Statistics, Data Mining, and Machine Learning in Astronomy 452 • Chapter 10 Time Series Analysis 0 6 0 7 0 8 0 9 1 0 1 1 a 0 0 0 5 1 0 1 5 φ 7 8 9 10 11 12 13 r0 3 8 3 9 4 0 4 1 4 2 ω 0 6 0 7 0 8 0 9[.]
• Chapter 10 Time Series Analysis 1.1 20 1.0 15 r(t) 452 a 0.9 10 0.8 0.7 0.6 10 t 1.5 φ 1.0 0.5 0.0 4.2 ω 4.1 4.0 3.9 3.8 10 11 12 13 0.6 r0 0.7 0.8 0.9 a 1.0 1.1 0.0 0.5 1.0 1.5 φ Figure 10.24 Modeling time-dependent flux based on arrival time data The top-right panel shows the rate r (t) = r [1 + a sin(ωt + φ)], along with the locations of the 104 detected photons The remaining panels show the model contours calculated via MCMC; dotted lines indicate the input parameters The likelihood used is from eq 10.83 Note the strong covariance between φ and ω in the bottom-right panel 10.4 Temporally Localized Signals A case frequently encountered in practice is a stationary signal with an event localized in time Astronomical examples include the magnification due to gravitational microlensing, and bursts of emission (where the source brightness increases and then decreases to the original level over a finite time interval), and the signature of a gravitational wave in data from LIGO (and other gravitational wave detectors) When the noise properties are understood, and the expected shape of the signal is known, a tool of choice is full forward modeling That is, here too the analysis includes model selection and parameter estimation steps, and is often called a matched filter search Even when the shape of the matched filter is not known, it can be treated in a nonparametrized form as was discussed in the context of arrival time data Similarly, even when a full understanding of the noise is missing, it is possible 10.4 Temporally Localized Signals • 453 to marginalize over unknown noise when some weak assumptions are made about its properties (recall the example from §5.8.5) We will discuss two simple parametric models here: a burst signal and a chirp signal In both examples we assume Gaussian known errors The generalization to nonparametric models and more complex models can be relatively easily implemented by modifying the code developed for these two examples 10.4.1 Searching for a Burst Signal Consider a model where the signal is stationary, y(t) = b0 +, and at some unknown time, T , it suddenly increases, followed by a decay to the original level b0 over some unknown time period Let us describe such a burst by y B (t|T, A, θ ) = A g B (t − T |θ), (10.85) where the function g B describes the shape of the burst signal (g B (t < 0) = 0) This function is specified by a vector of parameters θ and can be analytic, tabulated in the form of a template, or treated in a nonparametric form Typically, MCMC methods are used to estimate model parameters For illustration, we consider here a case with g B (t|α) = exp(−αt) Figure 10.25 shows the simulated data and projections of posterior pdf for the four model parameters (b0 , T , A, and α) Other models for the burst shape can be readily analyzed using the same code with minor modifications Alternatively, the burst signal could be treated in the case of arrival time data, using the approach outlined in §10.3.5 Here, the rate function is not periodic, and can be obtained as r (t) = ( t)−1 y(t), where y(t) is the sum of the stationary signal and the burst model (eq 10.85) 10.4.2 Searching for a Chirp Signal Here we consider a chirp signal, added to a stationary signal b0 , y(t) = b0 + A sin[ωt + βt ], (10.86) and analyze it using essentially the same code as for the burst signal Figure 10.26 shows the simulated data and projections of posterior pdf for the four model parameters (b0 , A, ω, and β) Note that here the second term in the argument of the sine function above (βt ) produces the effect of increasing frequency in the signal seen in the top-right panel The resulting fit shows a strong inverse correlation between β and ω This is expected because they both act to increase the frequency: starting from a given model, slightly increasing one while slightly decreasing the other leads to a very similar prediction Figure 10.27 illustrates a more complex ten-parameter case of chirp modeling The chirp signal is temporally localized and it decays exponentially for t > T : yC (t|T, A, φ, ω, β) = A sin[φ + ω(t − T ) + β(t − T )2 ] exp[−α(t − T )] (10.87) • Chapter 10 Time Series Analysis 12 hobs 10 A 17 16 15 14 13 12 11 10 20 40 60 80 100 t 54 T 52 50 48 46 0.25 0.20 0.15 α 454 0.10 0.05 0.00 9.5 10.0 10.5 b0 11.0 10 12 46 A 48 50 T 52 54 Figure 10.25 A matched filter search for a burst signal in time series data A simulated data set generated from a model of the form y(t) = b0 for t < T and y = b0 + A exp[−α(t − T )] for t > T , with homoscedastic Gaussian errors with σ = 2, is shown in the top-right panel The posterior pdf for the four model parameters is determined using MCMC and shown in the other panels The signal in the absence of chirp is taken as y(t) = b0 + b1 sin(1 t) sin(2 t) (10.88) Here, we can consider parameters A, ω, β, and α as “interesting,” and other parameters can be treated as “nuisance.” Despite the model complexity, the MCMCbased analysis is not much harder than in the first simpler case, as illustrated in figure 10.28 In both examples of a matched filter search for a signal, we assumed white Gaussian noise When noise power spectrum is not flat (e.g., in the case of LIGO data; see figure 10.6), the analysis becomes more involved For signals that are localized not only in time, but in frequency as well, the wavelet-based analysis discussed in §10.2.4 is a good choice A simple example of such an analysis is shown in figure 10.28 The two-dimensional wavelet-based PSD easily recovers the increase of characteristic ... the burst signal could be treated in the case of arrival time data, using the approach outlined in §10.3.5 Here, the rate function is not periodic, and can be obtained as r (t) = ( t)−1 y(t), where... effect of increasing frequency in the signal seen in the top-right panel The resulting fit shows a strong inverse correlation between β and ω This is expected because they both act to increase the... shown in the top-right panel The posterior pdf for the four model parameters is determined using MCMC and shown in the other panels The signal in the absence of chirp is taken as y(t) = b0 + b1 sin(1