Purpose of Time Series Analysis Autocovariance Function
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Kalman and Particle Filtering
Abstract: The Kalman and Particle filters are algorithms that recursively update an estimate of the state and find the innovations driving a stochastic process given a sequence of observations. The Kalman filter accomplishes this goal by linear projections, while the Particle filter does so by a sequential Monte Carlo method. With the state estimates, we can forecast and smooth the stochastic process. With the innovations, we can estimate the parameters of the model. The article discusses how to set a dynamic model in a state-space form, derives the Kalman and Particle filters, and explains how to use them for estimation. Kalman and Particle Filtering The Kalman and Particle filters are algorithms that recursively update an estimate of the state and find the innovations driving a stochastic process given a sequence of observations. The Kalman filter accomplishes this goal by linear projections, while the Particle filter does so by a sequential Monte Carlo method. Since both filters start with a state-space representation of the stochastic processes of interest, section 1 presents the state-space form of a dynamic model. Then, section 2 intro- duces the Kalman filter and section 3 develops the Particle filter. For extended expositions of this material, see Doucet, de Freitas, and Gordon (2001), Durbin and Koopman (2001), and Ljungqvist and Sargent (2004). 1. The state-space representation of a dynamic model A large class of dynamic models can be represented by a state-space form: Xt+1 = ϕ (Xt,Wt+1; γ) (1) Yt = g (Xt,Vt; γ) . (2) This representation handles a stochastic process by finding three objects: a vector that l describes the position of the system (a state, Xt X R ) and two functions, one mapping ∈ ⊂ 1 the state today into the state tomorrow (the transition equation, (1)) and one mapping the state into observables, Yt (the measurement equation, (2)). -
Lecture 19: Wavelet Compression of Time Series and Images
Lecture 19: Wavelet compression of time series and images c Christopher S. Bretherton Winter 2014 Ref: Matlab Wavelet Toolbox help. 19.1 Wavelet compression of a time series The last section of wavelet leleccum notoolbox.m demonstrates the use of wavelet compression on a time series. The idea is to keep the wavelet coefficients of largest amplitude and zero out the small ones. 19.2 Wavelet analysis/compression of an image Wavelet analysis is easily extended to two-dimensional images or datasets (data matrices), by first doing a wavelet transform of each column of the matrix, then transforming each row of the result (see wavelet image). The wavelet coeffi- cient matrix has the highest level (largest-scale) averages in the first rows/columns, then successively smaller detail scales further down the rows/columns. The ex- ample also shows fine results with 50-fold data compression. 19.3 Continuous Wavelet Transform (CWT) Given a continuous signal u(t) and an analyzing wavelet (x), the CWT has the form Z 1 s − t W (λ, t) = λ−1=2 ( )u(s)ds (19.3.1) −∞ λ Here λ, the scale, is a continuous variable. We insist that have mean zero and that its square integrates to 1. The continuous Haar wavelet is defined: 8 < 1 0 < t < 1=2 (t) = −1 1=2 < t < 1 (19.3.2) : 0 otherwise W (λ, t) is proportional to the difference of running means of u over successive intervals of length λ/2. 1 Amath 482/582 Lecture 19 Bretherton - Winter 2014 2 In practice, for a discrete time series, the integral is evaluated as a Riemann sum using the Matlab wavelet toolbox function cwt. -
Autocovariance Function Estimation Via Penalized Regression
Autocovariance Function Estimation via Penalized Regression Lina Liao, Cheolwoo Park Department of Statistics, University of Georgia, Athens, GA 30602, USA Jan Hannig Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA Kee-Hoon Kang Department of Statistics, Hankuk University of Foreign Studies, Yongin, 449-791, Korea Abstract The work revisits the autocovariance function estimation, a fundamental problem in statistical inference for time series. We convert the function estimation problem into constrained penalized regression with a generalized penalty that provides us with flexible and accurate estimation, and study the asymptotic properties of the proposed estimator. In case of a nonzero mean time series, we apply a penalized regression technique to a differenced time series, which does not require a separate detrending procedure. In penalized regression, selection of tuning parameters is critical and we propose four different data-driven criteria to determine them. A simulation study shows effectiveness of the tuning parameter selection and that the proposed approach is superior to three existing methods. We also briefly discuss the extension of the proposed approach to interval-valued time series. Some key words: Autocovariance function; Differenced time series; Regularization; Time series; Tuning parameter selection. 1 1 Introduction Let fYt; t 2 T g be a stochastic process such that V ar(Yt) < 1, for all t 2 T . The autoco- variance function of fYtg is given as γ(s; t) = Cov(Ys;Yt) for all s; t 2 T . In this work, we consider a regularly sampled time series f(i; Yi); i = 1; ··· ; ng. Its model can be written as Yi = g(i) + i; i = 1; : : : ; n; (1.1) where g is a smooth deterministic function and the error is assumed to be a weakly stationary 2 process with E(i) = 0, V ar(i) = σ and Cov(i; j) = γ(ji − jj) for all i; j = 1; : : : ; n: The estimation of the autocovariance function γ (or autocorrelation function) is crucial to determine the degree of serial correlation in a time series. -
LECTURES 2 - 3 : Stochastic Processes, Autocorrelation Function
LECTURES 2 - 3 : Stochastic Processes, Autocorrelation function. Stationarity. Important points of Lecture 1: A time series fXtg is a series of observations taken sequentially over time: xt is an observation recorded at a specific time t. Characteristics of times series data: observations are dependent, become available at equally spaced time points and are time-ordered. This is a discrete time series. The purposes of time series analysis are to model and to predict or forecast future values of a series based on the history of that series. 2.2 Some descriptive techniques. (Based on [BD] x1.3 and x1.4) ......................................................................................... Take a step backwards: how do we describe a r.v. or a random vector? ² for a r.v. X: 2 d.f. FX (x) := P (X · x), mean ¹ = EX and variance σ = V ar(X). ² for a r.vector (X1;X2): joint d.f. FX1;X2 (x1; x2) := P (X1 · x1;X2 · x2), marginal d.f.FX1 (x1) := P (X1 · x1) ´ FX1;X2 (x1; 1) 2 2 mean vector (¹1; ¹2) = (EX1; EX2), variances σ1 = V ar(X1); σ2 = V ar(X2), and covariance Cov(X1;X2) = E(X1 ¡ ¹1)(X2 ¡ ¹2) ´ E(X1X2) ¡ ¹1¹2. Often we use correlation = normalized covariance: Cor(X1;X2) = Cov(X1;X2)=fσ1σ2g ......................................................................................... To describe a process X1;X2;::: we define (i) Def. Distribution function: (fi-di) d.f. Ft1:::tn (x1; : : : ; xn) = P (Xt1 · x1;:::;Xtn · xn); i.e. this is the joint d.f. for the vector (Xt1 ;:::;Xtn ). (ii) First- and Second-order moments. ² Mean: ¹X (t) = EXt 2 2 2 2 ² Variance: σX (t) = E(Xt ¡ ¹X (t)) ´ EXt ¡ ¹X (t) 1 ² Autocovariance function: γX (t; s) = Cov(Xt;Xs) = E[(Xt ¡ ¹X (t))(Xs ¡ ¹X (s))] ´ E(XtXs) ¡ ¹X (t)¹X (s) (Note: this is an infinite matrix). -
Alternative Tests for Time Series Dependence Based on Autocorrelation Coefficients
Alternative Tests for Time Series Dependence Based on Autocorrelation Coefficients Richard M. Levich and Rosario C. Rizzo * Current Draft: December 1998 Abstract: When autocorrelation is small, existing statistical techniques may not be powerful enough to reject the hypothesis that a series is free of autocorrelation. We propose two new and simple statistical tests (RHO and PHI) based on the unweighted sum of autocorrelation and partial autocorrelation coefficients. We analyze a set of simulated data to show the higher power of RHO and PHI in comparison to conventional tests for autocorrelation, especially in the presence of small but persistent autocorrelation. We show an application of our tests to data on currency futures to demonstrate their practical use. Finally, we indicate how our methodology could be used for a new class of time series models (the Generalized Autoregressive, or GAR models) that take into account the presence of small but persistent autocorrelation. _______________________________________________________________ An earlier version of this paper was presented at the Symposium on Global Integration and Competition, sponsored by the Center for Japan-U.S. Business and Economic Studies, Stern School of Business, New York University, March 27-28, 1997. We thank Paul Samuelson and the other participants at that conference for useful comments. * Stern School of Business, New York University and Research Department, Bank of Italy, respectively. 1 1. Introduction Economic time series are often characterized by positive -
Time Series: Co-Integration
Time Series: Co-integration Series: Economic Forecasting; Time Series: General; Watson M W 1994 Vector autoregressions and co-integration. Time Series: Nonstationary Distributions and Unit In: Engle R F, McFadden D L (eds.) Handbook of Econo- Roots; Time Series: Seasonal Adjustment metrics Vol. IV. Elsevier, The Netherlands N. H. Chan Bibliography Banerjee A, Dolado J J, Galbraith J W, Hendry D F 1993 Co- Integration, Error Correction, and the Econometric Analysis of Non-stationary Data. Oxford University Press, Oxford, UK Time Series: Cycles Box G E P, Tiao G C 1977 A canonical analysis of multiple time series. Biometrika 64: 355–65 Time series data in economics and other fields of social Chan N H, Tsay R S 1996 On the use of canonical correlation science often exhibit cyclical behavior. For example, analysis in testing common trends. In: Lee J C, Johnson W O, aggregate retail sales are high in November and Zellner A (eds.) Modelling and Prediction: Honoring December and follow a seasonal cycle; voter regis- S. Geisser. Springer-Verlag, New York, pp. 364–77 trations are high before each presidential election and Chan N H, Wei C Z 1988 Limiting distributions of least squares follow an election cycle; and aggregate macro- estimates of unstable autoregressive processes. Annals of Statistics 16: 367–401 economic activity falls into recession every six to eight Engle R F, Granger C W J 1987 Cointegration and error years and follows a business cycle. In spite of this correction: Representation, estimation, and testing. Econo- cyclicality, these series are not perfectly predictable, metrica 55: 251–76 and the cycles are not strictly periodic. -
Generating Time Series with Diverse and Controllable Characteristics
GRATIS: GeneRAting TIme Series with diverse and controllable characteristics Yanfei Kang,∗ Rob J Hyndman,† and Feng Li‡ Abstract The explosion of time series data in recent years has brought a flourish of new time series analysis methods, for forecasting, clustering, classification and other tasks. The evaluation of these new methods requires either collecting or simulating a diverse set of time series benchmarking data to enable reliable comparisons against alternative approaches. We pro- pose GeneRAting TIme Series with diverse and controllable characteristics, named GRATIS, with the use of mixture autoregressive (MAR) models. We simulate sets of time series using MAR models and investigate the diversity and coverage of the generated time series in a time series feature space. By tuning the parameters of the MAR models, GRATIS is also able to efficiently generate new time series with controllable features. In general, as a costless surrogate to the traditional data collection approach, GRATIS can be used as an evaluation tool for tasks such as time series forecasting and classification. We illustrate the usefulness of our time series generation process through a time series forecasting application. Keywords: Time series features; Time series generation; Mixture autoregressive models; Time series forecasting; Simulation. 1 Introduction With the widespread collection of time series data via scanners, monitors and other automated data collection devices, there has been an explosion of time series analysis methods developed in the past decade or two. Paradoxically, the large datasets are often also relatively homogeneous in the industry domain, which limits their use for evaluation of general time series analysis methods (Keogh and Kasetty, 2003; Mu˜nozet al., 2018; Kang et al., 2017). -
Monte Carlo Smoothing for Nonlinear Time Series
Monte Carlo Smoothing for Nonlinear Time Series Simon J. GODSILL, Arnaud DOUCET, and Mike WEST We develop methods for performing smoothing computations in general state-space models. The methods rely on a particle representation of the filtering distributions, and their evolution through time using sequential importance sampling and resampling ideas. In particular, novel techniques are presented for generation of sample realizations of historical state sequences. This is carried out in a forward-filtering backward-smoothing procedure that can be viewed as the nonlinear, non-Gaussian counterpart of standard Kalman filter-based simulation smoothers in the linear Gaussian case. Convergence in the mean squared error sense of the smoothed trajectories is proved, showing the validity of our proposed method. The methods are tested in a substantial application for the processing of speech signals represented by a time-varying autoregression and parameterized in terms of time-varying partial correlation coefficients, comparing the results of our algorithm with those from a simple smoother based on the filtered trajectories. KEY WORDS: Bayesian inference; Non-Gaussian time series; Nonlinear time series; Particle filter; Sequential Monte Carlo; State- space model. 1. INTRODUCTION and In this article we develop Monte Carlo methods for smooth- g(yt+1|xt+1)p(xt+1|y1:t ) p(xt+1|y1:t+1) = . ing in general state-space models. To fix notation, consider p(yt+1|y1:t ) the standard Markovian state-space model (West and Harrison 1997) Similarly, smoothing can be performed recursively backward in time using the smoothing formula xt+1 ∼ f(xt+1|xt ) (state evolution density), p(xt |y1:t )f (xt+1|xt ) p(x |y ) = p(x + |y ) dx + . -
Lecture 1: Stationary Time Series∗
Lecture 1: Stationary Time Series∗ 1 Introduction If a random variable X is indexed to time, usually denoted by t, the observations {Xt, t ∈ T} is called a time series, where T is a time index set (for example, T = Z, the integer set). Time series data are very common in empirical economic studies. Figure 1 plots some frequently used variables. The upper left figure plots the quarterly GDP from 1947 to 2001; the upper right figure plots the the residuals after linear-detrending the logarithm of GDP; the lower left figure plots the monthly S&P 500 index data from 1990 to 2001; and the lower right figure plots the log difference of the monthly S&P. As you could see, these four series display quite different patterns over time. Investigating and modeling these different patterns is an important part of this course. In this course, you will find that many of the techniques (estimation methods, inference proce- dures, etc) you have learned in your general econometrics course are still applicable in time series analysis. However, there are something special of time series data compared to cross sectional data. For example, when working with cross-sectional data, it usually makes sense to assume that the observations are independent from each other, however, time series data are very likely to display some degree of dependence over time. More importantly, for time series data, we could observe only one history of the realizations of this variable. For example, suppose you obtain a series of US weekly stock index data for the last 50 years. -
Measuring Complexity and Predictability of Time Series with Flexible Multiscale Entropy for Sensor Networks
Article Measuring Complexity and Predictability of Time Series with Flexible Multiscale Entropy for Sensor Networks Renjie Zhou 1,2, Chen Yang 1,2, Jian Wan 1,2,*, Wei Zhang 1,2, Bo Guan 3,*, Naixue Xiong 4,* 1 School of Computer Science and Technology, Hangzhou Dianzi University, Hangzhou 310018, China; [email protected] (R.Z.); [email protected] (C.Y.); [email protected] (W.Z.) 2 Key Laboratory of Complex Systems Modeling and Simulation of Ministry of Education, Hangzhou Dianzi University, Hangzhou 310018, China 3 School of Electronic and Information Engineer, Ningbo University of Technology, Ningbo 315211, China 4 Department of Mathematics and Computer Science, Northeastern State University, Tahlequah, OK 74464, USA * Correspondence: [email protected] (J.W.); [email protected] (B.G.); [email protected] (N.X.) Tel.: +1-404-645-4067 (N.X.) Academic Editor: Mohamed F. Younis Received: 15 November 2016; Accepted: 24 March 2017; Published: 6 April 2017 Abstract: Measurement of time series complexity and predictability is sometimes the cornerstone for proposing solutions to topology and congestion control problems in sensor networks. As a method of measuring time series complexity and predictability, multiscale entropy (MSE) has been widely applied in many fields. However, sample entropy, which is the fundamental component of MSE, measures the similarity of two subsequences of a time series with either zero or one, but without in-between values, which causes sudden changes of entropy values even if the time series embraces small changes. This problem becomes especially severe when the length of time series is getting short. For solving such the problem, we propose flexible multiscale entropy (FMSE), which introduces a novel similarity function measuring the similarity of two subsequences with full-range values from zero to one, and thus increases the reliability and stability of measuring time series complexity. -
Time Series Analysis 1
Basic concepts Autoregressive models Moving average models Time Series Analysis 1. Stationary ARMA models Andrew Lesniewski Baruch College New York Fall 2019 A. Lesniewski Time Series Analysis Basic concepts Autoregressive models Moving average models Outline 1 Basic concepts 2 Autoregressive models 3 Moving average models A. Lesniewski Time Series Analysis Basic concepts Autoregressive models Moving average models Time series A time series is a sequence of data points Xt indexed a discrete set of (ordered) dates t, where −∞ < t < 1. Each Xt can be a simple number or a complex multi-dimensional object (vector, matrix, higher dimensional array, or more general structure). We will be assuming that the times t are equally spaced throughout, and denote the time increment by h (e.g. second, day, month). Unless specified otherwise, we will be choosing the units of time so that h = 1. Typically, time series exhibit significant irregularities, which may have their origin either in the nature of the underlying quantity or imprecision in observation (or both). Examples of time series commonly encountered in finance include: (i) prices, (ii) returns, (iii) index levels, (iv) trading volums, (v) open interests, (vi) macroeconomic data (inflation, new payrolls, unemployment, GDP, housing prices, . ) A. Lesniewski Time Series Analysis Basic concepts Autoregressive models Moving average models Time series For modeling purposes, we assume that the elements of a time series are random variables on some underlying probability space. Time series analysis is a set of mathematical methodologies for analyzing observed time series, whose purpose is to extract useful characteristics of the data. These methodologies fall into two broad categories: (i) non-parametric, where the stochastic law of the time series is not explicitly specified; (ii) parametric, where the stochastic law of the time series is assumed to be given by a model with a finite (and preferably tractable) number of parameters. -
STA 6857---Autocorrelation and Cross-Correlation & Stationary Time Series
Announcements Autocorrelation and Cross-Correlation Stationary Time Series Homework 1c STA 6857—Autocorrelation and Cross-Correlation & Outline Stationary Time Series (§1.4, 1.5) 1 Announcements 2 Autocorrelation and Cross-Correlation 3 Stationary Time Series 4 Homework 1c Arthur Berg STA 6857—Autocorrelation and Cross-Correlation & Stationary Time Series (§1.4, 1.5) 2/ 25 Announcements Autocorrelation and Cross-Correlation Stationary Time Series Homework 1c Announcements Autocorrelation and Cross-Correlation Stationary Time Series Homework 1c Homework Questions from Last Time We’ve seen the AR(p) and MA(q) models. Which one do we use? In scientific modeling we wish to choose the model that best Our TA, Aixin Tan, will have office hours on Thursdays from 1–2pm describes or explains the data. Later we will develop many in 218 Griffin-Floyd. techniques to help us choose and fit the best model for our data. Homework 1c will be assigned today and the last part of homework Is this white noise process in the models unique? 1, homework 1d, will be assigned on Friday. Homework 1 will be We will see later that any stationary time series can be described as collected on Friday, September 7. Don’t wait till the last minute to do a MA(1) + “deterministic part” by the Wold decomposition, where the homework, because more homework will follow next week. the white noise process in the MA(1) part is unique. So in short, the answer is yes. We will also see later how to estimate the white noise process which will aid in forecasting.