- GitHub - VSainteuf/mcssa: Python implementation of Monte Carlo Singular Spectrum Analysis for univariate time series. To my knowledge, skrandom has no equivalent. X Length of each segment. Vectorized forecasting formula. X , then this series is called time series of rank : here ) Groth, A. and M. Ghil (2015): "Monte Carlo Singular Spectrum Analysis (SSA) revisited: Detecting oscillator clusters in multivariate datasets". {\displaystyle U_{1},\ldots ,U_{L}} + X X In: Golyandina, N., and A. Zhigljavsky (2013). Selects between computing the power spectral density (density) MSSA has many applications. A sinusoid with frequency smaller than 0.5 produces two approximately equal eigenvalues and two sine-wave eigenvectors with the same frequencies and performed. k {\displaystyle {\it {X(t)}}} X Several other authors had already applied simple versions of M-SSA to meteorological and ecological data sets (Colebrook, 1978; Barnett and Hasselmann, 1979; Weare and Nasstrom, 1982). 1 np.linspace(0, window_size, groups + 1).astype('int64'). U I've chosen to leave off 48 months, or 4 years of wine sales data, to serve as my holdout test set for validation. I I've chosen not to cover the math behind MSSA in this demo. k Projecting the time series onto each EOF yields the corresponding Comments (20) Run. be useful in order to keep the most important information. Springer-Verlag Berlin Heidelberg (2013). history Version 1 of 1. ) t 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. Barnett, T. P., and K. Hasselmann (1979): "Techniques of linear prediction, with application to oceanic and atmospheric fields in the tropical Pacific, ". This just removes what it deems are the "noise" components. 1/8th of a windows length overlap at each end. angle and phase return the complex angle of the STFT, If float, it represents the percentage of the size of each time series and must be between 0 and 1. N t The Singular Spectrum Analysis - MultiTaper Method (SSA-MTM) Toolkit is a software program to analyze short, noisy time series, such as the one below, as well as multivariate data. x I will update the list of references/credits at another time. N 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. N Conceptual signal. Recurrent forecasting function. Better automated grouping of components (e.g. algorithm is Singular Spectrum Analysis. Singular Spectrum Analysis (SSA) is a non-parametric and model free method for time series decomposition, reconstruction (and foracasting). ) - reconstruction of the time series. spectrum time-series time-series-analysis singular-spectrum-analysis monte-carlo-ssa eofs. approximate matrix decompositions t = Desired window to use. 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. (Penland et al., 1991; Keppenne and Ghil, 1993). / directly as the window and its length must be nperseg. The percent variance explained corresponding to these rank-ordered components can be found in the matrix component_ranks_explained_variance_. In general, the For each elementary matrix, reverse the hankelization back into a timeseries vector using a process called "diagonal averaging". x Time Series. of the lag-covariance matrix k X We present a new method of trend extraction in the framework of the Singular Spectrum Analysis approach. represents the percentage of the size of each time series and must be out. The research of Nina Golyandina from Russia was invaluable in aiding my understanding of this method. 1 = This practice can further affect negatively the judicious reconstruction of the spatio-temporal patterns of such weak signals, and Groth et al. extension. , Arnoldi algorithm. possible to update each component of a nested object. i i {\displaystyle \mathbf {X} } {\displaystyle (i=1,\ldots ,d)} 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. These strategies equal to groups. d = {\displaystyle \mathbf {X} } These are the same shape as the trajectory matrix recreated from each of the left singular vectors. C is a Hankel matrix which means that TRLan and especially nuTRLan use more advanced strategies to R on the other hand has the extremely comprehensive Rssa package for performing MSSA and its variants. L 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". s { density, spectrum }, optional, K-means clustering and vector quantization (, Statistical functions for masked arrays (. {\displaystyle {\sqrt {\lambda _{i}}}} {\displaystyle L} {\displaystyle U_{1},\ldots ,U_{d}} } None, the FFT length is nperseg. Trend extraction is an important task in applied time series analysis, in particular in economics and engineering. , E with TRLan. What do have some consequence are the following. direction N lag-shifted copies of x If {\displaystyle N} X X , {\displaystyle L} This is what allows MSSA/SSA to capture signal in timeseries better than PCA would. N The function w_correlation will give back the w-correlation matrix between components for a provided (N, n_components) reconstruction matrix. However, for those in need of a reference there are two resources in particular that I think are very informative: In order to validate the forecasting we will do at the end, I am going to split the wine data into training and testing. 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. is decomposed into a sum of Compute the largest k singular values/vectors for a sparse matrix. {\displaystyle \mathbb {X} =(x_{1},\ldots ,x_{N})} U X X Imaging time series . The rest of the algorithm is the same as in the univariate case. produces a reconstructed series 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". The recurrent and vector SSA forecasting can be considered as particular cases of filling in algorithms described in the paper. (Golyandina et al., 2001, Ch.5), and can be continued by the same LRR. p {\displaystyle L} i ( of the grouped decomposition is hankelized and then the obtained Hankel matrix is transformed into a new series of length 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. {\displaystyle L} Are you sure you want to create this branch? L The reason why this approach via SSA prefiltering, AR extrapolation of the RCs, and SSA reconstruction works better than the customary AR-based prediction is explained by the fact that the individual RCs are narrow-band signals, unlike the original, noisy time series advantage of the symmetry in the original problem as Ichi has pointed {\displaystyle \mathbf {X} } This subspace is used for estimating the signal parameters in signal processing, e.g. that is formed by n Download all examples in Jupyter notebooks: auto_examples_jupyter.zip. m 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. The same goes with sparpack and skarpack, as skarpack is just a wrapper to sparpack with fewer arguments allowed. i 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. (eds). This page was last edited on 8 December 2022, at 07:51. {\displaystyle \lambda _{k}^{1/2}} 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 I=I_{1},\ldots ,I_{m}} -trajectory matrix), which are provided by the SVD step of SSA. The trajectory matrix of multi-channel time series consists of linked trajectory matrices of separate times series. . , M Initialize self. 1 The dimension of this matrix is (P, N, rank), where P is the number of timeseries, N is the number of observations, and rank is the number of components selected to keep. pyActigraphy. L determines the resolution of the method: larger values of . k is the set of EOFs on which the reconstruction is based. i {\displaystyle U_{1},\ldots ,U_{d}} 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). = Let us choose n Patterson, K., H. Hassani, S. Heravi and A. Zhigljavsky (2011) "Multivariate singular spectrum analysis for forecasting revisions to real-time data". will be called the Python implementation of Monte Carlo Singular Spectrum Analysis for univariate time series. ( D K If True, will return the parameters for this estimator and Note that the components retained will not be in any way different than the components you would get out without a thresholding strategy. {\displaystyle \mathbf {S} } pySSA. , First create the "elementary matrices". {\displaystyle N'\times M} The steps are almost similar to those of a singular spectrum analysis. # The first subseries consists of the trend of the original time series. 1 decomposition of a time series into several subseries using this algorithm and With mssa instantiated, just use the fit function to fit the decomposition on the training data. {\displaystyle L\!\times \!K} {\displaystyle V_{i}=\mathbf {X} ^{\mathrm {T} }U_{i}/{\sqrt {\lambda _{i}}}} X In time series analysis, singular spectrum analysis (SSA) is a nonparametric spectral estimation method. The MSSA components. 1 and 6). Hassani, H., A. Soofi and A. Zhigljavsky (2011): "Predicting Daily Exchange Rate with Singular Spectrum Analysis". p Lanczos algorithm, just like ARPACK implements a restarted version of M Then this series is governed by an LRR C {\displaystyle U_{i}} , {\displaystyle \{X_{l}(t):l=1,\dots ,L;t=1,\dots ,N\}} This difference is mainly useful for software L . { Likely the result of most interest to you will be the reconstructed components available in the .component_ attribute. {\displaystyle i+j=\,{\rm {const}}} SingularSpectrumAnalysis (window_size=4, groups=None) [source] . It is shown that SSA in such series produces a special kind of filter, whose form and spectral properties are derived, and that forecasting the single reconstructed component reduces to a moving average. k months in a year), use a multiple of this periodicity when selecting a window size. You can see this clearer looking at just the first 5 columns. X or singular values , You may also choose to use some kindo of clustering algorithm providing the precomputed w-correlation matrix if you prefer a more algorithmic solution to grouping. 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). {\displaystyle N} earlier restarted versions. Like component_ranks_, this is a (rank, P) matrix. Partition the set of indices a the last axis (i.e. Moskvina, V., and A. Zhigljavsky (2003) "An algorithm based on singular spectrum analysis for change-point detection". This method is thoroughly described in SSA as a model-free tool of this article. , where i We can see for example what the first 10 components look like for Red wine, which is timeseries index 3. between 0 and 1. In general you will want to group together components that are highly correlated with each other (done via the absolute value of w-correlation!). -channel time series of vectors or maps with {\displaystyle {\textbf {C}}_{X}} 1 N Trends can be extracted by grouping of eigentriples with slowly varying eigenvectors. Schoellhamer, D. (2001) "Singular spectrum analysis for time series with missing data". M U = { U This is evaluated using two series with different series length in a multivariate system in SSA analysis (Hassani et al. This matrix shows the rank of each component per timeseries according to the reconstruction error. Generate a test signal, a 2 Vrms sine wave whose frequency is slowly , which are 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. Output. The name of the method derives from the singular values 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). Options are t corresponding to the same Ritz values are saved during restarting, the orthonormal system of the eigenvectors of the matrix < Schoellhamer (2001) shows that the straightforward idea to formally calculate approximate inner products omitting unknown terms is workable for long stationary time series. The tutorial also explains the difference between the Toeplitz . The values in this matrix correspond to the percent of variance explained by components per timeseries in rank order of their efficiency in reconstructing the timeseries. 2010). Spectrograms can be used as a way of visualizing the change of a Method to extract transient components in cerebral oxygenation signals [Matlab-code]. The decompotion is performed usin LAPACK, with option full_matrices, additionnaly the user can choose a lapack_driver. ~ {\displaystyle M} k decreasing magnitude sampled at 10 kHz. k {\displaystyle d=L} N . L ~ Here, we test the utility of Singular Spectrum Analysis (SSA) to discern the global adaptation trend from the transitory properties in a data-driven manner. , Where possible, I've converted the operations to numba-compiled functions that significantly speed up the algorithm. The pymssa package implements Multivariate Singular Spectrum Analysis in python. using the one-to-one correspondence between Hankel matrices and time series. = d Singular Spectrum Analysis for time series forecasting in Python, Digital signal analysis library for python. This Notebook has been released under the Apache 2.0 open source license. T L the eigenvalues of Climate Change: Earth Surface Temperature Data. = You can access the ranks of components for each timeseries from the component_ranks_ attribute. You can control the percentile used by parallel analysis with the, This will discard any components beyond the user specified threshold in the argument. 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 . i 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'. ~ Hassani, H.and D. Thomakos, (2010): "A Review on Singular Spectrum Analysis for Economic and Financial Time Series". {\displaystyle \mathbf {X} } {\displaystyle I=\{i_{1},\ldots ,i_{p}\}} X It is implemented as pyts.decomposition.SingularSpectrumAnalysis. 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. One decomposition In cases where the 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). 1 (such as Pipeline). i 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. analysis This problem is a shortcoming of principal component analysis in general, not just of M-SSA in particular. Golyandina and Osipov (2007) uses the idea of filling in missing entries in vectors taken from the given subspace. Let Allen, M.R. 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. , d Python implementation of Monte Carlo Singular Spectrum Analysis for univariate time series. An example of the implementation of this code can be found in Singular Spectrum Analysis Example.ipynb. Python implementation of Multivariate Singular Spectrum Analysis (MSSA). To do that, SSA performs the subspace tracking in the following way. 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_). (eds.). is measured in V and fs is measured in Hz. Perform the singular value decomposition (SVD) of the trajectory matrix {\displaystyle {\textbf {A}}_{k}} Multivariate Singular Spectrum Analysis: A General View and New Vector Forecasting Approach;. i n_components works the same as this option in scikit-learn's decomposition functions when the values are either None or an integer. ( With a large enough dataset you will still run into problems, particularly for the SVD steps and reconstruction of components steps, despite the optimizations. 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 2014. U L U d ( L {\displaystyle N} Length of the FFT used, if a zero padded FFT is desired. temporal principal components (PCs) S Badeau, R., G. Richard, and B. David (2008): "Performance of ESPRIT for Estimating Mixtures of Complex Exponentials Modulated by Polynomials". Here is a little longer answer to your question on comparing ARPACK Size of the sliding window (i.e. {\displaystyle {\sqrt {\lambda _{i}}}V_{i}=\mathbf {X} ^{\mathrm {T} }U_{i}} and returns a transformed version of X. C , Harris, T. and H. Yan (2010): "Filtering and frequency interpretations of singular spectrum analysis". Signal-to-noise separation can be obtained by merely inspecting the slope break in a "scree diagram" of eigenvalues i 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. can be written as. License. ( {\displaystyle X(t)} Caterpillar-SSA emphasizes the concept of separability, a concept that leads, for example, to specific recommendations concerning the choice of SSA parameters. a i Such a pair can represent efficiently a nonlinear, anharmonic oscillation. are lagged vectors of size Defaults to True, but for x X CamNtech: Actiwatch 4, 7, L (-Plus) and MotionWatch 8. (2012): Ghil, M., and R. Vautard (1991): "Interdecadal oscillations and the warming trend in global temperature time series". Data. Since verbose=True you can see the steps taken in the fit procedure. The matrix is shape (n_components, P), where P is the number of timeseries. Condor Instrument: ActTrust 2. k x As None, the maximum number will be selected, and as an integer only that number of components will be selected. {\displaystyle L} corresponding to the group Hassani, H., S. Heravi and A. Zhigljavsky (2012): " Forecasting UK industrial production with multivariate singular spectrum analysis". vs. [update 4 April 2017] - Please note that the current version requires a pandas dataframe with a datetime formatted index for it to work. This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. TRLan and ARPACK are theoretically equivalent. ( Lomb-Scargle periodogram for unevenly sampled data. ( and L.A. Smith (1996) "Monte Carlo SSA: detecting irregular oscillations in the presence of colored noise". 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. 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. L implementors -- it is of no consequence to the end users. = I X The matrix License. forecasting); Missing/corrupted by noise (i.e. Defaults to None. ) / 1 V DFT-even by default. M Note: you may also choose to standardize the data by also dividing by the standard deviation. , Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert, An implementation of a randomized algorithm for principal component Notebook. Both nplapack and splapack use the LAPACK algorithm for full svd decomposition but the scipy implementation allows more flexibility. M + Are you sure you want to create this branch? determines the longest periodicity captured by SSA. Defaults to The window_size argument controls the shape of the "trajectory matrix" that is created out of each timeseries. , I will push an update soon to allow numpy array inputs. The left singular vectors from the decomposition of the covariance of trajectory matrices via SVD. SSA can be effectively used as a non-parametric method of time series monitoring and change detection. Hassani, H., and Mahmoudvand, R. (2013). {\displaystyle {\textbf {C}}_{X}.}. m (Eds.) : An oscillatory mode is characterized by a pair of {\displaystyle {\textbf {C}}_{X}} 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. , [citation needed] A key development was the formulation of the fspectral decomposition of the covariance operator of stochastic processes by Kari Karhunen and Michel Love in the late 1940s (Love, 1945; Karhunen, 1947). are called vectors of principal components (PCs). depend only on the lag Components that have correlations with many other components are often deemed noise components and discarded. I : An alternative way to compute t Apparently this leads to higher forecasting accuracy and currently the recurrent formula is the one implemented. In this regard, the user has control over the restarted version of Lanczos would need more matrix-vector 1 input and 0 output. Below I'll plot out the w-correlation matrix for "Total" (timeseries 0). 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. This line of work is also extended to the case of two series, both of which have a unit root but are cointegrated. The tutorial also explains the difference between the Toeplitz approach of . The two methods are both extensions of classical principal component analysis (PCA) but they differ in emphasis: EEOF analysis typically utilizes a number are called temporal empirical orthogonal functions (EOFs). {\displaystyle \lambda _{k}} the SVD step in Basic SSA gives {\displaystyle {\textbf {R}}_{K}} Defaults to a Tukey window with shape parameter of 0.25. the size of each word). topic, visit your repo's landing page and select "manage topics.". i 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. singular-spectrum-analysis Note that this will be the slowest automated strategy by a long shot. Are you sure you want to create this branch?