Lecture 1: Stationary Time Series∗
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Stationary Processes and Their Statistical Properties
Stationary Processes and Their Statistical Properties Brian Borchers March 29, 2001 1 Stationary processes A discrete time stochastic process is a sequence of random variables Z1, Z2, :::. In practice we will typically analyze a single realization z1, z2, :::, zn of the stochastic process and attempt to esimate the statistical properties of the stochastic process from the realization. We will also consider the problem of predicting zn+1 from the previous elements of the sequence. We will begin by focusing on the very important class of stationary stochas- tic processes. A stochastic process is strictly stationary if its statistical prop- erties are unaffected by shifting the stochastic process in time. In particular, this means that if we take a subsequence Zk+1, :::, Zk+m, then the joint distribution of the m random variables will be the same no matter what k is. Stationarity requires that the mean of the stochastic process be a constant. E[Zk] = µ. and that the variance is constant 2 V ar[Zk] = σZ : Also, stationarity requires that the covariance of two elements separated by a distance m is constant. That is, Cov(Zk;Zk+m) is constant. This covariance is called the autocovariance at lag m, and we will use the notation γm. Since Cov(Zk;Zk+m) = Cov(Zk+m;Zk), we need only find γm for m 0. The ≥ correlation of Zk and Zk+m is the autocorrelation at lag m. We will use the notation ρm for the autocorrelation. It is easy to show that γk ρk = : γ0 1 2 The autocovariance and autocorrelation ma- trices The covariance matrix for the random variables Z1, :::, Zn is called an auto- covariance matrix. -
Methods of Monte Carlo Simulation II
Methods of Monte Carlo Simulation II Ulm University Institute of Stochastics Lecture Notes Dr. Tim Brereton Summer Term 2014 Ulm, 2014 2 Contents 1 SomeSimpleStochasticProcesses 7 1.1 StochasticProcesses . 7 1.2 RandomWalks .......................... 7 1.2.1 BernoulliProcesses . 7 1.2.2 RandomWalks ...................... 10 1.2.3 ProbabilitiesofRandomWalks . 13 1.2.4 Distribution of Xn .................... 13 1.2.5 FirstPassageTime . 14 2 Estimators 17 2.1 Bias, Variance, the Central Limit Theorem and Mean Square Error................................ 19 2.2 Non-AsymptoticErrorBounds. 22 2.3 Big O and Little o Notation ................... 23 3 Markov Chains 25 3.1 SimulatingMarkovChains . 28 3.1.1 Drawing from a Discrete Uniform Distribution . 28 3.1.2 Drawing From A Discrete Distribution on a Small State Space ........................... 28 3.1.3 SimulatingaMarkovChain . 28 3.2 Communication .......................... 29 3.3 TheStrongMarkovProperty . 30 3.4 RecurrenceandTransience . 31 3.4.1 RecurrenceofRandomWalks . 33 3.5 InvariantDistributions . 34 3.6 LimitingDistribution. 36 3.7 Reversibility............................ 37 4 The Poisson Process 39 4.1 Point Processes on [0, )..................... 39 ∞ 3 4 CONTENTS 4.2 PoissonProcess .......................... 41 4.2.1 Order Statistics and the Distribution of Arrival Times 44 4.2.2 DistributionofArrivalTimes . 45 4.3 SimulatingPoissonProcesses. 46 4.3.1 Using the Infinitesimal Definition to Simulate Approx- imately .......................... 46 4.3.2 SimulatingtheArrivalTimes . 47 4.3.3 SimulatingtheInter-ArrivalTimes . 48 4.4 InhomogenousPoissonProcesses. 48 4.5 Simulating an Inhomogenous Poisson Process . 49 4.5.1 Acceptance-Rejection. 49 4.5.2 Infinitesimal Approach (Approximate) . 50 4.6 CompoundPoissonProcesses . 51 5 ContinuousTimeMarkovChains 53 5.1 TransitionFunction. 53 5.2 InfinitesimalGenerator . 54 5.3 ContinuousTimeMarkovChains . -
Stochastic Process - Introduction
Stochastic Process - Introduction • Stochastic processes are processes that proceed randomly in time. • Rather than consider fixed random variables X, Y, etc. or even sequences of i.i.d random variables, we consider sequences X0, X1, X2, …. Where Xt represent some random quantity at time t. • In general, the value Xt might depend on the quantity Xt-1 at time t-1, or even the value Xs for other times s < t. • Example: simple random walk . week 2 1 Stochastic Process - Definition • A stochastic process is a family of time indexed random variables Xt where t belongs to an index set. Formal notation, { t : ∈ ItX } where I is an index set that is a subset of R. • Examples of index sets: 1) I = (-∞, ∞) or I = [0, ∞]. In this case Xt is a continuous time stochastic process. 2) I = {0, ±1, ±2, ….} or I = {0, 1, 2, …}. In this case Xt is a discrete time stochastic process. • We use uppercase letter {Xt } to describe the process. A time series, {xt } is a realization or sample function from a certain process. • We use information from a time series to estimate parameters and properties of process {Xt }. week 2 2 Probability Distribution of a Process • For any stochastic process with index set I, its probability distribution function is uniquely determined by its finite dimensional distributions. •The k dimensional distribution function of a process is defined by FXX x,..., ( 1 x,..., k ) = P( Xt ≤ 1 ,..., xt≤ X k) x t1 tk 1 k for anyt 1 ,..., t k ∈ I and any real numbers x1, …, xk . -
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). -
Examples of Stationary Time Series
Statistics 910, #2 1 Examples of Stationary Time Series Overview 1. Stationarity 2. Linear processes 3. Cyclic models 4. Nonlinear models Stationarity Strict stationarity (Defn 1.6) Probability distribution of the stochastic process fXtgis invariant under a shift in time, P (Xt1 ≤ x1;Xt2 ≤ x2;:::;Xtk ≤ xk) = F (xt1 ; xt2 ; : : : ; xtk ) = F (xh+t1 ; xh+t2 ; : : : ; xh+tk ) = P (Xh+t1 ≤ x1;Xh+t2 ≤ x2;:::;Xh+tk ≤ xk) for any time shift h and xj. Weak stationarity (Defn 1.7) (aka, second-order stationarity) The mean and autocovariance of the stochastic process are finite and invariant under a shift in time, E Xt = µt = µ Cov(Xt;Xs) = E (Xt−µt)(Xs−µs) = γ(t; s) = γ(t−s) The separation rather than location in time matters. Equivalence If the process is Gaussian with finite second moments, then weak stationarity is equivalent to strong stationarity. Strict stationar- ity implies weak stationarity only if the necessary moments exist. Relevance Stationarity matters because it provides a framework in which averaging makes sense. Unless properties like the mean and covariance are either fixed or \evolve" in a known manner, we cannot average the observed data. What operations produce a stationary process? Can we recognize/identify these in data? Statistics 910, #2 2 Moving Average White noise Sequence of uncorrelated random variables with finite vari- ance, ( 2 often often σw = 1 if t = s; E Wt = µ = 0 Cov(Wt;Ws) = 0 otherwise The input component (fXtg in what follows) is often modeled as white noise. Strict white noise replaces uncorrelated by independent. Moving average A stochastic process formed by taking a weighted average of another time series, often formed from white noise. -
Regularity of Solutions and Parameter Estimation for Spde’S with Space-Time White Noise
REGULARITY OF SOLUTIONS AND PARAMETER ESTIMATION FOR SPDE’S WITH SPACE-TIME WHITE NOISE by Igor Cialenco A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2007 Copyright 2007 Igor Cialenco Dedication To my wife Angela, and my parents. ii Acknowledgements I would like to acknowledge my academic adviser Prof. Sergey V. Lototsky who introduced me into the Theory of Stochastic Partial Differential Equations, suggested the interesting topics of research and guided me through it. I also wish to thank the members of my committee - Prof. Remigijus Mikulevicius and Prof. Aris Protopapadakis, for their help and support. Last but certainly not least, I want to thank my wife Angela, and my family for their support both during the thesis and before it. iii Table of Contents Dedication ii Acknowledgements iii List of Tables v List of Figures vi Abstract vii Chapter 1: Introduction 1 1.1 Sobolev spaces . 1 1.2 Diffusion processes and absolute continuity of their measures . 4 1.3 Stochastic partial differential equations and their applications . 7 1.4 Ito’sˆ formula in Hilbert space . 14 1.5 Existence and uniqueness of solution . 18 Chapter 2: Regularity of solution 23 2.1 Introduction . 23 2.2 Equations with additive noise . 29 2.2.1 Existence and uniqueness . 29 2.2.2 Regularity in space . 33 2.2.3 Regularity in time . 38 2.3 Equations with multiplicative noise . 41 2.3.1 Existence and uniqueness . 41 2.3.2 Regularity in space and time . -
Stationary Processes
Stationary processes Alejandro Ribeiro Dept. of Electrical and Systems Engineering University of Pennsylvania [email protected] http://www.seas.upenn.edu/users/~aribeiro/ November 25, 2019 Stoch. Systems Analysis Stationary processes 1 Stationary stochastic processes Stationary stochastic processes Autocorrelation function and wide sense stationary processes Fourier transforms Linear time invariant systems Power spectral density and linear filtering of stochastic processes Stoch. Systems Analysis Stationary processes 2 Stationary stochastic processes I All probabilities are invariant to time shits, i.e., for any s P[X (t1 + s) ≥ x1; X (t2 + s) ≥ x2;:::; X (tK + s) ≥ xK ] = P[X (t1) ≥ x1; X (t2) ≥ x2;:::; X (tK ) ≥ xK ] I If above relation is true process is called strictly stationary (SS) I First order stationary ) probs. of single variables are shift invariant P[X (t + s) ≥ x] = P [X (t) ≥ x] I Second order stationary ) joint probs. of pairs are shift invariant P[X (t1 + s) ≥ x1; X (t2 + s) ≥ x2] = P [X (t1) ≥ x1; X (t2) ≥ x2] Stoch. Systems Analysis Stationary processes 3 Pdfs and moments of stationary process I For SS process joint cdfs are shift invariant. Whereby, pdfs also are fX (t+s)(x) = fX (t)(x) = fX (0)(x) := fX (x) I As a consequence, the mean of a SS process is constant Z 1 Z 1 µ(t) := E [X (t)] = xfX (t)(x) = xfX (x) = µ −∞ −∞ I The variance of a SS process is also constant Z 1 Z 1 2 2 2 var [X (t)] := (x − µ) fX (t)(x) = (x − µ) fX (x) = σ −∞ −∞ I The power of a SS process (second moment) is also constant Z 1 Z 1 2 2 2 2 2 E X (t) := x fX (t)(x) = x fX (x) = σ + µ −∞ −∞ Stoch. -
Stochastic Pdes and Markov Random Fields with Ecological Applications
Intro SPDE GMRF Examples Boundaries Excursions References Stochastic PDEs and Markov random fields with ecological applications Finn Lindgren Spatially-varying Stochastic Differential Equations with Applications to the Biological Sciences OSU, Columbus, Ohio, 2015 Finn Lindgren - [email protected] Stochastic PDEs and Markov random fields with ecological applications Intro SPDE GMRF Examples Boundaries Excursions References “Big” data Z(Dtrn) 20 15 10 5 Illustration: Synthetic data mimicking satellite based CO2 measurements. Iregular data locations, uneven coverage, features on different scales. Finn Lindgren - [email protected] Stochastic PDEs and Markov random fields with ecological applications Intro SPDE GMRF Examples Boundaries Excursions References Sparse spatial coverage of temperature measurements raw data (y) 200409 kriged (eta+zed) etazed field 20 15 15 10 10 10 5 5 lat lat lat 0 0 0 −10 −5 −5 44 46 48 50 52 44 46 48 50 52 44 46 48 50 52 −20 2 4 6 8 10 14 2 4 6 8 10 14 2 4 6 8 10 14 lon lon lon residual (y − (eta+zed)) climate (eta) eta field 20 2 15 18 10 16 1 14 5 lat 0 lat lat 12 0 10 −1 8 −5 44 46 48 50 52 6 −2 44 46 48 50 52 44 46 48 50 52 2 4 6 8 10 14 2 4 6 8 10 14 2 4 6 8 10 14 lon lon lon Regional observations: ≈ 20,000,000 from daily timeseries over 160 years Finn Lindgren - [email protected] Stochastic PDEs and Markov random fields with ecological applications Intro SPDE GMRF Examples Boundaries Excursions References Spatio-temporal modelling framework Spatial statistics framework ◮ Spatial domain D, or space-time domain D × T, T ⊂ R. -
Solutions to Exercises in Stationary Stochastic Processes for Scientists and Engineers by Lindgren, Rootzén and Sandsten Chapman & Hall/CRC, 2013
Solutions to exercises in Stationary stochastic processes for scientists and engineers by Lindgren, Rootzén and Sandsten Chapman & Hall/CRC, 2013 Georg Lindgren, Johan Sandberg, Maria Sandsten 2017 CENTRUM SCIENTIARUM MATHEMATICARUM Faculty of Engineering Centre for Mathematical Sciences Mathematical Statistics 1 Solutions to exercises in Stationary stochastic processes for scientists and engineers Mathematical Statistics Centre for Mathematical Sciences Lund University Box 118 SE-221 00 Lund, Sweden http://www.maths.lu.se c Georg Lindgren, Johan Sandberg, Maria Sandsten, 2017 Contents Preface v 2 Stationary processes 1 3 The Poisson process and its relatives 5 4 Spectral representations 9 5 Gaussian processes 13 6 Linear filters – general theory 17 7 AR, MA, and ARMA-models 21 8 Linear filters – applications 25 9 Frequency analysis and spectral estimation 29 iii iv CONTENTS Preface This booklet contains hints and solutions to exercises in Stationary stochastic processes for scientists and engineers by Georg Lindgren, Holger Rootzén, and Maria Sandsten, Chapman & Hall/CRC, 2013. The solutions have been adapted from course material used at Lund University on first courses in stationary processes for students in engineering programs as well as in mathematics, statistics, and science programs. The web page for the course during the fall semester 2013 gives an example of a schedule for a seven week period: http://www.maths.lu.se/matstat/kurser/fms045mas210/ Note that the chapter references in the material from the Lund University course do not exactly agree with those in the printed volume. v vi CONTENTS Chapter 2 Stationary processes 2:1. (a) 1, (b) a + b, (c) 13, (d) a2 + b2, (e) a2 + b2, (f) 1. -
Random Processes
Chapter 6 Random Processes Random Process • A random process is a time-varying function that assigns the outcome of a random experiment to each time instant: X(t). • For a fixed (sample path): a random process is a time varying function, e.g., a signal. – For fixed t: a random process is a random variable. • If one scans all possible outcomes of the underlying random experiment, we shall get an ensemble of signals. • Random Process can be continuous or discrete • Real random process also called stochastic process – Example: Noise source (Noise can often be modeled as a Gaussian random process. An Ensemble of Signals Remember: RV maps Events à Constants RP maps Events à f(t) RP: Discrete and Continuous The set of all possible sample functions {v(t, E i)} is called the ensemble and defines the random process v(t) that describes the noise source. Sample functions of a binary random process. RP Characterization • Random variables x 1 , x 2 , . , x n represent amplitudes of sample functions at t 5 t 1 , t 2 , . , t n . – A random process can, therefore, be viewed as a collection of an infinite number of random variables: RP Characterization – First Order • CDF • PDF • Mean • Mean-Square Statistics of a Random Process RP Characterization – Second Order • The first order does not provide sufficient information as to how rapidly the RP is changing as a function of timeà We use second order estimation RP Characterization – Second Order • The first order does not provide sufficient information as to how rapidly the RP is changing as a function -
Subband Particle Filtering for Speech Enhancement
14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIP SUBBAND PARTICLE FILTERING FOR SPEECH ENHANCEMENT Ying Deng and V. John Mathews Dept. of Electrical and Computer Eng., University of Utah 50 S. Central Campus Dr., Rm. 3280 MEB, Salt Lake City, UT 84112, USA phone: + (1)(801) 581-6941, fax: + (1)(801) 581-5281, email: [email protected], [email protected] ABSTRACT as those discussed in [16, 17] and also in this paper, the integrations Particle filters have recently been applied to speech enhancement used to compute the filtering distribution and the integrations em- when the input speech signal is modeled as a time-varying autore- ployed to estimate the clean speech signal and model parameters do gressive process with stochastically evolving parameters. This type not have closed-form analytical solutions. Approximation methods of modeling results in a nonlinear and conditionally Gaussian state- have to be employed for these computations. The approximation space system that is not amenable to analytical solutions. Prior work methods developed so far can be grouped into three classes: (1) an- in this area involved signal processing in the fullband domain and alytic approximations such as the Gaussian sum filter [19] and the assumed white Gaussian noise with known variance. This paper extended Kalman filter [20], (2) numerical approximations which extends such ideas to subband domain particle filters and colored make the continuous integration variable discrete and then replace noise. Experimental results indicate that the subband particle filter each integral by a summation [21], and (3) sampling approaches achieves higher segmental SNR than the fullband algorithm and is such as the unscented Kalman filter [22] which uses a small num- effective in dealing with colored noise without increasing the com- ber of deterministically chosen samples and the particle filter [23] putational complexity.