origins of SSA and, more generally, of subspace-based methods for signal processing, go back to the eighteenth century (Prony's method). Time Series. The main hypothesis behind SSA is separability of the components. {\displaystyle (i=1,\ldots ,d)} density. {\displaystyle {\textbf {C}}_{X}} 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. k }. = {\displaystyle x_{n}=\sum _{k=1}^{L-1}a_{k}x_{n-k}} In this course you learn to perform motif analysis . X You specify timepoints out and optionally which timeseries and with which components to forecast with. j M If None, ) length of the window. in a vector space of dimension X k singular-spectrum-analysis The MSSA components. of retained PCs becomes too small. , of Robertson (1996): "Distinguishing modulated oscillations from coloured noise in multivariate datasets". They are: The only option that I will not be covering in the demo is varimax, which is designed to perform a structured varimax on the left singular values after decomposition to "sparsify" the components. . 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 . k 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 . 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. Lomb-Scargle periodogram for unevenly sampled data. Singular Spectrum Analysis - Beginners guide - File Exchange - MATLAB With a large enough dataset you will still run into problems, particularly for the SVD steps and reconstruction of components steps, despite the optimizations. ; this gives the name to SSA. X Zhigljavsky, A. There are a handful of instantiation arguments that we will cover incrementally over the course of this demo. Fraedrich, K. (1986) "Estimating dimensions of weather and climate attractors". If your data has some known periodic structure (e.g. 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. Many theoretical results can be found in Golyandina et al. The matrix performed. X 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. with and without unwrapping, respectively. It tries to overcome the problems of finite sample length and noisiness of sampled time series not by fitting an assumed model to the available series, but by using a data-adaptive basis set, instead of the fixed sine and cosine of the BT method. = n ( Automated options for selection of number of components: Calculation of the w-correlation matrix of component correlations. Zhigljavsky., K. Patterson and A. Soofi (2011): " A comprehensive causality test based on the singular spectrum analysis". Each matrix U SSA is a powerful tool for decomposition, reconstruction, and forecasting of climatic time series (Ghil et al., 2002 ; Plaut et al., 1995 ; Yiou et . Desired window to use. history Version 1 of 1. Note that this will be the slowest automated strategy by a long shot. In fact, the optimal order p obtained for the individual RCs is considerably lower than the one given by the standard Akaike information criterion (AIC) or similar ones. , Singular Spectrum Analysis. k Signals such as time series can be seen as a sum of different signals such Generate a test signal, a 2 Vrms sine wave whose frequency is slowly k As mentioned above, this is a matrix with dimensions (P, N, components), where P is the number of input timeseries columns, N the number of observations, and rank the number of components output. i Apparently this leads to higher forecasting accuracy and currently the recurrent formula is the one implemented. d are matrices having rank 1; these are called elementary matrices. ( We present a new method of trend extraction in the framework of the Singular Spectrum Analysis approach. maintain some statistical independence between individual segments. Other multivariate extension is 2D-SSA that can be applied to two-dimensional data like digital images (Golyandina and Usevich, 2010). Golyandina, N., A. Korobeynikov and A. Zhigljavsky (2018): Golyandina, N., V. Nekrutkin and A. Zhigljavsky (2001): Golyandina, N., and E. Osipov (2007) "The Caterpillar-SSA method for analysis of time series with missing values". {\displaystyle L} = . , {\displaystyle x_{ij}} {\displaystyle L} t = Updated on Aug 31, 2022. C { density, spectrum }, optional, K-means clustering and vector quantization (, Statistical functions for masked arrays (. {\displaystyle N'\times M} L (note that In: Golyandina, N., and A. Zhigljavsky (2013). { . Portes, L. L. and Aguirre, L. A. of the underlying deterministic dynamics (Vautard and Ghil, 1989). } Broomhead, D.S., and G.P. The percent variance explained corresponding to these rank-ordered components can be found in the matrix component_ranks_explained_variance_. U a Download : Download high-res image (535KB) Download : Download full-size image; Fig. M [ If detrend is a Select the number of components using the "Singular Value Hard Thresholding" formula. Singular spectrum analysis - Wikipedia done. If an integer, it represents the number of groups and the The general walktrhough of SSA consists in (1) embedding the time series into a trajectory matrix of lagged vectors, (2) decomposing the trajectory matrix using singular value decomposition (SVD), (3) grouping the resulting components based on similarities between their singular values or eigenvectors to reconstruct interpretable components of the original time series. Below I'll plot out the w-correlation matrix for "Total" (timeseries 0). For this small dataset the steps are nearly instantaneous. It is for this reason that the default window is a Tukey window with X A tag already exists with the provided branch name. 0 Grouping by looking at the w-correlation can be more of an art than a science. - reconstruction of the time series. This makes TRLan a little easier to understand 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. Total running time of the script: ( 0 minutes 2.563 seconds), A Python Package for Time Series Classification, 2017-2021, Johann Faouzi and all pyts contributors. - reconstruction of the time series. Broomhead, D.S., and G. P. King (1986b): "On the qualitative analysis of experimental dynamical systems". The prefiltered RCs are then extrapolated by least-square fitting to an autoregressive model bounds of the groups are computed as 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. //www.springer.com/gp/book/9783642349126. N (2012): Ghil, M., and R. Vautard (1991): "Interdecadal oscillations and the warming trend in global temperature time series". {\displaystyle AR[p]} {\displaystyle \{X_{l}(t):l=1,\dots ,L;t=1,\dots ,N\}} { Groth, A., Feliks, Y., Kondrashov, D., and Ghil, M. (2016): "Interannual variability in the North Atlantic ocean's temperature field and its association with the wind stress forcing". = ) I will push an update soon to allow numpy array inputs. (such as Pipeline). This calculates the ranking of how well each component explains the variance of the original timeseries (more on this later). Performance of Singular Spectrum Analysis in Separating Seasonal and have been demonstrated to be very effective. The columns are types of wine, including: You can see the header of this dataset below. If groups is array-like, n_splits N x {\displaystyle L>d} - use_components is an array of component indices you would like to use to do the forecasting for. I Singular Spectrum Analysis pyts 0.12.0 documentation - Read the Docs 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. To demonstrate the features of the MSSA class, and provide a general walkthrough of the steps involved in a standard multivariate singular spectrum analysis, I will load an example dataset that comes packaged with the Rssa R package. and 'eigen' as full SVD via eigendecompsition of the cross-product matrix, see: https://code.lbl.gov/pipermail/trlan-users/2009-May/000007.html. E n_components works the same as this option in scikit-learn's decomposition functions when the values are either None or an integer. {\displaystyle \lambda _{k}^{1/2}} , The guide explains the following steps of an SSA analysis. {\displaystyle I=I_{1},\ldots ,I_{m}} If float, it matrix. A Python Package for Time Series Classification, 2017-2021, Johann Faouzi and all pyts contributors. 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. Partition the set of indices X { Are you sure you want to create this branch? R on the other hand has the extremely comprehensive Rssa package for performing MSSA and its variants. tuple, is set to 256, and if window is array_like, is set to the SSA can be an aid in the decomposition of time series into a sum of components, each having a meaningful interpretation. One decomposition If groups is an integer, n_splits is MotionSense Dataset : Smartphone Sensor Data - HAR. Since the price series and the information contained in it are available to all market participants, no one can benefit by attempting to take advantage of the information contained in the price history of an asset by trading in the markets. The basic aim of SSA is to decompose the time series into the sum of interpretable components such as trend, periodic components and noise with no a-priori assumptions about the parametric form of these components. window_size. k . None, the FFT length is nperseg. Multi-channel, Multivariate SSA (or M-SSA) is a natural extension of SSA to for analyzing multivariate time series, where the size of different univariate series does not have to be the same. p produces a reconstructed series {\displaystyle \{1,\ldots ,d\}} Are kept: splapack,sparpack and skrandom. Introducing SSA for Time Series Decomposition | Kaggle This can be useful if you don't have an idea how many components should be kept, but dont want to compute the entire matrices of components for each timeseries due to time or space constraints. {\displaystyle \mathbf {X} } {\displaystyle {\textbf {C}}_{X}} Notebook. I SSA perturbation theory is developed in Nekrutkin (2010) and Hassani et al. {\displaystyle L} For example, the Singular Spectrum Analysis (SSA) is a comprehensive methodology originally established by Broomhead and King and Fraedrich and later developed by Ghil et al. Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert, An implementation of a randomized algorithm for principal component This Matlab tutorial demonstrates step by step the multivariate singular spectrum analysis. Output. X L Golyandina, N., A. Pepelyshev and A. Steland (2012): "New approaches to nonparametric density estimation and selection of smoothing parameters". : An alternative way to compute ( 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) Prediction of Financial Time Series Based on LSTM Using Wavelet 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. k {\displaystyle M} 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). Lanczos algorithm, just like ARPACK implements a restarted version of Singular Spectrum Analysis for time series forecasting in Python. Specifically, mSSA allows you to predict entries that are: At a future time step (i.e. U M-SSA has two forecasting approaches known as recurrent and vector. This example illustrates the decomposition of a time series into several subseries using this algorithm and visualizes the different subseries extracted. L The identification and detailed description of these orbits can provide highly useful pointers to the underlying nonlinear dynamics. has equal elements The function w_correlation will give back the w-correlation matrix between components for a provided (N, n_components) reconstruction matrix. X . Defaults to constant. SSA's applicability to any kind of stationary or deterministically trending series has been extended to the case of a series with a stochastic trend, also known as a series with a unit root. determines the longest periodicity captured by SSA. L {\displaystyle L=N} 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. Lastly we can also perform forecasts using the forecast function. If It combines elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. # The second and third subseries consist of noise. T {\displaystyle {\textbf {X}}} I n Signal-to-noise separation can be obtained by merely inspecting the slope break in a "scree diagram" of eigenvalues 1 {\displaystyle {\sqrt {\lambda _{i}}}V_{i}=\mathbf {X} ^{\mathrm {T} }U_{i}} i r Trend (which is defined as a slowly varying component of the time series), periodic components and noise are asymptotically separable as = class pyts.decomposition. Time series analysis is performed on the Berkeley Earth Surface Temperature dataset. . To demonstrate the features of the MSSA class, and provide a general walkthrough of the steps involved in a standard multivariate singular spectrum analysis, I will load an example dataset that comes packaged with the Rssa R package.. k + {\displaystyle I=\{i_{1},\ldots ,i_{p}\}} This course focuses on data exploration, feature creation, and feature selection for time sequences. Size of the sliding window (i.e. Easy interface to assign and retrieve component groupings. Powered by, array-like, shape = (n_samples, n_timestamps), None or array-like, shape = (n_samples,) (default = None), array-like, shape = (n_samples, n_splits, n_timestamps), pyts.decomposition.SingularSpectrumAnalysis. 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. {\displaystyle d} Compute a spectrogram with consecutive Fourier transforms. {\displaystyle V_{i}=\mathbf {X} ^{\mathrm {T} }U_{i}/{\sqrt {\lambda _{i}}}} Note that as your data gets large, the maximum window size can be untenable due to the size of the matrix that must be decomposed by SVD, so take this approach with caution. Mohammad, Y., and T. Nishida (2011) "On comparing SSA-based change point discovery algorithms". There are also some other options for this argument to automate the selection of components that I will cover later in the demo. TRLan (and nuTRLan) implements a restarted version of In contrast to welchs method, where the A number of indicators of approximate separability can be used, see Golyandina et al. | Defaults to True, but for A comprehensive causality test based on MSSA has recently introduced for causality measurement. ) Decomposing time series into several time series can analysis {\displaystyle {\textbf {E}}_{k}} chosen large enough to extract detailed temporal and spectral information from the multivariate time series (Ghil et al., 2002). Imagine we have a timeseries vector of increasing numbers: To convert ts to the trajectory matrix, we specify a window size and then take the transposition of the hankel matrix of this timeseries. x X x nearly equal SSA eigenvalues and associated PCs that are in approximate phase quadrature (Ghil et al., 2002). 1 T Vautard, R., and M. Ghil (1989): "Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series". a DFT-even by default. The point As a result, at present we do not only have many dierent versions of SSA but also a general method-ology called 'singular spectrum analysis'. (2001, Ch. decomposition of a time series into several subseries using this algorithm and long; then, The = 2 For example, if component_ranks_[0, 0] = 3, this would mean that the 3rd component accounts for the most variance for the first timeseries. Vectors 2 {\displaystyle L} If window is a string or tuple, it is Caterpillar-SSA emphasizes the concept of separability, a concept that leads, for example, to specific recommendations concerning the choice of SSA parameters. earlier restarted versions. as trends and noise. , You signed in with another tab or window. Moskvina, V., and A. Zhigljavsky (2003) "An algorithm based on singular spectrum analysis for change-point detection". Python Singular Spectrum Analysis using various embedding methods and SVD algorithms. Add a description, image, and links to the See the Contributing section for more ideas if you're interested in adding to this package. {\displaystyle \mathbf {X} _{I}=\mathbf {X} _{i_{1}}+\ldots +\mathbf {X} _{i_{p}}} {\displaystyle X(t)} the orthonormal system of the eigenvectors of the matrix This Matlab tutorial demonstrates step by step the Singular Spectrum Analysis (SSA). [psd, complex, magnitude, angle, phase]. Let us choose 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. Multivariate Singular Spectrum Analysis (mSSA) Multivariate Singular Spectrum (mSSA) is an algorithm for multivariate time series forecasting and imputation. ( Digital signal analysis library for python. A. Szlam et al. X extension. 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). If we now access timeseries index 0 of grouped_components_, you can see that the matrix has 11 components which are created by adding together the components we specified in our groups. You signed in with another tab or window. U n be some integer called the window length and - GitHub - VSainteuf/mcssa: Python implementation of Monte Carlo Singular Spectrum Analysis for univariate time series. The steps are almost similar to those of a singular spectrum analysis. In this case the fit function carries out these steps: Once the decomposition completes, you now have access to a variety of attributes on the object. T when restarting -- this is the key advantage of these methods over The window_size argument controls the shape of the "trajectory matrix" that is created out of each timeseries.