Topic 7: Random Processes Random Processes
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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 . -
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. -
5. the Student T Distribution
Virtual Laboratories > 4. Special Distributions > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5. The Student t Distribution In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of a standardized version of the sample mean when the underlying distribution is normal. The Probability Density Function Suppose that Z has the standard normal distribution, V has the chi-squared distribution with n degrees of freedom, and that Z and V are independent. Let Z T= √V/n In the following exercise, you will show that T has probability density function given by −(n +1) /2 Γ((n + 1) / 2) t2 f(t)= 1 + , t∈ℝ ( n ) √n π Γ(n / 2) 1. Show that T has the given probability density function by using the following steps. n a. Show first that the conditional distribution of T given V=v is normal with mean 0 a nd variance v . b. Use (a) to find the joint probability density function of (T,V). c. Integrate the joint probability density function in (b) with respect to v to find the probability density function of T. The distribution of T is known as the Student t distribution with n degree of freedom. The distribution is well defined for any n > 0, but in practice, only positive integer values of n are of interest. This distribution was first studied by William Gosset, who published under the pseudonym Student. In addition to supplying the proof, Exercise 1 provides a good way of thinking of the t distribution: the t distribution arises when the variance of a mean 0 normal distribution is randomized in a certain way. -
1 One Parameter Exponential Families
1 One parameter exponential families The world of exponential families bridges the gap between the Gaussian family and general dis- tributions. Many properties of Gaussians carry through to exponential families in a fairly precise sense. • In the Gaussian world, there exact small sample distributional results (i.e. t, F , χ2). • In the exponential family world, there are approximate distributional results (i.e. deviance tests). • In the general setting, we can only appeal to asymptotics. A one-parameter exponential family, F is a one-parameter family of distributions of the form Pη(dx) = exp (η · t(x) − Λ(η)) P0(dx) for some probability measure P0. The parameter η is called the natural or canonical parameter and the function Λ is called the cumulant generating function, and is simply the normalization needed to make dPη fη(x) = (x) = exp (η · t(x) − Λ(η)) dP0 a proper probability density. The random variable t(X) is the sufficient statistic of the exponential family. Note that P0 does not have to be a distribution on R, but these are of course the simplest examples. 1.0.1 A first example: Gaussian with linear sufficient statistic Consider the standard normal distribution Z e−z2=2 P0(A) = p dz A 2π and let t(x) = x. Then, the exponential family is eη·x−x2=2 Pη(dx) / p 2π and we see that Λ(η) = η2=2: eta= np.linspace(-2,2,101) CGF= eta**2/2. plt.plot(eta, CGF) A= plt.gca() A.set_xlabel(r'$\eta$', size=20) A.set_ylabel(r'$\Lambda(\eta)$', size=20) f= plt.gcf() 1 Thus, the exponential family in this setting is the collection F = fN(η; 1) : η 2 Rg : d 1.0.2 Normal with quadratic sufficient statistic on R d As a second example, take P0 = N(0;Id×d), i.e. -
Random Variables and Probability Distributions 1.1
RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 1. DISCRETE RANDOM VARIABLES 1.1. Definition of a Discrete Random Variable. A random variable X is said to be discrete if it can assume only a finite or countable infinite number of distinct values. A discrete random variable can be defined on both a countable or uncountable sample space. 1.2. Probability for a discrete random variable. The probability that X takes on the value x, P(X=x), is defined as the sum of the probabilities of all sample points in Ω that are assigned the value x. We may denote P(X=x) by p(x) or pX (x). The expression pX (x) is a function that assigns probabilities to each possible value x; thus it is often called the probability function for the random variable X. 1.3. Probability distribution for a discrete random variable. The probability distribution for a discrete random variable X can be represented by a formula, a table, or a graph, which provides pX (x) = P(X=x) for all x. The probability distribution for a discrete random variable assigns nonzero probabilities to only a countable number of distinct x values. Any value x not explicitly assigned a positive probability is understood to be such that P(X=x) = 0. The function pX (x)= P(X=x) for each x within the range of X is called the probability distribution of X. It is often called the probability mass function for the discrete random variable X. 1.4. Properties of the probability distribution for a discrete random variable. -
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. -
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 -
11. Parameter Estimation
11. Parameter Estimation Chris Piech and Mehran Sahami May 2017 We have learned many different distributions for random variables and all of those distributions had parame- ters: the numbers that you provide as input when you define a random variable. So far when we were working with random variables, we either were explicitly told the values of the parameters, or, we could divine the values by understanding the process that was generating the random variables. What if we don’t know the values of the parameters and we can’t estimate them from our own expert knowl- edge? What if instead of knowing the random variables, we have a lot of examples of data generated with the same underlying distribution? In this chapter we are going to learn formal ways of estimating parameters from data. These ideas are critical for artificial intelligence. Almost all modern machine learning algorithms work like this: (1) specify a probabilistic model that has parameters. (2) Learn the value of those parameters from data. Parameters Before we dive into parameter estimation, first let’s revisit the concept of parameters. Given a model, the parameters are the numbers that yield the actual distribution. In the case of a Bernoulli random variable, the single parameter was the value p. In the case of a Uniform random variable, the parameters are the a and b values that define the min and max value. Here is a list of random variables and the corresponding parameters. From now on, we are going to use the notation q to be a vector of all the parameters: Distribution Parameters Bernoulli(p) q = p Poisson(l) q = l Uniform(a,b) q = (a;b) Normal(m;s 2) q = (m;s 2) Y = mX + b q = (m;b) In the real world often you don’t know the “true” parameters, but you get to observe data. -
4.2 Autoregressive (AR) Moving Average Models Are Causal Linear Processes by Definition. There Is Another Class of Models, Based
4.2 Autoregressive (AR) Moving average models are causal linear processes by definition. There is another class of models, based on a recursive formulation similar to the exponentially weighted moving average. Definition 4.11 (Autoregressive AR(p)). Suppose φ1,...,φp ∈ R are constants 2 2 and (Wi) ∼ WN(σ ). The AR(p) process with parameters σ , φ1,...,φp is defined through p Xi = Wi + φjXi−j, (3) Xj=1 whenever such stationary process (Xi) exists. Remark 4.12. The process in Definition 4.11 is sometimes called a stationary AR(p) process. It is possible to consider a ‘non-stationary AR(p) process’ for any φ1,...,φp satisfying (3) for i ≥ 0 by letting for example Xi = 0 for i ∈ [−p+1, 0]. Example 4.13 (Variance and autocorrelation of AR(1) process). For the AR(1) process, whenever it exits, we must have 2 2 γ0 = Var(Xi) = Var(φ1Xi−1 + Wi)= φ1γ0 + σ , which implies that we must have |φ1| < 1, and σ2 γ0 = 2 . 1 − φ1 We may also calculate for j ≥ 1 j γj = E[XiXi−j]= E[(φ1Xi−1 + Wi)Xi−j]= φ1E[Xi−1Xi−j]= φ1γ0, j which gives that ρj = φ1. Example 4.14. Simulation of an AR(1) process. phi_1 <- 0.7 x <- arima.sim(model=list(ar=phi_1), 140) # This is the explicit simulation: gamma_0 <- 1/(1-phi_1^2) x_0 <- rnorm(1)*sqrt(gamma_0) x <- filter(rnorm(140), phi_1, method = "r", init = x_0) Example 4.15. Consider a stationary AR(1) process. We may write n−1 n j Xi = φ1Xi−1 + Wi = ··· = φ1 Xi−n + φ1Wi−j. -
Random Variables and Applications
Random Variables and Applications OPRE 6301 Random Variables. As noted earlier, variability is omnipresent in the busi- ness world. To model variability probabilistically, we need the concept of a random variable. A random variable is a numerically valued variable which takes on different values with given probabilities. Examples: The return on an investment in a one-year period The price of an equity The number of customers entering a store The sales volume of a store on a particular day The turnover rate at your organization next year 1 Types of Random Variables. Discrete Random Variable: — one that takes on a countable number of possible values, e.g., total of roll of two dice: 2, 3, ..., 12 • number of desktops sold: 0, 1, ... • customer count: 0, 1, ... • Continuous Random Variable: — one that takes on an uncountable number of possible values, e.g., interest rate: 3.25%, 6.125%, ... • task completion time: a nonnegative value • price of a stock: a nonnegative value • Basic Concept: Integer or rational numbers are discrete, while real numbers are continuous. 2 Probability Distributions. “Randomness” of a random variable is described by a probability distribution. Informally, the probability distribution specifies the probability or likelihood for a random variable to assume a particular value. Formally, let X be a random variable and let x be a possible value of X. Then, we have two cases. Discrete: the probability mass function of X specifies P (x) P (X = x) for all possible values of x. ≡ Continuous: the probability density function of X is a function f(x) that is such that f(x) h P (x < · ≈ X x + h) for small positive h.