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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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