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 ID: 551072
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Slide1
Statistical properties ofRandom time series (“noise”)Slide2
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)Slide3
From time series to Gaussian parameters
N=50: <
z(t
)>=5.57 (11%); <(z(t)-<z
>)2>=3.10N=500: <z(t
)>=6.23 (4%); <(z(t)-<z
>)2>=3.03N=104: <z(t)>=6.05 (0.8%); <(
z(t)-<z>)
2>=3.06Slide4
Divide and conquer
Treat N=10
4
points as 20 sets of 500 pointsCalculate:
mean of means: E{m
}=<mk
>=5.97 std of means: sm=<(
m-E{
m})2
k>=0.13Compare with N=500: <z(t)>=6.23; <z
2
(t)>=3.03
N=10
4
: <
z(t
)>=6.05; <z
2
(t)>=3.06
1/√500=0.04; 2
s
m
/
E{
m}=0.04Slide5
Generic definitions (for any kind of
ergodic
, stationary noise)
Auto-correlation function
For normal distributions:Slide6
Autocorrelation function of a normal distribution (boring)Slide7
Autocorrelation function of a normal distribution (boring)Slide8
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.
IFSlide9Slide10
Discrete and finite time series
Slide11
Take a time series of total time T, with sampling
D
t
Divide it in N segments of length T/NCalculate FT of each segment, for
Df=N/TCalculate
S(f) the average of the ensemble of FTsWe can have few long segments (more uncertainty, more frequency resolution), or many short segments (less uncertainty, coarser frequency resolution)Slide12