Signal Processing - January 2017 - 107
with correlated noise, ML periodogram
methods have been developed to circumvent these issues [13], [14]. The ML
periodogram method does not generate
a periodogram that shows power on the
y-axis, but instead it shows the log-likelihood of the model that is compared to
the data at each step. In this way, any
model can be compared to the data directly across a grid of frequencies or orbital periods, and the log-likelihood can
be calculated for each, with a detected
signal having the maximum likelihood
L (m | i) =
N
%
f=1
+
2
vl )
2
exp )
- [m f - o l (t f )]
3 . (3)
2 (v 2f + v 2l )
The likelihood to be maximized
can be described as in (3) with L (m | i)
being the likelihood of the data m given
the model parameters i. v f and v l
represent the stellar and instrumental
white noise components, respectively,
and o l (t f ) is the Keplerian model to
fit, similar to (2) but with correlated
noise terms included. Maximization of
this likelihood function allows signal
detection to be performed and probabilities can be calculated directly from
the log-likelihood values. Although
in practice this method is slower than
the aforementioned methods, it has the
desired effect of allowing multiple signals to be detected at the same time
(i.e., a global model approach), and it
also means the model can include correlated noise components, along with
the white noise component(s). Therefore, given the continuing increase in
computer processing power, the extra
information and flexibility of ML periodograms outweigh the inefficiency of
its application to real radial velocity
data. However, as with all model fitting methods, one must be careful not
to overfit the data by adding unnecessary terms to the applied model, which
is where proper model comparison statistical tests should be applied.
q
+ f f,d +
/ c z p z, f, d
d
z=1
p
+
/ z z, d exp ' t f - xz - t f f z, d 1 .
d
z=1
(4)
Here model m for a given Keplerian
k and velocity data point f, previous
measurement z, and data set d can be
described by the Keplerian model as a
function of time (F (t)), a systemic offset velocity c, a linear trend as a function of time ct, a Gaussian noise model
to describe the random noise f, a red
noise component described by a moving average (MA) model with exponential smoothing (parameters z and x),
and a set of linear correlations c with
activity indicators that parameterise
the activity state of the star at the time
of the observation p. The Bayesian
approach is the least efficient of these
signal detection methods, since long
chains are required to properly search
the multidimensional parameter space
in a robust manner. However, currently
this method is the most flexible, allowing the user to assess the parameter
space in many different ways. It also
allows visualization of the full parameter space after the chains are complete, meaning nonlinear correlations
between parameters can be scrutinized. Finally, this method was shown
to be the most robust signal detection
and false-positive suppression method
currently used, given the results of an
International Challenge (Extreme Precision Radial Velocities, Yale 2015)
[30] issued to the radial velocity planet
detection community.
Bayesian analysis
Like the ML approach, Bayesian analysis applies a global model to the data,
including correlated noise components,
In [18]-[20], the independent sinusoidal components in nonuniformly
sampled radial velocity data are determined by means of the MMSE method
or its direct extension, the ML estimation scheme. According to [19], significance tests are employed to filter out
the parasitic solutions appearing on the
way. In [18], the MMSE-based method
applies a trellis-based optimal global
search and returns the optimal number
of sinusoidal components including
their frequencies, phases, and amplitudes. This technique employs the
MMSE criterion as an objective function in all the analysis.
If C i is the ith sinusoidal component, and N C is their number, each
component may be written in the form
C i = (~ i, a i, z i), where ~ i, a i, z i are the
frequency, amplitude, and phase of the
ith component, respectively.
The MMSE technique tries to find
the set S = " (~ i, a i, z i) ,iN=C 1 that minimizes the mean square error between
the original signal and S by optimizing
~ i, a i, and z i of each component [18].
First, the target frequency bandwidth
is divided into K ~ levels. Each level
~ k is represented by ~ k = r # k/K ~,
where 1 # k # K ~ . For each ~ k an
optimal amplitude and phase, a ~ k, z ~ k
are obtained by performing an MMSEbased Fourier analysis: for each ~ k,
a ~ k and z ~ k are optimized to minimize the mean square error between
the original signal and the components
a k cos (~ k t + z ~ k) .
The number of components to analyze,
N, is then estimated for all the frequencies having local minimum of MMSE
values and/or higher amplitudes with
respect to a defined threshold. Therefore, a subset S min = " (~ i, a ~ i, z ~ i) ,
is constructed out of the set S P , which
includes only these components. Next,
a neighborhood band Vi is defined
for each component, C i in Smin as, Vi =
" (~, a ~, z ~) ! S P /~ ! [~ i - d, ~ i + d] ,,
m f , d = c d + ct f + Fk (t f )
1
2r (v 2f
MMSE-based method
and assesses the parameter space using
Markov chains (e.g., [15), where the
model is assessed by covering a given
frequency/period domain. The maximum of the posterior density distribution can be used to detect a signal in the
data (e.g., [16] and [17])
j
j
j
(~t i , at i , zt i ) iN=C 1 = arg
T
j
j
|
January 2017
(~ i , a i , z i ) 1 # i # N C t = 1
IEEE Signal Processing Magazine
NC
2
/ e x (t) - / a ij cos (~ ij t + z ij) o
min
j
|
(5)
i=1
107
Table of Contents for the Digital Edition of Signal Processing - January 2017
Signal Processing - January 2017 - Cover1
Signal Processing - January 2017 - Cover2
Signal Processing - January 2017 - 1
Signal Processing - January 2017 - 2
Signal Processing - January 2017 - 3
Signal Processing - January 2017 - 4
Signal Processing - January 2017 - 5
Signal Processing - January 2017 - 6
Signal Processing - January 2017 - 7
Signal Processing - January 2017 - 8
Signal Processing - January 2017 - 9
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Signal Processing - January 2017 - 17
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Signal Processing - January 2017 - 19
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Signal Processing - January 2017 - 21
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Signal Processing - January 2017 - 25
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Signal Processing - January 2017 - 28
Signal Processing - January 2017 - 29
Signal Processing - January 2017 - 30
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Signal Processing - January 2017 - 32
Signal Processing - January 2017 - 33
Signal Processing - January 2017 - 34
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Signal Processing - January 2017 - 36
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Signal Processing - January 2017 - 101
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Signal Processing - January 2017 - 103
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Signal Processing - January 2017 - 105
Signal Processing - January 2017 - 106
Signal Processing - January 2017 - 107
Signal Processing - January 2017 - 108
Signal Processing - January 2017 - 109
Signal Processing - January 2017 - 110
Signal Processing - January 2017 - 111
Signal Processing - January 2017 - 112
Signal Processing - January 2017 - 113
Signal Processing - January 2017 - 114
Signal Processing - January 2017 - 115
Signal Processing - January 2017 - 116
Signal Processing - January 2017 - Cover3
Signal Processing - January 2017 - Cover4
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