Some EAGLE Algorithms

5 .3 Temporal Spectroscopy: The Investigation of Multiperiodic Signals

5.3.2 Some EAGLE Algorithms

To determine initial values for the frequencies, I used the technique of prewhitening;

the Fourier transform of a dataset is calculated to determine starting values of the frequency, amplitude and phase of the highest amplitude signal. I then use a least- squares routine to fit a sinusoid with these parameters to the data set, which is then subtracted from the data, point by point. Removing a frequency removes its entire window pattern. The Fourier transform of the residuals is then calculated, and the next highest frequency found and subtracted.

When working with partially-resolved, closely-spaced signals, it soon becomes

Period (s)

6100 1100 600 400 300

0.015

0.010

0.005

0

0.015

0.010

0.005

0

0 1000 2000 3000

Frequency (microHz)

250

4000

200

0.015

0.010

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0

0.015

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Figure 5.2: The amplitude spectrum of GW Librae in (a) March '97, (b) April '97, (c) September '97, and (d) May/June '98.

clear that least squares prewhitening of signals one by one ceases to give meaning- ful results after the second iteration (often sooner). This is because the algorithm cannot completely deconvolve one signal from the others, due to the phenomenon of spectral leakage: window patterns of frequencies interfere with each other. If a dataset covers 90% or more of their beat period⁵, the spectral leakage between two signals will be very small, but for datasets that only partially sample beat periods, separating one frequency from another becomes very difficult or impossible (Kurtz 1998, O'Donoghue 1998, Kleinman 1998a, Yuille 1998).

The only way to hope to analyse a transform of partially-resolved, closely-spaced frequencies is to fit the frequencies simultaneously. One starts by finding the first two signals using the method described above, and then fitting them simultaneously with a nonlinear fitting routine that looks for the best possible fit, and then prewhitening the original (unprewhitened) data with these refined parameters. A third signal is then located in the residuals, and all three are fitted simultaneously to the data, first with a linear least squares routine to find better values for their parameters, which are then refined by fitting with the nonlinear algorithm. Only when all the signals present in the data can be extracted, parameterized and simultaneously fitted to the data successfully, can a scientific analysis of them begin.

The EAGLE routines that perform these fits are lstsq (for least squares fitting) and nonlin (for nonlinear fitting). I will give a brief outline of the mathematics behind these routines, from Press et al. (1992).

A linear least squares routine finds the best-fitting values of the parameters ak of a set of functions

M

Yi(xi) =

L

^{akXk(xi) (5.5)}

k=l

fitted to a set of data points. The basis functions, X₁(x), ... , XM(x), are arbitrary fixed functions of x. In our case, the functions are sines and cosines, the YiS are the magnitudes, the xis are the times, and the ak are the amplitudes and phases (in the

5The beat period P of two closely-spaced signals with periods P1 and P2 is :}5 =

t, - A .

^The

more closely spaced two frequencies are in period, the longer their beat period is.

lstsq routine, the frequencies are fixed).

We define the merit function

(5.6)

The best values for the parameters ak are those that minimize

x

^2. To minimize (5.6), we construct a matrix A whose N x M components are theM basis functions evaluated at the N abscissas xi, or

(5.7)

called the design matrix of the fitting problem.

The minimum of (5.6) occurs where the derivative of

x

^{2 with respect} to all M parameters ^ak vanishes. This condition yields the M equations

k=l, ... ,M, (5.8)

which, by interchanging the order of summations, we can write as

M

L akjaj

=

f3k or equivalently

[a]· a= [/3]

^(5.9)

j==l

where

N

akj

=

L Xj(xi)Xk(xi) or equivalently [a]=AT·A (5.10)

i==l

an M x M matrix, and

N

f3k

=

LYiXk(xi) or equivalently

[/3]

=AT. b (5.11)

i==l

a vector of length M. We have defined a vector b of length N by bi = Yi, and denoted theM vector whose components are the parameters to be fitted, a1, . . . , aM, by a.

Equations (5.8) or (5.9) are called the normal equations of the least squares

problem. They can be solved for the vector a by any of a selection of standard methods well-suited to computers, see for example Chapter 2 of Press et al. (1992).

In the nonlin routine, the frequencies as well as the amplitudes and phases are allowed to vary as the routine hunts for the best fit. I will now outline the mathemat- ical techniques underlying nonlinear fitting. The approach is the same as outlined above - defining a

x

² merit function and determining the best-fit parameters by its minimization - except that now the model depends nonlinearly on the parameters a1 , . . . , aM. With nonlinear deependencies, the minimization must proceed itera- tively. Given a set of starting values for ^ak (I obtain this by first invoking the lstsq command to obtain a simultaneous linear fit), one needs a procedure that improves the initial solution. The procedure is repeated until

x

² stops (or effectively stops) decreasing.

The procedure amounts to taking a step down the gradient, towards a "minimum energy solution", in other words,

(5.12) where the constant is small enough not to exhaust the downhill direction. Derivations and detailed discussion can be found in chapters 10 and 15 of Press et al. (1992).

Translating this equation into a computer-usable form, and fitting a large set of sinusoids to the vast datasets generated by multisite campaigns, generates very large matrices, which are very demanding in terms of computer memory and processing power. Prior to 1998, we ran these routines on an alpha machine, but this year Darragh O'Donoghue helped me recompile the code to enable them to be run on a Pentium II PC, whose greater power allowed a refinement of nonlin, which has increased its ability to find solutions by a factor of 5.