Statistical properties ofRandom time series (“noise”)

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)

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

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

Generic definitions (for any kind of ergodic, stationary noise)

• Auto-correlation function

For normal distributions:

Autocorrelation function of a normal distribution (boring)

Autocorrelation function of a normal distribution (boring)

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

Discrete and finite time series

• 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)