MA4081 PENGANTAR PROSES STOKASTIK Topik Khusus: M

Diskusi Hasil Ujian Silabus dan Tujuan Perkuliahan Struktur Perkuliahaan Pendahuluan Model Autoregresif Model Autoregresif Bernilai Integer

MA4081 PENGANTAR PROSES STOKASTIK Topik Khusus: Model AR dan INAR Dosen: Khreshna I.A. Syuhada, MSc. PhD.

“Cerdas dan Stokastik”

Dosen: Khreshna I.A. Syuhada, MSc. PhD.

MA4081 PENGANTAR PROSES STOKASTIK Topik Khusus: M


Setelah rantai Markov, distribusi eksponensial, lalu apa?




Proses Bernoulli, Proses Poisson, Proses Stokastik lain?




Mhs dengan 2 jawaban BENAR: 037, 039, 053, 089, 094




Mhs dengan 3 jawaban BENAR: 073




Mhs dengan 4 jawaban BENAR: 107




Silabus Tujuan

Silabus

Proses autoregresif (AR), proses AR bernilai integer, kestasioneran, struktur momen (tidak) bersyarat, metode likelihood maksimum, metode kuadrat terkecil, penaksiran parameter.




Silabus Tujuan

Tujuan

1

Mempelajari proses autoregresif (AR)

2

Mempelajari proses AR bernilai integer (INAR)

3

Menentukan daerah kestasioneran

4

Menurunkan struktur momen bersyarat dan tidak bersyarat

5

Mempelajari metode penaksiran parameter




Struktur Perkuliahaan

Kuliah Pengantar Tugas kelompok: 4 orang/kelompok (dipilih acak) Presentasi: 15/11, 17/11, 22/11, 24/11 (jadwal presentasi ditentukan tanggal 10/11 dst.)




Tentang tugas: Membahas 1-2 bagian dalam suatu artikel ilmiah Artikel tentang model AR atau INAR dan variannya (konfirmasikan terlebih dahulu artikel yang dipilih) Kajian dapat berupa teoritis atau komputasi




Penilaian presentasi: Dilakukan oleh dosen dan 2 kelompok lain yang terpilih (acak) Kelompok penilai harus memberikan pertanyaan Dosen dapat bertanya atau meminta seseorang/sekelompok untuk bertanya dan menjawab Cakupan penilaian: - Materi dan akurasi - Tingkat kesulitan - Cara presentasi Nilai: 0-10 (bobot 15%)




Contoh.




Formulir: Daftar kelompok Jadwal presentasi Penilaian




Model Random Walk

Pandang suatu rantai Markov dengan keadaaan-keadaan 0, ±1, ±2 dan peluang transisi Pi,i+1 = p = 1 − Pi,i−1 , dimana 0 < p < 1.




Kita dapat menuliskan m.p.t: (−2, −1, 0, 1, 2) × (−2, −1, 0, 1, 2)    P=  

0 p 0 0 1−p 0 p 0 0 1−p 0 p 0 0 1−p 0 0 0 0 1−p


0 0 0 p 0

     



Model diatas disebut Random Walk.




Model AR

Consider a stationary zero-mean Gaussian first-order autoregressive process {Yt } satisfying Yt = ρYt−1 + εt where |ρ| < 1 and the εt are independent and identically N(0, v ) distributed. Let θ = (ρ, v ). Suppose that the data is Y1 , . . . , Yn and that we wish to find a prediction interval for Yn+1 .




b We employ the following estimators Θ. b is equal to Θ e = ρ˜, V e where (i) The estimator Θ n−1 X

ρ˜ =

Yt Yt+1

t=1 n−1 X

n−1

and 2 Yt+1

e = V

1 X (Yt − ρ˜ Yt+1 )2 . n−1 t=1

t=1

These estimators are obtained by least squares from the backward representation of the process {Yt }.




b is equal to Θ ¯ = ρ¯, V ¯ where (ii) The estimator Θ n X

ρ¯ =

Yt Yt−1

t=2 n X

n

and 2 Yt−1

¯ = V

1 X (Yt − ρ¯ Yt−1 )2 . n−1 t=2

t=2

These estimators are obtained by maximizing the loglikelihood function conditional on Y1 = y1 .




The estimators ρ¯ and ρe differ by only a small amount. Yet their asymptotic biases conditional on Yn = yn are quite different. These asymptotic conditional biases are described as follows. E ρe − ρ|Yn = yn = −2 ρ n−1 + · · · E ρ¯ − ρ|Yn = yn = yn2 (1 − ρ2 )ρ (σ 2 )−1 − 3 ρ n−1 + · · ·




Model INAR Suppose that {Yt } is a discrete-time stationary non-negative INAR(1) process satisfying Yt−1

Yt =

X

Vti + εt , t ≥ 1

(1)

i=1

where Vti ’s denote i.i.d. random variables following certain (discrete) distribution and εt ’s be uncorrelated non-negative integer-valued random variables. The first term in r.h.s. may be presented as “θ ◦ Yt−1 ”, where ◦ is the thinning operator. The θ is the probability of ‘success’ of random variables Vti . Dosen: Khreshna I.A. Syuhada, MSc. PhD.



Consider the INAR(1) process as in (1). We assume that Vti ’s are Bernoulli random variables with probability of ‘success’ θ i.e. P(Vti = 1) = 1 − P(Vti = 0) = θ and εt follows a Poisson distribution with parameter (1 − θ)λ. Thus, Yt has a Poisson distribution with parameter λ. The process (1) is known as a Poisson INAR(1) process.




The fact that the distribution of εt implies the distribution of Yt has shown us the same role of the distribution of εt in the usual stationary Gaussian AR(1) process.




Conditional on Yn = yn , the probability mass function (pmf) of Z is given by p(z | yn ; θ, λ) = P Z = z | Yn = yn min(z,yn )

=

X

Ckyn θk (1 − θ)yn −k

k=0

1 e −(1−θ)λ {(1 − θ)λ}z−k (z − k)!

for z = 0, 1, 2, . . ..




We have used the following Yule-Walker estimators n−1 X

θˆ =

Yt − Y¯

Yt+1 − Y¯

t=1 n X

n

ˆ= and λ Yt − Y¯

2

1 X Yt −θˆ Yt−1 n−1 t=2

t=1

to estimate θ and λ, respectively.



MA4081 PENGANTAR PROSES STOKASTIK Topik Khusus: M

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