(t) , , t . , X(t) xk(t), . xk(t) X(t). X(t). X(t) . 17.1.1. .
. 17.1.1. .
, xk(t) . , . N- (xn;tn). , N- , . - .
.
. 17.1.2. X(t). |
, X(t) {x1(t), x2(t), xk(t),}. t1 {x1(t1), x2(t1), xk(t1),}. X(t1) X(t). 100 X(t) t1 t2 (. 17.1.1) . 17.1.2.
(x,ti) , ti X(ti) x:
F(x,ti) = P{X(ti)≤x}.
, 0 1 F(x,t) F(-¥,t)=0 F(¥,t)=1. F(x,t) , X(ti) [a, b] :
P{a<X(ti)≤b} = F(b,ti) F(a,ti).
p(x,t) (t) (ti) ti. :
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p(x,ti) = dF(x,ti)/dx.
ti X(t) p(x,ti) X(ti) . p(x,ti)dx X(ti) dx x, , .
. 17.1.3.
. 17.1.3 X(t) t1 (. 17.1.1). N=1000 N ¥.
X(ti) [a, b] :
P(a<X(ti)≤b) = p(x,ti) dx.
1, .. - , :
p(x,ti) dx =1.
:
F(x,ti) = p(x,ti) dx.
( ), :
(mean value) X(ti), ti . , :
mx(t) º M{(t)}º = x p(x;t) dx, (17.1.1)
mx(t) X(t). . 17.1.1. 17.1.2 m(t) X(t) N ¥.
(variance) (t)-mx(t), :
Dx(t) = M{[(t)-mx(t)]2} = M{X2(t)} - mx2(t) = [xo(t)]2 p(x;t) dx, (17.1.2)
xo(t) = x(t)-mx(t).
(standard deviation) :
sx(t) = . (17.1.3)
. 17.1.4. |
, sx2.
. 17.1.4 X(t) (. 17.1.1) s m(t).
. - .
p(x1,x2; t1,t2) (t1) (t2) t1 t2 - . {X(ti), X(tj)} X(ti) dxi xi ti , tj X(tj) dxj xj:
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p(xi,xj; ti,tj) dxi dxj = P{|X(ti-xi|≤dxi/2, |X(tj-xj|≤dxj/2}.
{X(ti), X(tj)} , :
RX(ti,tj) = M{X(t1) X(t2)}.
X(t) ti tj ti tj, . .
. 17.1.5 , .
. 17.1.5. |
, , . , , , . t ( ) , , , . - (t) t1 t2 . X(t) :
R(ti,tj) = x(ti)x(tj) p(xi,tj; xi,tj) dxi dxj, (17.1.4)
. 17.1.6. X(t).
. 17.1.6 X(t) . 0- (=100) Dt=1.
tj t , :
R(t,t+t) = M{(t)(t+t)}. (17.1.4')
, , .
. (), . X(t)-mx(t) ti tj :
K(ti,tj) = (x(ti)-mx(ti)) (x(tj)-mx(tj)) p(xi,tj; xi,tj) dxi dxj, (17.1.5)
. . mx :
KX(t,t+t) = RX(t,t+t) - mx2(t).
( ):
r(t,t+t) = K(t,t+t)/[s(t)s(t+t)]. (17.1.6)
t = 0 r 1, :
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K(t) = D(t).
, (). .
. 17.1.7. .
. 17.1.7.
.
1. t = 0. , .. t = 0 , . .
2. : RX(t) = RX(-t). : X(t)X(t+t) = X(t-t)X(t) t = t-t. , X(t1) X(t2) , , : Rx(t1,t2) = Rx(t2,t1), Kx(t1,t2) = Kx(t2,t1).
3. t Þ ¥ , , , . tmax - , . , :
Tk =2 |rx(t)| dt º (2/Kx(0) |Kx(t)| dt. (17.1.7)
() , Tk, .
, t Þ ¥ Tk 2, , 1. :
Tk =2 |rx(t)| dt - 1 º (2/Kx(0) |Kx(t)| dt - 1. (17.1.7')
4. X(t) f(t), .
Y(t)=X(t)+f(t). : = + f(t). , Y(t) - = X(t) - , Ky(t1,t2) = Kx(t1,t2).
5. X(t) f(t), Rx(t1,t2) f(t1)×f(t2). , .
6. 2 .
X(t) Y(t) xk(t) yk(t).
X(t) Y(t) , - . , t1 = t t2 = t+t:
RXY(t,t+t) = M{(X(t)(Y(t+t)}. (17.1.8)
KXY(t,t+t) = M{(X(t)-mx(t))(Y(t+t)-my(t+t))}. (17.1.9)
( ), :
Rxy(-t) = Ryx(t), (17.1.10)
|Rxy(t)|2 £ Rx(0)Ry(0).
, Rxy(t) = Kxy(t).
( ), t , :
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rxy(t) = Kxy(t)/(sxsy). (17.1.11)
X Y. , , . :
p(x,y) = p(x) p(y).
. . , . :
rxy = [M{XY} M{X)M{Y}]/ .
rxy -1 +1. , x=ay+b, 1 . rxy=0, rxy :
M{XY} = M{X)M{Y}.
. . , , x=cos j y=sin j, j - 02p, , .
. ( ).
. , t, .. . .
. , , t = t2-t1, .e.:
m(t1) = m(t2) = m = const, (17.1.12)
D(t1) = D(t2) = D = const,
R(t1,t1+t) º Rx(t2-t,t2) = R(t) º R(-t),
rx(t) = Rx(t)/Dx, rx(0) = 1, |rx(t)| ≤ 1, rx(-t) = rx(t).
( ) . :
|Rx(t)| £ Rx(0), |Kx(t)| £ Kx(0) º Dx.
t Rx(t) rx(t), , .
, , , ( , ).
. : , , . , , , .
. ( ). . , , . , , . (ergodic). :
mX(t) = M{x(t)} = x(t) dt, (17.1.13)
D(t) = M{x(t) - m(t)]2} = (x(t) - m(t))2 dt, (17.1.14)
R(t) = M{x(t)x(t+t)} = x(t)x(t+t) dt. (17.1.15)
, . , . , . , , (17.1.15), ( ) :
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R(t) = (1/T) x(t) * x(t+t).
, . :
K(t) dt = 0. (17.1.16)
(t), , .
. Z(t)=X(t)+Y, X(t) - , Y- , X(t). Z(t)?
mz(t) = mz(x)+my, Kz(t) = Kx(t)+Dy.
Z(t) , , t Þ ¥ Kz(t) Þ Dy.