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Statistical properties of Random time series (“noise”)

Statistical properties of Random time series (“noise”). Normal (Gaussian) distribution. Probability density: . A realization (ensemble element) as a 50 point “time series” . Another realization with 500 points (or 10 elements of an ensemble). From time series to Gaussian parameters.

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Statistical properties of Random time series (“noise”)

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  1. Statistical properties ofRandom time series (“noise”)

  2. Normal (Gaussian) distribution Probability density: A realization (ensemble element) as a 50 point “time series” Another realization with 500 points (or 10 elements of an ensemble)

  3. From time series to Gaussian parameters • N=50: <z(t)>=5.57 (11%); <(z(t)-<z>)2>=3.10 • N=500: <z(t)>=6.23 (4%); <(z(t)-<z>)2>=3.03 • N=104: <z(t)>=6.05 (0.8%); <(z(t)-<z>)2>=3.06

  4. Divide and conquer • Treat N=104 points as 20 sets of 500 points • Calculate: • mean of means: E{m}=<mk>=5.97 • std of means: sm=<(m-E{m})2k>=0.13 • Compare with • N=500: <z(t)>=6.23; <z2(t)>=3.03 • N=104: <z(t)>=6.05; <z2(t)>=3.06 • 1/√500=0.04; 2sm/E{m}=0.04

  5. Generic definitions (for any kind of ergodic, stationary noise) • Auto-correlation function For normal distributions:

  6. Autocorrelation function of a normal distribution (boring)

  7. Autocorrelation function of a normal distribution (boring)

  8. Frequency domain • Fourier transform (“FFT” nowadays): • Not true for random noise! • Define (two sided) power spectral density using autocorrelation function: • One sided psd: only for f>0, twice as above. IF

  9. Discrete and finite time series

  10. Take a time series of total time T, with sampling Dt • Divide it in N segments of length T/N • Calculate FT of each segment, for Df=N/T • Calculate S(f) the average of the ensemble of FTs • We can have few long segments (more uncertainty, more frequency resolution), or many short segments (less uncertainty, coarser frequency resolution)

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