for s.e. of flux multiply cv by mean flux over time period damage: penetration depends on size
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For s.e. of flux multiply cv by mean flux over time period Damage: penetration depends on size. sagtu17.pdf Ascona12.pdf. Filtering/smoothing . 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) - PowerPoint PPT PresentationTRANSCRIPT
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Relative standard error of flux
Includes extra Poisson variation multiplier of 1.16Observation time (hr)
coef
ficie
nt o
f var
iatio
n
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
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sagtu17.pdf
Ascona12.pdf
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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.
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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
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Bank of bandpass filters
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Fourier series.
(*) )()(A Approx
)( Series
)(21)( tsCoefficien
|)(|
(n) uae
uae
dAeua
dA
nn
iu
u
iu
iu
How close is A(n)() to A() ?
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By substitution
nn
uin
nn
en
D
dADA
21sin
)21sin(
)(2
)()()()(
negative becan but 0,near edConcentrat
2 Period
1)(
dDn
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Error
phenomenon sGibb'always )(approach t doesn' )(
|)(|||1
|)(||)()(|
)(
||
)(
AA
uaun
uaAA
n
kk
nu
n
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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
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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"() + ...
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Lowpass filter.
u
utXu
unuhtY )(sin)()(
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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)
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Kernel smoother.
S(t) = wb(t-s)Y(s) / wb(t-s)
wb(t) = w(t/b)
b: bandwidth
ksmooth()
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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
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Running median
medt-kst+k Y(s)
Repeat til no change
Other things: parametric model, splines, ...
choice of bandwidth/binwidth
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Finite Fourier transforms. Considered
(*) )()(A Approx
)( Series
)(21)( tsCoefficien
|)(|
(n) uae
uae
dAeua
dA
nn
iu
u
iu
iu
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Empirical Fourier analysis.
Uses.
Estimation - parameters and periods
Unification of data types
Approximation of distributions
System identification
Speeding up computations
Model assessment
...
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)()()(
)()(
)()2(
)()(
1,...,0 ),( .)(
10
TY
TX
TYX
TX
TX
TX
TX
Tt
tiTX
ddd
dd
dd
etXd
TttXDataVector
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Examples. 1. Constant. X(t)=1
)-(D2X(t)
.polynomial ricTrigonomet 3.
)cos(,...2,at peaks
)(2)( Cosine. .2
)( :Definition
)(2)2/sin(
)2/)12sin((
kn
)(
10
k
tik
nn
nti
ti
TTt
ti
nn
nti
ke
t
DeetX
e
Dne
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Inversion.
ansformFourier tr discrete :)/2(
)/2(}/2exp{
)()2()(
10
1
20
1
Tsd
TstdTstiT
ddtX
TX
Ts
TX
TX
fft()
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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
}2exp{)2()2(10
1
Ssti
SsA
SsdS S
sTX
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Periodogram. |dT ()|2
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Chandler wobble.
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Interpretation of frequency.
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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 Tj 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}
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2. M.p.p. (sampled time series).
{j , Mj } {Y(j )}
j Mj exp{-ij}
j Y(j ) exp{-ij}
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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 ikk )(2
0
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Other processes.
process on sphere, line process, generalized process, vector-valued time, LCA group