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Threshold strategy of an
estimation in a problem
of choice of the best
variant
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Task:
It is necessary to choose
the most suitable variant
from some set of objects
by those or other criteria.
Example 1.
Coding and decoding in transfer
systems of the discrete information.
By means of a choice
of an appropriate
method of coding it is
necessary to reduce
probability of
erroneous acceptance
as much as possible.
Example 2.
Choice of the best supplier.
Adaptive control methods
It is required to solve a problem not for
concrete, certain object, and for any
object from some set    .
The algorithm should "adapt", should
"be arranged" to an object and after
"training" should provide purpose
achievement.
The control theory
Object of control
(controlled random process)
Control
purpose
Control
algorithm (strategy)
Models of controlled objects and operating
systems
• The object O is described by an element x of phase
space Х.
• Y = {y} – set of variants.
• Dynamics of phase space:
xt  f ( x , y , ) , t  1   
t 1
t
t 1
x  ( x1 , x2 ,..., xt 1 )
t 1
y  ( y1 , y2 ,.., yt 1 )
control strategy is characterized by a rule:
t 1
t 1
yt  g ( x , y )
Controlled random processes
Definition.
Controlled random process is the pair ( , )
formed by object of control (family of
conditional probabilities  t , t  1 ) and a class
of admissible strategy   { },

where

 ( M x t 1 )  P( t  M x t 1 )
- family of the
distributions having a final measure
t
- family of random variables with values in space Х.
The control purpose
- optimization of value of some functional which depending
on a condition of controlled process accepts this or that
value.
The most typical kinds of functionals:
1)
2)
where
1 t
1 ( x ; )   x k the average income received for time of
control till the moment t inclusive.
t k 1
t
1 t
 2 ( x ; )  w ( )   x k
t k 1
t
w( )  max wl ( )
l 1,.. K
the average loss of the income in
comparison with highest
possible expected income at the
moment of time t.
- highest possible expected income at a
concrete choice of parameter of environment.
It is necessary to choose thus control strategy
 
, that for any admissible parameter
conditions was carried out:
 
one of
t
lim
E

(
x
;

)

w
(

)

; 1
t 
or
lim E ;  2 ( x ; )  0
t
t 
where
strategy
is fixed.
E ;
- the expectation value sign provided that
   is chosen and concrete parameter of environment 
Threshold strategy of control
We use (k-1)-stage strategy of rejecting of the worst variant according
to the results of estimation.
Let are available: phase space Х={x},
class of admissible strategy
  { }
and set of variants Y={1,2,..К}.
1
Random variables  t  
, t=1,…,T
0

- incomes of applied at present
moment of time variants
P{ t  1 | yt  l}  pl ,
P{ t  0 | yt  l}  ql ,
pl  ql  1,
l  1,...K .
The control purpose consists in minimization of the guaranteed size of an
expectation value of the full income losses, that is in minimization of the
functional
n
 n  n  max p k    t
k 1, 2 ,..., K
t 1
We assume that all probabilities p1 , p 2 ,..., p k
are unknown.
Rejection procedure
Let M variants (1≤M≤K) with numbers
and with initial incomes
S l1 ,..., S lM
Let's apply variants cyclically,
l1 , l 2 ,..., l M
be available.
y (t )  li
at
t  M  i
accumulating relevant full incomes
n 1
n 1
 0
 0
S l1 (n)  S l1    M 1 ,..., S lM (n)  S lM    M  M
until then time of modeling yet won't end or for some variable n 0 the
inequality won't be executed
max Sli (n0 )  min Sli (n0 )  aM  0
i
i
Function of losses
If the object of control is known it is possible to receive maximum
expectation income equal to T max pl .
l
You should apply a variant corresponding to the greatest value of
If object of control is unknown, the real expected income for
application of the threshold strategy is equal to
T
 E (
t 1
t
| a 2 ,..., a K ; p1 ,..., p K ) .
Function of losses is equal to
T
LT ( , )  T max pl   E ( t | a 2 ,..., a K ; p1 ,..., p K )
l
t 1
The goal is to determine
inf sup LT ( , )


pi .
Property of invariancy for threshold strategy of estimation:
Let
D  p1 (1  p1 )
1
2
D
pi  p1  bi ; bi   i    ,  i  0
T 
1
ai   i  DT  2
then the equality holds:

1
2
lim ( DT ) LT (a2 ,.., ak ; p1 ,.., pk )  L( 2 ,..,  k ; 1 ,..,  k )
T 
On the other hand the aproximate equality holds
1
2
LT (a2 ,.., ak ; p1 ,.., pk )  ( DT ) L( 2 ,..,  k ; 1 ,..,  k )
where
 i  ai  DT 

D
 i  bi   
T 

1
2
1
2
LT (a2 ,.., ak ; p1 ,.., pk )  E (losses till the first rejection  losses for
expected losses K  1 variants)
Optimization of calculation of value of the minimum
losses at a choice of the best variant.
Let K=2 and
1
2
D
p1  0.5 , p 2  0.5  b ; b      ,   0 , D  p1 (1  p1 )  0.25
T 
The goak e consists in a value finding:
min max LT (a, b)
a
b
In the game formulation this size corresponds to the top price of game.
Here it is necessary to make calculations
then to find
min max LT (ai , b j )
a
b
(n  m) of values LT (ai , b j )
,
On the other hand, at К=2
0.5b
p1  0.5 , p2  0.5 
T
it is possible to present function of losses LN (a, b) in the form:

 

LT (a, b)  E 0.5n0  S1,n0  0.5n0  S 2,n0  ET  2n0 0.5  pi* 
Here only the first composed needs modeling, and the second one
is calculated directly.
Therefore it becomes possible at once, for one run of algorithm to
calculate sequence of values of function LT (ai , b j ) for some
increasing set of values a1 , a 2 ,..., a n using earlier found sums
S1,n and S2,n , corresponding to the previous value of ai 1
Therefore at once we find value
L (b j )  min LT (ai , b j ) and
ai
The problem dares for
m steps .
.
In the latter case calculations are reduced to a value finding
max min LT (a, b) ,
b
a
that corresponds to the bottom price of game.
“ Add your company slogan ”
Shelonina Tatyana,
the associate professor of the
applied mathematics department
Novgorod State University of a
name of Yaroslav the Wise
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