Outline. STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) Summary Statistics of Midterm Scores

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1 STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) Outline 1 Midterm Results 2 Multiple Series and Cross-Spectra 3 Linear Filters Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 2/ 25 Summary Statistics of Midterm Scores Histrogram of Midterm Scores Average = 36.0 (72%) Median = 38.0 (76%) Max = 50 (100%) Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 4/ 25 Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 5/ 25

2 Your Lucky Day! Homework 4 We will curve the midterm scores by 4 points uniformly. Therefore the new summary statistics are: Average = 40 (80%) Median = 42 (84%) Max = 54 (108%) The midterm counts for 35% of your grade. There s still homework 4, the final project, and the R test. Homework 4 is due Friday, October 26 in class. Make sure you start thinking about your final project. I ll send around a paper on Monday collecting your initial project titles. Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 6/ 25 Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 7/ 25 Cross Spectrum Squared Coherence Function Recall the cross-covariance function γ xy (h) for a jointly stationary series x t and y t as defined by γ xy (h) = E [(x t+h µ x )(y t µ y )] Taking the Fourier transform, we have the cross-spectrum f xy (ω) = = γ xy (h)e 2πiωh γ xy (h) cos(2πωh) i } {{ } c xy (ω) γ xy (h) sin(2πωh) }{{} qxy (ω) The squared coherence function measures the the strength of the relationship between x t and y t in the frequency domain. Definition (Squared Coherence Function) The squared coherence function is defined as ρ 2 yx(ω) = f yx(ω) 2 f x (ω)f y (ω) Note the analogy to the conventional square correlation given by ρ 2 yx = σ2 yx σ 2 xσ 2 y Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 9/ 25 Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 10/ 25

3 Spectral Matrix Testing for Significant Coherence For a p-dimensional stationary vector time series with autocovariance matrix given by Γ(h) = E [ (x t+h µ)(x t µ) ] The spectral matrix is the term-by-term Fourier transform of the autocovariance matrix which can be simply written as f (ω) = Γ(h)e 2πiωh, 1/2 ω 1/2 Under the hypothesis ρ 2 yx(ω) = 0 the statistic ρ 2 yx(ω) 1 ρ 2 yx(ω) follows an F distribution which allows one to test for significance against the null. Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 11/ 25 Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 12/ 25 Coherence Function Between SOI and Recruitment Series > x = ts(cbind(soi,rec)) > s = spec.pgram(x, kernel("daniell",9), taper=0) > s$df # df = [1] > f = qf(.999, 2, s$df-2) # f = > c = f/(18+f) # c = > plot(s, plot.type = "coh", ci.lty = 2) > abline(h = c, lwd=3, col="purple") Linear Filter Definition A linear filter is a linear transformation of a process x t given as y t = r= a r x t r The coefficients a r are collectively called the impulse response function and it is assumed that they are absolutely summable, i.e. t= a t < Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 13/ 25 Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 15/ 25

4 Output Spectrum of Filtered Series Two Filters of SOI The frequency response function, A yx (ω), is defined as Fourier transform of the impulse response function, i.e. Theorem A yx (ω) = t= a t e 2πiωt The spectrum of the filtered output y t satsifies f y (ω) = A yx 2 f x (ω) Note the relation to classical statistics where multiplying a random variable X by a constant c changes its variance to c 2 var(x). Let x t represent the SOI values. Consider the two filters: The difference filter y t = x t = x t x t 1 Here a 0 = 1 and a 1 = 1. This is an example of a high-pass filter. The symmetric moving average filter y t = 1 24 (x t 6 + x t+6 ) r= 5 x t r Here a 6 = a 6 = 1/24, a k = 1/12 for 5 k 5, and a k = 0 otherwise. This is an example of a low-pass filter. Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 16/ 25 Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 17/ 25 Two Filters Applied to SOI Frequency Response Functions of the Two Examples > w = seq(0,.5, length=1000) #-- frequency response > FR = abs(1-exp(2i*pi*w))^2 > FR2<-double(length(w)) > count<-1 > for(j in w){ + FR2[count]=abs(1/12*(1+cos(12*pi*j)+2*sum(cos(2*pi*j*1:5))))^2 + count<-count+1 + } > plot(w,fr2,type="l",lwd=3,col="blue") > windows() > plot(w, FR, type="l",lwd=3,col="orange") Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 18/ 25 Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 19/ 25

5 Frequency Response Functions of the Two Examples Here we can see why differencing is referred to as a high-pass filter and the moving average as a low-pass filter. Parametric Spectral Density Estimation Recall for an ARMA(p, q) process φ(b)x t = θ(b)w t, we would have f x (ω) = σ 2 θ(e 2πiω ) 2 φ(e 2πiω ) 2 ( ) General approach to parametric spectral density estimation: 1 Use a model selection criterion (like AIC, AICc, BIC etc.) to estimate the best p when fitting an AR(p) model. 2 Fit an AR(p) model. 3 Compute the spectral density of the fitted AR(p) model (use ( )). Alternative approach: Fit an ARMA(p,q) model and compute the spectral density using ( ). Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 20/ 25 Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 21/ 25 Theorem Validating the Parametric Approach Back to the SOI Theorem Let g(ω) be the spectral density of a stationary process. Then, given ε > 0, there is causal AR(p) time series, i.e. a time series with the representation p x t = φ k x t k + w t k=1 where w t is white noise with variance σ 2, such that The nonparametric estimate of the spectral density: f x (ω) g(ω) < ε for all ω [ 1/2, 1/2] Moreover p is finite and the roots of φ(z) = 1 p k=1 φ kz k are outside the unit circle. Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 22/ 25 Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 23/ 25

6 AIC Selection for SOI Parametric Spectral Density Estimate of SOI > ar(soi, order.max=30)$aic The parametric estimate of the spectral density: Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 24/ 25 Arthur Berg STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) 25/ 25