The Empirical FT continued. What is the large sample distribution of the EFT?

Slides:

Advertisements

Similar presentations

Statistical Time Series Analysis version 2

Advertisements

Change-Point Detection Techniques for Piecewise Locally Stationary Time Series Michael Last National Institute of Statistical Sciences Talk for Midyear.

The Simple Linear Regression Model Specification and Estimation Hill et al Chs 3 and 4.

The General Linear Model. The Simple Linear Model Linear Regression.

Spectral analysis for point processes. Error bars. Bijan Pesaran Center for Neural Science New York University.

Multivariate distributions. The Normal distribution.

The Simple Linear Regression Model: Specification and Estimation

Linear Regression Models Based on Chapter 3 of Hastie, Tibshirani and Friedman Slides by David Madigan.

Descriptive statistics Experiment  Data  Sample Statistics Experiment  Data  Sample Statistics Sample mean Sample mean Sample variance Sample variance.

The Fourier transform. Properties of Fourier transforms Convolution Scaling Translation.

Inference. Estimates stationary p.p. {N(t)}, rate p N, observed for 0

7. Nonparametric inference  Quantile function Q  Inference on F  Confidence bands for F  Goodness- of- fit tests 1.

The Multivariate Normal Distribution, Part 2 BMTRY 726 1/14/2014.

1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking.

Diffusion Maps and Spectral Clustering

Maximum Likelihood Estimation

Simulation Output Analysis

Chapter 7 Estimation: Single Population

Marginal and Conditional distributions. Theorem: (Marginal distributions for the Multivariate Normal distribution) have p-variate Normal distribution.

Time Series Spectral Representation Z(t) = {Z 1, Z 2, Z 3, … Z n } Any mathematical function has a representation in terms of sin and cos functions.

Borgan and Henderson:. Event History Methodology

Estimation of the spectral density function. The spectral density function, f( ) The spectral density function, f(x), is a symmetric function defined.

ELEC 303 – Random Signals Lecture 18 – Classical Statistical Inference, Dr. Farinaz Koushanfar ECE Dept., Rice University Nov 4, 2010.

CHEE825 Fall 2005J. McLellan1 Spectral Analysis and Input Signal Design.

Week 21 Stochastic Process - Introduction Stochastic processes are processes that proceed randomly in time. Rather than consider fixed random variables.

Basic Time Series Analyzing variable star data for the amateur astronomer.

Estimation of the spectral density function. The spectral density function, f( ) The spectral density function, f(x), is a symmetric function defined.

SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart.

Confidence Interval & Unbiased Estimator Review and Foreword.

Chapter 1 Random Process

Review of Probability. Important Topics 1 Random Variables and Probability Distributions 2 Expected Values, Mean, and Variance 3 Two Random Variables.

ETHEM ALPAYDIN © The MIT Press, Lecture Slides for.

Stochastic Process Theory and Spectral Estimation Bijan Pesaran Center for Neural Science New York University.

Multivariate Time Series Analysis

Geology 6600/7600 Signal Analysis 09 Oct 2015 © A.R. Lowry 2015 Last time: A Periodogram is the squared modulus of the signal FFT! Blackman-Tukey estimates.

Statistics Sampling Distributions and Point Estimation of Parameters Contents, figures, and exercises come from the textbook: Applied Statistics and Probability.

Stochastic Process Theory and Spectral Estimation

SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory.

Week 21 Order Statistics The order statistics of a set of random variables X 1, X 2,…, X n are the same random variables arranged in increasing order.

Geology 6600/7600 Signal Analysis 05 Oct 2015 © A.R. Lowry 2015 Last time: Assignment for Oct 23: GPS time series correlation Given a discrete function.

Chapter 8 Estimation ©. Estimator and Estimate estimator estimate An estimator of a population parameter is a random variable that depends on the sample.

Spinal cord stimulation restores locomotion in animal models of Parkinson's disease "... led us to hypothesize that stimulation... could alleviate motor.

Week 21 Statistical Model A statistical model for some data is a set of distributions, one of which corresponds to the true unknown distribution that produced.

ETHEM ALPAYDIN © The MIT Press, Lecture Slides for.

Locating a Shift in the Mean of a Time Series Melvin J. Hinich Applied Research Laboratories University of Texas at Austin

Time Series Spectral Representation

Tatiana Varatnitskaya Belаrussian State University, Minsk

STATISTICAL INFERENCE

Estimation of the spectral density function

The Simple Linear Regression Model: Specification and Estimation

Outline Introduction Signal, random variable, random process and spectra Analog modulation Analog to digital conversion Digital transmission through baseband.

Parameter Estimation 主講人：虞台文.

Time series power spectral density.

Inference. Estimates stationary p.p. {N(t)}, rate pN , observed for 0

The Empirical FT. What is the large sample distribution of the EFT?

Mixing. Stationary case unless otherwise indicated

Power spectral density. frequency-side, , vs. time-side, t

6-1 Introduction To Empirical Models

Statistical Assumptions for SLR

The Empirical FT. What is the large sample distribution of the EFT?

Second order stationary p.p.

Summarizing Data by Statistics

Stat Oct 2008 D. R. Brillinger Chapter 7 - Spectral analysis 7.1 Fourier analysis Xt = μ + α cos ωt + βsin ωt + Zt Cases ω known versus.

Multivariate Time Series Analysis

FT makes the New Yorker, October 4, 2010 page 71

Inference. Estimates stationary p.p. {N(t)}, rate pN , observed for 0

Ch11 Curve Fitting II.

The Empirical FT. What is the large sample distribution of the EFT?

Networks. partial spectra - trivariate (M,N,O)

16. Mean Square Estimation

Presentation transcript:

The Empirical FT continued. What is the large sample distribution of the EFT?

The complex normal.

Theorem. Suppose p.p. N is stationary mixing, then

Evaluate first and second-order cumulants Bound higher cumulants Normal is determined by its moments Proof. Write

Consider

Comments. Already used to study rate estimate Tapering makes Get asymp independence for different frequencies The frequencies 2  r/T are special, e.g.  T (2  r/T)=0, r  0 Also get asymp independence if consider separate stretches p-vector version involves p by p spectral density matrix f NN ( )

Estimation of the (power) spectrum. An estimate whose limit is a random variable

Some moments. The estimate is asymptotically unbiased Final term drops out if = 2  r/T  0 Best to correct for mean, work with

Periodogram values are asymptotically independent since d T values are - independent exponentials Use to form estimates

Approximate marginal confidence intervals

More on choice of L

Approximation to bias

Indirect estimate Could approximate p.p. by 0-1 time series and use t.s. estimate choice of cells?

Estimation of finite dimensional . approximate likelihood (assuming I T values independent exponentials)

Bivariate case.

Crossperiodogram. Smoothed periodogram.

Complex Wishart

Predicting N via M

Plug in estimates.

Large sample distributions. var log|A T |  [|R| -2 -1]/L var argA T  [|R| -2 -1]/L

Networks. partial spectra - trivariate (M,N,O) Remove linear time invariant effect of M from N and O and examine coherency of residuals,

Point process system. An operation carrying a point process, M, into another, N N = S[M] S = {input,mapping, output} Pr{dN(t)=1|M} = {  +  a(t-u)dM(u)}dt System identification: determining the charcteristics of a system from inputs and corresponding outputs {M(t),N(t);0  t<T} Like regression vs. bivariate

Realizable case. a(u) = 0 u<0 A( ) is of special form e.g. N(t) = M(t-  ) arg A( ) = - 

Advantages of frequency domain approach. many stationary processes look the same approximate i.i.d sample values assessing models (character of departure)...

Muscle spindle example

Published by AAAS R. Fuentes et al., Science 323, (2009) Fig. 2. DCS restores locomotion and desynchronizes corticostriatal activity