Euler predicted free nutation of the rotating Earth in 1755

Discovered by Chandler in 1891

Data from International Latitude Observatories setup in 1899

Rotating solid

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=0

T-1 exp {-it}[Z(t+1)-Z(t)]

= 2s/T , s = 0, 1, 2, …, T-1

Periodogram

IZZT() = (2T)-1|dZ

T()|2

variance

Appendix C. Spectral Domain Theory

4.3 Spectral distribution function

Cp. rv’s

f is non-negative, symmetric(, periodic)

White noise. (h) = cov{x t+h, xt} = w

2 h=0 and otherwise = 0 f() = w

2

dF()/d = f() if differentiable

dF() = f()d

Cramer representation/Spectral representation

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

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}

Example. Bay of Fundy

flattened

Periodogram “sample spectral density”

Mean“correction”

Non parametric spectral estimation.

L = 2m+1

Fire video

Comb5

start about 13:00

Weighted average.

Expected value ( K( /B) /B) f(-) d

Kernel(“modified daniel”, c(3,3))

Bivariate series.

Two-sided case as well

AKA

Bivariate example. Gas furnace

Linear filters

Transfer function. amplitude, phase

A() = |A()| exp{ ()}

Impulse response: {aj}

Cramer representations

Xt = exp {i t}dZx ()

Yt = exp {i t} dZy()

= at-u exp{i u} dZx ()

= A() exp {i t} dZx ()

dZ y() = A() dZx ()

Cov{ dZx (), dZx() ] = ( – } fxx () d d

f yy() = |A()|2 fxx()

Interpretation of power spectrum

ARMA process

f yy () = |A()|2 fxx ( ) z = exp{ -I )

Xt = exp {i t}dZx ()

d() =