Relative standard error of flux

Includes extra Poisson variation multiplier of 1.16Observation time (hr)

coe

ffici

en

t of v

aria

tion

10 50 100 500 1000

0.05

0.10

0.50

1.00

1-2cm

2-4cm

8-16cm

4-8cm

For s.e. of flux multiply cv by mean flux over time period

Damage: penetration depends on size

sagtu17.pdf

Ascona12.pdf

Use of A(,). bandpass filtering

Suppose X(x,y) j,k jk exp{i(j x + k y)}

Y(x,y) = A[X](x,y)

j,k A(j,k) jk exp{i(j x + k y)}

e.g. If A(,) = 1, | ± 0|, |±0|

= 0 otherwise

Y(x,y) contains only these terms

Repeated xeroxing

Filtering/smoothing.

Approximating an ideal low-pass filter.

Transfer function

A() = 1 ||

Ideal

Y(t) = a(u) X(t-u) t,u in Z

A() = a(u) exp{-i u) - <

a(u) = exp{iu}A()d / 2

= |lamda|<Omega exp{i u}d/2

= / u=0

= sin u/u u 0

Bank of bandpass filters

Fourier series.

(*) )()(A

Approx

)( Series

)(21

)( tsCoefficien

|)(|

(n) uae

uae

dAeua

dA

nn

iu

u

iu

iu

How close is A(n)() to A() ?

By substitution

nn

ui

n

n

n

en

D

dADA

21

sin

)21

sin()(2

)()()()(

negative becan but

0,near edConcentrat

2 Period

1)(

dDn

Error

phenomenon sGibb'

always )(approach t doesn' )(

|)(|||1

|)(||)()(|

)(

||

)(

AA

uaun

uaAA

n

k

k

nu

n

Convergence factors. Fejer (1900)

Replace (*) by

dAn

uaenun

nui

)(2/sin2/sin

n21

)()||

1(

2

-

Fejer kernel

integrates to 1

non-negative

approximate Dirac delta

General class. h(u) = 0, |u|>1

h(u/n) exp{-iu} a(u)

= H(n)() A(-) d (**)

with

H(n)() = (2)-1 h(u/n) exp{-iu}

h(.): convergence factor, taper, data window, fader

(**) = A() + n-1 H()d A'()

+ ½n-22H()d A"() + ...

Lowpass filter.

u

utXu

unu

htY )(sin

)()(

Smoothing/smoothers.

goal: retain smooth/low frequency components of signal while reducing the more irregular/high frequency ones

difficulty: no universal definition of smooth curve

Example. running mean

avet-kst+k Y(s)

Kernel smoother.

S(t) = wb(t-s)Y(s) / wb(t-s)

wb(t) = w(t/b)

b: bandwidth

ksmooth()

Local polynomial.

Linear case

Obtain at , bt OLS intercept and slope of points

{(s,Y(s)): t-k s t+k}

S(t) = at + btt

span: (2k+1)/n

lowess(), loess(): WLS

can be made resistant

Running median

medt-kst+k Y(s)

Repeat til no change

Other things: parametric model, splines, ...

choice of bandwidth/binwidth

Finite Fourier transforms. Considered

(*) )()(A

Approx

)( Series

)(21

)( tsCoefficien

|)(|

(n) uae

uae

dAeua

dA

nn

iu

u

iu

iu

Empirical Fourier analysis.

Uses.

Estimation - parameters and periods

Unification of data types

Approximation of distributions

System identification

Speeding up computations

Model assessment

...

)()()(

)()(

)()2(

)()(

1,...,0 ),( .)(

10

T

Y

T

X

T

YX

T

X

T

X

T

X

T

X

Tt

tiT

X

ddd

dd

dd

etXd

TttXDataVector

Examples. 1. Constant. X(t)=1

)-(D2

X(t)

.polynomial ricTrigonomet 3.

)cos(

,...2,at peaks

)(2

)( Cosine. .2

)( :Definition

)(2)2/sin(

)2/)12sin((

kn

)(

10

k

ti

k

nn

nti

ti

TTt

ti

nn

nti

ke

t

De

etX

e

Dn

e

Inversion.

ansformFourier tr discrete :)/2(

)/2(}/2exp{

)()2()(

10

1

20

1

Tsd

TstdTstiT

ddtX

T

X

Ts

T

X

T

X

fft()

Convolution.

Lemma 3.4.1. If |X(t)M, a(0) and |ua(u)| A,

Y(t) = a(t-u)X(u) then,

|dYT() – A() dY

T() | 4MA

Application. Filtering

Add S-T zeroes

}2

exp{)2

()2

(10

1

Sst

iS

sA

Ss

dS Ss

T

X

Periodogram. |dT ()|2

Chandler wobble.

Interpretation of frequency.

Some other empirical FTs.

1. Point process on the line.

{0j <T}, j=1,...,N

N(t), 0t<T

dN(t)/dt = j (t-j)

j

jNj

T T

j dtttitdNtii )(}exp{)(}exp{}exp{1 0 0

Might approximate by a 0-1 time series

Yt = 1 point in [0,t)

= 0 otherwise

j Yt exp{-it}

2. M.p.p. (sampled time series).

{j , Mj } {Y(j )}

j Mj exp{-ij}

j Y(j ) exp{-ij}

3. Measure, processes of increments

T ti tdYe0 )(

4. Discrete state-valued process

Y(t) values in N, g:NR

t g(Y(t)) exp{-it}

5. Process on circle

Y(), 0 <

Y() = k k exp{ik}

dYe ik

k )(2

0

Other processes.

process on sphere, line process, generalized process, vector-valued time, LCA group