Márton Ispány

(University of Debrecen, Hungary)

https://s3-seminar.github.io/seminars/marton-ispany

Title — Poisson INAR processes with serial and seasonal correlation

Abstract — Recently, there has been considerable interest in integer-valued time series models. Motivation to include discrete data models comes from the need to account for the discrete nature of certain data sets, often counts of events, objects or individuals. Among the most successful integer-valued time series models proposed in the literature we mention the INteger-valued AutoRegressive model of order p (INAR(p)). However, seasonal count processes have not been investigated yet, except one of our new papers. In the talk, we study INAR processes which possess serial and seasonal structure as well. The main properties of the models will be derived such as the stationarity and the autocorrelation function. The conditional least squares and conditional maximum likelihood estimators of the model parameters will be studied and their asymptotical properties will be established. In addition, we would like to discuss the case in which the marginal distributions are Poisson in detail. Monte Carlo experiments will be conducted to evaluate and compare the performance of various estimators for finite sample sizes. Real data set on the area of insurance will be applied to evaluate the model performance.

Biography — Márton Ispány received the M.Sc.(1989) and PhD (summa cum laude) in Statistics (1997) from University of Debrecen. Since 2007 he has been with the Department of Information Technology, Faculty of Informatics, University of Debrecen. Since 2012 he has been the head of the department. Márton Ispány 's recent research interests are in branching processes (functional limit theorems, asymptotics for conditional least squares estimation, integer valued autoregression), statistical modelling(generalized SVD, contaminated statistical models, EM algorithm), data mining (decision trees, stochastic algorithms, MCMC, web mining), and applied statistics: econometrics and insurance, cross-country modelling, statistical genetics.

Transcript

1. Poisson INAR processes with serial and
   seasonal correlation
   Márton Ispány
   University of Debrecen, Faculty of Informatics
   Joint result with Marcelo Bourguignon, Klaus L. P. Vasconcellos,
   and Valdério A. Reisen
   Workshop on Time series and counting processes
   with application to environmental and networking problems
   Supélec January 30, 2015
   The research was supported by the TÁMOP-4.2.2.C-11/1/KONV-2012-0001
   project. The project has been supported by the European Union, co-financed by
   the European Social Fund.
   Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
   

   View Slide

2. Outline
   • Integer valued autoregression and INAR(1) model
   • Comparison of AR, INAR, and branching processes
   • The purely seasonal INAR(1) model
   • Estimation methods
   • Simulation and real data examples
   • INAR process with serial and seasonal correlation
   • Stationarity and second order properties
   • Estimation methods
   Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
   

   View Slide

3. Integer valued autoregression (INAR)
   INAR(1) model (Al-Osh and Alzaid (1987))
   Xt =
   Xt−1
   j=1
   ξt,j + εt , t ∈ Z,
   {ξt,j, : t ∈ Z, j ∈ N} and {εt : t ∈ Z} are independent,
   non-negative, integer-valued, identically distributed r.v.’s
   P(ξ1,1 ∈ {0, 1}) = 1, i.e., ξ1,1
   has Bernoulli distribution
   Parameters: α := E ξ1,1
   , λ := E ε1
   , b2 := Var ε1
   Reformulation:
   Xt = α ◦ Xt−1 + εt
   Classification:
   α < 1
   stable
   α = 1
   unstable
   Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
   

   View Slide

4. Branching process with immigration (BPI)
   1 . . . ξk,1
   7
   7
   7
   ¤
   ¤
   ¤
   t
   t
   t
   1
   1 . . . ξk,2
   7
   7
   7
   ¤
   ¤
   ¤
   t
   t
   t
   2
   . . .
   1 . . . ξk,Xk−1
   
   
   
   
   t
   t
   t
   Xk−1
   offsprings
   1 2 . . . εk
   immigration
   Xk =
   Xk−1
   j=1
   ξk,j + εk , X0 = 0
   {ξk,j, εk : j ∈ N, k ∈ Z+} independent
   {ξk,j : j ∈ N, k ∈ Z+} identically distributed
   {εk : k ∈ Z+} identically distributed with P(ε1 = 0) > 0
   Parameters: m := E ξ1,1
   , σ2 = Var ξ1,1
   , λ := E ε1
   , b2 := Var ε1
   Classification:
   m < 1
   subcritical
   m = 1
   critical
   m > 1
   supercritical
   Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
   

   View Slide

5. Conditional structure
   Filtration: Fk := σ(X0, X1, . . . , Xk ), k ∈ Z+
   Conditional expectation: E(Xk | Fk−1) = mXk−1 + λ
   Mk := Xk − E(Xk | Fk−1) = Xk − mXk−1 − λ, k ∈ N
   martingale differences, and we have
   Xk = λ + mXk−1 + Mk
   Conditional variance: E(M2
   k
   | Fk−1) = σ2Xk−1 + b2
   since
   Mk = Xk − mXk−1 − λ =
   Xk−1
   j=1
   (ξk,j − m) + (εk − λ)
   Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
   

   View Slide

6. Autoregressive process (AR)
   AR(1) model
   Xt = µ + αXt−1 + εt , t ∈ Z
   µ ∈ R is the drift, α ∈ R is the autoregressive parameter, and
   {εt , t ∈ Z} is a sequence of martingale differences
   Classification:
   α < 1
   stable
   α = 1
   unstable
   α > 1
   explosive
   Connection
   • All INAR(1) process is a branching process with immigration.
   • All branching process with immigration is an AR(1)
   processes with drift and conditionally heteroscedasticity.
   Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
   

   View Slide

7. INAR(1) process with a seasonal structure
   INAR(1)s model (Bourguignon, Vasconcellos, Reisen, I (2014))
   Yt =
   Yt−s
   j=1
   ξt,j + εt , t ∈ Z,
   {ξt,j : t ∈ Z, j ∈ N} and {εt : t ∈ Z} are independent,
   non-negative, integer-valued, identically distributed r.v.’s
   P(ξ1,1 ∈ {0, 1}) = 1, i.e., ξ1,1
   has Bernoulli distribution
   s ∈ N denotes the seasonal period
   Parameters: φ := E ξ1,1
   , λ := E ε1
   Reformulation:
   Yt = φ ◦ Yt−s + εt
   Classification:
   φ < 1
   stable
   φ = 1
   unstable
   Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
   

   View Slide

8. Stationarity and second order properties
   If φ ∈ [0, 1), the unique stationary marginal distribution of
   INAR(1)s model can be expressed in terms of {εt : t ∈ Z} as
   Yt
   d
   =
   ∞
   k=0
   φk ◦ εt−ks = εt +
   ∞
   k=1
   εt−sk
   j=1
   Zt,k,j, t ∈ Z,
   where d
   = stands for equality in distribution and Zt,k,j ∼ Be(φk ).
   Let {εt : t ∈ Z} be an i.i.d. sequence of Poisson distributed
   variables with mean λ ∈ R+ and let φ ∈ [0, 1). Then the unique
   stationary solution satisfies Yt ∼ Po(λ/(1 − φ)) and the
   autocorrelation function is given by
   ρ(k) =
   φk/s, if k is a multiple of s,
   0, otherwise.
   Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
   

   View Slide

9. Sample path and its sample ACF
   100 simulated values of the INAR(1)s process and its sample
   autocorrelation function for φ = 0.5, λ = 1 and s = 12.
   Time
   yt
   0 20 40 60 80 100
   0 1 2 3 4 5
   0 10 20 30 40 50
   −0.2 0.0 0.2 0.4 0.6 0.8 1.0
   Lag
   ACF
   Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
   

   View Slide

10. Estimation methods: conditional least squares (CLS)
    The conditional least squares estimator of θ = (φ, λ)T is given
    by
    θCLS
    := arg min
    θ
    n
    t=s+1
    [Yt − Eθ(Yt |Ft−1)]2
    with Eθ(Yt |Ft−1) = Eθ(Yt |Yt−s) = g(θ, Yt−s), where
    g(θ, y) := φy + λ. Solving the normal equations we have
    φCLS
    :=
    (n−s)
    n
    t=s+1
    Yt Yt−s−
    n
    t=s+1
    Yt
    n
    t=s+1
    Yt−s
    (n−s)
    n
    t=s+1
    Y2
    t−s−
    n
    t=s+1
    Yt−s
    2
    λCLS
    := 1
    n−s
    n
    t=s+1
    Yt −φCLS
    n
    t=s+1
    Yt−s
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

11. Asymptotic result for conditional least squares
    √
    n
    φCLS − φ
    λCLS − λ
    d
    → N
    0
    0
    , Σ
    where
    Σ :=
    λ−1φ(1 − φ)2 + (1 − φ2) −(1 + φ)λ
    −(1 + φ)λ λ + (1 + φ)(1 − φ)−1λ2
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

12. Estimation methods: conditional maximum likelihood
    (CML)
    The INAR(1)s process consists of s mutually independent
    INAR(1) processes, thus it is an s-step Markov chain.
    Hence, the conditional log-likelihood function is given by
    (θ) = log Pθ(Yn, . . . , Ys|Ys−1, . . . , Y0) =
    n
    t=s
    log[Pθ(Yt |Yt−s)],
    where
    Pθ(Yt |Yt−s) = [Bi(Yt−s, φ) ∗ Po(λ)] (Yt )
    =e−λ min(Yt ,Yt−s)
    i=0
    λYt −i
    (Yt −i)!
    (Yt−s
    i
    )φi (1−φ)Yt−s−i
    Asymptotic result:
    √
    n
    φCML − φ
    λCML − λ
    d
    → N(0, I−1(θ)),
    where I(θ) is a 2 × 2 Fisher information matrix.
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

13. Monte Carlo simulation study
    Table: Biases of estimators for λ = 1 (MSE in parenthesis)
    Bias(φ)/MSE(φ) Bias(λ)/MSE(λ)
    n φ YW CLS CML YW CLS CML
    0.30 −0.0178 −0.0307 −0.0067 0.0315 0.0508 0.0040
    (0.0125) (0.0133) (0.0114) (0.0353) (0.0365) (0.0291)
    100 0.50 −0.0240 −0.0334 −0.0081 0.0500 0.0691 0.0064
    (0.0100) (0.0116) (0.0063) (0.0549) (0.0560) (0.0304)
    0.80 −0.0267 −0.0362 −0.0031 0.1385 0.1854 0.0113
    (0.0058) (0.0078) (0.0012) (0.1583) (0.1921) (0.0289)
    0.30 −0.0115 −0.0156 −0.0067 0.0221 0.0282 0.0115
    (0.0045) (0.0044) (0.0035) (0.0130) (0.0133) (0.0104)
    250 0.50 −0.0106 −0.0146 −0.0029 0.0254 0.0337 0.0057
    (0.0037) (0.0040) (0.0023) (0.0174) (0.0184) (0.0109)
    0.80 −0.0143 −0.0166 −0.0016 0.0700 0.0823 0.0028
    (0.0019) (0.0022) (0.0004) (0.0511) (0.0572) (0.0113)
    0.30 −0.0058 −0.0079 −0.0023 0.0063 0.0093 −0.0008
    (0.0022) (0.0022) (0.0018) (0.0055) (0.0056) (0.0045)
    500 0.50 −0.0033 −0.0056 −0.0007 0.0102 0.0148 0.0029
    (0.0018) (0.0018) (0.0010) (0.0087) (0.0089) (0.0052)
    0.80 −0.0086 −0.0098 −0.0003 0.0468 0.0500 0.0043
    (0.0009) (0.0009) (0.0002) (0.0237) (0.0255) (0.0055)
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

14. Real data example (Freeland)
    Monthly counts of claims of short-term disability benefits
    reported to the Richmond, BC Workers Compensation Board.
    Time
    Claims count
    0 20 40 60 80 100 120
    5 10 15 20
    5 10 15 20
    −0.2 0.0 0.2 0.4
    Lag
    ACF
    5 10 15 20
    −0.2 0.0 0.2 0.4
    Lag
    PACF
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

15. Fitted models
    Model CML estimates CLS estimates AIC BIC
    INAR(1)12 φ 0.1746 (0.0036) 0.2410 (0.0899) 530.613 536.013
    λ 5.1391 (0.1951) 4.7554 (0.5897)
    INAR(1) φ 0.4418 (0.0029) 0.5510 (0.0783) 538.469 543.869
    λ 3.5224 (0.1364) 2.8526 (0.5079)
    The model fitted by CML estimation is
    Yt = 0.1746 ◦ Yt−12 + t , t ∼ Po(5.1391)
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

16. INAR(1) process with serial and seasonal structure
    Seasonal INAR({1,s}) model (I and Reisen (2014))
    Zt =
    Zt−1
    j=1
    ξt,j +
    Zt−s
    j=1
    ηt,j + εt , t ∈ Z,
    {ξt,j : t ∈ Z, j ∈ N}, {ηt,j : t ∈ Z, j ∈ N} and {εt : t ∈ Z} are
    independent, non-negative, integer-valued, i.d. r.v.’s
    ξ1,1
    and η1,1
    have Bernoulli distribution
    s ∈ N denotes the seasonal period
    Parameters: α := E ξ1,1
    , φ := E η1,1
    , λ := E ε1
    Reformulation:
    Zt = α ◦ Zt−1 + φ ◦ Zt−s + εt
    Classification:
    α + φ < 1
    stable
    α + φ = 1
    unstable
    α + φ > 1
    explosive
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

17. State space representation
    Zt = A ◦ Zt−1 + εt
    where
    A :=
    
    
    
    
    
    
    α 0 · · · 0 φ
    1 0 · · · 0 0
    ...
    0 · · · 1 0
    
    
    
    
    
    
    Zt :=
    
    
    
    
    
    
    Zt
    Zt−1
    .
    .
    .
    Zt−s+1
    
    
    
    
    
    
    εt :=
    
    
    
    
    
    
    εt
    0
    .
    .
    .
    0
    
    
    
    
    
    
    The characteristic polynomial of A is given by
    det(xI − A) = xsP(x−1)
    where P denotes the autoregressive polynomial defined by
    P(x) := 1 − αx − φxs
    The INAR({1,s}) model is called primitive if the matrix A is
    primitive which holds iff α > 0 and φ > 0
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

18. Stationarity
    Lemma
    The roots of a primitive autoregressive polynomial P lie outside
    of the complex unit circle iff α + φ < 1. Then, for |x| 1,
    P(x)−1 =
    ∞
    j=0
    γj
    xj with
    ∞
    j=0
    γj < ∞.
    The non-negative sequence {γj : j ∈ Z+} satisfies the recursion
    γ0 = 1, γj = αγj−1
    , j = 1, . . . , s − 1, γj = αγj−1 + φγj−s
    , j ≥ s.
    If α + φ < 1, the unique stationary marginal distribution of
    INAR({1,s}) model can be expressed in terms of {εt : t ∈ Z} as
    Zt
    d
    =
    ∞
    k=0
    γk ◦ εt−ks = εt +
    ∞
    k=1
    εt−sk
    j=1
    Ut,k,j, Ut,k,j ∼ Be(γk )
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

19. Second order properties
    Let {εt : t ∈ Z} be an i.i.d. sequence of Poisson distributed
    variables with mean λ ∈ R+ and let φ ∈ [0, 1). Then the unique
    stationary solution satisfies Yt ∼ Po(λ/(1 − α − φ)).
    The autocorrelation function satisfies the recursion
    ρ(k) = αρ(k − 1) + φρ(k − s), k ∈ Z
    Recursive computation of the autocorrelation function starting
    from initial values ρ(0) = 1 and
    ρ(k) = αρ(k − 1) + φρ(s − k), k = 1, . . . , s − 1
    The partial autocorrelation function satisfies
    τ(k)
    = 0, if k = 0, 1, . . . , s
    = 0, otherwise.
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

20. Sample ACF and PACF (α = 0.3, φ = 0.5 and s = 12)
    0 10 20 30 40 50 60 70 80 90 100
    −0.2
    0
    0.2
    0.4
    0.6
    0.8
    Lag
    Sample Autocorrelation
    Sample Autocorrelation Function (ACF)
    0 10 20 30 40 50 60 70 80 90 100
    −0.2
    0
    0.2
    0.4
    0.6
    0.8
    Lag
    Sample Partial Autocorrelations
    Sample Partial Autocorrelation Function
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

21. Estimation methods: conditional least squares (CLS)
    The conditional least squares estimator of θ = (α, φ, λ)T is
    given by
    θCLS
    := arg min
    θ
    n
    t=s+1
    [Yt − Eθ(Yt |Ft−1)]2
    with Eθ(Yt |Ft−1) = Eθ(Yt |Yt−1, Yt−s) = αYt−1 + φYt−s + λ.
    The normal equations are given by
    n
    t=s+1
    
    
    
    Yt−1
    Yt−s
    1
    
    
    
    Yt−1
    Yt−s 1
    
    
    
    α
    φ
    λ
    
    
    
    =
    n
    t=s+1
    Yt
    
    
    
    Yt−1
    Yt−s
    1
    
    
    
    Asymptotic result:
    √
    n(θCLS − θ) d
    → N(0, Σ)
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

22. Estimation methods: conditional maximum likelihood
    (CML)
    The conditional log-likelihood function is given by
    (θ) = log Pθ(Yn, . . . , Ys|Ys−1, . . . , Y0) =
    n
    t=s
    log[Pθ(Yt |Yt−1, Yt−s)],
    where
    Pθ(Yt |Yt−1, Yt−s) = [Bi(Yt−1, α) ∗ Bi(Yt−s, φ) ∗ Po(λ)] (Yt )
    Asymptotic result:
    √
    n
    
    
    
    αCML − α
    φCML − φ
    λCML − λ
    
    
    
    d
    → N(0, I−1(θ)),
    where I(θ) is a 3 × 3 Fisher information matrix.
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

23. Real data example revisited
    The CLS estimates of parameters by solving the normal
    equations are
    α = 0.5388 φ = 0.1561 λ = 1.8011
    Thank you!
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide

24. References
    AL-OSH, M.A., ALZAID, A.A.:
    First-order integer valued autoregressive (INAR(1)) process.
    J. Time Ser. Anal. 8 (1987) 261–275.
    BARCZY, M. ISPÁNY, M., PAP, G.:
    Asymptotic behavior of unstable INAR(p) processes
    Stoch. Proc. Appl. 121 (2011) 583–608.
    BOURGUIGNON, B., ISPÁNY, M., REISEN, V., VASCONCELLOS:
    A Poisson INAR(1) process with a seasonal structure.
    J. Stat. Comp. Simul. accepted
    DU, J., LI, Y.:
    The integer-valued autoregressive (INAR(p)) model.
    J. Time Ser. Anal. 12 (1991) 129–142.
    Supélec January 30, 2015 Márton Ispány Poisson INAR processes with serial and seasonal correlation
    

    View Slide