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[(Templeton,)-1002(JAAVSO)-258(Volume)-246(32,)-258(2004)]TJ
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(Time-Series Analysis of Variable Star Data)Tj
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(Matthew Templeton)Tj
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(AAVSO, 25 Birch Street, Cambridge, MA 02138)Tj
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(Based on a workshop session at the 92nd Spring Meeting of the AAVSO, April 25,)Tj
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(2003; revised August 2004)Tj
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(Abstract)Tj
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(Time-series analysis is a rich field of mathematical and statistical)Tj
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(analysis, in which physical understanding of a time-varying system can be gained)Tj
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(through the analysis of time-series measurements. There are several different)Tj
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(techniques of time-series analysis that can be usefully applied to variable star data)Tj
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[(sets. Some of these techniques are particularly useful for data found in the AA)79(VSO)]TJ
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(International Database. In this paper, I give a broad overview of time-series analysis)Tj
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(techniques useful for variable star data, along with some practical suggestions for)Tj
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(the application of different techniques to different types of variables. Included are)Tj
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(elementary discussions of traditional Fourier methods, along with wavelet and)Tj
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(autocorrelation analysis.)Tj
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[(1.)-245(Introduction)]TJ
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(Time-series analysis is the application of mathematical and statistical tests to)Tj
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(time- varying data, both to quantify the variation itself and to learn something about)Tj
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(the behavior of the system. Ultimately, the goal of time-series analysis is to gain)Tj
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(some physical understanding of the system under observation: )Tj
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(what makes the)Tj
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(system time-variable?; what makes the system similar to or different from other)Tj
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(systems?; is the system predictable?; and can we place reliable limits on the)Tj
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(behavior of the system?)Tj
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(Clearly, simple forms of \223time-series analysis\224 were known in ancient times,)Tj
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(since many ancient civilizations made accurate predictions of various cyclical)Tj
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(celestial phenomena. The birth of modern time-series analysis dates to the early 19th)Tj
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(century, with Joseph Fourier\222s description of the Fourier series, and later, the Fourier)Tj
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(transform. \(Carl Friedrich Gauss first derived the Fast Fourier Transform around)Tj
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[(1805, before Fourier published his work, but Gauss did not publish his results.\) The)]TJ
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(greatest advances in time-series analysis coincided with the development of)Tj
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(computing machines and the digital computer in the mid-20th century. The digital)Tj
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(computer made possible the statistical analysis of large amounts of data in much less)Tj
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(time than would be possible by human calculators. Along with this came the)Tj
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(development of more efficient algorithms for time- series analysis \(like the rediscovery)Tj
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(of Gauss\222 Fast Fourier Transform by Danielson and Lanczos in the 1940s\), and the)Tj
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(development of new ideas such as wavelet analysis and chaos theory. Today, time-)Tj
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(series analysis is regularly applied to a wide variety of problems in the real world,)Tj
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(from radio and telecommunications engineering to financial forecasting.)Tj
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[(Templeton,)-1002(JAAVSO)-258(Volume)-246(32,)-258(2004)]TJ
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( ,)Tj
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(In this paper, I will give a basic overview of several topics in time-series and)Tj
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(statistical analysis relevant to variable star research. I will begin with a discussion)Tj
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(of the Fourier transform and its many implementations and uses. Then I will discuss)Tj
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(two other important types of time-series analysis: )Tj
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(, which is useful)Tj
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(, which)Tj
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(is useful for systems which may not exhibit coherent periodic behavior but)Tj
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(nevertheless have characteristic periods. I conclude with a summary and a list of)Tj
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[(2.)-245(Fourier analysis and the Fourier transform)]TJ
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[(Fourier analysis is the technique of using an infinite number of sine and cosine)]TJ
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(functions with different periods, amplitudes, and phases to represent a given set of)Tj
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(numerical data or analytic function. In so doing, you can estimate the period \(or)Tj
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(periods\) of variability by determining which of these functions are statistically)Tj
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(.)Tj
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(If we have a set of time-varying data, given by )Tj
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[(x)-46(\(t\))]TJ
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(,)Tj
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(, is given by the integral)Tj
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(where )Tj
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[( is )29(the )]TJ
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(, defined as )Tj
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[( is the imaginary square root of)-39( )]TJ
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[(\))-250(=)-247(cos\()]TJ
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(\) .)Tj
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(\(For a more complete discussion of the mathematics behind the Fourier transform,)Tj
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(see Bracewell \(2000\).\))Tj
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(If a set of data, )Tj
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(')Tj
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(value of the Fourier transform, )Tj
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(, should reach a local maximum at )Tj
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(n)Tj
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(')Tj
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(. If the data)Tj
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(contain several signals with different frequencies, then )Tj
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(F\()Tj
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(\))Tj
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[( should have local)]TJ
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(maxima at each, with the global maximum at the frequency having the largest)Tj
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(amplitude.)Tj
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(The Fourier transform is an extremely powerful yet elegant technique that is used)Tj
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(in many areas of mathematical analysis and the physical sciences. However, as one)Tj
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(might expect, its power is finite, limited by the amount and quality of data that are)Tj
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(transformed. The data place several limits on the usefulness of the transform,)Tj
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(including the maximum and minimum periods testable, the accuracy of the period)Tj
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(As an example, consider the following case: you have a data set spanning 5000)Tj
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[(Templeton,)-1002(JAAVSO)-258(Volume)-246(32,)-258(2004)]TJ
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(43)Tj
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(2.4 hours\). The maximum period detectable in this case is 5000 days, since the data)Tj
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(should cover one complete cycle. However, this detection would be very unreliable)Tj
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(since without additional data you have no idea whether the variation detected was)Tj
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(truly periodic or simply a short-term fluctuation that merely looks like a 5000-day)Tj
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(period. A more reliable limit is 5000/2 or 2500 days, since you could detect two)Tj
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(complete cycles at that period within the data set. The span of the data set also)Tj
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(determines the resolution of the Fourier transform, which is the precision to which)Tj
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(the frequency \(or period\) may be determined. The sampling theorem defines the set)Tj
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(of frequencies that may be measured by a given data set, and the separation between)Tj
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(two adjacent frequencies defines the resolution of the transform. The resolution is)Tj
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(defined by)Tj
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[( is the total number of samples \(50000\), and )]TJ
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(D)Tj
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[( is the space between the)]TJ
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(samples \(0.1 day\). In terms of frequency, the resolution is simply the inverse of the)Tj
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(span of the data; if the data span 5000 days, the frequency resolution is 1/5000 d)Tj
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(1)Tj
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(.)Tj
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[(An example of this is shown in Figure 1)73(,)0( where the peak of the Fourier transform of)]TJ
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(R CVn is shown for two different data sets, having different spans. The data set with)Tj
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(the longer span clearly provides a much more precise determination of the period)Tj
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(when studying long period variables.)Tj
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(The minimum period detectable by our example data set is 0.2 day, corresponding)Tj
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(to a maximum frequency of 5 cycles per day. This is because your )Tj
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(sampling)Tj
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(frequency)Tj
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[( of 10 points per day would \(potentially\) allow you to detect the object)]TJ
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(at maximum and minimum once each cycle. This is known as the )Tj
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(Nyquist frequency)Tj
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(.)Tj
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0.2 Tw
(The Nyquist frequency is important not only because it defines the highest)Tj
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(frequency \(and shortest period\) detectable with a given dataset, but also because)Tj
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(it defines the maximum sampling rate you need in order to fully describe variations)Tj
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(up to the maximum frequency.)Tj
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0.035 Tw
(In certain circumstances, it )Tj
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(is)Tj
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[( possible to detect frequencies higher than the)]TJ
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(sampling rate. The transform will suffer from )Tj
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(aliasing)Tj
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(, in which several different)Tj
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(peaks appear in the transform, along with the real one. The alias peaks are separated)Tj
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(from the true frequency by integer multiples of the sampling frequency, such that)Tj
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(the transform will look like a \223picket fence\224 when plotted. In the case of regular)Tj
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(sampling, the alias peaks will have equal statistical significance to the real peak, and)Tj
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(it is therefore impossible to tell which peak in the power spectrum is the correct one.)Tj
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(In the case of uneven sampling, you will still have aliasing, but the strengths of the)Tj
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(alias peaks will generally be lower than that of the dominant one. An example of this)Tj
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(can be seen in Figure 2, showing the Fourier transform of )Tj
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(MACHO)Tj
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[( \(Massive Compact)]TJ
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(Halo Objects\) observations of a )Tj
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[( Scuti star \(Alcock )]TJ
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(et al.)Tj
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[( 2000\). Although the )]TJ
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(sampling rate was very low \(one observation per day, on average\), )Tj
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(d)Tj
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[( Scuti variations)]TJ
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(could be detected because of the uneven sampling and nearly complete phase coverage.)Tj
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(low-amplitude stars even if the data are noisy, while in other cases, the available data)Tj
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T*
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(in proportion to the square of the number of data points, )Tj
/F6 1 Tf
21.936 0 TD
(N)Tj
/F9 1 Tf
0.672 0 TD
-0.095 Tw
(. However, as mentioned)Tj
-22.608 -1.2 TD
0.116 Tw
(in the introduction, Gauss invented a very fast implementation of the Fourier)Tj
T*
-0.117 Tw
(transform, which was later re-invented by Danielson and Lanczos in the 1940s, and)Tj
T*
0.099 Tw
(is now known as the )Tj
/F6 1 Tf
9.012 0 TD
(fast Fourier transform)Tj
/F9 1 Tf
9.324 0 TD
[( or FFT. This method permits the)]TJ
-18.336 -1.2 TD
-0.049 Tw
(extremely fast computation of Fourier transforms, since the calculation time only)Tj
T*
-0.123 Tw
(increases as )Tj
/F6 1 Tf
4.824 0 TD
-0.12 Tw
(N log)Tj
5.8 0 0 5.8 213.24 398.16 Tm
(2)Tj
10 0 0 10 216.24 401.52 Tm
0 Tc
(\(N\))Tj
/F9 1 Tf
1.332 0 TD
0.006 Tc
-0.124 Tw
(, rather than the )Tj
/F6 1 Tf
6.144 0 TD
(N)Tj
5.8 0 0 5.8 297.6 404.88 Tm
(2)Tj
/F9 1 Tf
10 0 0 10 300.6 401.52 Tm
-0.123 Tw
[( of discrete methods. The FFT is frequently)]TJ
-15.66 -1.2 TD
0 Tc
-0.125 Tw
(used in large-scale data analysis, and in \223real- time\224 Fourier analysis \(like laboratory)Tj
T*
0.005 Tc
(spectrum analyzers\). Its major drawback is that it )Tj
/F6 1 Tf
19.212 0 TD
(requires)Tj
/F9 1 Tf
3.372 0 TD
0.004 Tc
[( even data-sampling; you)]TJ
-22.584 -1.2 TD
0.006 Tc
-0.108 Tw
(must either sample your data evenly \(a rarity in long- term variable star observing\),)Tj
T*
-0.106 Tw
(or else re-grid your data \(which introduces errors\). Given the computational power)Tj
T*
-0.061 Tw
(available with even basic home computers, the use of the fast Fourier transform is)Tj
T*
-0.019 Tw
(no longer a necessity in time-series analysis, even for relatively large data sets.)Tj
1.572 -1.2 TD
0.188 Tw
(The Fourier transform has many uses in variable star research. Its most)Tj
-1.572 -1.2 TD
0.027 Tw
(fundamental use is in finding periods in data. In the case of monoperiodic data,)Tj
T*
-0.05 Tw
(Fourier analysis should reveal the dominant period if the amplitude is a sufficient)Tj
T*
-0.007 Tc
-0.125 Tw
(fraction of the noise level. However, it is rare that real stars have purely monoperiodic,)Tj
T*
0 Tc
(sinusoidal light curves, and Fourier analysis can reveal additional information about)Tj
T*
0.006 Tc
0.059 Tw
(the variability. For example, Fourier analysis is useful for analyzing stars with)Tj
T*
0.179 Tw
(multiple periods, as many types of pulsating variables have. Or, if a star is)Tj
T*
0.155 Tw
(monoperiodic, but has a light curve that is non- sinusoidal, then the Fourier)Tj
T*
0.108 Tw
(transform can provide the amplitudes and phases of the )Tj
/F6 1 Tf
23.616 0 TD
0.11 Tw
(Fourier harmonics)Tj
/F9 1 Tf
7.788 0 TD
(\227)Tj
-31.404 -1.2 TD
-0.004 Tc
-0.125 Tw
(signals at integer multiples of the fundamental frequency that distort the fundamental)Tj
T*
0.005 Tc
-0.046 Tw
[(sinusoid\227which can in turn provide information about the physical properties of)]TJ
T*
-0.063 Tw
(the star \(see Simon and Lee 1981\). The amplitude and phase information from the)Tj
T*
0.002 Tc
-0.125 Tw
(Fourier harmonics are commonly used in the analysis of pulsators like Cepheids and)Tj
T*
0.005 Tc
0.052 Tw
(RR Lyrae stars, providing information on the pulsation mode type, metallicity,)Tj
T*
0.131 Tw
(evolutionary state, and luminosity \(see Morgan 2003 and references therein\).)Tj
T*
-0.074 Tw
(Fourier analysis can provide other important information, such as the evolution of)Tj
ET
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BT
/F6 1 Tf
10 0 0 10 306 669.36 Tm
0 0 0 rg
BX /GS2 gs EX
0.006 Tc
[(Templeton,)-1002(JAAVSO)-258(Volume)-246(32,)-258(2004)]TJ
-16.2 0 TD
0.008 Tc
(48)Tj
/F9 1 Tf
0 -2.94 TD
-0.002 Tc
-0.125 Tw
(periods and amplitudes over time \(Foster 1995\), and the physical origin of variability)Tj
0 -1.2 TD
0.006 Tc
-0.08 Tw
(in aperiodic variables like accretion-powered sources \(e.g., van der Klis 1995\).)Tj
/F8 1 Tf
0 -2.4 TD
0.018 Tc
-0.125 Tw
[(4.)-246(Wavelet analysis)]TJ
/F9 1 Tf
1.572 -2.196 TD
0.006 Tc
-0.118 Tw
[(Wavelet analysis and the wavelet transform are relatively recent developments)]TJ
-1.572 -1.2 TD
-0.039 Tw
(in time-series analysis. The development of the wavelet transform came from the)Tj
T*
0.026 Tw
(need to analyze signals that were transient and/or non-sinusoidal in nature. The)Tj
T*
-0.02 Tw
(wavelet transform of a set of time-series data, )Tj
/F6 1 Tf
18.612 0 TD
0.001 Tc
(x\(t\))Tj
/F9 1 Tf
1.404 0 TD
0.005 Tc
-0.022 Tw
(, is given by)Tj
-16.296 -1.8 TD
-0.005 Tc
(W\()Tj
/F10 1 Tf
10 0 2.1 10 193.92 528 Tm
(w)Tj
/F9 1 Tf
10 0 0 10 201.72 528 Tm
(,)Tj
/F10 1 Tf
10 0 2.1 10 204.24 528 Tm
(t)Tj
/F9 1 Tf
10 0 0 10 210.24 528 Tm
0.007 Tc
-0.05 Tw
[(; x)-69(\()]TJ
/F6 1 Tf
1.416 0 TD
(t)Tj
/F9 1 Tf
0.324 0 TD
0.006 Tc
[(\)\))-246(=)]TJ
/F10 1 Tf
10 0 2.1 10 245.16 528 Tm
-0.002 Tc
(w )Tj
/F9 1 Tf
5.8 0 0 5.8 253.2 531.36 Tm
(\275)Tj
/F10 1 Tf
10 0 0 10 260.04 528 Tm
(\362)Tj
/F9 1 Tf
0.528 0 TD
0.007 Tc
(x\()Tj
/F6 1 Tf
0.852 0 TD
(t)Tj
/F9 1 Tf
0.324 0 TD
0.254 Tc
-0.308 Tw
[(\)f)251( *\()]TJ
/F10 1 Tf
10 0 2.1 10 299.28 528 Tm
(w)Tj
/F9 1 Tf
10 0 0 10 306.12 528 Tm
0.005 Tc
(,\()Tj
/F6 1 Tf
0.6 0 TD
(t)Tj
/F10 1 Tf
0.276 0 TD
(-)Tj
10 0 2.1 10 320.4 528 Tm
(t)Tj
/F9 1 Tf
10 0 0 10 326.04 528 Tm
0.008 Tc
-0.06 Tw
(\)\) d)Tj
/F6 1 Tf
1.38 0 TD
(t)Tj
/F9 1 Tf
1.284 0 TD
0.006 Tc
-0.058 Tw
(\(eq. 3; Foster 1996\))Tj
-20.868 -1.8 TD
[(wher)6(e)]TJ
/F10 1 Tf
10 0 2.1 10 169.56 510 Tm
(w)Tj
/F9 1 Tf
10 0 0 10 178.92 510 Tm
-0.114 Tw
(is a test frequency, )Tj
/F10 1 Tf
10 0 2.1 10 252.96 510 Tm
(t)Tj
/F9 1 Tf
10 0 0 10 257.4 510 Tm
[( )-87(is a \223lag\224 time or a position within the light curve, and)]TJ
-11.34 -1.2 TD
-0.044 Tw
(the function,)Tj
/F6 1 Tf
5.328 0 TD
(f)Tj
/F9 1 Tf
0.288 0 TD
-0.049 Tw
[(, is called the \223mother wavelet\224)41(\227a function which determines how)]TJ
-5.616 -1.2 TD
0 Tc
-0.125 Tw
(the signal should vary with time, frequency, and position within the light curve. \(The)Tj
T*
0.006 Tc
0.135 Tw
(\223*\224 indicates the )Tj
/F6 1 Tf
7.428 0 TD
(complex conjugate)Tj
/F9 1 Tf
7.752 0 TD
[( of the function, )]TJ
/F6 1 Tf
7.272 0 TD
(f)Tj
/F9 1 Tf
0.288 0 TD
0.137 Tw
(, is used.\) The wavelet)Tj
-22.74 -1.2 TD
0.113 Tw
(transform is )Tj
/F6 1 Tf
5.352 0 TD
0.11 Tw
(extremely flexible)Tj
/F9 1 Tf
7.296 0 TD
0.114 Tw
[( because the mother wavelet can be nearly any)]TJ
-12.648 -1.2 TD
-0.086 Tw
(function at all. This means, for example, one can include both a specific waveform)Tj
T*
-0.038 Tw
(\(e.g., sinusoid\) to search for a periodicity, and a time-varying weighting function)Tj
T*
-0.102 Tw
(\(like a sliding window\) to study the time-dependence of the signal. In this way, one)Tj
T*
-0.08 Tw
(could study both the frequency spectrum of a given signal, as well as the evolution)Tj
T*
-0.026 Tw
(of that spectrum as a function of time.)Tj
1.572 -1.2 TD
0.163 Tw
(This analysis method has great utility in several areas of astronomy and)Tj
-1.572 -1.2 TD
-0.045 Tw
(astrophysics, since many objects have varying periods, or have no fixed period at)Tj
T*
-0.074 Tw
(all and instead show transient periodicities or quasiperiodicities. For example, the)Tj
T*
-0.007 Tw
(long-period Mira stars have long been known to exhibit slightly varying periods)Tj
T*
-0.045 Tw
(from cycle to cycle, while a few of these stars \(like R Aquilae\) are known to have)Tj
T*
-0.064 Tw
(strongly varying periods indicative of evolutionary changes within the star. Other)Tj
T*
-0.068 Tw
(stars, like the semiregular variables and the RV Tauri stars, do not have a constant)Tj
T*
0.077 Tw
[(period but instead vary with one or more characteristic periods which become)]TJ
T*
-0.124 Tw
(incoherent when viewed over the full light curve. Still other stars exhibit temporary)Tj
T*
-0.118 Tw
(periods or quasiperiods, like accreting dwarf novae stars having superhumps, or X-)Tj
T*
-0.054 Tw
(ray binaries with high-frequency quasiperiodicities. In all cases, wavelet analysis)Tj
T*
-0.056 Tw
(enables you to look for transient or time-varying behavior within a given data set.)Tj
1.572 -1.2 TD
-0.01 Tc
-0.125 Tw
(However, like more traditional Fourier analysis techniques, the wavelet transform)Tj
-1.572 -1.2 TD
0.006 Tc
-0.07 Tw
(also has limitations. The major limitation, as with Fourier analysis, is that the data)Tj
T*
0.101 Tw
(set must be long enough and well-sampled enough to adequately measure the)Tj
T*
-0.076 Tw
(periods of interest. If the data only span 1000 days, it would be meaningless to test)Tj
T*
-0.034 Tw
[(periods longer than 500 days. Additionally, when the wavelet contains a window)]TJ
T*
-0.093 Tw
(function, the data should span a length of time such that the window is meaningful.)Tj
T*
-0.06 Tw
(For example, if the data span 1000 days, the period of interest is 200 days, and the)Tj
T*
-0.062 Tw
(wavelet window covers five cycles, the wavelet analysis will not give meaningful)Tj
T*
-0.018 Tw
[(information about the time evolution of the signal)-17(\227nearly all of the data will lie)]TJ
T*
-0.019 Tw
(within the window for any chosen value of )Tj
/F10 1 Tf
10 0 2.1 10 318.48 138 Tm
(t)Tj
/F9 1 Tf
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(.)Tj
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[(Templeton,)-1002(JAAVSO)-258(Volume)-246(32,)-258(2004)]TJ
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(49)Tj
/F9 1 Tf
-29.82 -2.94 TD
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[(The A)-8(A)126(VSO has a very powerful tool available for computing wavelet transforms)]TJ
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0.006 Tc
(called )Tj
7 0 0 7 171 627.96 Tm
0.01 Tc
(WWZ)Tj
10 0 0 10 188.64 627.96 Tm
0.005 Tc
0.025 Tw
(, for )Tj
/F6 1 Tf
2.004 0 TD
(weighted wavelet z-transform)Tj
/F9 1 Tf
12.036 0 TD
(. The algorithm was developed by)Tj
-18.504 -1.2 TD
-0.009 Tc
-0.124 Tw
[(Foster \(1996\) specifically with AA)125(VSO data in mind, and may be downloaded in )]TJ
7 0 0 7 447.72 615.96 Tm
-0.012 Tc
(BASIC)Tj
10 0 0 10 144 603.96 Tm
0.005 Tc
(and )Tj
7 0 0 7 160.2 603.96 Tm
0.009 Tc
[(FORTRAN)9(77)]TJ
10 0 0 10 200.4 603.96 Tm
0.005 Tc
-0.09 Tw
[( versions from our website. Foster\222s algorithm performs the wavelet)]TJ
-5.64 -1.2 TD
0.072 Tw
(transform given in equation 3, using a wavelet function which includes both a)Tj
T*
-0.12 Tw
(periodic, sinusoidal test function of the form )Tj
/F6 1 Tf
17.46 0 TD
0.008 Tc
[(exp)8(\()]TJ
/F10 1 Tf
10 0 2.1 10 335.64 579.96 Tm
0.043 Tc
[(iw)33(t)]TJ
/F6 1 Tf
10 0 0 10 351.6 579.96 Tm
0.001 Tc
(\(t)Tj
/F10 1 Tf
10 0 2.1 10 357.72 579.96 Tm
0.003 Tc
(-t)Tj
/F6 1 Tf
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-0.006 Tc
(\)\))Tj
/F9 1 Tf
0.66 0 TD
0.005 Tc
[( )123(and a Gaussian window)]TJ
-23.112 -1.2 TD
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(function of the form )Tj
/F6 1 Tf
8.208 0 TD
(exp\()Tj
/F10 1 Tf
10 0 2.1 10 243.48 567.96 Tm
(-)Tj
/F6 1 Tf
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(c)Tj
/F10 1 Tf
10 0 2.1 10 255.12 567.96 Tm
(w)Tj
/F6 1 Tf
5.8 0 0 5.8 263.64 571.32 Tm
(2)Tj
10 0 0 10 266.64 567.96 Tm
0.001 Tc
(\(t)Tj
/F10 1 Tf
10 0 2.1 10 272.76 567.96 Tm
0.015 Tc
(-t)Tj
/F6 1 Tf
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(\))Tj
5.8 0 0 5.8 287.64 571.32 Tm
(2)Tj
10 0 0 10 290.64 567.96 Tm
(\))Tj
/F9 1 Tf
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0.005 Tc
(, where both the frequency, )Tj
/F10 1 Tf
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(w)Tj
/F9 1 Tf
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-0.057 Tw
(, and the user-)Tj
-26.82 -1.2 TD
0.125 Tw
(defined constant, )Tj
/F6 1 Tf
7.428 0 TD
(c)Tj
/F9 1 Tf
0.492 0 TD
(, determine the width of the window. The algorithm fits a)Tj
-7.92 -1.2 TD
0.098 Tw
(sinusoidal wavelet to the data, but as it does so, it weights the data points by)Tj
T*
0.068 Tw
(applying the sliding window function to the data; points near the center of the)Tj
T*
-0.125 Tw
(window have the heaviest weights in the fit, while those near the edges have smaller)Tj
T*
0.016 Tw
(weights. The window slides along the data set, giving us a representation of the)Tj
T*
-0.051 Tw
(spectral content of the signal at times corresponding to the center of that window.)Tj
1.572 -1.2 TD
-0.106 Tw
[(When analyzing A)-7(A)127(VSO data with )]TJ
7 0 0 7 298.92 483.96 Tm
-0.002 Tc
[(WW)-19(Z)]TJ
10 0 0 10 316.92 483.96 Tm
0.005 Tc
-0.107 Tw
(, there are a few things to keep in mind.)Tj
-17.292 -1.2 TD
0.023 Tw
(For one, the data set should have a reasonably long time-span, preferably much)Tj
T*
-0.123 Tw
(longer than the expected period of the star. If you were interested in studying period)Tj
T*
-0.031 Tw
(evolution in a variable, it would be best to have a span of data many times longer)Tj
T*
-0.052 Tw
(\(perhaps by a factor of 50 or more\) than the mean period of the variable. This will)Tj
T*
-0.097 Tw
(allow the algorithm to slide the window over a large span of data and determine the)Tj
T*
-0.011 Tw
(best-fitting period over completely independent regions of the light curve.)Tj
1.572 -1.2 TD
-0.069 Tw
(Another thing to note is that )Tj
7 0 0 7 272.28 399.96 Tm
0.01 Tc
(WWZ)Tj
10 0 0 10 289.92 399.96 Tm
0.005 Tc
[( allows you to select the width of the window)]TJ
-14.592 -1.2 TD
-0.102 Tw
(\(via the constant, )Tj
/F6 1 Tf
6.912 0 TD
(c)Tj
/F9 1 Tf
0.492 0 TD
(\), which gives you some flexibility in the timescales you wish to)Tj
-7.404 -1.2 TD
0.09 Tw
(investigate for period changes. However, there are tradeoffs when making the)Tj
T*
-0.068 Tw
(window narrower or wider. Recall from the discussion about Fourier analysis that)Tj
T*
-0.072 Tw
[(the span of the data affects the period range and resolution. Since the data window)]TJ
T*
0.002 Tc
-0.125 Tw
(acts to change the span of the data, making the window narrower effectively reduces)Tj
T*
0.005 Tc
-0.014 Tw
(the span of the data and consequently makes the period resolution worse. But by)Tj
T*
-0.118 Tw
(doing so, you can study period changes over very short timescales. Likewise, if you)Tj
T*
-0.12 Tw
(widen the window to improve the period resolution, you worsen the time resolution)Tj
T*
-0.063 Tw
(of the transform, making it difficult to detect short-term variations.)Tj
1.572 -1.2 TD
0 Tc
-0.125 Tw
(As an example, the wavelet transforms of the semiregular variable Z Aurigae are)Tj
-1.572 -1.2 TD
(shown in Figure 3. The star is believed to undergo \223mode switches\224 where the period)Tj
T*
0.005 Tc
-0.052 Tw
(of the star suddenly changes from one period to another\227in this case from a 110-)Tj
T*
-0.09 Tw
(day to a 137-day period. Z Aurigae has made the switch between these two periods)Tj
T*
-0.063 Tw
(a few times over the past century of observations, and these mode switches can be)Tj
T*
-0.043 Tw
(easily detected with )Tj
7 0 0 7 225.84 219.96 Tm
0.01 Tc
(WWZ)Tj
10 0 0 10 243.48 219.96 Tm
(.)Tj
-8.376 -1.2 TD
0.005 Tc
0.058 Tw
(In the top panel, I\222ve chosen a wide window, which gives very fine period)Tj
-1.572 -1.2 TD
-0.048 Tw
(resolution, but has smoothed out much of the temporal variation. In fact, so much)Tj
T*
-0.12 Tw
(smoothing is used, that the transform missed two mode switches at JD 2429000 and)Tj
T*
-0.112 Tw
(2439000. The amplitude variation \(not shown\) is also greatly reduced, since it, too,)Tj
T*
-0.085 Tw
(is calculated as a weighted average of the amplitude over the entire window. In the)Tj
T*
-0.102 Tw
(bottom panel, I chose a much narrower window, which brings out the time-varying)Tj
T*
-0.074 Tw
[(nature of the spectrum. In particular, the transform has detected the two very short)]TJ
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[( )-37(to analyze your data, be aware that both the data and your)]TJ
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(toward both the limitations of your data, and the information you wish to obtain. No)Tj
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[(\(x\(t\),)-249(x\(t +)]TJ
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(t)Tj
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(. There are other methods as well, including the )Tj
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(\). Below, I discuss)Tj
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[(\223what does the light curve look like at times separated by some interval,)-6( )]TJ
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(t?)Tj
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(\224)Tj
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(What autocorrelation does is take each data point, measured at time t, and then)Tj
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(t+)Tj
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(. If you perform this test)Tj
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[(for each pair of data points separated by an interval,)6( )]TJ
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[(. If they\222re very different, there will be a trough.)8( )73(We would expect points separated)]TJ
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[( to be very similar if the data contained some variability with period)-6( )]TJ
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T*
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T*
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T*
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[(5.2)-246(Other statistical methods)]TJ
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(Autocorrelation is by no means the only statistical method for performing time-)Tj
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[( or )]TJ
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[( procedure,)]TJ
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[( program)]TJ
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[(of the AA)114(VSO\222s )]TJ
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[( kit, so you may be familiar with it already.)]TJ
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[( \(Stellingwerf)]TJ
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[(PD)-9(M)]TJ
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(,)Tj
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[(Figure 4. Two thousand days of A)-6(A)128(VSO data \(top panel\) and autocorrelation)]TJ
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(general, statistical methods are a very rich and varied field of time-series analysis,)Tj
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(be useful to you.)Tj
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[(6.)-246(Summary)]TJ
/F9 1 Tf
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(The goal of this paper was to introduce the casual reader to some basic principles)Tj
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(of time-series analysis, as well as to outline some of the considerations that go into)Tj
T*
-0.045 Tw
(selecting appropriate methods for analyzing a given set of data. The few methods)Tj
T*
-0.026 Tw
(outlined in detail here by no means give a comprehensive description of this rich)Tj
T*
-0.116 Tw
(field of research, but will hopefully serve as a starting point for your investigations.)Tj
T*
-0.065 Tw
(As with any field of research, you should investigate other methods of time-series)Tj
T*
0.198 Tw
(analysis and their applications to see whether there might be more suitable)Tj
T*
-0.058 Tw
[(alternatives to what I\222ve presented here.)-6( )42(While several different analysis methods)]TJ
T*
0.021 Tw
(may provide \223correct\224 results for a given project, there are likely other analysis)Tj
T*
-0.016 Tw
(methods unsuitable for your data, and these may give misleading results. A little)Tj
T*
-0.013 Tw
(research in advance may save you time and trouble later on.)Tj
1.572 -1.2 TD
-0.069 Tw
(Another point I wish to stress is that you should be aware of both the strengths)Tj
-1.572 -1.2 TD
-0.096 Tw
(and limitations of the data you wish to analyze )Tj
/F6 1 Tf
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(before)Tj
/F9 1 Tf
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[( you begin your analysis. The)]TJ
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(amount and quality of a given set of data may not be sufficient to detect the type)Tj
T*
-0.003 Tc
-0.125 Tw
(of stellar variability you are investigating. Measurement errors and noise place limits)Tj
T*
0.006 Tc
-0.069 Tw
(on the amplitudes detectable in a given data set, while the span and sampling rates)Tj
T*
0.048 Tw
(of data limit the precision to which periods can be defined. This is particularly)Tj
T*
-0.002 Tc
-0.125 Tw
[(important to remember when dealing with visual data from the AA)120(VSO; while visual)]TJ
T*
0.005 Tc
-0.018 Tw
(data may not be suitable for detecting small-amplitude variability, they are often)Tj
T*
-0.079 Tw
(ideally suited for studying long-term changes in pulsation behavior. Again, a little)Tj
T*
-0.119 Tw
(foreknowledge of the goals of your analysis and the limits of your data can save you)Tj
T*
-0.052 Tw
(much time and trouble later on.)Tj
/F8 1 Tf
0 -2.4 TD
0.015 Tc
-0.125 Tw
[(7.)-249(Additional resources)]TJ
/F9 1 Tf
1.572 -2.196 TD
0.006 Tc
0.19 Tw
(Beyond the information given in this paper, there are several additional)Tj
-1.572 -1.2 TD
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(resources for those interested in time-series analysis, including publicly available)Tj
T*
0 Tc
-0.125 Tw
(computer programs useful for astronomy. The best starting point for those interested)Tj
T*
-0.003 Tc
[(in analyzing AA)131(VSO data would be the publicly-available time-series analysis codes)]TJ
T*
0.005 Tc
0.002 Tw
[(available from the AA)115(VSO itself\227the Fourier analysis code )]TJ
7 0 0 7 390.96 224.04 Tm
0.016 Tc
(TS)Tj
10 0 0 10 399.24 224.04 Tm
0.006 Tc
(, and the wavelet)Tj
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(analysis code )Tj
7 0 0 7 200.28 212.04 Tm
0.01 Tc
(WWZ)Tj
10 0 0 10 217.92 212.04 Tm
0.005 Tc
-0.031 Tw
(. Both codes now exist in )Tj
7 0 0 7 321.12 212.04 Tm
(BASIC)Tj
10 0 0 10 342 212.04 Tm
-0.032 Tw
[( versions suitable for Microsoft)]TJ
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(Windows machines and in )Tj
7 0 0 7 254.76 200.04 Tm
0.009 Tc
[(FORTRAN)9(77)]TJ
10 0 0 10 294.96 200.04 Tm
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0.035 Tw
[( versions for any computer with an ANSI-)]TJ
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(compliant )Tj
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(FORTRAN)Tj
10 0 0 10 219 188.04 Tm
0.005 Tc
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[( compiler. Both are available free-of-charge from our website:)]TJ
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0.007 Tc
(http://www.aavso.org/cdata/software.stm)Tj
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(Another excellent, publicly available program is the )Tj
7 0 0 7 364.68 158.04 Tm
0.009 Tc
[(PERIOD)9(98)]TJ
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0.005 Tc
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[( software package)]TJ
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(available from the )Tj
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(d)Tj
/F9 1 Tf
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[( )-88(Scuti Network of the University of Vienna, Austria. This program)]TJ
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(was designed for the analysis of multi-periodic stars \(like )Tj
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(54)Tj
/F9 1 Tf
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(perfectly on monoperiodic variables. The website of this software package is:)Tj
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0.007 Tc
(http://www.astro.univie.ac.at/~dsn/dsn/Period98/current/)Tj
-3.048 -1.608 TD
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(Two other resources for time-series \(and other statistical analysis methods\) are the)Tj
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(Penn State University Astronomy Department\222s \223StatCodes\224 archive, available at)Tj
8.772 -1.392 TD
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(http://www.astro.psu.edu/statcodes/)Tj
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(and John Percy\222s \223Astrolab\224 page at the University of Toronto:)Tj
6.072 -1.392 TD
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(http://www.astro.utoronto.ca/~percy/analysis.htm)Tj
-6.072 -1.608 TD
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(which includes a \223self-correlation\224 analysis program, a slight variation on the)Tj
0 -1.2 TD
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(autocorrelation method described above.)Tj
1.572 -1.2 TD
0.001 Tc
-0.125 Tw
(Finally, to explore the relatively new field of non-linear time-series analysis and)Tj
-1.572 -1.2 TD
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(chaos theory, I strongly recommend investigating the )Tj
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(TISEAN)Tj
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[( \(Hegger, Kantz, and)]TJ
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(Schrieber 1999\) analysis package and its accompanying documentation. The)Tj
T*
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(package may be found at)Tj
7.2 -1.392 TD
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(http://www.mpipks-dresden.mpg.de/~tisean/)Tj
-5.628 -1.608 TD
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(The application of chaos and non-linear theory to variable star analysis is a)Tj
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[(relatively new and unexplored field, but data from the AA)127(VSO have already been)]TJ
T*
-0.056 Tw
(used to study the applicability of chaos theory to variable star research \(see Jevtic)Tj
/F6 1 Tf
T*
-0.1 Tw
(et al.)Tj
/F9 1 Tf
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[( 2003, Kollath )]TJ
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(et al.)Tj
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[( 1998, and Buchler )]TJ
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(et al.)Tj
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[( 1996\).)]TJ
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(References)Tj
/F9 1 Tf
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(Alcock, C., )Tj
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[(et)9( al.)]TJ
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[( 2000, )]TJ
/F6 1 Tf
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(, )Tj
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(536)Tj
/F9 1 Tf
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(, 798.)Tj
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(Box, G. E. P., and Jenkins, G. 1976, )Tj
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(,)Tj
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(Holden-Day Series in Time-Series Analysis, Holden-Day, San Francisco.)Tj
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/F6 1 Tf
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(The Fourier Transform and Its Applications)Tj
/F9 1 Tf
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(, 3rd edition,)Tj
-25.68 -1.2 TD
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(McGraw-Hill, Boston.)Tj
-1.572 -1.2 TD
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-0.115 Tw
(Buchler, J. R., Kollath, Z., Serre, T., and Mattei, J. A. 1996, )Tj
/F6 1 Tf
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(, )Tj
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(Fahlman, G. G., and Ulrych, T. J. 1982, )Tj
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(, )Tj
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(Ferraz-Mello, S. 1981, )Tj
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(, )Tj
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(86)Tj
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(, 619.)Tj
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(Foster, G. 1996, )Tj
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(, )Tj
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(112)Tj
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(, 1709.)Tj
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(Foster, G. 1995, )Tj
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(109)Tj
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(Hegger, R., Kantz, H., and Schreiber, T. 1999, )Tj
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(, )Tj
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[(Morgan, S. 2003, )]TJ
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(, )Tj
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(van der Klis, M. 1995, \223Rapid aperiodic variability in X-ray binaries,\224 in )Tj
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