Coded data generation or conversion – Digital code to digital code converters – To or from code based on probability
Reexamination Certificate
2000-11-03
2002-08-13
Young, Brian (Department: 2819)
Coded data generation or conversion
Digital code to digital code converters
To or from code based on probability
C704S240000, C707S793000
Reexamination Certificate
active
06433710
ABSTRACT:
BACKGROUND TO THE INVENTION
This invention relates to a method of data compression for the estimation of a plurality of parameters on which a dataset is dependent.
There are many instances where objects consist of many data, whose values are determined by a small number of parameters. Often, it is only these parameters which are of interest. An aim of this invention is to find linear combinations of the data which are focussed on estimating the physical parameters with as small an error as possible.
Such a problem is very general, and has been attacked in the case of parameter estimation in large-scale structure and the microwave background (e.g. Tegmark M., Taylor A., Heavens A., 1997 Astrophysical Journal 480, 22. [hereafter TTH], Tegmark M., 1997a. Astrophysical Journal (Letters) 480, L87 and Tegmark M., 1997b Physical Review D, 55, 5895.) The entire disclosure of these three documents is incorporated herein by way of reference. Previous work has concentrated largely on the estimation of a single parameter; the main advance of this invention is that it provides a method for the estimation of multiple parameters. The method provides one projection per parameter, with the consequent possibility of a massive data compression factor. Furthermore, if the noise in the data is independent of the parameters, then the method is entirely lossless. i.e. the compressed dataset contains as much information about the parameters as the full dataset, in the sense that the Fisher information matrix is the same for the compressed dataset as the entire original dataset. An equivalent statement is that the mean likelihood surface is at the peak locally identical when the full or compressed data are used.
The method will be illustrated herein with the case of galaxy spectra, for which there are surveys underway which will provide ~10
6
objects. In this application, the noise is generally not independent of the parameters, as there is a photon shot-noise component which depends on how many photons are expected. We take a spectrum with poor signal-to-noise, whose noise is approximately from photon counting alone, and investigate how the method fares. In this case, the method is not lossless, but the increase in error bars is shown to be minimal, and superior in this respect to an alternative compression system PCA (Principal Component Analysis).
One advantage such radical compression offers is speed of analysis. One major scientific goal of galaxy spectral surveys is to determine physical parameters of the stellar component of the galaxies, such as the age, star formation history, initial mass function and so on. Such a process can, in principle, be achieved by generating model galaxy spectra by stellar population synthesis techniques, and finding the best-fitting model by maximum-likelihood techniques. This can be very time-consuming, and must inevitably be automated for so many galaxies. In addition, one may have a large parameter space to explore, so any method which can speed up this process is worth investigation. One possible further application of the data compression method is that the handful of numbers might provide the basis of a classification scheme which is based on the physical properties one wants to measure.
SUMMARY OF THE INVENTION
The present invention provides a method of compressing a dataset which is dependent on a plurality of parameters, as well as on noise, comprising forming a plurality of sets of weights, one set for each parameter, from an initial guess for the parameters, multiplying all the data in the dataset by each set of weights in turn turn and summing the products in each case to give compressed data for estimating the parameters, the number of compressed data being equal to the number of parameters.
In an embodiment of the invention, the dataset is represented by a vector x
i
, i=1, . . . N (e.g. a set of fluxes at different wavelengths). These measurements include a signal part, which we denote by &mgr;, and noise, n:
x=&mgr;+n
(1)
Assuming the noise has zero mean, <x>=&mgr;. The signal will depend on a set of parameters {&thgr;
&agr;
}, which we wish to determine. For galaxy spectra, the parameters may be, for example, age, magnitude of source, metallicity and some parameters describing the star formation history. Thus, &mgr; is a noise-free spectrum of a galaxy with certain age, metallicity etc.
The noise properties are described by the noise covariance matrix, C, with components C
ij
=<n
i
n
j
>. If the noise is gaussian, the statistical properties of the data are determined entirely by &mgr; and C. In principle, the noise can also depend on the parameters. For example, in galaxy spectra, one component of the noise will come from photon counting statistics, and the contribution of this to the noise will depend on the mean number of photons expected from the source.
The aim is to derive the parameters from the data. If we assume uniform priors for the parameters, then the a posteriori probability for the parameters is the likelihood, which for gaussian noise is
ℒ
(
θ
α
)
=
1
(
2
π
)
N
/
2
det
(
C
)
×
exp
[
-
1
2
∑
i
,
j
(
x
i
-
μ
i
)
C
ij
-
1
(
x
j
-
μ
j
)
]
.
(
2
)
One approach is simply to find the (highest) peak in the likelihood, by exploring all parameter space, and using all N pixels. The position of the peak gives estimates of the parameters which are asymptotically (low noise) the best unbiased estimators (see TTH). This is therefore the best we can do. The maximum-likelihood procedure can, however, be time-consuming if N is large, and the parameter space is large. An aim of this invention is to reduce the N values to a smaller number, without increasing the uncertainties on the derived parameters &thgr;
&agr;
. To be specific, we try to find a number N′<N of linear combinations of the spectral data x which encompass as much as possible of the information about the physical parameters. We have found that this can be done lossless in some circumstances; the spectra can be reduced to a handful of numbers without loss of information. The speed-up in parameter estimation is by a factor ~100.
In general, reducing the dataset in this way will lead to larger error bars in the parameters. To assess how well the compression is doing, consider the behavior of the (logarithm of the) likelihood function near the peak. Performing a Taylor expansion and truncating at the second-order terms,
ln
ℒ
=
ln
ℒ
peak
+
1
2
∂
2
ln
ℒ
∂
θ
α
∂
θ
β
Δθ
α
Δθ
β
.
(
3
)
Truncating here assumes that the likelihood surface itself is adequately approximated by a gaussian everywhere, not just at the maximum-likelihood point. The actual likelihood surface will vary when different data are used; on average, though, the width is set by the (inverse of the) Fisher information matrix:
F
αβ
≡
-
⟨
∂
2
ln
ℒ
∂
θ
α
∂
θ
β
⟩
(
4
)
where the average is over an ensemble with the same parameters but different noise.
For a single parameter, the Fisher matrix F is a scalar F, and the error on the parameter can be no smaller than F
½
. If the data depend on more than one parameter, and all the parameters have to be estimated from the data, then the error is larger. The error on one parameter a (marginalized over the others) is at least [(F
−1
)
&agr;&agr;
]
−½
see Kendall M. G., Stuart A., 1969
The Advanced Theory of Statistics,
London: Griffin. There is a little more discussion of the Fisher matrix in TTH. The Fisher matrix depends on the signal and noise terms in the following way (TTH, equation 15)
F
αβ
=
1
2
Tr
[
C
Heavens Alan F.
Jimenez Raul
Lahav Ofer
Dann Dorfman Herrell and Skillman, P.C.
Nguyen John
The University Court of the University of Edinburgh
Young Brian
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