DOCSLIB.ORG

Appendix a Large Sample Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix a Large Sample Theory Appendix A Large Sample Theory A.1 Convergence Modes The study of the optimality properties of various estimators (such as the sample autocorrelation function) depends, in part, on being able to assess the large-sample behavior of these estimators. We summarize briefly here the kinds of convergence useful in this setting, namely, mean square convergence, convergence in probability, and convergence in distribution. We consider first a particular class of random variables that plays an important role in the study of second-order time series, namely, the class of random variables belonging to the space L2, satisfying E|x|2 < ∞. In proving certain properties of the class L2 we will often use, for random variables x, y ∈ L2,theCauchy–Schwarz inequality, |E(xy)|2 ≤ E(|x|2)E(|y|2), (A.1) and the Tchebycheff inequality, E(|x|2) Pr{|x|≥a}≤ , (A.2) a2 for a > 0. Next, we investigate the properties of mean square convergence of random vari- ables in L2. 2 Definition A.1 A sequence of L random variables {xn} is said to converge in mean square to a random variable x ∈ L2, denoted by ms xn → x, (A.3) if and only if 2 E|xn − x| → 0(A.4) as n →∞. © Springer International Publishing AG 2017 473 R.H. Shumway, D.S. Stoffer, Time Series Analysis and Its Applications, Springer Texts in Statistics, DOI 10.1007/978-3-319-52452-8 474 Appendix A: Large Sample Theory Example A.1 Mean Square Convergence of the Sample Mean Consider the white noise sequence wt and the signal plus noise series xt = μ + wt . Then, because σ2 E|x¯ − μ|2 = w → 0 n n →∞ = −1 n →ms μ as n , wherex ¯n n t=1 xt is the sample mean, we havex ¯n . We summarize some of the properties of mean square convergence as follows. If ms ms xn → x, and yn → y, then, as n →∞, E(xn)→E(x);(A.5) 2 2 E(|xn | )→E(|x| );(A.6) E(xn yn)→E(xy). (A.7) We also note the L2 completeness theorem known as the Riesz–Fischer Theorem. 2 2 Theorem A.1 Let {xn} be a sequence in L . Then, there exists a x in L such that ms xn → x if and only if lim sup E|x − x |2 = 0 . (A.8) →∞ n m m n≥m Often the condition of Theorem A.1 is easier to verify to establish that a mean square limit x exists without knowing what it is. Sequences that satisfy (A.8) are said to be Cauchy sequences in L2 and (A.8) is also known as the Cauchy criterion for L2. Example A.2 Time Invariant Linear Filter As an important example of the use of the Riesz–Fisher Theorem and the properties of mean square convergent series given in (A.5)–(A.7), a time-invariant linear filter is defined as a convolution of the form ∞ yt = aj xt−j (A.9) j=−∞ for each t = 0, ±1, ±2,..., where xt is a weakly stationary input series with mean μx and autocovariance function γx(h), and aj ,forj = 0, ±1, ±2,...are constants satisfying ∞ |aj | < ∞. (A.10) j=−∞ The output series yt defines a filtering or smoothing of the input series that changes the character of the time series in a predictable way. We need to know the conditions under which the outputs yt in (A.9) and the linear process (1.31) exist. A.1 Convergence Modes 475 Considering the sequence n n = , yt aj xt−j (A.11) j=−n = , ,... n n 1 2 , we need to show first that yt has a mean square limit. By Theorem A.1, it is enough to show that n − m2 → E yt yt 0 as m, n →∞.Forn > m > 0, 2 n − m2 = E yt yt E aj xt−j <| |≤ m j n = aj ak E(xt−j xt−k ) <| |≤ ≤| |≤ m j n m k n ≤ |aj ||ak ||E(xt−j xt−k )| <| |≤ ≤| |≤ m j n m k n 2 1/2 2 1/2 ≤ |aj ||ak |(E|xt−j | ) (E|xt−k | ) m<| j |≤n m≤|k |≤n 2 = γ ( ) + μ2 | | → x 0 x aj 0 m≤|j |≤n as m, n →∞, because γx(0) is a constant and {aj } is absolutely summable (the second inequality follows from the Cauchy–Schwarz inequality). { n} Although we know that the sequence yt given by (A.11) converges in mean square, we have not established its mean square limit. If S denotes the mean square n | − |2 = | − n |2 ≤ limit of yt , then using Fatou’s Lemma, E S yt E lim infn→∞ S yt | − n |2 = n lim infn→∞ E S yt 0, which establishes that yt is the mean square limit of yt . Finally, we may use (A.5) and (A.7) to establish the mean, μy and autocovariance function, γy(h) of yt . In particular we have, ∞ μy = μx aj, (A.12) j=−∞ and ∞ ∞ γy(h) = E aj (xt+h−j − μx)aj (xt−k − μx) j=−∞ k=−∞ ∞ ∞ = aj γx(h − j + k)ak . (A.13) j=−∞ k=−∞ A second important kind of convergence is convergence in probability. 476 Appendix A: Large Sample Theory Definition A.2 The sequence {xn},forn = 1, 2,..., converges in probability to a random variable x, denoted by p xn → x, (A.14) if and only if Pr{|xn − x| >}→0 (A.15) for all >0,asn →∞. An immediate consequence of the Tchebycheff inequality, (A.2), is that E(|x − x|2) Pr{|x − x|≥}≤ n , n 2 so convergence in mean square implies convergence in probability, i.e., ms p xn → x ⇒ xn → x. (A.16) This result implies, for example, that the filter (A.9) exists as a limit in probability because it converges in mean square [it is also easily established that (A.9) exists with probability one]. We mention, at this point, the useful Weak Law of Large Numbers which states that, for an independent identically distributed sequence xn of random variables with mean μ,wehave p x¯n → μ (A.17) →∞ = −1 n as n , wherex ¯n n t=1 xt is the usual sample mean. We also will make use of the following concepts. Definition A.3 For order in probability we write xn = op(an) (A.18) if and only if x p n → 0. (A.19) an The term boundedness in probability, written xn = Op(an), means that for every >0, there exists a δ() > 0 such that xn Pr >δ() ≤ (A.20) an for all n. p Under this convention, e.g., the notation for xn → x becomes xn − x = op(1). The definitions can be compared with their nonrandom counterparts, namely, for a fixed sequence xn = o(1) if xn → 0 and xn = O(1) if xn,forn = 1, 2,...is bounded. Some handy properties of op(·) and Op(·) are as follows. A.1 Convergence Modes 477 (i) If xn = op(an) and yn = op(bn), then xn yn = op(anbn) and xn + yn = op(max(an, bn)). (ii) If xn = op(an) and yn = Op(bn), then xn yn = op(anbn). (iii) Statement (i) is true if Op(·) replaces op(·). Example A.3 Convergence and Order in Probability for the Sample Mean 2 For the sample mean,x ¯n, of iid random variables with mean μ and variance σ ,by the Tchebycheff inequality, E[(x¯ − μ)2] Pr{|x¯ − μ| >}≤ n n 2 σ2 = → 0, n2 p as n →∞. It follows thatx ¯n → μ,or¯xn − μ = op(1). To find the rate, it follows that, for δ() > 0, √ σ2/n σ2 Pr n |x¯ − μ| >δ() ≤ = n δ2()/n δ2() √ by Tchebycheff’s inequality, so taking = σ2/δ2() shows that δ() = σ/ does the job and −1/2 x¯n − μ = Op(n ). p For k × 1 random vectors xn, convergence in probability, written xn → x or xn − x = op(1) is defined as element-by-element convergence in probability, or equivalently, as convergence in terms of the Euclidean distance p xn − x → 0, (A.21) p = 2 → where a j aj for any vector a. In this context, we note the result that if xn x and g(xn) is a continuous mapping, p g(xn) → g(x). (A.22) Furthermore, if xn − a = Op(δn) with δn → 0 and g(·) is a function with continuous first derivatives continuous in a neighborhood of a = (a1, a2,...,ak ) ,we have the Taylor series expansion in probability ∂ ( ) ( ) = ( ) + g x ( − ) + (δ ), g xn g a ∂ xn a Op n (A.23) x x=a where ∂ ( ) ∂ ( ) ∂ ( ) g x = g x ,..., g x ∂ ∂ ∂ x x=a x1 x=a xk x=a denotes the vector of partial derivatives with respect to x1, x2,...,xk , evaluated at a. This result remains true if Op(δn) is replaced everywhere by op(δn). 478 Appendix A: Large Sample Theory Example A.4 Expansion for the Logarithm of the Sample Mean With the same conditions as Example A.3, consider g(x¯n) = logx ¯n, which has a −1/2 derivative at μ,forμ>0. Then, becausex ¯n − μ = Op(n ) from Example A.3, the conditions for the Taylor expansion in probability, (A.23), are satisfied and we have −1 −1/2 logx ¯n = log μ + μ (x¯n − μ) + Op(n ). The large sample distributions of sample mean and sample autocorrelation func- tions defined earlier can be developed using the notion of convergence in distribution. Definition A.4 A sequence of k × 1 random vectors {xn} is said to converge in distribution, written d xn → x (A.24) if and only if Fn(x)→F(x) (A.25) at the continuity points of distribution function F(·).
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • Power Comparisons of Five Most Commonly Used Autocorrelation Tests
    Pak.j.stat.oper.res. Vol.16 No. 1 2020 pp119-130 DOI: http://dx.doi.org/10.18187/pjsor.v16i1.2691 Pakistan Journal of Statistics and Operation Research Power Comparisons of Five Most Commonly Used Autocorrelation Tests Stanislaus S. Uyanto School of Economics and Business, Atma Jaya Catholic University of Indonesia, stan.uyanto@atmajaya.ac.id Abstract In regression analysis, autocorrelation of the error terms violates the ordinary least squares assumption that the error terms are uncorrelated. The consequence is that the estimates of coefficients and their standard errors will be wrong if the autocorrelation is ignored. There are many tests for autocorrelation, we want to know which test is more powerful. We use Monte Carlo methods to compare the power of five most commonly used tests for autocorrelation, namely Durbin-Watson, Breusch-Godfrey, Box–Pierce, Ljung Box, and Runs tests in two different linear regression models. The results indicate the Durbin-Watson test performs better in the regression model without lagged dependent variable, although the advantage over the other tests reduce with increasing autocorrelation and sample sizes. For the model with lagged dependent variable, the Breusch-Godfrey test is generally superior to the other tests. Key Words: Correlated error terms; Ordinary least squares assumption; Residuals; Regression diagnostic; Lagged dependent variable. Mathematical Subject Classification: 62J05; 62J20. 1. Introduction An important assumption of the classical linear regression model states that there is no autocorrelation in the error terms. Autocorrelation of the error terms violates the ordinary least squares assumption that the error terms are uncorrelated, meaning that the Gauss Markov theorem (Plackett, 1949, 1950; Greene, 2018) does not apply, and that ordinary least squares estimators are no longer the best linear unbiased estimators.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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
    [Show full text]
  • 3 Autocorrelation
    3 Autocorrelation Autocorrelation refers to the correlation of a time series with its own past and future values. Autocorrelation is also sometimes called “lagged correlation” or “serial correlation”, which refers to the correlation between members of a series of numbers arranged in time. Positive autocorrelation might be considered a specific form of “persistence”, a tendency for a system to remain in the same state from one observation to the next. For example, the likelihood of tomorrow being rainy is greater if today is rainy than if today is dry. Geophysical time series are frequently autocorrelated because of inertia or carryover processes in the physical system. For example, the slowly evolving and moving low pressure systems in the atmosphere might impart persistence to daily rainfall. Or the slow drainage of groundwater reserves might impart correlation to successive annual flows of a river. Or stored photosynthates might impart correlation to successive annual values of tree-ring indices. Autocorrelation complicates the application of statistical tests by reducing the effective sample size. Autocorrelation can also complicate the identification of significant covariance or correlation between time series (e.g., precipitation with a tree-ring series). Autocorrelation implies that a time series is predictable, probabilistically, as future values are correlated with current and past values. Three tools for assessing the autocorrelation of a time series are (1) the time series plot, (2) the lagged scatterplot, and (3) the autocorrelation function. 3.1 Time series plot Positively autocorrelated series are sometimes referred to as persistent because positive departures from the mean tend to be followed by positive depatures from the mean, and negative departures from the mean tend to be followed by negative departures (Figure 3.1).
    [Show full text]
  • Spatial Autocorrelation: Covariance and Semivariance Semivariance
    Spatial Autocorrelation: Covariance and Semivariancence Lily Housese ­P eters GEOG 593 November 10, 2009 Quantitative Terrain Descriptorsrs Covariance and Semivariogram areare numericc methods used to describe the character of the terrain (ex. Terrain roughness, irregularity) Terrain character has important implications for: 1. the type of sampling strategy chosen 2. estimating DTM accuracy (after sampling and reconstruction) Spatial Autocorrelationon The First Law of Geography ““ Everything is related to everything else, but near things are moo re related than distant things.” (Waldo Tobler) The value of a variable at one point in space is related to the value of that same variable in a nearby location Ex. Moranan ’s I, Gearyary ’s C, LISA Positive Spatial Autocorrelation (Neighbors are similar) Negative Spatial Autocorrelation (Neighbors are dissimilar) R(d) = correlation coefficient of all the points with horizontal interval (d) Covariance The degree of similarity between pairs of surface points The value of similarity is an indicator of the complexity of the terrain surface Smaller the similarity = more complex the terrain surface V = Variance calculated from all N points Cov (d) = Covariance of all points with horizontal interval d Z i = Height of point i M = average height of all points Z i+d = elevation of the point with an interval of d from i Semivariancee Expresses the degree of relationship between points on a surface Equal to half the variance of the differences between all possible points spaced a constant distance apart
    [Show full text]
  • Modeling and Generating Multivariate Time Series with Arbitrary Marginals Using a Vector Autoregressive Technique
    Modeling and Generating Multivariate Time Series with Arbitrary Marginals Using a Vector Autoregressive Technique Bahar Deler Barry L. Nelson Dept. of IE & MS, Northwestern University September 2000 Abstract We present a model for representing stationary multivariate time series with arbitrary mar- ginal distributions and autocorrelation structures and describe how to generate data quickly and accurately to drive computer simulations. The central idea is to transform a Gaussian vector autoregressive process into the desired multivariate time-series input process that we presume as having a VARTA (Vector-Autoregressive-To-Anything) distribution. We manipulate the correlation structure of the Gaussian vector autoregressive process so that we achieve the desired correlation structure for the simulation input process. For the purpose of computational efficiency, we provide a numerical method, which incorporates a numerical-search procedure and a numerical-integration technique, for solving this correlation-matching problem. Keywords: Computer simulation, vector autoregressive process, vector time series, multivariate input mod- eling, numerical integration. 1 1 Introduction Representing the uncertainty or randomness in a simulated system by an input model is one of the challenging problems in the practical application of computer simulation. There are an abundance of examples, from manufacturing to service applications, where input modeling is critical; e.g., modeling the time to failure for a machining process, the demand per unit time for inventory of a product, or the times between arrivals of calls to a call center. Further, building a large- scale discrete-event stochastic simulation model may require the development of a large number of, possibly multivariate, input models. Development of these models is facilitated by accurate and automated (or nearly automated) input modeling support.
    [Show full text]
  • 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
    [Show full text]
  • On the Use of the Autocorrelation and Covariance Methods for Feedforward Control of Transverse Angle and Position Jitter in Linear Particle Beam Accelerators*
    '^C 7 ON THE USE OF THE AUTOCORRELATION AND COVARIANCE METHODS FOR FEEDFORWARD CONTROL OF TRANSVERSE ANGLE AND POSITION JITTER IN LINEAR PARTICLE BEAM ACCELERATORS* Dean S. Ban- Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439 ABSTRACT It is desired to design a predictive feedforward transverse jitter control system to control both angle and position jitter m pulsed linear accelerators. Such a system will increase the accuracy and bandwidth of correction over that of currently available feedback correction systems. Intrapulse correction is performed. An offline process actually "learns" the properties of the jitter, and uses these properties to apply correction to the beam. The correction weights calculated offline are downloaded to a real-time analog correction system between macropulses. Jitter data were taken at the Los Alamos National Laboratory (LANL) Ground Test Accelerator (GTA) telescope experiment at Argonne National Laboratory (ANL). The experiment consisted of the LANL telescope connected to the ANL ZGS proton source and linac. A simulation of the correction system using this data was shown to decrease the average rms jitter by a factor of two over that of a comparable standard feedback correction system. The system also improved the correction bandwidth. INTRODUCTION Figure 1 shows the standard setup for a feedforward transverse jitter control system. Note that one pickup #1 and two kickers are needed to correct beam position jitter, while two pickup #l's and one kicker are needed to correct beam trajectory-angle jitter. pickup #1 kicker pickup #2 Beam Fast loop Slow loop Processor Figure 1. Feedforward Transverse Jitter Control System It is assumed that the beam is fast enough to beat the correction signal (through the fast loop) to the kicker.
    [Show full text]
  • Lecture 10: Lagged Autocovariance and Correlation
    Lecture 10: Lagged autocovariance and correlation c Christopher S. Bretherton Winter 2014 Reference: Hartmann Atm S 552 notes, Chapter 6.1-2. 10.1 Lagged autocovariance and autocorrelation The lag-p autocovariance is defined N X ap = ujuj+p=N; (10.1.1) j=1 which measures how strongly a time series is related with itself p samples later or earlier. The zero-lag autocovariance a0 is equal to the power. Also note that ap = a−p because both correspond to a lag of p time samples. The lag-p autocorrelation is obtained by dividing the lag-p autocovariance by the variance: rp = ap=a0 (10.1.2) Many kinds of time series decorrelate in time so that rp ! 0 as p increases. On the other hand, for a purely periodic signal of period T , rp = cos(2πp∆t=T ) will oscillate between 1 at lags that are multiples of the period and -1 at lags that are odd half-multiples of the period. Thus, if the autocorrelation drops substantially below zero at some lag P , that usually corresponds to a preferred peak in the spectral power at periods around 2P . More generally, the autocovariance sequence (a0; a1; a2;:::) is intimately re- lated to the power spectrum. Let S be the power spectrum deduced from the DFT, with components 2 2 Sm = ju^mj =N ; m = 1; 2;:::;N (10.1.3) Then one can show using a variant of Parseval's theorem that the IDFT of the power spectrum gives the lagged autocovariance sequence. Specif- ically, let the vector a be the lagged autocovariance sequence (acvs) fap−1; p = 1;:::;Ng computed in the usual DFT style by interpreting indices j + p outside the range 1; ::; N periodically: j + p mod(j + p − 1;N) + 1 (10.1.4) 1 Amath 482/582 Lecture 10 Bretherton - Winter 2014 2 Note that because of the periodicity an−1 = a−1, i.
    [Show full text]