euler predicted free nutation of the rotating earth in 1755 discovered by chandler in 1891

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() =