Euler predicted free nutation of the rotating Earth in 1755 Discovered by Chandler in 1891

3. Rotating solid Euler predicted free nutation of the rotating Earth in 1755 Discovered by Chandler in 1891 Data from International Latitude Observatories setup in 1899

5. Monthly data, t = 1 month. Work with complex-values, Z(t) = X(t) + iY(t). Compute the location differences, Z(t), and then the finite FT dZT() = t=0T-1exp {-it}[Z(t+1)-Z(t)]  = 2s/T , s = 0, 1, 2, …, T-1 Periodogram IZZT() = (2T)-1|dZT()|2

7. variance

8. Appendix C. Spectral Domain Theory

9. 4.3 Spectral distribution function Cp. rv’s

10. f is non-negative, symmetric(, periodic) White noise. (h) = cov{xt+h,xt} = w2 h=0 and otherwise = 0 f() = w2

11. dF()/d = f() if differentiable dF() = f()d Cramer representation/Spectral representation

12. Dirac delta function, () generalized function simplifies many t.s. manipulations r.v. X Prob{X = 0} = 1 P(x) = Prob{X  x} = 1 if x  0 = 0 if x < 0 = H(x) Heavyside E{g(X)} = g(0) =  g(x) dP(x) =  g(x) (x) dx (x)  density function = dH(x)/dx

13. Approximant X  N(0,2 ) (x/)/ with  small E{g(X)}  g(0) cov{dZ(1),dZ(2)} = (1 – 2) f(1) d 1 d 2 Means 0 cov{X,Y} = E{X conjg(Y)} var{X} = E{|X|2}

14. Example. Bay of Fundy

16. flattened

18. Periodogram “sample spectral density” Mean“correction”

19. Non parametric spectral estimation. L = 2m+1

23. Fire video Comb5 start about 13:00

26. Weighted average. Expected value  ( K( /B) /B) f(-) d

27. Kernel(“modified daniel”, c(3,3))

28. Bivariate series. Two-sided case as well AKA

29. Bivariate example. Gas furnace

31. Linear filters Impulse response: {aj} Transfer function. amplitude, phase A() = |A()| exp{ ()}

32. Cramer representations Xt = exp {i t}dZx () Yt = exp {i t} dZy() =  at-uexp{i u} dZx() =  A() exp {i t} dZx() dZy() = A() dZx() Cov{ dZx(), dZx() ] =  ( – } fxx () d d fyy() = |A()|2fxx() Interpretation of power spectrum

33. ARMA process f yy () = |A()|2fxx ( ) z = exp{ -I  )

34. Xt = exp {i t}dZx () d() =