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-1 exp {-it}[Z(t+1)-Z(t)]
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{x t+h, xt} = w2 h= and otherwise = 0
f() = w2

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

12. Dirac delta function, ()
a 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 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