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Национальный исследовательский университет «МИЭТ»

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Estimation of Probability Distribution and Density Functions of Stochastic Process										503
	σ{F*(x)}	1
		1	2
0.4			2			1	p << 1
0.4						1	p << 1
			3			2	p = 0.1
0.3			3			3	p = 1.0
0.3						3	p = 1.0
						4	p = 10
0.2						5	p = 100
0.1			4
0.1			5
			5
									z
0	0.4		0.8	1.2	1.6	2.0		2.4	2.8

FIGURE 14.5 Maximal value of σ{F*(x)} as a function of p. Rayleigh stochastic process.

The root-mean-square deviation σ{F*(x)} of measurements of the probability distribution function of Rayleigh stochastic process as a function of the normalized level z for various values of p is presented in Figure 14.5. As we can see from Figure 14.5, the maximal value of σ{F*(x)} = 0.5 is obtained about the point z = 0.83.

The root-mean-square deviation σ{F*(x)} at z = 1 as a function of the normalized observation interval p is shown in Figure 14.6. Based on Figure 14.6 and (14.89), we can conclude that starting from p ≥ 10, in the case of Rayleigh stochastic process, the root-mean-square deviation σ{F*(x)} is inversely proportional to the ratio p = T/τcor. In discrete case, when the sample size is equal to N, the variance of the probability distribution function estimations F*(x) of Rayleigh stochastic process in accordance with (14.41) and (14.80) can be presented in the following form:

∞

Var{F (x)} =

+ ∑dv′bv,

exp −

− exp −

2σ

v=1

where

dv′ =	2	N −1	1	−	k	2v	(kTp ).


	N ∑				N
		k=1

Applied to the Rayleigh stochastic process with uncorrelated samples, we obtain

	1			x	2					x	2
Var{F (x)} =	1		−	x				− exp	−	x
		exp					1						.
		exp				2	1					2	.
	N			2σ		2				2σ		2
	N			2σ						2σ

(14.90)

(14.91)

(14.92)

σ {F *(x)}
1.0
0.1
0.01
0.001						p
0.001	10–1	1.0	10	102	103	104
10–2	10–1	1.0	10	102	103	104

FIGURE 14.6 σ{F*(x)} as a function of the normalized observation interval p. Rayleigh stochastic process.

504	Signal Processing in Radar Systems

TABLE 14.4

Normalized Correlation Function (14.68)

as a Function of the Level z

					z2
z1	0.1	0.2	0.3	0.5	0.7	1.0	1.5	2.0
0.1	1.00	0.50	0.33	0.19	0.13	0.08	0.03	0.01
0.2	0.50	1.00	0.66	0.38	0.25	0.15	0.07	0.03
0.3	0.33	0.66	1.00	0.58	0.39	0.23	0.11	0.04
0.5	0.19	0.38	0.58	1.00	0.67	0.41	0.18	0.07
0.7	0.13	0.25	0.39	0.67	1.00	0.61	0.27	0.11
1.0	0.08	0.15	0.23	0.41	0.61	1.00	0.45	0.18
1.5	0.03	0.07	0.11	0.18	0.27	0.45	1.00	0.40
2.0	0.01	0.03	0.04	0.07	0.11	0.18	0.40	1.00

Table 14.4 represents the normalized correlation function given by (14.68) in the case of probability distribution function estimation F*(x) of Rayleigh stochastic process at different values of the level z under investigations by independent samples (14.67).

14.4 CHARACTERISTICS OF THE PROBABILITY DENSITY FUNCTION ESTIMATE

Determine the bias and variance of the pdf estimate of ergodic stochastic process based on an investigation of the continuous realization x(t) within the limits of the observation interval [0, T] in accordance with (14.25). Deviation of the measured pdf p*(x) from the true value p(x) can be presented in the following form:

p(x) = p (x) − p(x) = T1 x ∫χ(t)dt − p(x),

where

χ(t) is given by (14.23)

xis the interval between the adjacent levels xk and xk+1

x= xk + xk +1 . 2

(14.93)

(14.94)

In other words, the pdf is measured at the point corresponding to the middle of adjacent fixed levels. The bias of the pdf estimate of ergodic stochastic process can be presented in the following form:

	1	T
	1		(14.95)
b{p (x)} =	T x ∫ χ(t) dt − p(x).
		0

The average value by realizations process is within the limits of the

χ(t) is the probability of the event that the investigated stochastic interval x ± 0.5 x, i.e.,

χ(t) = x+ 0∫.5	x p(x)dx = F(x + 0.5 x) − F(x − 0.5 x).	(14.96)
x − 0.5	x

Estimation of Probability Distribution and Density Functions of Stochastic Process

505

Based on (14.95) and (14.96), we are able to define the bias of the pdf estimate of ergodic stochastic process

b{p (x)} =

[F(x + 0.5

x) − F(x − 0.5

x)] − p(x).

(14.97)

We can use the Taylor series expansion for the functions F(x + 0.5 x) and F(x − 0.5

x) about the

point x:

∞

dFµ (x)

F(x + 0.5 x) = ∑

(0.5 x)µ ;

(14.98)

µ!

µ =0

∞ dFµ (x) (−1)µ

F(x − 0.5

x) = ∑

(0.5

x) .

(14.99)

µ!

µ =0

The bias of the pdf estimate of stochastic process can be presented in the following form:

∞	d2 +1F(x)			(0.5 x)2
b{p (x)} = ∑			×		.	(14.100)
b{p (x)} = ∑	dx	2 +1	×	(2µ + 1)!	.	(14.100)
=1	dx			(2µ + 1)!
=1

In practice, we can use the following approximation:

b{p (x)} ≈	x2	×	d2 p(x)	+		x4	×	d4 p(x)	.	(14.101)
	24		dx2		1920
								dx4

As we can see from (14.97) through (14.101), the bias of the pdf estimate of stochastic process decreases proportionally to a decrease in the interval x between two adjacent fixed levels. It is


clear from the physical viewpoint. In the limiting case, as				x → 0, the bias of the pdf estimate of
stochastic process tends to approach zero. In other words, as				x → 0, the pdf estimate of stochastic
process is unbiased. Actually, as		x → 0
lim	F(x + 0.5	x) − F(x − 0.5 x)	− p(x) = dF(x) − p(x) = 0.		(14.102)

x→0		x		dx

With a decrease in the interval x, the variance of the pdf estimate of stochastic process increases since with a decrease in the interval x we observe a high deviation of total time values of stochastic process within the limits of the interval x from realization to realization.

Apply (14.100) to the Gaussian and Rayleigh stochastic processes. Since

	dv {1 − Q(x/σ)}		=	1 − Qv (z)	,	(14.103)
	dxv		=	σ0.5v	,	(14.103)
	dxv			σ0.5v
where
	z =	x	,			(14.104)
	z =	σ	,			(14.104)
		σ

506	Signal Processing in Radar Systems

the bias of the pdf estimate of Gaussian stochastic process takes the following form:

			∞		2σ}		2v
b{p (x)} =	1	∑{ x/			2σ}
b{p (x)} =	σ	∑{ x/
	σ	v=1		(2v +1)!
		v=1
	1		{	x/σ}	2
≈	1		{	x/σ}			−
						1
	σ			24		1
	σ			24

The minimal bias under the condition

1	− Q2v+1	x


			σ

Q	3		x	+	{ x/σ}4
					1920

			σ
	x	< 1
σ

	x
1 − Q5	x	.
1 − Q5		.

	σ

(14.105)

(14.106)

corresponds to the value z ≈ 1 at which Q3(z) ≈ 0.

Applied to the Rayleigh stochastic process, at the first approximation the bias of the pdf estimate of the Rayleigh stochastic process (the first term in (14.101)) can be presented in the following form:

b{p (x)} ≈

( x)2

/σ2

−2

−1

− 3)xσ

exp

−

(14.107)

24σ

2σ

	The relative bias
	\| b{p (x)}\|	(14.108)
	p(x)	(14.108)
	p(x)

of the pdf estimates of Gaussian (the solid line) and Rayleigh (the dashed line) stochastic processes as a function of the normalized interval

z =	x
	σ

at various values of the normalized level z are shown in Figure 14.7. As we can see from Figure 14.7, the relative bias of pdf estimate does not exceed the level 0.03 at z ≤ 0.5, which is acceptable in practice.

		\|b{ p*(x)}\|					5	1	3
		p (x)						1

0.08
0.08

			1	z = 0
0.06			2	z = 0.5					4
0.06			3	z = 1.0					4
			4	z = 1.5
0.04			5	z = 2.0				1	5
									4
									2
0.02
										z
	0			0.2	0.4	0.6	0.8		1.0

FIGURE 14.7 Relative bias of pdf estimate versus the normalized observation interval p.

Estimation of Probability Distribution and Density Functions of Stochastic Process

507

Determine the variance of the pdf estimate of stochastic process. According to (14.25), we can write

			1			T T		1	T	2
			1					1
Var{p (x)} =						∫∫χ(t1)χ(t2 ) dt1dt2	−		∫ χ(t) dt		,	(14.109)
Var{p (x)} =	T	2	(	x)	2	∫∫χ(t1)χ(t2 ) dt1dt2	−	x	∫ χ(t) dt		,	(14.109)
	T		(	x)			T	x
						0 0			0

where

	x + 0.5	x x + 0.5	x	′ ′
χ(t1)χ(t2 ) =	∫	∫	′ ′	′ ′	≡ (x ± 0.5 x; t1, t2 )
χ(t1)χ(t2 ) =			f (x1, x2	; t1, t2 )dx1dx2	≡ (x ± 0.5 x; t1, t2 )	(14.110)
	x − 0.5	x x − 0.5	x

is the probability of the event that the stochastic process ξ(t) lies within the limits of the interval x ± 0.5 x at the instants t1 and t2. In the case of stationary stochastic process, the following condition is satisfied:

(x ± 0.5 x; τ) = (x ± 0.5 x; −τ),

(14.111)

where τ = t2 − t1. By introducing new variables τ = t2 − t1 and t = t1, the double integral in (14.109) is transformed into a single integral, i.e., we obtain

Var{p (x)} =	2		T			τ		2
				1	−		(x ± 0.5	x)dτ − ( p (x) )	.	(14.112)
	T ( x)	2		1	−		(x ± 0.5	x)dτ − ( p (x) )	.	(14.112)
	T ( x)		∫			T
			0

In practice, at the first approximation, we can suppose that the oneand two-dimensional pdfs are constant within the limits of the interval x ± 0.5 x. Then p*(x) ≈ p(x) and

(x ± 0.5 x; τ) ≈ p2 (x, x; τ) × ( x)2 .

(14.113)

In this case, the variance of probability density function estimate of stochastic process can be presented as

Var{p (x)} ≈	2	T	1	−	τ	p2 (x, x; τ)dτ − p	2	(x).	(14.114)

	T
	T	∫			T
		0

In the case of Gaussian and Rayleigh stochastic processes, the two-dimensional pdfs are given by (14.50) and (14.71), respectively.

In the case of discrete stochastic process, in accordance with (14.26) the variance of the pdf estimate can be presented in the following form:

Var{p (x)} =

∑

χiχ j −

∑ χi

(14.115)

(

i=1, j =1

N x

i=1

508 Signal Processing in Radar Systems

where χi χj is given by (14.110). When the stochastic process samples are uncorrelated, we can write

χi	at	i = j,
χi χ j =		i =	(14.116)
χi χ j	at	i =	j.

As a result, the variance of the pdf estimate of stochastic process takes the following form:

Var{p (x)} = [F(x + 0.5 x) − F(x − 0.5	x)][1 − F(x + 0.5 x) + F(x − 0.5 x)] .			(14.117)
	N( x)2
Applying the Taylor series expansion for the function F(x ± 0.5			x) about the point x and limiting by
the first three terms, we obtain
Var{p (x)} ≈	p(x)	[1 − xp(x)].		(14.118)

	N x
For the considered x, we have the following restriction p(x)			x < 1. As we can see from (14.118),

with an increase in the interval x the variance of the pdf estimate of stochastic process is decreased, which is confirmed by physical representation.

As applied to the Gaussian stochastic process, the variance of the pdf estimate of stochastic pro-

cess defined by uncorrelated samples can be presented in the following form:
Var{p (x)} = [Q(z − 0.5 z) − Q(z + 0.5 z)][Q(z − 0.5 z) − Q(z + 0.5 z)]2 .	(14.119)
N( z)σ2
The dependences Var{p (x)}N p(x) versus the normalized interval between the levels	z and var-

ious values of the normalized level z under investigation of the Gaussian (the solid line) and Rayleigh (the dashed line) stochastic processes in the case of uncorrelated samples are shown in Figure 14.8.

σ [ p*(x)]N

5	p (x)
	p (x)

4
4			1	z= 0
			2	z = 1.0
			3	z = 2.0
3
2						2
						3
1						1
1						2
						2
						z

0	0.2	0.4	0.6		0.8	1.0

FIGURE 14.8 Var{p (x)}N p(x) versus the normalized interval between the levels z.

Estimation of Probability Distribution and Density Functions of Stochastic Process			509
{D[p*(x)]N }1/2
p (x)
0.06
1	1	N = 104
1	2 N = 105
0.04	2 N = 105
0.04

0.022

0 0.2 0.4 0.6 0.8 1.0

FIGURE 14.9 Normalized error of pdf measurement versus the normalized interval between the levels z.

As we can see from Figure 14.8, as opposed to the bias of the pdf estimate, the variance is decreased with an increase in the interval between levels. However, under experimental measurement of the pdf the interval between the levels x must be defined based on the condition of minimization of dispersion of the pdf estimate, namely,

D{p (x)} = Var{p (x)}+ b2{p (x)}.

(14.120)

The normalized resulting errors of measurement of the pdf of Gaussian stochastic process

D{p (x)} p(x) as a function of the normalized interval between levels z at z = 0 and N = 104 and N = 105 are shown in Figure 14.9. As we can see from Figure 14.9, for the given number of samples and at z = 0, the minimal magnitude of the resulting error is decreased with a decrease in the values z under an increase in the number of samples.

14.5 PROBABILITY DENSITY FUNCTION ESTIMATE BASED ON EXPANSION IN SERIES COEFFICIENT ESTIMATIONS

The one-dimensional stochastic process pdf can be presented in the form of series using the previously given orthogonal functions φk(x) [2]:

	∞
	p(x) = ∑ckϕk (x),		(14.121)
	k =1
where the unknown coefficients are determined as
	ck = ∫∞ ϕk ( x) p( x)dx.		(14.122)
	−∞
In the case of normalized orthogonal functions, we can use the following relations:
∞		1,	k = l,
∫	ϕk (x)ϕl (x)dx =		(14.123)
	ϕk (x)ϕl (x)dx =	δkl =	(14.123)
		0,	k ≠ l.
−∞

510	Signal Processing in Radar Systems

Practically, the number of terms in expansion in series given by (14.121) is limited by some value v. Therefore, under approximation of the pdf by expansion in series, we will have forever an error

v
ε(x) = p(x) − ∑ckϕk (x)	(14.124)

k =1

that can be reduced to the given negligible value under the corresponding choice of the orthogonal functions φk(x) and the number of expansions in series terms.

Further, it is convenient to characterize the approximation accuracy by the root-mean-square deviation from the true value of the pdf p(x) for all possible values of the argument x

	2	∞		2	∞
	2	= ∫		2
ε		= ∫	ε		(x)dx = ∫ p(x)
		−∞			−∞

− ∑ckϕk

k =1

(x) dx. (14.125)

Taking into consideration (14.122) and (14.123), the formula (14.125) takes the following form:

∞	v
ε2 = ∫ p2 (x)dx − ∑ck2.		(14.126)
−∞	k =1

As we can see from the definition, the coefficients ck of expansion in series represent, in general, the mathematical expectations of random variables subjected to the nonlinear transformation φk(x). As applied to ergodic stochastic process, the coefficients ck can be defined by averaging the realization of stochastic process φk[x(t)] transformed in time (continuously or discretely). Taking into consideration these statements, the method to measure the pdf based on generation of orthogonal functions and estimation of coefficients of expansion in series of measured pdf was supposed in Ref. [3].

The estimate of the pdf p*(x) can be presented in the following form:

v
p (x) = ∑ck ϕk (x),	(14.127)

k =1

where the estimations of coefficients ck in the cases of continuous and discrete methods of investigation of the realization x(t) can be presented in the following form:

ck	=	1		∫T ϕk[ x(t)]dt;			(14.128)
		T
					0
						N
ck	=	1				∑ϕk (xi ),	(14.129)
			N
						i=1

where T and N are the observation time and number of samples, respectively.

A flowchart of the considered pdf measurer functioning in real time is shown in Figure 14.10. At the measurer output, the pdf estimate is formed as a function of time

v
p (t) = ∑ck ϕk (t).	(14.130)

k =1

Estimation of Probability Distribution and Density Functions of Stochastic Process

511

1[x(t)]	c1*
	c1*
2[x(t)]	c2*
	c2*
t(x[)]				Σ	p*(t)
x(t)
v[x(t)]	cv*
	cv*
	v(t)	. . .	2(t)	1(t)
Synchronizer		Generator

FIGURE 14.10 Measurer of pdf.

The measurer operates in the following way. The input realization x(t) of the stochastic process is subjected to multichannel inertialess transformation φ[x(t)] and is averaged after that by ideal integrators. As a result, the estimations of the expansion in series coefficients ck are formed. These coefficients ck come in at the mixer inputs. The orthogonal functions of time φk(t) come in at the second mixer input. The estimate of the pdf p*(t) as a function of time is formed at the summator output. The orthogonal time functions φk(t) coming at the mixers are delayed on the time T. This time is required to obtain the estimations of coefficients ck .

Determine the characteristics of the pdf estimations. Since

ϕk[ x(t)] = ∫∞ ϕk (x) p(x)dx,	(14.131)
−∞

we can see from (14.128) and (14.129) that the estimations of coefficients ck are unbiased, i.e.,

ck = ck .

(14.132)

The resulting dispersion of the pdf estimate can be presented in the following form:

∞					v				(14.133)
									(14.133)
D{p } = ∫	[ p(x) − p (x)] dx		= ε			2	− ck	.
D{p } = ∫	[ p(x) − p (x)] dx		= ε	2	+ ∑ ck		− ck	.
		2		2			2
−∞					k =1

The second term

v	2	2		(14.134)
Var{p } = ∑	ck	− ck		(14.134)

k =1

is the variance of the pdf estimate caused by random errors under estimation of the coefficients ck .

512	Signal Processing in Radar Systems

Determine the second initial moment of the coefficient of expansion in series applied to the continuous method given by (14.128). We obtain

2		1	T T
ck	=		∫∫ϕk[ x(t1)]ϕk[ x(t2 )] dt1dt2.		(14.135)
ck	=	T 2	∫∫ϕk[ x(t1)]ϕk[ x(t2 )] dt1dt2.		(14.135)
			0	0
The mathematical expectation of the integrand can be presented in the following form:
∞	∞

ϕk [ x(t1 )]ϕk [ x(t2 )] = ∫ ∫ ϕk (x1 )ϕk (x2 )p2 (x1, x2 ; τ)dx1dx2 = Ψ(τ = \| t2 − t1 \| ),	(14.136)

−∞ −∞

where p2(x1, x2; τ) is the two-dimensional pdf of (14.136) into (14.135), introducing new variables integration, we obtain

the investigated stochastic process. Substituting τ = t2 − t1 and t = t1, and changing the order of

2		2	T			τ
ck	=			1	−		Ψ(τ)dτ.	(14.137)
ck	=	T		1	−		Ψ(τ)dτ.	(14.137)
		T	∫			T
			0

As applied to investigation of stochastic process realization at discrete instants

ck	= N 2	N	N	(14.138)
ck	= N 2	∑∑ϕk (xi )ϕk (x j ) ,		(14.138)
2	1
		i=1	j =1

where φk(xi)φk(xj) is defined by analogous way as in (14.136).

Under analysis of stochastic process realizations at independent instants, the components

Δϕk (xi ) = ϕk (xi ) − ck

(14.139)

will be independent random variables. Therefore, the mathematical expectation of the square of

the estimate of the expansion in series coefficients ck 2 given by (14.138) can be presented in the following form:

		1	N
2		1	∑ Δϕ2k (xi ) .
ck	= ck2 +			(14.140)
ck	= ck2 +	N 2		(14.140)
			i=1

The variance of random component Δφk(xi) is defined by the well-known relationship

	∞
2	2		(14.141)
Var{ ϕk} = ϕk (xi )	= ∫[ϕk (x) − ck ]	p(x)dx	(14.141)
Var{ ϕk} = ϕk (xi )	= ∫[ϕk (x) − ck ]	p(x)dx

−∞

and does not depend on the number of sample. Because of this, we can write the variance of the pdf estimate in the following form:

Var{p } = 1 ∑Var{Δϕk}. (14.142)

N k=1

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