EXSTAT

Calculation of statistical moments.

[SPU EXSTAT X PX NORM OUT M1 M2 M3 M4 N]

input	description	data type	value type	default value
X	data vector	number, vector	variable	0,1,..
PX	probability vector	number, vector	variable	1, 1, ...
NORM	normalization flag	number (int.), string	constant	0 (= NO)
output	description	data type	value type	comment
M1	1st moment (mean)	number	variable
M2	2nd moment (variance or spread)	number	variable
M2	3rd moment (skewness)	number	variable
M4	4th moment (kurtosis)	number	variable
N	number of data samples	number	constant

Note:

• At least one of the data vectors X and PX must be supplied!
• The number of data points N is set to the length of the vector X or PX.
• If X is a not connected, the x-data are initialized with x_i = i.
• If X is a number, the x-data are initialized with x_i = X+i.
• If PX is not a vector, the probabilies px_i are set to 1.

Description

μ = ∑ i = 0 N − 1 x i p x i ∑ i = 0 N − 1 p x i σ = ∑ i = 0 N − 1 ( x i − M 1 ) 2 p x i ∑ i = 0 N − 1 p x i μ = ∑ i = 0 N − 1 x i p x i ∑ i = 0 N − 1 p x i σ = ∑ i = 0 N − 1 ( x i − M 1 ) 2 p x i ∑ i = 0 N − 1 p x i {\displaystyle {\begin{matrix}\mu ={\frac {\sum _{i=0}^{N-1}x_{i}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}&\sigma ={\frac {\sum _{i=0}^{N-1}(x_{i}-M1)^{2}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}\\\mu ={\frac {\sum _{i=0}^{N-1}x_{i}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}&\sigma ={\frac {\sum _{i=0}^{N-1}(x_{i}-M1)^{2}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}\end{matrix}}} ${\begin{matrix}\mu ={\frac {\sum _{i=0}^{N-1}x_{i}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}&\sigma ={\frac {\sum _{i=0}^{N-1}(x_{i}-M1)^{2}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}\\\mu ={\frac {\sum _{i=0}^{N-1}x_{i}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}&\sigma ={\frac {\sum _{i=0}^{N-1}(x_{i}-M1)^{2}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}\end{matrix}}$

output	NORM=0	NORM=1
M1	∑ i = 0 N − 1 x i p x i ∑ i = 0 N − 1 p x i {\displaystyle {\frac {\sum _{i=0}^{N-1}x_{i}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}} ${\frac {\sum _{i=0}^{N-1}x_{i}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}$	∑ i = 0 N − 1 x i p x i ∑ i = 0 N − 1 p x i {\displaystyle {\frac {\sum _{i=0}^{N-1}x_{i}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}} ${\frac {\sum _{i=0}^{N-1}x_{i}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}$
M1	∑ i = 0 N − 1 ( x i − M 1 ) 2 p x i ∑ i = 0 N − 1 p x i {\displaystyle {\frac {\sum _{i=0}^{N-1}(x_{i}-M1)^{2}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}} ${\frac {\sum _{i=0}^{N-1}(x_{i}-M1)^{2}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}$	1 M 1 ∑ i = 0 N − 1 ( x i − M 1 ) 2 p x i ∑ i = 0 N − 1 p x i {\displaystyle {\frac {1}{M1}}{\frac {\sum _{i=0}^{N-1}(x_{i}-M1)^{2}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}} ${\frac {1}{M1}}{\frac {\sum _{i=0}^{N-1}(x_{i}-M1)^{2}px_{i}}{\sum _{i=0}^{N-1}px_{i}}}$

This SP-atom applies a non-linear magnitude weighting (= limiter function) to the signal. The limiter function is only applied if the absolute value of the signal magnitude is higher than the specified limiter start magnitude LIM. For the limiter function, the following algorithm is used:

y i = { x i  if  | x i | ⩽ L I M s i g n ( x i ) . f ( | x i | M A X )  otherwise {\displaystyle y_{i}={\begin{cases}x_{i}&{\mbox{ if }}|x_{i}|\leqslant LIM\\sign(x_{i}).f\left({\frac {|x_{i}|}{MAX}}\right)&{\mbox{ otherwise}}\end{cases}}} $y_{i}={\begin{cases}x_{i}&{\mbox{ if }}|x_{i}|\leqslant LIM\\sign(x_{i}).f\left({\frac {|x_{i}|}{MAX}}\right)&{\mbox{ otherwise}}\end{cases}}$

The absolute magitude of the limited signal is always lower than MAX. The limiter function is selected by the input TYPE.

TYPE	limiter function f(zi)
0 or RECTANGLE	k {\displaystyle k\!} $k\!$
1 or ATAN	k + ( 1 − k ) ⋅ 2 π ⋅ a t a n ( z − k 1 − k ⋅ π 2 ) {\displaystyle k+(1-k)\cdot {\frac {2}{\pi }}\cdot atan\left({\frac {z-k}{1-k}}\cdot {\frac {\pi }{2}}\right)} $k+(1-k)\cdot {\frac {2}{\pi }}\cdot atan\left({\frac {z-k}{1-k}}\cdot {\frac {\pi }{2}}\right)$
2 or EXPONENTIAL	1 − ( 1 − k ) ⋅ e − z − k 1 − k {\displaystyle 1-(1-k)\cdot e^{-{\frac {z-k}{1-k}}}} $1-(1-k)\cdot e^{-{\frac {z-k}{1-k}}}$

with: z i = | x i | M A X , k = L I M M A X {\displaystyle z_{i}={\frac {|x_{i}|}{MAX}},k={\frac {LIM}{MAX}}} $z_{i}={\frac {|x_{i}|}{MAX}},k={\frac {LIM}{MAX}}$

The output Q is set to the relative number of limited (changed) samples.

Q = c h a n g e d S a m p l e s p r o c e s s e d S a m p l e s {\displaystyle Q={\frac {changedSamples}{processedSamples}}} $Q={\frac {changedSamples}{processedSamples}}$

See also

<SP-atoms>