Compulsory Exercise no. 1 Deadline: 1 May 2014

Transcription

1 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 to write the source code for the software you need to follow the detailed description given below. Time Series Analysis: The Power Spectrum In this first compulsory exercise (the first of three) the aim is to construct a computer programme that will be used throughout the rest of the course. The software needs to be flexible such that it can serve several different types of time series analysis problems. As we discussed during the lectures we define a time series as: 1. A time series is a data set that is ordered in time.. In the present course we will only consider finite time series data (N) that are discrete with data taken at the following times: t(1), t(), t(3),, t(n). 3. t(i+1) - t(i) may not be a constant. Analysing a data set that fulfils the above criterion is the aim of the present exercise. In practise we may give some requirements for the input to the software, for example one may require that the input will be an ascii-file with two columns; one containing the time and one containing the data values The software that you need to develop should be able to calculate the power spectrum of a time series (as given above). In order to allow this the user will be - as a minimum requirement - asked to specify the frequency interval for the power

2 Side spectrum as well as the frequency resolution in the power spectrum. The software may in fact suggest a series of good values for frequency band and resolution (based on the properties of the time series). As described during the lectures we will use the following algorithm for calculating the power spectrum: P( ) ( ) ( ) ( ) s cc c sc, sscc sc ( ) c ss s sc sscc sc where ν is the angular frequency and s, c, ss, cc and sc are given as (in each case you need to sum over all data points (f(t) and t) ): s f ( t) sin( t) c f ( t) cos( t) ss sin ( t) cc cos ( t) sc sin( t) cos( t) Note that the frequency ν is given as angular frequency. In the software (input and output) one may find it more natural to use the cyclic frequency f, that is given as f = 1/P, where P is the period. You are free to choose whatever software you prefer for constructing the computer programme. It may be natural to use IDL or MatLab, but FORTRAN or C will also be

3 Side 3 an excellent langue for the present software package. PASCAL, MS/EXCEL etc. are also allowed - if you prefer! The software may be build such that it runs through a looped subroutine that for each frequency calculate the power using the above equations. The Subroutine may be constructed as: In order to avoid the so-called zero-frequency that may dominate the power at a broad frequency range it is recommended to subtract the mean value of the time series before one calculate the power spectrum. It is assumed that the user specifies the frequency interval that is relevant for calculating the power spectrum (including a specification of the frequency resolution). The frequencies are specified in cyclic frequencies (f = 1/P). Calculate the power for a given frequency f (This part will then run for all user defined frequencies). o The frequency is changed from cyclic to angular frequency: ν = π f o Calculate s, c, ss, cc and sc for the frequency ν. o Calculate α and β for the frequency ν. o Calculate the power from α and β. The output from the software will be a file containing all the values for the power (or amplitude) for all the user defined frequencies. In case you use IDL or MatLab it is obvious to let the programme plot the power spectrum on the screen. In the present version of the software you are not asked to consider the phase information (calculated from α and β). When the software is ready you need to test and verify the performance of the programme. This should be done use the following three steps: 1. Calculate the power spectrum for a simple harmonic oscillator (try to vary the frequency and the phase and vary also the length of the series). In this part of the test you only need to consider an artificial time series.. Calculate the power spectrum for stellar data. At the homepage ( you will find a series of data sets that can be used to test the software.

4 Side 4 3. Plot the spectrum where you zoom in on some of the frequencies. Show that the stars are oscillating in modes that can not be characterized by a number of simple harmonic oscillations (each mode is not described by a single sinc-like function). Contents of the report of the exercise In connection to the exercise you need to prepare a report describing the results of your work. The report need to contain: A description of the software that you developed. The source code (as an appendix) and a short description of how the software was tested and verified including the power spectrum of a simple harmonic oscillator. A short description on how the software will be run. A short description of the results of analyzing the stellar data. At maximum you should use 10 pages for the report (not including the appendixes). The report may be in English or in Danish. Handing in the report The deadline for handing in the report from exercise no.1 is 1 May 014. You should submit a printed version (in one copy). Each of you will need to submit an individual report and the reports need to be independent from other students (due to marking requirements). It is however recommended that you work together in groups of or 3 students such that you can help each other.

5 Side 5 Learning Objectives The learning Objectives are of course relevant to consider when you make your software and write the report. The learning objectives are: When the course is finished the student is expected to be able to: Describe the theoretical background for a number of the techniques and methods used in time series analysis in Astrophysics. Discuss and evaluate the possibilities and applications of the methods and techniques used in time series analysis and describe the boundaries and limitations for each of the used techniques. Calculate the signal-to-noise ratios and evaluate the significance of the measured time series parameters (such as frequencies, phases and amplitudes). Discuss and describe the noise sources related to time series analysis. Develop and test a series of software routines that may be used to apply simple time series analysis. The software shall be able to do Fourier analysis, CLEAN and cross-correlation. Use the developed software for analysis of specific time series data Supervision throughout the project The supervisor for the project will be: hans@phys.au.dk Good luck with your work.