Mathematics - 386 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Linear Prediction of Sum of Two Poisson Process S. H. Raheem Faculty of Human Science and Physical Education /School of Human Science,University of Garmian/Iraq Received in: 17 March 2012 Accepted in: 21 May 2012 Abstract Our goal from this work is to find the linear prediction of the sum of two Poisson process )()()( tYtXtZ += at the future time 0),( ≥+ ττtZ and that is when we know the values of )(tZ in the past time and the correlation function )(τβ z Key words: Poisson process, linear Prediction, sum of two Poisson process. Introduction The Poisson process has long tradition in applied probability and stochastic process theory, in (1903) thesis Fillip Lundberg already exploited it as a model for the claim number process N later on in the 1930s , Gramer the famous statistician extensively developed collective risk theory by using the total claim amount process S with arrivals Ti which are generated by Poisson process for historical reasons but also since it has very attractive mathematical properties the Poisson process plays a central role in insurance mathematics[1,p7].Bellow we will give a definition of Poisson process. A Poisson process with parameter or rate λ > 0 is an integer- valued, continuous time stochastic process }0),({ ≥ttX satisfying :- 1. 0)0( =X 2. for all t0 = t1 < t2 < … < tn the increment X(t1) – X(t0), X(t2) – X(t1), …, X(tn) – X(tn – 1 ) are independent random variables. 3. for ,0≥t s > 0 and non- negative integers k, the increments have the Poisson distribution k t( t) e Pr[X(t s) X(s) k] k! λλ − + − = = , k = 0, 1, 2, … …(1) It is convenient to view the Poisson process X(t) as a special counting process of events in any interval of length t specified via condition (3), [2,p120]. We have to know that Poisson process is on of the most important examples of Marcov process that plays an important role in both theory and in variety of applications, [3,p346]. An arrival is simply an occurrence of some event-like a phone call, job offer or whatever that happens at a particular point in time [4]. Our goal from this work is to find the linear prediction of the sum of two independent homogenous Poisson process }),({},),({ TttYTttX ∈∈ with parameters λ1, λ2, (i.e.) to find the linear prediction of: }),({}),({}),({ TttYTttXTttZ ∈+∈=∈ at the future time ( t + τ ) , τ ≥ 0. Mathematics - 387 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Theorem [2] Let X(t), Y(t) be two independent Poisson process with parameters λ1 and λ2 respectively, then X(t)+Y(t) is also a Poisson process with density 21 λλ + Proof: Since )()( tYtX + is an additive Markov process we let: )()()( tYtXtZ += ,so ∑ = −=−+⋅=−+==⋅+ n oj jnsYtsYjsXtsXsZtsZ })()(Pr{})()(Pr{})()(Pr{ η )!( )( ! )( 2 0 1 21 jn t e j t e jn t n j j t − ⋅= − = ∑ λλ λλ = ∑ = − +− − n j jnj t tt jnj n n e 0 21 )( )()( )!(! ! ! 21 λλ λλ 1 2( ) t n 1 2 e Z(t) ( t t) n! λ λ λ λ − + = + …(2) ∴ Z(t) is homogenous Poisson process Moments of Z(t) We can find the first moment of Z(t) as follows ∑ ∞ = = 0)( )()(]),([ tZ z tftZttZE = ∑ ∞ = +− + 0)( )( 21)( )!( ])[( )( 21 tZ tZ t tZ t etZ λλλλ = ∑ ∞ = +− +− − + 1)( 11)( 21)( ]!1)([ ])[( 21 tZ tZ t tZ t e λλλλ ∑ ∞ = − +− − + += 1)( 1)( 21)( 21 ]!1)([ ])[( )( 21 tZ tZ t tZ t te λλ λλ λλ t)( 21 λλ += Know to find the second moment ∑ ∞ = = 0)( 22 )()(]),([ tZ z tftZttZE = ∑ ∞ = +− + 0)( )( 21)(2 )!( ])[( )( 21 tZ tZ t tZ t etZ λλλλ ∑ ∞ = − +− − + += 1)( 1)( 21)( 21 ]!1)([ ])[( )()( 21 tZ tZ t tZ t tZte λλ λλ λλ Let 1)()( −= tZtK then we get ∑ ∞ = +− +++= 1)( )( 21)( 21 )!( ])[( )1)(()( 21 tZ tK t tK t tKte λλ λλ λλ Mathematics - 388 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 = )( ]!1)([ ])( ])[( 21 0)( 11)( 21)( 21 21 λλ λλ λλ λλ ++ − + + ∑ ∞ = +− +− tK tK t tK t et 2 21 2 1 2E[Z (t), t] [( )t] ( )tλ λ λ λ∴ = + + + …(3) So that: 22 )]),(([]),([]),([ ttZEttZEttZVar −= = 1 2( )t (4)λ λ+  Correlation Function of Z(t) We have: )()()( tYtXtZ += The correlation function of Z(t) is: )()()( τβτβτβ YXz += Or: ),(),(),( τβτβτβ +++=+ tttttt YXz )]()([)]()([ ττ +++= tYtYEtXtXE )]}([)]([)]()({[)]}([)]([)()({ ττττ +−+++−+= tYEtYEtYtYEtXEtXEtXtXE 2 2 2E[X (t)] E[X(t)X(t ] E[X (t)] E[X(t)]E[X(t )]] E[Y (t)]τ τ= + + − − + + + 2E[Y(t)Y(t )] E[Y (t)] E[Y(t)]E[Y(t )]τ τ+ − − + 2 2E[X (t)] E[X(t)]E[X(t ) X(t)] E[X(t)]E[X(t )] E[Y (t)]τ τ= + + − − + + + E[Y(t)E[Y(t ) Y(t)] E[Y(t)]E[Y(t )]τ τ+ − − + 2222 )]]([[)]([)]]([[)]([ tYEtYEtXEtXE +++= = )()( tVarYtVarX + ttz 21)( λλτβ +=∴ = 1 2( )t ...(5)λ λ+ The Formulation of The Status When Finite Numbers of Values of Z(t) Are Known The idea of best linear prediction is very important linear prediction theory has important application in standard linear models and the analysis of special data the theory has traditionally been taught as part of multivariate analysis. It is important for general stochastic process, time series and it is the basis for Linear Bayesian method [5, p134] Linear prediction is a mathematical operation where future values of discrete – time signals are estimated as a linear function of previous sample [6]. We shall express here how to predict Z(t) for the future time )( τ+t and for that we need to know the finite values of the past time and the correlation function )(τβz . Suppose we know finite value of Z(t) of the past time: )(,),(),( )1(2 )1( 1 )1( nstZstZstZ −−−  ; i.e. ~ (1) (1) (1) 1 2 nZ(t ) g{Z (t s ), Z (t s ), , Z (t s )}τ+ = − − − …(6) And the prediction is equivalent to the determining value of coefficients α1, α2 , …, αn of the linear combination : 1 1 2 2 n nZ(t s ) Z(t s ) Z(t s )α α α− + − + + − …(7) Mathematics - 389 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 [7, p.97] so: n(1) (1) (1) 1 2 n k k k 1 g{Z (t s ), Z (t s ), , Z (t s )} Z(t s )α = ∑− − − = − …(8) Substitute (8) in (6) we get : ~ n k k k 1 Z(t ) Z(t s )τ α = ∑+ = − …(9) The difference between the point Z(1)(t + τ) and its predicted value ~ Z (t + τ) represents the prediction error and denoted by : )( ~ )()1(2, ττστ +−+= tZtZn We shall consider here only one realization Z(1)(t–sk) ,k =1,2,….,n and that is because we can not take the absolute value of the prediction error │ζτ,n│as an index of the quality of prediction formula since it will be different for different realization of Z(t) and : 2n 2 ,n k k k 1 E Z(t ) Z(t s )τσ τ α = ∑= − − − …(10) takes its minimum value and the formula (10) called the mean square prediction error see[4, p145] n n 2 ,n k k k k k 1 k 1 [Z(t ) Z(t s )][Z(t ) Z(t s )]τσ τ α τ α = = ∑ ∑= − − − − − − …(11) ])([)()([])()([])()([ 111 ∑∑∑ === −+−−−−+−++= n k kk n k kk n k kk stZEtZstZEstZtZEtZtZE αταατττ Hence: ( ) )([)]([ 1 2 , k n k n stZtZEtt −+−+−+= ∑ = ατττβστ ∑∑∑ === −−++−− n k kk n k kk n k kk stZstZEtZstZE 111 )()([)()([ αατα Therefore: στ 2 ,n = ∑∑∑∑ = === −++−+− n n k kk n k kk n k ssstt 1 111 )(])([)()0(  βααβατβαβ Then: n n n 2 ,n k k k k k 1 1 k 1 (0) 2 Re (t s ) (s s )τσ β α β α α β = = = ∑ ∑ ∑= − + + −   …(12) Now we have to find the value of α1 = a1, α2 = a2,…,αn = an in which στ 2 ,n takes its Minimum value, by writing (12) as follows: ∑∑∑∑ = === −+−−−+−= n n k kk n k kk n k n ssstt 1 111 2 , )()()()0(  βααβατβαβστ Then by: 0 0....,, , 2211 = ∂ ∂ ==== aaa nnk n ααα α στ Mathematics - 390 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 We can fined the values αααα nnaa === ,....,, 2211 where στ 2 ,n takes its minimum value and: 0)s(10-)s(-0 k n 1 k 0....,, , 2211 =−+= ∂ ∂ ∑ =====   s aaa nnk n βατβ ααα α στ ∑ = =−++−= n kk ssas 1 0)()(  βτβ n k k 1 ( s ) a (s s )β τ β = ∑+ = −   …(13) then we can write (9) by : n k k 1 Z(t ) a Z(t )τ τ = ∑+ = + …(14) From (13) and since we have the correlation function of the sum of tow Poisson process: τλλβ τ )( 21)( +=Z we get the following system of equation )()()()( 11212111 sssassassa nn +=−++−+− τββββ  )()()()( 22222121 sssassassa nn +=−++−+− τββββ   )()()()( 2211 nnnnnn sssassassa +=−++−+− τββββ  Or: )()()(0 11212 sssassa nn +=−++−+ τ )()(0)( 21111 sssassa nn +=−+++− τ  )(0)()()( 1111212111 sssassassa nn +=+−++−+− −− τβ where this system can be solved to obtain the values of the coefficients a1, a2,…, an after that we can write the beast prediction formula (1.4.12) by using a1, a2,…, an and the know observed values )(,),(),( 21 nstZstZstZ −−−  by 0;)()()()( ~ 2211 ≥−++−+−=+ ττ nn stZastZastatZ  The Mean Square Error of Prediction Since we have ∑∑∑∑ = === −+−−−+−= n n k kk n k kkk n k kn ssstst 1 111 2 , )()()()0(  βααβαβαβστ And by (2.2.11) we get: ∑∑∑ = == −=+ n n k kkk n k k ssst 1 11 )()(  βααβα see [1,p101] Mathematics - 391 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 so : ∑∑∑∑∑ = === = −++−−−= n n k kk n k kk n n k kkn sssass 1 111 1 2 , )()()()0(     βαατββααβστ ∴ ∑∑ = = −−= n n k kkn ss 1 1 2 , )()0(  βααβστ And by using τλλτβ )()( 21 +=Z ∑ = ++= n k kkn sa 1 21, ))(( τλλστ = ∑∑ = = − n n k kk ss 1 1 )(  βαα References 1. Micosch, Th. (2009) Non–Life Insurance Mathematic, second edition-Verlag Barlin Heidelberg. 2. WWW.nas.its.tudelft.nl/people/piet/cupboockapters/pacup. 3. Gnedenko, B.V. (1962) the theory of probability, New York. 4. Yaglom, A.M, (1962) An Introduction to The Theory of Stationary Random Function, Prentice Hall. 5. Christensen,R. (2011) Plane Answers to Complex Question, fourth edition, science Business Media, LLC. 6. en.wikipedia.org/wiki/Linear_prediction 7. Winner .N. (1999) Extrapolation, Interpolation and Smoothing of Stationary Time series, New York. 8. WWW.ssc.upenn.edu/rwight/cours/poisson. http://www.nas.its.tudelft.nl/people/piet/cupboockapters/pacup http://www.ssc.upenn.edu/rwight/cours/poisson Mathematics - 392 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 التنبؤ الخطي لمجموع عمليتي بوايسون سيران حمزة كريم كلية العلوم االنسانية والرياضية الجامعة، جامعة كرميان 2012ايار 21قبل البحث: 2012اذار 17استلم البحث: الخالصة في زمن المستقبل Z(t)=X(t)+y(t) ان الهدف من عملنا هذا هو ايجاد التنبوء الخطي لمجموع عمليتي بوايسون Z(t+τ) , τ ≥ 0 وذلك عند معرفتنا لقيمZ(t) في الزمن الماضي ومعامل االرتباط لهذا المجموع βz(t). عمليات بوايسون، التنبؤ الخطي، جمع عمليتي بوايسون.الكلمات المفتاحية: