Apresentaremos agora os limites de controle ajustados aos dados sem `a presen¸ca de outlier.
Figura 5: Dados de IP sem outlier
(a) Gr´afico da S´erie com os limites do gr´afico c
modificado usando os estimadores usuais
Amostra 0 50 100 150 200 250 0 1 2 3 4 5 λ =0.91 α =0.29
(b) Gr´afico da S´erie com os limites do gr´afico
c modificado usando o estimador de (Bourguig- non e Vasconcellos, 2014) Amostra y 0 50 100 150 200 250 0 1 2 3 4 5 λ ~= 0.84 α ~=0.34
Fonte: Elaborada pelo autor
Novamente, a mesma interpreta¸c˜ao vale para a Figura 5. Contudo, podemos com- parar os estimadores n˜ao s´o pelos limites de controle obtidos, mas sim pelas estimativas do modelo Poisson INAR(1), que foram diferentes para os estimadores que utilizamos. Optamos por analisar a s´erie sem a presen¸ca de outlier, ent˜ao, obtemos as estimativas dos parˆametros do modelo INAR(1) com os estimadores usuais foramα = 0.29 e bb λ = 0.91, j´a as estimativas dos parˆametros do modelo INAR(1) utilizando o estimador de Bourguig- non e Vasconcellos (2014), foram α = 0.34 e ee λ = 0.84. Como o α representa a taxa de sobreviventes no tempo t − 1 e o λ representa a taxa de imigrantes, ou seja, a estimativa do erro, em que foi visto na Equa¸c˜ao (7). Ent˜ao, um usu´ario observado no per´ıodo t estava tamb´em ativo no per´ıodo t − 1 com probabilidade 0.29, ou seja, est´a novamente acessando o servidor, visto que seria a taxa de usu´arios ativos no per´ıodo t est´a tamb´em ativo no per´ıodo t − 1. Com o estimador de Bourguignon e Vasconcellos (2014), obtemos uma probabilidade maior de usu´arios ativos no per´ıodo t−1, sendo 0.34. J´a o λ refere-se a taxa de imigrante, referente ao erro, ou seja, o n´umero de usu´arios que chegar˜ao a acessar
a p´agina do servidor em t com probabilidade 0.91. Usando o outro estimador obtemos uma probabilidade de 0.84, menor em rela¸c˜ao aos estimadores usuais.
9
Considera¸c˜oes Finais
Este trabalho procurou apresentar uma metodologia alternativa `as t´ecnicas tradi- cionais de CEP, atrav´es de gr´aficos de controle modificados para an´alise do gr´afico c para n˜ao-conformidades na amostra para dados de contagem autocorrelacionados submetidos ao modelo INAR(1).
Dessa forma, foi realizado uma revis˜ao do modelo INAR(1) atrav´es do operador thinning binomial, o que tornou-se necess´ario, pois os modelos usuais de s´eries temporais n˜ao s˜ao adequados para modelar dados de contagem. Atrav´es do modelo INAR(1) pode- mos modificar os gr´aficos existentes de controle para dados de contagem para o gr´afico c, na qual chegamos as express˜oes dos limites de controle.
O INAR(1) ´e um modelo simples e ajust´avel para muitas situa¸c˜oes pr´aticas que sur- gem no contexto de CEP. Utilizando a simula¸c˜ao computacional foi poss´ıvel criar cen´arios que representavam estes gr´aficos sob diferentes situa¸c˜oes de autocorrela¸c˜ao, usamos per- turba¸c˜oes de ±10% e ±20% para as diferentes combina¸c˜oes dos parˆametros e medimos o seu desempenho, atrav´es do n´umero m´edio de amostras at´e detectar mudan¸ca (NMA).
Em fun¸c˜ao das vari´aveis serem discretas n˜ao ´e poss´ıvel encontrar um NMAF ideal, assim eles apenas coincidem grosseiramente.
O gr´afico n˜ao ´e ´util para detectar mudan¸ca quando λ0
1−α0 ´e fixo. J´a para mu-
dan¸cas apenas em λ0 ou apenas em α0, o gr´afico ´e ´util para detectar mudan¸cas positivas
nos parˆametros, que ´e a situa¸c˜ao mais interessante, em geral, buscando um aumento no n´umero m´edio de n˜ao conformidades do processo. Por´em, n˜ao ´e ´util para detectar mudan¸ca que diminuam esses parˆametros o que indicariam melhoria do processo.
Diante do que foi exposto, pode-se concluir que autocorrela¸c˜ao provoca nos gr´aficos de controle c uma sinaliza¸c˜ao mais tardia quando este sofre alguma perturba¸c˜ao de −20% e −10%, no entanto quando estamos lidando com uma perturba¸c˜ao de +20% na autocor- rela¸c˜ao, a medida que deslocamento aumenta, o gr´afico detecta r´apido a mudan¸ca situa¸c˜ao semelhante ocorre para mudan¸ca em λ.
A aplica¸c˜ao do modelo INAR(1), usando os estimadores usuais da literatura e o estimador de Bourguignon e Vasconcellos (2014), obtiveram resultados semelhantes para a obten¸c˜ao dos limites de controle para o gr´afico c modificado para dados de contagem autocorrelacionados. Enquanto para a estimativa do α de Bourguignon de Vasconcel-e los (2014), obteve uma probabilidade maior, ou seja, uma taxa maior em rela¸c˜ao aos estimadores usuais.
9.1
Sugest˜oes para trabalhos futuros
Uma extens˜ao deste trabalho seria trabalho com gr´afico de controle para dados de contagem autocorrelacionados, referente a dados com sobredispers˜ao, cuja variˆancia ´e maior do que a m´edia. A ideia, seria usar os gr´aficos e atrav´es de simula¸c˜oes, verificar o comportamento para diferentes combina¸c˜oes dos parˆametros.
Quando analisamos dados reais uma alternativa a ser implementada seria a ob- ten¸c˜ao dos limites de controle via processo de reamostragem, bootstrap.
Referˆencias
[1] Al-Osh, M. A.; Alzaid, A. A. First-order integer-valued autoregressive (inar (1)) process. Journal of Time Series Analysis,1987, 8, 261-275.
[2] Alzaid, A.;Al-Osh, M. An integer-valued pth-order autoregressive structure(Inar (p)) process. Journal of Applied Probability,1990, pp. 314-324.
[3] Bourguignon, M. Modelos Inar Sazonais e de Ra´ızes Unit´arias. Tese de Doutorado, Universidade Federal de Pernambuco, 2011.
[4] Bourguignon, M.;Vasconcellos, K. Improved estimation for Poisson Inar (1) models. Journal of Statistical Computation and Simulation, 2014, pp. 1-17.
[5] da Silva, I. M. M. Contributions to the analysis of discrete-valued time series. Tese de Doutorado, Universidade do Porto, 2005.
[6] Donadelli, J. Fun¸c˜ao geradora de probabilidade e soma de vari´aveis independentes. https://anotacoesdeaula.wordpress.com/2013/05/21/impe bc1414-fgp/, 2013, acesso em 03/12/2015.
[7] Franke, J.;Seligmann, T. Conditional maximum likelihood estimates for inar (1) pro- cesses and their application to modelling epileptic seizure counts. Developments in time series analysis, 1993, pp. 310-330.
[8] Freeland, R.;McCabe, B. P. Analysis of low count time series data by Poisson auto- regression. Journal of Time Series Analysis, 2004, 25, pp. 701-722.
[9] Lima, T. A. C. Modelos INAR e RCINAR, estima¸c˜ao e aplica¸c˜ao. Disserta¸c˜ao de Mestrado,Universidade de S˜ao Paulo, 2013.
[10] Montgomery, D. C. Introdu¸c˜ao ao controle estat´ıstico da qualidade. LTC, 2004. [11] Russo, S. L. Gr´aficos de controle para vari´aveis n˜ao-conformes autocorrelacionadas.
Tese de Doutorado, Universidade Federal de Santa Catarina, 2002.
[12] Sheaffer, R.;Leavenworth, R. The negative binomial model for counts in units of varying size. Journal of Quality Technology, 8 ,1976.
[13] Silva, L. A. O Efeito da Autocorrela¸c˜ao no Poder de Detec¸c˜ao do Gr´afico de M´edias(X) com Modelos Autorregressivos de ordem 1 e 2, Univerisidade Federal do Rio Grande do Norte, 2012.
[14] Steutel, F.;Van Harn, K. Discrete analogues of self-decomposability and stability. The Annals of Probability, 1979, pp. 893-899.
[15] Weiβ, C. H. Controlling correlated processes of poisson counts. Quality and reliability engineering international, 2007, v.23,pp. 741-754.
Apˆendice A
Algumas demonstra¸c˜oes do Operador Thinning Binomial
A seguir as demonstra¸c˜oes das propriedades do LEMA 3.1.1, da p´agina 17. 1. 0 ◦ N1 = N1 X i=1 tal que P (Xi = 1) = α = 0 ⇒ P (Xi = 0) = 1 − α = 1. Dessa forma, 0 ◦ N1 = N1 X i=1 X1 = N1 X i=1 0 = 0. 2. 1 ◦ N1 = N1 X i=1 Xi tal que P (Xi = 1) = α = 1 assim, 1 ◦ N1 = N1 X i=1 Xi = N1 X i=1 1 = N1. 3. α1◦ (α2◦ N2) = (α1α2) ◦ N2
Seja GN(t) = E(tN) a fun¸c˜ao geradora de probabilidade de N . Se Y = α1◦ N = N
X
j=1
zj, temos que
GY(t) = Gα1◦N(t) = GN(Gz(t)) = G(α1t + 1 − α1),
em que Gz(t) ´e a fun¸c˜ao geradora de probabilidade da Bernoulli(α1). Fazendo
Y = α2◦ N = N X j=1 zj e H = α1◦ Y = Y X j=1 hj, temos que GH(t) = GY(Gh(t)) = GN(Gz(Gh(t))) = GN(α1α2t + 1 − α1α2) Portanto, α1◦ (α2◦ N ) = (α1α2) ◦ N .
4. α1◦ (N1+ N2) = α1◦ N1+ α2◦ N2 α1◦(N1+N2) = N1+N2 X i=1 Xi = N1 X i=1 Xi+ N2 X i=N1+1 = α1◦N1+ N2 X i=1 Xi+N1 = α1◦N1+α1◦N2. 5. E[α1◦ N1] = E[E[(α1◦ N1)|N1]] = E " E " N 1 X i=1 Xi|N1 ## = E "N 1 X i=1 E[Xi|N1] # = E "N 1 X i=1 E[X1] # = E "N 1 X i=1 α1 # = E[α1N1] = α1E[N1]. 6. E[(α1◦ N1)N2] = E " N1 X i=1 Xi ! N2 # = E[X1+ X2+ · · · + XN1N2] = E[X1N2+ X2N2+ · · · + XN1N2]
= E[X1]E[N2] + E[X2]E[N2] + · · · + E[XN1E[N2]]
= E[N2] [E[N1] + E[X2] + · · · + E[XN1]]
= E[N2] [E[N1+ X2+ · · · + XN1]]
= E[N2]E[α1 ◦ N1]
= α1E[N1]E[N2]
8. V ar(α1◦ N1) = α21V ar(N1) + α1(1 − α1)E(N1).
Notemos que, a variˆancia pode ser escrita da seguinte forma: V ar(Y ) = E(V ar(Y |X))+ V ar(E(Y |X)).
V ar(α1◦ N1) = V ar(E(α1◦ N1|N1)) + E(V ar(α1◦ N1|N1))
= V ar(α1N1) + E(α1(1 − α1)N1)
= α21V ar(N1) + α1(1 − α1)E(N1).
Apˆendice B - Rotina Computacional
########## Fun¸c~oes ######################### ########## INAR(1) ############
##Modelo INAR(1) -> Xt = \alpha o Xt-1 + It #It => vari´avel discreta poisson(\lambda) Inar1 <- function(alpha,lambda,n){
y0 <- lambda/(1-alpha) #Esperan¸ca de Xt => E(Xt) b <- 5*n
y <- numeric() y[1] <-round(y0,0) k <- 2
while(k <= (n + b)){
opbin <- rbinom(1, y[k-1], alpha) poisson <- rpois(1, lambda)
y[k] <- opbin + poisson k <- k + 1
}
y[(b + 1):(n + b)]
}
######### Fun¸c~ao que gera o NMA ############
NMA<-function(alfa,lambida,k,mud2,mud){ nma <- NULL
LSC<-lambida/(1-alfa)+k*sqrt(lambida/(1-alfa))
LIC<-max(0,lambida/(1-alfa)-k*sqrt(lambida/(1-alfa))) rep <- 10000
for(i in 1:rep){ #print(i)
j <- 0 inar <- LSC r <- 10000
while((inar <= LSC) && (inar >= LIC) && (j < r)){ inar = Inar1(alfa*mud2,lambida*mud,1) j <- j + 1 } nma[i] <- j } return(nma) }
#################NMAs para diferentes situa¸c~oes dos par^ametros############ # REMOVENDO TODOS OS OBJETOS DA ´AREA DE TRABALHO
rm(list = ls()) args(NMA)
#Sem autocorrela¸c~ao
M <- NMA(0,1,3,1,1) ; mean(M)
mu0 <- NMA(0.2,1,3,1,1) ; mean(mu0) #=107.1711 mu01 <- NMA(0.4,1,3,1,1) ; mean(mu01) #=134.5859 mu02 <- NMA(0.6,1,3,1,1) ; mean(mu02) #263.3284
########Suponha o processo em controle com \mu0 = 3 mu03 <- NMA(0.2,3,3,1,1) ; mean(mu03)
mu04 <- NMA(0.4,3,3,1,1) ; mean(mu04) mu05 <- NMA(0.6,3,3,1,1) ; mean(mu05)
######Suponha o processo em controle com \mu0 = 5 mu06 <- NMA(0.2,5,3,1,1) ; mean(mu06)
mu07 <- NMA(0.4,5,3,1,1) ; mean(mu07) mu08 <- NMA(0.6,5,3,1,1) ; mean(mu08)
######Suponha o processo em controle com \mu0 = 7 mu09 <- NMA(0.2,7,3,1,1) ; mean(mu09)
mu10 <- NMA(0.4,7,3,1,1) ; mean(mu10) mu11 <- NMA(0.6,7,3,1,1) ; mean(mu11)
######Suponha o processo em controle com \mu0 = 9 mu12 <- NMA(0.2,9,3,1,1) ; mean(mu12)
mu13 <- NMA(0.4,9,3,1,1) ; mean(mu13) mu14 <- NMA(0.6,9,3,1,1) ; mean(mu14)
##Sem autocorrela¸c~ao
x1 <- NMA(0,3,3,1,1) ; mean(x1) x2 <- NMA(0,5,3,1,1) ; mean(x2) x3 <- NMA(0,7,3,1,1) ; mean(x3) x4 <- NMA(0,9,3,1,1) ; mean(x4)
##############MUDANC¸A NA M´EDIA DO PROCESSO (AUMENTO DE 20%) ############# ###### Mudan¸ca na m´edia \lambda = 1.2 * \lambda0 , alfa0 = 0.2
args(NMA)
lamb0<- NMA(0.0,1,3,1,1.2) ; mean(lamb0) #Sem autocorrela¸c~ao lam1 <- NMA(0.2,1,3,1,1.2) ; mean(lam1)
lam2 <- NMA(0.4,1,3,1,1.2) ; mean(lam2) lam3 <- NMA(0.6,1,3,1,1.2) ; mean(lam3)
###### Mudan¸ca na m´edia \lambda = 1.2 * \lambda0 , alfa0 =0.2 lam4 <- NMA(0.2,3,3,1,1.2) ; mean(lam4)
lam5 <- NMA(0.4,3,3,1,1.2) ; mean(lam5) lam6 <- NMA(0.6,3,3,1,1.2) ; mean(lam6)
###### Mudan¸ca na m´edia \lambda = 1.2 * \lambda0 , alfa0 =0.2 lam7 <- NMA(0.2,5,3,1,1.2) ; mean(lam7)
lam8 <- NMA(0.4,5,3,1,1.2) ; mean(lam8) lam9 <- NMA(0.6,5,3,1,1.2) ; mean(lam9)
###### Mudan¸ca na m´edia \lambda = 1.2 * \lambda0 , alfa0 =0.2 lam10 <- NMA(0.2,7,3,1,1.2) ; mean(lam10)
lam11 <- NMA(0.4,7,3,1,1.2) ; mean(lam11) lam12 <- NMA(0.6,7,3,1,1.2) ; mean(lam12)
###### Mudan¸ca na m´edia \lambda = 1.2 * \lambda0 , alfa0 =0.2
lam13 <- NMA(0.2,9,3,1,1.2) ; mean(lam13) lam14 <- NMA(0.4,9,3,1,1.2) ; mean(lam14) lam15 <- NMA(0.6,9,3,1,1.2) ; mean(lam15)
args(NMA) #Sem autocorrela¸c~ao a1 <- NMA(0,3,3,1,1.2) ; mean(a1) a2 <- NMA(0,5,3,1,1.2) ; mean(a2) a3 <- NMA(0,7,3,1,1.2) ; mean(a3) a4 <- NMA(0,9,3,1,1.2) ; mean(a4) ################################################################## ########## Mudan¸ca na m´edia = lambda = 1.1 * lambda0 ########## args(NMA) ##### lambda = 1 x0<- NMA(0,1,3,1,1.1) ; mean(x0) x1 <- NMA(0.2,1,3,1,1.1) ; mean(x1) x2 <- NMA(0.4,1,3,1,1.1) ; mean(x2) x3 <- NMA(0.6,1,3,1,1.1) ; mean(x3) ##### lambda = 3 x4 <- NMA(0,3,3,1,1.1) x4 <- NMA(0.2,3,3,1,1.1) ; mean(x4) x5 <- NMA(0.4,3,3,1,1.1) ; mean(x5) x6 <- NMA(0.6,3,3,1,1.1) ; mean(x6) ##### lambda = 5 x7 <- NMA(0.2,5,3,1,1.1) ; mean(x7) x8 <- NMA(0.4,5,3,1,1.1) ; mean(x8) x9 <- NMA(0.6,5,3,1,1.1) ; mean(x9) ##### lambda = 7 x10 <- NMA(0.2,7,3,1,1.1) ; mean(x10) x11 <- NMA(0.4,7,3,1,1.1) ; mean(x11) x12 <- NMA(0.6,7,3,1,1.1) ; mean(x12) ##### lambda = 9 x13 <- NMA(0.2,9,3,1,1.1) ; mean(x13) x14 <- NMA(0.4,9,3,1,1.1) ; mean(x14) x15 <- NMA(0.6,9,3,1,1.1) ; mean(x15)
#Sem autocorrela¸c~ao alfa = 0 b <- NMA(0,1,3,1,1.1) ; mean(b) b1 <- NMA(0,3,3,1,1.1) ; mean(b1) b2 <- NMA(0,5,3,1,1.1) ; mean(b2) b3 <- NMA(0,7,3,1,1.1) ; mean(b3) b4 <- NMA(0,9,3,1,1.1) ; mean(b4)
########## Mudan¸ca na m´edia = lambda = 0.9 * lambda0 ########## xx0 <- NMA(0,1,3,1,0.9) ; mean(xx0) #Sem autocorrela¸c~ao
xx1 <- NMA(0.2,1,3,1,0.9) ; mean(xx1) # lambda = 1 xx2 <- NMA(0.4,1,3,1,0.9) ; mean(xx2) # lambda = 1 xx3 <- NMA(0.6,1,3,1,0.9) ; mean(xx3) # lambda = 1
xx4 <- NMA(0.2,3,3,1,0.9) ; mean(xx4) # lambda = 3 xx5 <- NMA(0.4,3,3,1,0.9) ; mean(xx5) # lambda = 3 xx6 <- NMA(0.6,3,3,1,0.9) ; mean(xx6) # lambda = 3
xx7 <- NMA(0.2,5,3,1,0.9) ; mean(xx7) # lambda = 5 xx8 <- NMA(0.4,5,3,1,0.9) ; mean(xx8) # lambda = 5 xx9 <- NMA(0.6,5,3,1,0.9) ; mean(xx9) # lambda = 5
xx10 <- NMA(0.2,7,3,1,0.9); mean(xx10) # lambda = 7 xx11 <- NMA(0.4,7,3,1,0.9); mean(xx11) # lambda = 7 xx12 <- NMA(0.6,7,3,1,0.9); mean(xx12) # lambda = 7
xx13 <- NMA(0.2,9,3,1,0.9); mean(xx13) # lambda = 9 xx14 <- NMA(0.4,9,3,1,0.9); mean(xx14) # lambda = 9 xx15 <- NMA(0.6,9,3,1,0.9); mean(xx15) # lambda = 9
#Sem autocorrela¸c~ao c1 <- NMA(0,3,3,1,0.9) ; mean(c1) c2 <- NMA(0,5,3,1,0.9) ; mean(c2) c3 <- NMA(0,7,3,1,0.9) ; mean(c3) c4 <- NMA(0,9,3,1,0.9) ; mean(c4) ######################################################################### ############ MUDANC¸A NA M´EDIA (DECR´ECIMO DE 20%) ##############
###### Mudan¸ca na m´edia \lambda = 0.8 * \lambda0 , alfa0 = 0.2, 0.4 e 0.6 la1 <- NMA(0.2,1,3,1,0.8) ; mean(la1)
la2 <- NMA(0.4,1,3,1,0.8) ; mean(la2) la3 <- NMA(0.6,1,3,1,0.8) ; mean(la3)
###### Mudan¸ca na m´edia \lambda = 0.8 * \lambda0 , alfa0 = 0.2, 0.4 e 0.6 la4 <- NMA(0.2,3,3,1,0.8) ; mean(la4)
la5 <- NMA(0.4,3,3,1,0.8) ; mean(la5) la6 <- NMA(0.6,3,3,1,0.8) ; mean(la6)
###### Mudan¸ca na m´edia \lambda = 0.8 * \lambda0 , alfa0 = 0.2, 0.4 e 0.6 la7 <- NMA(0.2,5,3,1,0.8) ; mean(la7)
la8 <- NMA(0.4,5,3,1,0.8) ; mean(la8) la9 <- NMA(0.6,5,3,1,0.8) ; mean(la9)
###### Mudan¸ca na m´edia \lambda = 0.8 * \lambda0 , alfa0 = 0.2, 0.4 e 0.6 la10 <- NMA(0.2,7,3,1,0.8) ; mean(la10) #lambida = 7
la11 <- NMA(0.4,7,3,1,0.8) ; mean(la11) #lambida = 7 la12 <- NMA(0.6,7,3,1,0.8) ; mean(la12) #lambida = 7
###### Mudan¸ca na m´edia \lambda = 0.8 * \lambda0 , alfa0 = 0.2, 0.4 e 0.6 la13 <- NMA(0.2,9,3,1,0.8) ; mean(la13) #lambida = 9
la14 <- NMA(0.4,9,3,1,0.8) ; mean(la14) #lambida = 9 la15 <- NMA(0.6,9,3,1,0.8) ; mean(la15) #lambida = 9
#Sem autocorrela¸c~ao, ou seja, alfa = 0 => Gr´afico c tradicional d1 <- NMA(0,3,3,1,0.8) ; mean(d1)
d2 <- NMA(0,5,3,1,0.8) ; mean(d2) d3 <- NMA(0,7,3,1,0.8) ; mean(d3) d4 <- NMA(0,9,3,1,0.8) ; mean(d4)
############################################################################ #Mudan¸ca na autocorrela¸c~ao \alpha = 0.8 * \aplha0 , alfa0 = 0.2, 0.4 e 0.6
aut1 <- NMA(0.2,1,3,0.8,1) ; mean(aut1) #lambida = 1 aut2 <- NMA(0.4,1,3,0.8,1) ; mean(aut2) #lambida = 1 aut3 <- NMA(0.6,1,3,0.8,1) ; mean(aut3) #lambida = 1
aut4 <- NMA(0.2,3,3,0.8,1) ; mean(aut4) #lambida = 3 aut5 <- NMA(0.4,3,3,0.8,1) ; mean(aut5) #lambida = 3 aut6 <- NMA(0.6,3,3,0.8,1) ; mean(aut6) #lambida = 3
aut7 <- NMA(0.2,5,3,0.8,1) ; mean(aut7) #lambida = 5 aut8 <- NMA(0.4,5,3,0.8,1) ; mean(aut8) #lambida = 5 aut9 <- NMA(0.6,5,3,0.8,1) ; mean(aut9) #lambida = 5
aut10 <- NMA(0.2,7,3,0.8,1) ; mean(aut10) #lambida = 7 aut11 <- NMA(0.4,7,3,0.8,1) ; mean(aut11) #lambida = 7 aut12 <- NMA(0.6,7,3,0.8,1) ; mean(aut12) #lambida = 7
aut13 <- NMA(0.2,9,3,0.8,1) ; mean(aut13) #lambida = 9 aut14 <- NMA(0.4,9,3,0.8,1) ; mean(aut14) #lambida = 9 aut15 <- NMA(0.6,9,3,0.8,1) ; mean(aut15) #lambida = 9
##########Mudan¸ca na autocorrela¸c~ao do processo ############# ##### Em controle
####### Em controle alfa = alfa0 , alfa0 = 0.2 ####### lambda0 = 1
al1 <- NMA(0.2,1,3,1,1) ; mean(al1) al2 <- NMA(0.4,1,3,1,1) ; mean(al2) al3 <- NMA(0.6,1,3,1,1) ; mean(al3)
############################################################################# ########## MUDANC¸A NA AUTOCORRELAC¸~AO (ACR´ECIMO DE 20% )#########
#Mudan¸ca no alfa <=> alfa = 1.2 alfa0 , lambda = 1 , alfa0 = 0.2. 0.4 e 0.6 alp1 <- NMA(0.2,1,3,1.2,1) ; mean(alp1)
alp2 <- NMA(0.4,1,3,1.2,1) ; mean(alp2) alp3 <- NMA(0.6,1,3,1.2,1) ; mean(alp3)
#Mudan¸ca no alfa <=> alfa = 1.2 alfa0 , lambda = 3 , alfa0 = 0.2. 0.4 e 0.6 alp4 <- NMA(0.2,3,3,1.2,1) ; mean(alp4)
alp5 <- NMA(0.4,3,3,1.2,1) ; mean(alp5) alp6 <- NMA(0.6,3,3,1.2,1) ; mean(alp6)
alp7 <- NMA(0.2,5,3,1.2,1) ; mean(alp7) alp8 <- NMA(0.4,5,3,1.2,1) ; mean(alp8) alp9 <- NMA(0.6,5,3,1.2,1) ; mean(alp9)
#Mudan¸ca no alfa <=> alfa = 1.2 alfa0 , lambda = 7 , alfa0 = 0.2. 0.4 e 0.6 alp10 <- NMA(0.2,7,3,1.2,1) ; mean(alp10)
alp11 <- NMA(0.4,7,3,1.2,1) ; mean(alp11) alp12 <- NMA(0.6,7,3,1.2,1) ; mean(alp12)
#Mudan¸ca no alfa <=> alfa = 1.2 alfa0 , lambda = 9 , alfa0 = 0.2. 0.4 e 0.6 alp13 <- NMA(0.2,9,3,1.2,1) ; mean(alp13)
alp14 <- NMA(0.4,9,3,1.2,1) ; mean(alp14) alp15 <- NMA(0.6,9,3,1.2,1) ; mean(alp15)
#Sem autocorrela¸c~ao alfa = 0.
e <- NMA(0,1,3,1.2,1) ; mean(e) #273 e1 <- NMA(0,3,3,1.2,1) ; mean(e1) e2 <- NMA(0,5,3,1.2,1) ; mean(e2) e3 <- NMA(0,7,3,1.2,1) ; mean(e3) e4 <- NMA(0,9,3,1.2,1) ; mean(e4) ############################################################################# ########### Mudan¸ca na autocorrela¸c~ao alfa = 1.1. alfa0 ###############
#lambda = 1
al1 <- NMA(0.2,1,3,1.1,1) ; mean(al1) al2 <- NMA(0.4,1,3,1.1,1) ; mean(al2) al3 <- NMA(0.6,1,3,1.1,1) ; mean(al3)
#lambda = 3
al4 <- NMA(0.2,3,3,1.1,1) ; mean(al4) al5 <- NMA(0.4,3,3,1.1,1) ; mean(al5) al6 <- NMA(0.6,3,3,1.1,1) ; mean(al6)
#lambda = 5
al7 <- NMA(0.2,5,3,1.1,1) ; mean(al7) al8 <- NMA(0.4,5,3,1.1,1) ; mean(al8) al9 <- NMA(0.6,5,3,1.1,1) ; mean(al9)
al10 <- NMA(0.2,7,3,1.1,1) ; mean(al10) al11 <- NMA(0.4,7,3,1.1,1) ; mean(al11) al12 <- NMA(0.6,7,3,1.1,1) ; mean(al12)
#lambda = 9
al13 <- NMA(0.2,9,3,1.1,1) ; mean(al13) al14 <- NMA(0.4,9,3,1.1,1) ; mean(al14) al15 <- NMA(0.6,9,3,1.1,1) ; mean(al15)
#Sem autocorrela¸c~ao args(NMA) d <-NMA(0,1,3,1.1,1) ; mean(d) d1 <-NMA(0,3,3,1.1,1) ; mean(d1) d2 <-NMA(0,5,3,1.1,1) ; mean(d2) d3 <-NMA(0,7,3,1.1,1) ; mean(d3) d4 <-NMA(0,9,3,1.1,1) ; mean(d4) ####################################################################### ########## MUDANC¸A NA AUTOCORRELAC¸~AO (DECR´ECIMO DE 20%) ########
args(NMA)
#Mudan¸ca no alfa alfa = 0.8 * alfa0 , lambida = 1, alfa0 = 0.2. 0.4 e 0.6 aut1 <-NMA(0.2,1,3,0.8,1) ; mean(aut1)
aut2 <-NMA(0.4,1,3,0.8,1) ; mean(aut2) aut3 <-NMA(0.6,1,3,0.8,1) ; mean(aut3)
#Mudan¸ca no alfa alfa = 0.8 * alfa0, lambida = 3,alfa0 = 0.2. 0.4 e 0.6 aut4 <-NMA(0.2,3,3,0.8,1) ; mean(aut4)
aut5 <-NMA(0.4,3,3,0.8,1) ; mean(aut5) aut6 <-NMA(0.6,3,3,0.8,1) ; mean(aut6)
#Mudan¸ca no alfa alfa = 0.8 * alfa0, lambida = 5,alfa0 = 0.2. 0.4 e 0.6
aut7 <- NMA(0.2,5,3,0.8,1) ; mean(aut7) aut8 <- NMA(0.4,5,3,0.8,1) ; mean(aut8) aut8 <- NMA(0.6,5,3,0.8,1) ; mean(aut9)
#Mudan¸ca no alfa alfa = 0.8 * alfa0, lambida = 7,alfa0 = 0.2. 0.4 e 0.6
aut11 <- NMA(0.4,7,3,0.8,1) ; mean(aut11) aut12 <- NMA(0.6,7,3,0.8,1) ; mean(aut12)
#Mudan¸ca no alfa alfa = 0.8 * alfa0, lambida = 9,alfa0 = 0.2. 0.4 e 0.6 aut13 <- NMA(0.2,9,3,0.8,1) ; mean(aut13)
aut14 <- NMA(0.4,9,3,0.8,1) ; mean(aut14) aut15 <- NMA(0.6,9,3,0.8,1) ; mean(aut15)
##Sem autocorrela¸c~ao f <- NMA(0,1,3,0.8,1) ; mean(f) f1 <- NMA(0,3,3,0.8,1) ; mean(f1) f2 <- NMA(0,5,3,0.8,1) ; mean(f2) f3 <- NMA(0,7,3,0.8,1) ; mean(f3) f4 <- NMA(0,9,3,0.8,1) ; mean(f4) ############## alfa = 0.9 alfa0 ################ ######### lambda = 1
auto1 <- NMA(0.2,1,3,0.9,1) ; mean(auto1) auto2 <- NMA(0.4,1,3,0.9,1) ; mean(auto2) auto3 <- NMA(0.6,1,3,0.9,1) ; mean(auto3)
######### lambda = 3
auto4 <- NMA(0.2,3,3,0.9,1) ; mean(auto4) auto5 <- NMA(0.4,3,3,0.9,1) ; mean(auto5) auto6 <- NMA(0.6,3,3,0.9,1) ; mean(auto6)
######### lambda = 5
auto7 <- NMA(0.2,5,3,0.9,1) ; mean(auto7) auto8 <- NMA(0.4,5,3,0.9,1) ; mean(auto8) auto9 <- NMA(0.6,5,3,0.9,1) ; mean(auto9)
######### lambda = 7
auto10 <- NMA(0.2,7,3,0.9,1) ; mean(auto10) auto11 <- NMA(0.4,7,3,0.9,1) ; mean(auto11) auto12 <- NMA(0.6,7,3,0.9,1) ; mean(auto12)
######### lambda = 9
auto14 <- NMA(0.4,9,3,0.9,1) ; mean(auto14) auto15 <- NMA(0.6,9,3,0.9,1) ; mean(auto15)
#Sem autocorrela¸c~ao g <- NMA(0,1,3,0.9,1) ; mean(g) g1 <- NMA(0,3,3,0.9,1) ; mean(g1) g2 <- NMA(0,5,3,0.9,1) ; mean(g2) g3 <- NMA(0,7,3,0.9,1) ; mean(g3) g4 <- NMA(0,9,3,0.9,1) ; mean(g4)
##########Distribui¸c~ao marginal "lambida/(1-alfa)" constante ###### ##Espera-se encontrar NMAs pr´oximos
##NMAs considerando uma mudan¸ca em alfa e lambda, mas
#lambida/(1-alfa) constante = 1
c1 <- NMA(0.2,0.8,3,1,1) ; mean(c1) #lambda = 0.8 = 274.5897 c2 <- NMA(0.4,0.6,3,1,1) ; mean(c2) #lambda = 0.6 = 274.7449 c3 <- NMA(0.6,0.4,3,1,1) ; mean(c3) #lambda = 0.4 = 277.3347
#lambida/(1-alfa) constante = 3 #args(NMA)
const1 <- NMA(0.2,2.4,3,1,1) ; mean(const1) ##para o lambda = 2.4 const2 <- NMA(0.4,1.8,3,1,1) ; mean(const2) ##para o lambda = 1.8 const3 <- NMA(0.6,1.2,3,1,1) ; mean(const3) ##para o lambda = 1.2
#lambida/(1-alfa) constante = 6
const4 <- NMA(0.2,4.8,3,1,1) ; mean(const4) ##para o lambda = 4.8 const5 <- NMA(0.4,3.6,3,1,1) ; mean(const5) ##para o lambda = 3.6 const6 <- NMA(0.6,2.4,3,1,1) ; mean(const6) ##para o lambda = 2.4
#lambida/(1-alfa) constante = 9
const7 <- NMA(0.2,7.2,3,1,1) ; mean(const7) ##para o lambda = 7.2 const8 <- NMA(0.4,5.4,3,1,1) ; mean(const8) ##para o lambda = 5.4 const9 <- NMA(0.6,3.6,3,1,1) ; mean(const9) ##para o lambda = 3.6
const10 <- NMA(0.2,9.6,3,1,1) ; mean(const10) ##para o lambda = 9.6 const11 <- NMA(0.4,7.2,3,1,1) ; mean(const11) ##para o lambda = 7.2 const12 <- NMA(0.6,4.8,3,1,1) ; mean(const12) ##para o lambda = 4.8
#lambida/(1-alfa) constante = 15
const13 <- NMA(0.2,12,3,1,1) ; mean(const13) ##para o lambda = 12 const14 <- NMA(0.4,9,3,1,1) ; mean(const14) ##para o lambda = 9 const15 <- NMA(0.6,6,3,1,1) ; mean(const15) ##para o lambda = 6
#Sem autocorrela¸c~ao args(NMA) h <- NMA(0,3,3,1,1) ; mean(h) h1 <- NMA(0,6,3,1,1) ; mean(h1) h2 <- NMA(0,9,3,1,1) ; mean(h2) h3 <- NMA(0,12,3,1,1) ; mean(h3) h4 <- NMA(0,15,3,1,1) ; mean(h4)
################ Considerando alfa = 1.1 * alfa0 ################ ####### lambda/(1-alfa) = 1 ########
u1 <- NMA(0.22,0.78,3,1,1) ; mean(u1) #lambda = 0.78 = 279.8315 u2 <- NMA(0.44,0.56,3,1,1) ; mean(u2) #lambda = 0.56 = 273.1792 u3 <- NMA(0.66,0.34,3,1,1) ; mean(u3) #lambda = 0.34 = 295.5982
####### lambda/(1-alfa) = 3 ######### m1 <- NMA(0.22,2.34,3,1,1) ; mean(m1) m2 <- NMA(0.44,1.68,3,1,1) ; mean(m2) m3 <- NMA(0.66,1.02,3,1,1) ; mean(m3) ###### lambda/(1-alfa) = 6 ######### m4 <- NMA(0.22,4.68,3,1,1) ; mean(m4) m5 <- NMA(0.44,3.36,3,1,1) ; mean(m5) m6 <- NMA(0.66,2.04,3,1,1) ; mean(m6) ####### lambda/(1-alfa) = 9 ######### m7 <- NMA(0.22,7.02,3,1,1) ; mean(m7) m8 <- NMA(0.44,5.04,3,1,1) ; mean(m8)
m9 <- NMA(0.66,3.06,3,1,1) ; mean(m9) ####### lambda/(1-alfa) = 12 ######### m10 <- NMA(0.22,9.36,3,1,1) ; mean(m10) m11 <- NMA(0.44,6.72,3,1,1) ; mean(m11) m12 <- NMA(0.66,4.08,3,1,1) ; mean(m12) ####### lambda/(1-alfa) = 15 ######### m13 <- NMA(0.22,11.7,3,1,1) ; mean(m13) m14 <- NMA(0.44,8.4,3,1,1) ; mean(m14) m15 <- NMA(0.66,5.1,3,1,1) ; mean(m15) #Sem autocorrela¸c~ao args(NMA) i <- NMA(0,3,3,1,1) ; mean(i) i1 <- NMA(0,6,3,1,1) ; mean(i1) i2 <- NMA(0,9,3,1,1) ; mean(i2) i3 <- NMA(0,12,3,1,1) ; mean(i3) i4 <- NMA(0,15,3,1,1) ; mean(i4)
############### Considerando alfa = 1.2 * alfa0 ################## ##### alfa = 1.2 * alfa0 , lambda/(1-alfa) = 3
#args(NMA)
malf1 <- NMA(0.24,2.28,3,1,1) ; mean(malf1) ##para o lambda = 2.28 malf2 <- NMA(0.48,1.56,3,1,1) ; mean(malf2) ##para o lambda = 1.56 malf3 <- NMA(0.72,0.84,3,1,1) ; mean(malf3) ##para o lambda = 0.84
##### alfa = 1.2 * alfa0 ,lambda/(1-alfa) constante = 6
malf4 <- NMA(0.24,4.56,3,1,1) ; mean(malf4) ##para o lambda = 4.56 malf5 <- NMA(0.48,3.12,3,1,1) ; mean(malf5) ##para o lambda = 3.12 malf6 <- NMA(0.72,1.68,3,1,1) ; mean(malf6) ##para o lambda = 1.68
##### alfa = 1.2 * alfa0 , lambida/(1-alfa) constante = 9
malf7 <- NMA(0.24,6.84,3,1,1) ; mean(malf7) ##para o lambda = 6.84 malf8 <- NMA(0.48,4.68,3,1,1) ; mean(malf8) ##para o lambda = 4.68 malf9 <- NMA(0.72,2.52,3,1,1) ; mean(malf9) ##para o lambda = 2.52
malf10 <- NMA(0.24,9.12,3,1,1) ; mean(malf10) #316
malf11 <- NMA(0.48,6.24,3,1,1) ; mean(malf11) #323
malf12 <- NMA(0.72,3.36,3,1,1) ; mean(malf12) #381
##### alfa = 1.2 * alfa0 , lambida/(1-alfa) constante = 15 malf13 <- NMA(0.24,11.4,3,1,1) ; mean(malf13)
#
malf14 <- NMA(0.48,7.8,3,1,1) ; mean(malf14) #
malf15 <- NMA(0.72,4.2,3,1,1) ; mean(malf15)
#Sem autocorrela¸c~ao args(NMA) j <- NMA(0,3,3,1,1) ; mean(j) j1 <- NMA(0,6,3,1,1) ; mean(j1) j2 <- NMA(0,9,3,1,1) ; mean(j2) j3 <- NMA(0,12,3,1,1) ; mean(j3) j4 <- NMA(0,15,3,1,1) ; mean(j4)
#################### Considerando alfa = 0.8 * alfa0 ################## ####### alfa = 0.8 * alfa0 , lambida/(1-alfa) constante = 3
muda1 <- NMA(0.16,2.52,3,1,1) ; mean(muda1) #lambda = 2.52 muda2 <- NMA(0.32,2.04,3,1,1) ; mean(muda2) #lambda = 2.04 muda3 <- NMA(0.48,1.56,3,1,1) ; mean(muda3) #lambda = 1.56
####### alfa = 0.8 * alfa0 , lambida/(1-alfa) constante = 6 muda4 <- NMA(0.16,5.04,3,1,1) ; mean(muda4) #lambda = 5.04 muda5 <- NMA(0.32,4.08,3,1,1) ; mean(muda5) #lambda = 4.08 muda6 <- NMA(0.48,3.12,3,1,1) ; mean(muda6) #lambda = 3.12
####### alfa = 0.8 * alfa0 , lambida/(1-alfa) constante = 9 muda7 <- NMA(0.16,7.56,3,1,1) ; mean(muda7) #lambda = 7.56 muda8 <- NMA(0.32,6.12,3,1,1) ; mean(muda8) #lambda = 6.12
muda9 <- NMA(0.48,4.68,3,1,1) ; mean(muda9) #lambda = 4.68
####### alfa = 0.8 * alfa0 , lambida/(1-alfa) constante = 12 muda10 <- NMA(0.16,10.8,3,1,1) ; mean(muda10) #lambda = 10.8 muda11 <- NMA(0.32,8.16,3,1,1) ; mean(muda11) #lambda = 8.16 muda12 <- NMA(0.48,6.24,3,1,1) ; mean(muda12) #lambda = 6.24
####### alfa = 0.8 * alfa0 , lambida/(1-alfa) constante = 15 muda13 <- NMA(0.16,12.6,3,1,1) ; mean(muda13) #lambda = 12.6 muda14 <- NMA(0.32,10.8,3,1,1) ; mean(muda14) #lambda = 10.8 muda15 <- NMA(0.48,7.8,3,1,1) ; mean(muda15) #lambda = 7.8
#Sem autocorrela¸c~ao args(NMA) k <- NMA(0,3,3,1,1) ; mean(k) k1 <- NMA(0,6,3,1,1) ; mean(k1) k2 <- NMA(0,9,3,1,1) ; mean(k2) k3 <- NMA(0,12,3,1,1) ; mean(k3) k4 <- NMA(0,15,3,1,1) ; mean(k4) ####################################################################### #################### Considerando alfa = 0.9 * alfa0 ################## ####### alfa = 0.9 * alfa0 , lambda/(1-alfa) constante = 3
mu1 <- NMA(0.18,2.46,3,1,1) ; mean(mu1) mu2 <- NMA(0.36,1.92,3,1,1) ; mean(mu2) mu3 <- NMA(0.54,1.38,3,1,1) ; mean(mu3)
####### alfa = 0.9 * alfa0 , lambda/(1-alfa) constante = 6 mu4 <- NMA(0.18,4.92,3,1,1) ; mean(mu4)
mu5 <- NMA(0.36,3.84,3,1,1) ; mean(mu5) mu6 <- NMA(0.54,2.76,3,1,1) ; mean(mu6)
####### alfa = 0.9 * alfa0 , lambda/(1-alfa) constante = 9 mu7 <- NMA(0.18,7.38,3,1,1) ; mean(mu7)
mu8 <- NMA(0.36,5.76,3,1,1) ; mean(mu8) mu9 <- NMA(0.54,4.14,3,1,1) ; mean(mu9)
mu10 <- NMA(0.18,9.84,3,1,1) ; mean(mu10) mu11 <- NMA(0.36,7.68,3,1,1) ; mean(mu11) mu12 <- NMA(0.54,5.52,3,1,1) ; mean(mu12)
####### alfa = 0.9 * alfa0 , lambda/(1-alfa) constante = 15 mu13 <- NMA(0.18,12.3,3,1,1) ; mean(mu13)
mu14 <- NMA(0.36,9.6,3,1,1) ; mean(mu14) mu15 <- NMA(0.54,6.9,3,1,1) ; mean(mu15)
#Sem autocorrela¸c~ao args(NMA) l <- NMA(0,3,3,1,1) ; mean(l) l1 <- NMA(0,6,3,1,1) ; mean(l1) l2 <- NMA(0,9,3,1,1) ; mean(l2) l3 <- NMA(0,12,3,1,1) ; mean(l3) l4 <- NMA(0,15,3,1,1) ; mean(l4)
# Gr´afico NMAs considerando uma mudan¸ca em \lambda0 x<-c(0.8,0.9,1,1.1,1.2)
y<-c(712,436,271,183,127)
plot(x,y,type = "b",col=1,main=expression(paste("NMA considerando \n uma mudan¸ca em " , lambda)) ,xlim = c(0.8,1.2),ylim = c(54,954),axes=F,
xlab = expression(lambda[0]), ylab = "NMA",pch=16, lty = 1) box()
axis(1,at=c(0.8,0.9,1,1.1,1.2)) axis(2,at=seq(0,1000,100))
y1<-c(272,171,107,76,54) #alfa = 0.2
points(x,y1,type = "b",col=2,pch = 15, lty = 2)
y2<-c(406,219,136,88,60) #alfa = 0.4
points(x,y2,type = "b",col=3,pch = 15, lty = 3)
y3<-c(954,477,263,134,86) #alfa = 0.6
#y3<- c(712,436,271,183,127) #alfa=0.0
#points(x,y3,type = "b",col=4, pch = 18, lty =4)
labels<-c(expression(paste(lambda[0]==1," , " , alpha[0]==0, " ")), expression(paste(lambda[0]==1," , " , alpha[0]==0.2, " ")),
expression(paste(lambda[0]==1 ," , ", alpha[0]==0.4, " ")), expression(paste(lambda[0]==1 ," , ", alpha[0]==0.6, " ")))
legend("topright",labels,lwd=1,col=c(1,2,3,4),lty=c(1,2,3,4), pch=c(16,15,17,18),bty = "n") abline(v=1, col = "black", lty = 3)
## Para lambda = 3 e alfa = 0.2, 0.4 e 0.6 y3 <-c(902,390,187,100,60) #alfa = 0.2
plot(x,y3,type = "b",col=1,main=expression(paste("NMA considerando \n uma mudan¸ca em " , lambda)) ,xlim = c(0.8,1.2),ylim = c(46,2059),axes=F,
xlab = expression(lambda[0]), ylab = "NMA",pch=16, lty = 1) box()
#at=seq(from=0,to=1.2,by=0.3) axis(1,at=c(0.8,0.9,1,1.1,1.2)) axis(2,at=seq(0,3000,100))
y4 <- c(1048,413,181,90,50) #alfa = 0.4
points(x,y4 , type = "b", col=2, pch = 15, lty = 2)
y5 <- c(2059,591,251,97,46) #alfa = 0.6
points(x,y5 , type = "b", col=3, pch = 17, lty = 3)
y6<- c(1186,530,265,146,84) #alfa=0.0
points(x,y6,type = "b",col=4, pch = 18, lty =4)
labels<-c(expression(paste(lambda[0]==3," , " , alpha[0]==0.0, " ")), expression(paste(lambda[0]==3," , " ,alpha[0]==0.2," ")),
expression(paste(lambda[0]==3 ," , ", alpha[0]==0.4," ")), expression(paste(lambda[0]==3," , ", alpha[0]==0.6," ")))
legend("topright",labels,lwd=1,col=c(1,2,3,4),lty=c(1,2,3,4),bty = "n",pch = c(15,16,17,18)) abline(v=1, col = "black", lty = 3)