In this lecture we consider four important properties of time series analysis. 1. Determination of the oscillation phase.

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Domenic Hopkins
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1 In this lecture we consider four important properties of time series analysis. 1. Determination of the oscillation phase. 2. The accuracy of the determination of phase, frequency and amplitude. 3. Issues related to the data sampling, e.g. gaps in the data. 4. The use of statistical weights and lowering the noise by use of statistical weights, The phase information can be understood if we consider a vector that represent an oscillation. The amplitude (a) is the length of the oscillation, and α and β are the two orthogonal vectors representing the oscillation vector a r The phase φ is the angel at t=0. The vector a r is a function of time. We may calculate α and β as the RMS minimization (fit to data) for any and given cyclic frequency f or angular frequency ν

2 Since we may represent the oscillation by vector a r (t) we may understand the process of determining the phase, frequency and amplitude for a given oscillation as a process where we minimize the spread (error) between a model for a r (t) with frequency f, phase φ and amplitude a and the observed data points. Let us consider the following example of an oscillation with measurement noise attached. The amplitude spectrum clearly shows the oscillation as a single strong peak. Scaling the spectrum such that we may see the background indicate that there are noise peaks present (with random phase and amplitude).

3 In order to estimate the errors for RMS (least squares fitting) fitting (amplitude/power spectrum extraction) we may now consider the model above and calculate the errors on amplitude, phase and frequency from the vector model. With no noise the least squares solution can be seen in the spectrum as: Amplitude: high of peak Frequency: Position of peak Phase: the angel of vector (α,β) at time equal zero. We will now consider a normal distributed noise source with mean zero and standard deviation σ. Such a distribution will have mean power equal 4σ 2 /N and mean amplitude equal σ π / N This needs to be compared to the amplitude a (power a 2 ). In the paper by M. H. Montgomery and D. O'Donoghue, 1999, DSSN 13. A derivation of the errors for least squares fitting to time series data the errors are calculated for a simple normal distributed noise source.

4 We will not go through the detailed calculations in that paper, but we will discuss the implications of this paper. As one can see there is a link between the error on the determination of the amplitude and the mean noise level in the background. Even if the noise is not strictly normal distributed we may use this as a way to estimate the error on the amplitude determination. The error on the phase is simply the ratio between the noise on the amplitude and the amplitude. This is simple to understand if one remember the vector model discussed above. Finally the error on the frequency is related to the error on the phase (which is obvious from the vector model) and the length of time. The longer we measure the more accurate we may measure. Note that since σ ( ϕ) 1/ T One will find that σ ( f ) T 3 / 2 The next property we will discuss is related to data sampling e.g. gaps in the data.

5 We first consider a simple harmonic oscillation. As described before we can measure α and β and from this we can calculate the phase and the power as α 2 + β 2 The amplitude spectrum is the calculated as 2 2 α + β

If we look at the oscillator and compare this to an oscillation that shows a slightly different frequency, we will find that the two oscillations fit each other quite well at some periods in time.

6 If we look at the oscillator and compare this to an oscillation that shows a slightly different frequency, we will find that the two oscillations fit each other quite well at some periods in time. and are out of phase at other times. Let us assume that we only observe the harmonic oscillator at the periods where the two oscillations agree and that no data exist in between (i.e. we have gabs in the data). In this case we will find a good fit even for frequencies that are far from the correct one.

7 The amplitude spectrum will then show peaks not only for the main frequency but also for frequencies that are separated by 1/d where d is the time between two observing windows. The result is that the amplitude spectrum for a single harmonic oscillator will show a series of peaks. If we assume that we observe once per day (which is true for an observer at a single observatory) the distance between the peak structure is mhz.

8 The final aspect of time series analysis we will discuss in this lecture is statistical weighting. If we again consider a time series consisting of a simple harmonic oscillator and noise on each data point, we may give higher weights to data points that show lower noise and lower weights to bad data points. By doing this we may lower the noise but keep the power in the signal. If we assume that we know the statistical quality of each data point we can then simply include this quality as a statistical weight that one shall include in the calculation of s, c, ss, cc and sc.

Compulsory Exercise no. 1 Deadline: 1 May 2014

Side 1 hans@phys.au.dk 6 April 014 Compulsory Exercise no. 1 Deadline: 1 May 014 The goal of the present compulsory exercise is to construct software that can be used for time series analysis. In order