![Page 1: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/1.jpg)
Statistical properties ofRandom time series (“noise”)
![Page 2: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/2.jpg)
Normal (Gaussian) distributionProbability density:
A realization (ensemble element) as a 50 point “time series”
Another realization with 500 points(or 10 elements of an ensemble)
![Page 3: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/3.jpg)
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
![Page 4: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/4.jpg)
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{ }=0.04m
![Page 5: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/5.jpg)
Generic definitions (for any kind of ergodic, stationary noise)
• Auto-correlation function
For normal distributions:
![Page 6: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/6.jpg)
Autocorrelation function of a normal distribution (boring)
![Page 7: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/7.jpg)
Autocorrelation function of a normal distribution (boring)
![Page 8: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/8.jpg)
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
![Page 9: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/9.jpg)
![Page 10: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/10.jpg)
Discrete and finite time series
![Page 11: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/11.jpg)
• 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)
![Page 12: Statistical properties of Random time series (“noise”)](https://reader035.vdocuments.us/reader035/viewer/2022081516/56649cef5503460f949bdbd8/html5/thumbnails/12.jpg)