Post on 07-Jan-2016
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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() =