J u m p t o c o n t e n t
M a i n m e n u
M a i n m e n u
N a v i g a t i o n
● M a i n p a g e
● C o n t e n t s
● C u r r e n t e v e n t s
● R a n d o m a r t i c l e
● A b o u t W i k i p e d i a
● C o n t a c t u s
● D o n a t e
C o n t r i b u t e
● H e l p
● L e a r n t o e d i t
● C o m m u n i t y p o r t a l
● R e c e n t c h a n g e s
● U p l o a d f i l e
S e a r c h
Search
A p p e a r a n c e
● C r e a t e a c c o u n t
● L o g i n
P e r s o n a l t o o l s
● C r e a t e a c c o u n t
● L o g i n
P a g e s f o r l o g g e d o u t e d i t o r s l e a r n m o r e
● C o n t r i b u t i o n s
● T a l k
( T o p )
1
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
G e o m e t r i c p r o c e s s
A d d l a n g u a g e s
A d d l i n k s
● A r t i c l e
● T a l k
E n g l i s h
● R e a d
● E d i t
● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d
● E d i t
● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e
● R e l a t e d c h a n g e s
● U p l o a d f i l e
● S p e c i a l p a g e s
● P e r m a n e n t l i n k
● P a g e i n f o r m a t i o n
● C i t e t h i s p a g e
● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
In probability , statistics and related fields, the geometric process is a counting process, introduced by Lam in 1988.[1] It is defined as
The geometric process. Given a sequence of non-negative random variables :
{
X
k
,
k
=
1
,
2
,
…
}
{\displaystyle \{X_{k},k=1,2,\dots \}}
, if they are independent and the cdf of
X
k
{\displaystyle X_{k}}
is given by
F
(
a
k
−
1
x
)
{\displaystyle F(a^{k-1}x)}
for
k
=
1
,
2
,
…
{\displaystyle k=1,2,\dots }
, where
a
{\displaystyle a}
is a positive constant, then
{
X
k
,
k
=
1
,
2
,
…
}
{\displaystyle \{X_{k},k=1,2,\ldots \}}
is called a geometric process (GP ).
The GP has been widely applied in reliability engineering [2]
Below are some of its extensions.
The α- series process.[3] Given a sequence of non-negative random variables:
{
X
k
,
k
=
1
,
2
,
…
}
{\displaystyle \{X_{k},k=1,2,\dots \}}
, if they are independent and the cdf of
X
k
k
a
{\displaystyle {\frac {X_{k}}{k^{a}}}}
is given by
F
(
x
)
{\displaystyle F(x )}
for
k
=
1
,
2
,
…
{\displaystyle k=1,2,\dots }
, where
a
{\displaystyle a}
is a positive constant, then
{
X
k
,
k
=
1
,
2
,
…
}
{\displaystyle \{X_{k},k=1,2,\ldots \}}
is called an α- series process.
The threshold geometric process.[4] A stochastic process
{
Z
n
,
n
=
1
,
2
,
…
}
{\displaystyle \{Z_{n},n=1,2,\ldots \}}
is said to be a threshold geometric process (threshold GP), if there exists real numbers
a
i
>
0
,
i
=
1
,
2
,
…
,
k
{\displaystyle a_{i}>0,i=1,2,\ldots ,k}
and integers
{
1
=
M
1
<
M
2
<
⋯
<
M
k
<
M
k
+
1
=
∞
}
{\displaystyle \{1=M_{1}<M_{2}<\cdots <M_{k}<M_{k+1}=\infty \}}
such that for each
i
=
1
,
…
,
k
{\displaystyle i=1,\ldots ,k}
,
{
a
i
n
−
M
i
Z
n
,
M
i
≤
n
<
M
i
+
1
}
{\displaystyle \{a_{i}^{n-M_{i}}Z_{n},M_{i}\leq n<M_{i+1}\}}
forms a renewal process.
The doubly geometric process.[5] Given a sequence of non-negative random variables :
{
X
k
,
k
=
1
,
2
,
…
}
{\displaystyle \{X_{k},k=1,2,\dots \}}
, if they are independent and the cdf of
X
k
{\displaystyle X_{k}}
is given by
F
(
a
k
−
1
x
h
(
k
)
)
{\displaystyle F(a^{k-1}x^{h(k )})}
for
k
=
1
,
2
,
…
{\displaystyle k=1,2,\dots }
, where
a
{\displaystyle a}
is a positive constant and
h
(
k
)
{\displaystyle h(k )}
is a function of
k
{\displaystyle k}
and the parameters in
h
(
k
)
{\displaystyle h(k )}
are estimable, and
h
(
k
)
>
0
{\displaystyle h(k )>0}
for natural number
k
{\displaystyle k}
, then
{
X
k
,
k
=
1
,
2
,
…
}
{\displaystyle \{X_{k},k=1,2,\ldots \}}
is called a doubly geometric process (DGP).
The semi-geometric process.[6] Given a sequence of non-negative random variables
{
X
k
,
k
=
1
,
2
,
…
}
{\displaystyle \{X_{k},k=1,2,\dots \}}
, if
P
{
X
k
<
x
|
X
k
−
1
=
x
k
−
1
,
…
,
X
1
=
x
1
}
=
P
{
X
k
<
x
|
X
k
−
1
=
x
k
−
1
}
{\displaystyle P\{X_{k}<x|X_{k-1}=x_{k-1},\dots ,X_{1}=x_{1}\}=P\{X_{k}<x|X_{k-1}=x_{k-1}\}}
and the marginal distribution of
X
k
{\displaystyle X_{k}}
is given by
P
{
X
k
<
x
}
=
F
k
(
x
)
(
≡
F
(
a
k
−
1
x
)
)
{\displaystyle P\{X_{k}<x\}=F_{k}(x )(\equiv F(a^{k-1}x))}
, where
a
{\displaystyle a}
is a positive constant, then
{
X
k
,
k
=
1
,
2
,
…
}
{\displaystyle \{X_{k},k=1,2,\dots \}}
is called a semi-geometric process
The double ratio geometric process.[7] Given a sequence of non-negative random variables
{
Z
k
D
,
k
=
1
,
2
,
…
}
{\displaystyle \{Z_{k}^{D},k=1,2,\dots \}}
, if they are independent and the cdf of
Z
k
D
{\displaystyle Z_{k}^{D}}
is given by
F
k
D
(
t
)
=
1
−
exp
{
−
∫
0
t
b
k
h
(
a
k
u
)
d
u
}
{\displaystyle F_{k}^{D}(t )=1-\exp\{-\int _{0}^{t}b_{k}h(a_{k}u)du\}}
for
k
=
1
,
2
,
…
{\displaystyle k=1,2,\dots }
, where
a
k
{\displaystyle a_{k}}
and
b
k
{\displaystyle b_{k}}
are positive parameters (or ratios) and
a
1
=
b
1
=
1
{\displaystyle a_{1}=b_{1}=1}
. We call the stochastic process the double-ratio geometric process (DRGP).
References
[ edit ]
^ Lam, Y. (2007). Geometric process and its applications. World Scientific, Singapore MATH. ISBN 978-981-270-003-2 .
^ Braun, W. J., Li, W., & Zhao, Y. Q. (2005). Properties of the geometric and related processes . Naval Research Logistics (NRL), 52(7 ), 607–616.
^ Chan, J.S., Yu, P.L., Lam, Y. & Ho, A.P. (2006). Modelling SARS data using threshold geometric process . Statistics in Medicine . 25 (11 ): 1826–1839.
^ Wu, S. (2018). Doubly geometric processes and applications . Journal of the Operational Research Society , 69(1 ) 66-77. doi :10.1057/s41274-017-0217-4 .
^ Wu, S., Wang, G. (2017). The semi-geometric process and some properties . IMA J Management Mathematics , 1–13.
^ Wu, S. (2022) The double ratio geometric process for the analysis of recurrent events . Naval Research Logistics , 69(3 ) 484-495.
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Geometric_process&oldid=1083074202 "
C a t e g o r i e s :
● P o i n t p r o c e s s e s
● M a r k o v p r o c e s s e s
● P o i s s o n p o i n t p r o c e s s e s
H i d d e n c a t e g o r i e s :
● W i k i p e d i a a r t i c l e s t h a t a r e t o o t e c h n i c a l f r o m A u g u s t 2 0 2 0
● A l l a r t i c l e s t h a t a r e t o o t e c h n i c a l
● T h i s p a g e w a s l a s t e d i t e d o n 1 6 A p r i l 2 0 2 2 , a t 2 1 : 4 0 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w