statistics

II.DETECTION, MEASUREMENT AND PROBABILITY —— Lee S. Finn 1992

7.1 Random Processes for Probability Theory 269 —— Jolien D. E. Creighton and Warren G. Anderson Gravitational-Wave Physics and Astronomy

As a consequence of stationarity, the covariance of the noise in the frequency domain can be expressed by (Wiener et al. 1930; Khintchine 1934) —— 2104.01897

  • Wiener N., et al., 1930, Acta mathematica, 55, 117
  • Khintchine A., 1934, Mathematische Annalen, 109, 604

optimal statistics

As the noise nptq is stationary and Gaussian, the Whittle (log) likelihood can be used (Whittle 1957) - 2104.01897

  • Whittle P., 1957, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 19, 38

PE

Finally, to quantify the precision of measurements on parameters, we will make use of the the linear signal approx- imation (LSA) (Finn 1992). By considering a small pertur- bation θ “ θtr ` ∆θ, one can expand the waveform model in the vicinity of the best-fit parameters as —— 2104.01897

  • Finn L. S., 1992, Phys. Rev. D, 46, 5236

(Flanagan & Hughes 1998), (Miller 2005) and (Cutler & Vallisneri 2007), in which expressions are provided for the error on parameters due to the presence of noise and due to waveform errors. —— 2104.01897

  • Flanagan E. E., Hughes S. A., 1998, Phys. Rev. D, 57, 4566
  • Miller M., 2005, Phys. Rev. D, 71, 104016
  • Cutler C., Vallisneri M., 2007, Phys. Rev. D, 76, 104018

GEOMETRICAL INTERPRETATION OF PARAMETER ERRORS —— 2104.01897

Overlaping paper: We qualitatively confirm one of the main results of (Samajdar et al. 2021; Pizzati et al. 2021; Himemoto, Nishizawa & Taruya 2021; Relton & Raymond 2021) in Sec. (5.2), showing that biases arise when the difference between the coalescence times of two overlap- ping signals is smaller than a fraction of a second. —— 2104.01897

  • Samajdar A., Janquart J., Van Den Broeck C., Dietrich T., 2021
  • Pizzati E., Sachdev S., Gupta A., Sathyaprakash B., 2021
  • Himemoto Y., Nishizawa A., Taruya A., 2021
  • Relton P., Raymond V., 2021

REF

Books:

  1. Jolien D. E. Creighton and Warren G. Anderson Gravitational-Wave Physics and Astronomy
  2. Sophocles J. Orfanidis. Introduction to Signal Processing
  3. ANDRE ́S ASENSIO RAMOS and IN ̃ IGO ARREGUI. BAYESIAN ASTROPHYSICS
  4. Steven M. Kay. Fundamentals of Statistical Signal Processing: Estimation Theory
  5. Steven M. Kay. Fundamentals of Statistical Signal Processing: Detection Theory

Articles:

  1. An introduction to Bayesian inference in gravitational-wave astronomy: Parameter estimation, model selection, and hierarchical models. Publications of the Astronomical Society of Australia (2019), 36, e010, 12 pages doi:10.1017/pasa.2019.2

Gravitational Waves from Coalescing Binaries - Stanislav Babak (B)

others

  • 7.1 Random Processes for Probability Theory 269

    • 7.1.1 Power Spectrum
    • 7.1.2 Gaussian Noise 273

  • 7.2 Optimal Detection Statistic 275

    • 7.2.1 Bayes’s Theorem 275
    • 7.2.2 Matched Filter 276
    • 7.2.3 Unknown Matched Filter Parameters 277
    • 7.2.4 Statistical Properties of the Matched Filter 279
    • 7.2.5 Matched Filter with Unknown Arrival Time 281
    • 7.2.6 Template Banks of Matched Filters 282

  • 7.3 Parameter Estimation 286

    • 7.3.1 Measurement Accuracy 286
    • 7.3.2 Systematic Errors in Parameter Estimation 289
    • 7.3.3 Confidence Intervals 291

  • 7.4 Detection Statistics for Poorly Modelled Signals 293

    • 7.4.1 Excess-Power Method 293

  • 7.5 Detection in Non-Gaussian Noise 295

  • 7.6 Networks of Gravitational-Wave Detectors 298

    • 7.6.1 Co-located and Co-aligned Detectors 298
    • 7.6.2 General Detector Networks 300
    • 7.6.3 Time-Frequency Excess-Power Method for a Network of Detectors
    • 7.6.4 Sky Position Localization for Gravitational-Wave Bursts 305

  • 7.7 Data Analysis Methods for Continuous-Wave Sources 307

    • 7.7.1 Search for Gravitational Waves from a Known, Isolated Pulsar 309
    • 7.7.2 All-Sky Searches for Gravitational Waves from Unknown Pulsars 316

  • 7.8 Data Analysis Methods for Gravitational-Wave Bursts 317

    • 7.8.1 Searches for Coalescing Compact Binary Sources 318
    • 7.8.2 Searches for Poorly Modelled Burst Sources 332

  • 7.9 Data Analysis Methods for Stochastic Sources 333

    • 7.9.1 Stochastic Gravitational-Wave Point Sources 344

  • Basic Signal Processing

    • Signal
      • Analog vs. Digital
      • Sampling Frequency
    • Fourier Transform
    • Convolution
      • Windowing
    • Filtering
      • LTI System
      • Transfor Function
    • Discrete Time Filtering
    • Nyquist Sampling Theorem
    • Discrete Fourier Transform
      • Fast Fourier Transform
    • DFT & Discrete Time Convolution
    • Digital Filtering
      • Digital Filter Design
    • Linear Algebra Approach to Signal Processing
      • Signal and Vector Space
      • Fourier Basis
      • Wavelet Basis
    • Time-frequency Analysis

  • Appendix Introduction to Signal Processing

    • Sampling and Reconstruction
    • Discrete-Time Systems
    • FIR Filtering and Convolution
    • z-Transforms
    • Transfer Functions
    • Digital Filter Realizations
    • Signal Processing Applications
    • DFT/FFT Algorithms
    • FIR Digital Filter Design
    • IIR Digital Filter Design

  • Appendix Bayesian Inference and Computation

    • Markov Chain Monte Carlo
    • Model Selection and Nested Sampling
    • Assigning Prior Distributions

  • Appendix Gravitational-Wave Detector Data

    • Gravitational-Wave Detector Site Data
    • Idealized Initial LIGO Model