spectral density estimation via autoregressive modeling

Spectral Density Estimation via Autoregressive Modeling

Marie Forbelska

Masaryk University Brno, Department of Mathematics and Statistics

Podlesı, 3. 9. – 6. 9. 2013

Introduction Characteristics of Time Series

Characteristics of Time Series

Time Series (Discrete–Time Stochastic Processes)

A time series is a sequence of random variables{Yt , t = 0,±1,±2, . . .}

Beveridge Wheat Price Index, 1500-1869

n = 370

1500 1550 1600 1650 1700 1750 1800 1850

Marie Forbelska (MU – UMS) Spectral Density Estimation via AR Modeling Podlesı, 3. 9. – 6. 9. 2013 2 / 83

Introduction Weak Stationarity

Second Order Statistical Description

Definition

Stochastic process {Yt , t ∈ T} is said to be a second-order process ifEY 2

t <∞ for all t ∈ T .

Mean EYt = µt <∞ for all t ∈ T .

Autocovariance and Autocorelation

Autocovariance CY (t, s) of a random process {Yt , t ∈ Z} is definedas the covariance of Yt and Ys :

CY (t, s) = E (Yt − EYt)(Ys − EYs)

In particular, when t = s, we haveCY (t, t) = E (Yt − EYt)

2 = DYt

Autocorrelation coefficient is defined asRY (t, s) = CY (t,s)√

DYtDYs

Weak Stationarity

We introduce weak stationarity which require that time series exhibitcertain time-invariant behavior.

Definition

A time series {Yt , t ∈ Z} is (weak) stationary if EYt <∞ for each t,and

(i) EYt = µ is a constant, independent of t, and(ii) CY (t, t + k) is independent of t for each k .

Notation

If {Yt , t ∈ Z} is (weak) stationary denote byγY (k) = CY (t, t + k)ρY (k) = RY (t, t + k)

for all t.

Spectral theory

Spectral density

Let {Yt , t ∈ Z} be a zero mean stationary random sequence with theautocovariance function satisfying

∞∑t=−∞

|γ(t)| <∞.

Then the spectral density function is the continuous function f (λ) givenby the uniformly convergent series

f (λ) =∞∑

t=−∞γ(t)e−iλt

(see Doob 1953, p. 476).

White Noise

Definition

The process {εt , t ∈ T} is said to be an White Noise

if εt are uncorrelated random variables,

each with zero mean and variance σ2ε > 0

Notation: εt ∼WN(0, σ2ε).

Definition

If εt are also independent and identically distributed, then the process{εt , t ∈ T} is said to be an IID process.

Notation: εt ∼ IID(0, σ2ε).

Gaussian White Noise

εt = ηt − µ ∼ WN(0, σ2ε)

where ηt ∼ N(µ = 1, σ2 = 1)

density: fη(x) = 1√2πσ2

exp {−12

(x−µ)2

σ2 } pro x ∈ R

mean: Eηt = µ

variance: Dηt = σ2 = σ2ε

−2 0 2 4

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0 50 100 150 200 250 300

Exponential White Noise

εt = ηt − µ ∼ WN(0, σ2ε)

where ηt ∼ Exp(µ = 1)

density: fη(x) = 1µ exp {− 1

µx} pro x ≥ 0

mean: Eηt = µ

variance: Dηt = µ2 = σ2ε

0 2 4 6

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0 50 100 150 200 250 300

Beta-distributed White Noise

εt = ηt − µ ∼ WN(0, σ2ε)

where ηt ∼ Beta(a = 1.25, b = 1.25)

density: fη(x) = Γ(a+b)Γ(a)Γ(b) xa−1(1− x)b−1 pro x ∈ (0, 1)

mean: Eηt = µ = aa+b

variance: Dηt = ab(a+b)2(a+b+1)

= σ2ε

−0.4 −0.2 0.0 0.2 0.4

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0 50 100 150 200 250 300

ARMA Process

Definition

The process {Yt , t ∈ Z} is said to be an ARMA(p, q) process

if {Yt , t ∈ Z} is stationary and

if for every t ∈ Z,

Yt − ϕ1Yt−1 − · · · − ϕpYt−p = εt + θ1εt−1 + · · ·+ θqεt−q

where εt ∼WN(0, σ2ε).

We say that {Yt , t ∈ Z} is an ARMA(p, q) process with mean µ

if {Yt − µ, t ∈ Z} is an ARMA(p, q) process.

Special cases

If p = 0 then Yt is said to be moving average process MA(q).

If q = 0 then Yt is said to be autoregressive AR(p).

Backshift Operators and Characteristic Polynomials

Backshift Operator B such that

BYt = BYt−1 and BkYt = BYt−k for all k ∈ Z

ARMA notation using backshift operators

Yt ∼ ARMA(p, q) : Φ(B)Yt = Θ(B)εt

Characteristic polynomials

AR part Φ(z) = 1− ϕ1z − · · · − ϕpzp

MA part Θ(z) = 1 + θ1z + · · ·+ θqzp

defined on |z | < 1.

AR(2) : Yt = 0.5Yt−1 + 0.2Yt−2 + εt , εt ∼ N(0, 1)

0 50 100 150 200 250 300−4

MA(2) : Yt = εt − 0.5εt−1 − 0.2εt−1, εt ∼ N(0, 1)

0 50 100 150 200 250 300−4

ARMA(2, 2) : Yt = 0.5Yt−1 + 0.2Yt−2 + εt − 0.4εt−1 + 0.3εt−1, εt ∼ N(0, 1)

0 50 100 150 200 250 300−4

Causality and Invertibility of ARMA Processes

Definition

An ARMA(p, q) process is said to be causal (relative to {εt}) if there

exists a sequence of constants {ψi} such that∞∑i=0

|ψi | <∞ and

Yt =∞∑i=0

ψiεt−i , t ∈ Z

Which is equivalent to the condition

Φ(z) = 1− ϕ1z − . . .− ϕpzp 6= 0, ∀ |z | < 1

A similar definition for the invertibility of an ARMA(p, q) process relativeto εt can be presented if we interchange the role of {Yt} with {εt}. Thenthe invertibility is equivalent to the condition

Θ(z) = 1 + θ1z + . . .+ θpzq 6= 0, ∀ |z | < 1

Spectral density of ARMA process

Spectral density of a MA(q) process

fY (ω) = σ2ε

∣∣Θ (e−iω)∣∣2 for ω ∈ 〈−π, π〉

Spectral density of a AR(p) process

fY (ω) = σ2ε

|Φ(e−iω)|2for ω ∈ 〈−π, π〉

Spectral density of a ARMA(p, q) process

fY (ω) = σ2ε

2π|Θ(e−iω)|2|Φ(e−iω)|2

for ω ∈ 〈−π, π〉

Θ(z) = 1 + θ1z + . . .+ θqzq and Φ(z) = 1− ϕ1z − . . .− ϕpzp.

Moments of the AR(p) process

To calculate the mean we need causal AR(p) process:

EYt = E∑∞

j=0 ψjεt−j =∑∞

j=0 ψjEεt−j = 0.

Calculation of the autocovariance function is complicated:first equation Yt = ϕ1Yt−1 + · · ·+ ϕpYt−p + εtmultiplied by a term Yt−k and calculate the mean values of both sides, i.e.

EYtYt−k︸︷︷︸=γ(k)

= ϕ1 EYt−1Yt−k︸︷︷︸=γ(k−1)

+ · · ·+ ϕp EYt−pYt−k︸︷︷︸=γ(k−p)

+EεtYt−k .

then we compute

EYt−kεt = E (∞∑j=0

ψjεt−j−k)εt =∞∑j=0

ψjEεt−j−kεt =∞∑j=0

ψjσ2εδj+k

{σ2ε k = 0,

0 otherwise

Yule–Walker equations

By simple modifications of the previous equations we get Yule–Walkerequations

for the autocovariance function

for k = 0: γ(0) − ϕ1γ(1) − · · ·− ϕpγ(p) = σ2ε

for k 6= 0: γ(k) − ϕ1γ(k−1) − · · ·− ϕpγ(k−p) = 0

for the autocorrelation function

for k = 0: ρ(0)︸︷︷︸=1

− ϕ1ρ(1) − · · ·− ϕpρ(p) = σ2ε

for k 6= 0: ρ(k) − ϕ1ρ(k−1) − · · ·− ϕpρ(k−p) = 0 (YW∗)

Yule–Walker equation is a widely used method to estimate the coefficientsof the AR(p) models.

Limit properties of the ρ(k) of the AR(p) process

Solution of the homogeneous differential equation, which we marked witha (YW∗), we get in addition to recurrent relationship too explicit form ofthe autocorrelation function

ρAR(p)(k) =m∑j=1

(pj−1∑s=0

)λkj =

m∑j=1

(pj−1∑s=0

)rkj e ikθj ,

where cjs are constants determined by the initial conditions and λj = rjeiθj

are the inverse of the roots of the Φ(z) = 1− ϕ1z − . . .− ϕpzp withmultiplicities pj . Because holds

|λj | = rj < 1, kde Φ(z0j) = 0 pro z0j = 1λj,

we get here, that ρ(k) decreases for k →∞ exponentially to zero, i.e.

ρ(k) −−−→k→∞

which is a very important property identification autoregressive AR(p)processes.

AR(1) : Yt = 0.5Yt−1 + εt , εt ∼ N(0, 1)

0 50 100 150 200 250 300

−10 −8 −6 −4 −2 0 2 4 6 8 10

1fAR(ω) = σ2

1|Φ(e−iω)|2

−3 −2 −1 0 1 2 3

AR(1) : Yt = −0.5Yt−1 + εt , εt ∼ N(0, 1)

0 50 100 150 200 250 300

−10 −8 −6 −4 −2 0 2 4 6 8 10

−0.6

−0.4

−0.2

fAR(ω) = σ2ε

|Φ(e−iω)|2

−3 −2 −1 0 1 2 3

AR(1) : Yt = 0.95Yt−1 + εt , εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−30 −20 −10 0 10 20 30

fAR(ω) = σ2ε

|Φ(e−iω)|2

−3 −2 −1 0 1 2 3

AR(1) : Yt = −0.95Yt−1 + εt , εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−30 −20 −10 0 10 20 30

−1.0

−0.5

fAR(ω) = σ2ε

|Φ(e−iω)|2

−3 −2 −1 0 1 2 3

AR(2) : Yt = −0.75Yt−1 − 0.75Yt−2 + εt , εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−10 −5 0 5 10

−0.2

fAR(ω) = σ2ε

|Φ(e−iω)|2

−3 −2 −1 0 1 2 3

MA(2) : Yt = εt − 0.2279εt−1 + 0.2488εt−2, εt ∼ N(0, 1)

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0 50 100 150 200 250 300−3

−30 −20 −10 0 10 20 30

−0.2

1.0fMA(ω) = σ2

∣∣Θ (e−iω)∣∣2

−3 −2 −1 0 1 2 3

MA(4) : Yt = εt + 0.8εt−1 + 0.2εt−4, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−30 −20 −10 0 10 20 30

1.0fMA(ω) = σ2

∣∣Θ (e−iω)∣∣2

−3 −2 −1 0 1 2 3

MA(12) : Yt = εt + 0.8εt−1 + 0.2εt−12, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−30 −20 −10 0 10 20 30

1.0fMA(ω) = σ2

∣∣Θ (e−iω)∣∣2

−3 −2 −1 0 1 2 3

ARMA(1, 1) : Yt = 0.5Yt−1 + εt + 0.5εt−1, εt ∼ N(0, 1)

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−30 −20 −10 0 10 20 30

1.0fARMA(ω) = σ2

ε2π|Θ(e−iω)|2|Φ(e−iω)|2

−3 −2 −1 0 1 2 3

ARMA(2, 1) : Yt = 0.2Yt−1 + 0.7Yt−1 + εt + 0.5εt−1, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−30 −20 −10 0 10 20 30

1.0fARMA(ω) = σ2

−3 −2 −1 0 1 2 3

ARMA(1, 2) : Yt = −0.75Yt−1 + εt − εt−1 + 0.25εt−2, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−30 −20 −10 0 10 20 30

−0.5

1.0fARMA(ω) = σ2

−3 −2 −1 0 1 2 3

ARMA(1, 3) : Yt = −0.6Yt−1 + εt − 0.7εt−1 + 0.4εt−2 + 0.4εt−3, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−30 −20 −10 0 10 20 30

−0.5

1.0fARMA(ω) = σ2

−3 −2 −1 0 1 2 3

ARMA(2, 2) : Yt = 0.8897Yt−1 − 0, 4858Yt−2 + εt − 0.2279εt−1 + 0.2488εt−2, εt ∼ N(0, 1)

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−30 −20 −10 0 10 20 30

−0.2

1.0fARMA(ω) = σ2

−3 −2 −1 0 1 2 3

Introduction Stationary seasonal linear models

Seasonal linear models

So far we have discussed the links between neighboring random variables

. . . ,Yt ,Yt+1,Yt+2, . . . .

If a random process also includes seasonal fluctuations, it is necessary tonotice the dependencies between random variables, which divides seasonlength L.

. . . ,Yt ,Yt+L,Yt+2L, . . . .

First, we introduce seasonal differential operator of length L > 0:∆LYt =Yt − Yt−L = (1− BL)Yt

∆2LYt =∆L(∆LYt) = ∆L(Yt−Yt−L)

=(Yt−Yt−L)−(Yt−L−Yt−2L)=Yt−2Yt−L+Yt−2L = (1−BL)2Yt

...∆D

L Yt =(1− BL)DYt

Construction seasonal models

To better understand the structure of seasonal patterns in the B–Jmethodology, divide, for example, monthly data (L = 12) for r yearsin the following table.

Year January February · · · December

1 Y1 Y2 · · · Y12

2 Y13 Y14 · · · Y24...

......

r Y1+12(r−1) Y2+12(r−1) · · · Y12+12(r−1)

For each column j ∈ {1, . . . , 12} separately consider a ARMA(P,Q)model of the same type:

Yj+12t = π1Yj+12(t−1) + · · ·+ πPYj+12(t−1)+

ηj+12t + ψ1ηj+12(t−1) + · · ·+ ψQηj+12(t−1)

Because all 12 random processes is of the same type, we can writeπ(B12)Yt = Ψ(B12)ηt .

Remap white noise to the new process

When 12 white noise of the same type {η1+12t} ∼ WN(0, σ2η)

{η2+12t} ∼ WN(0, σ2η)

......

...{η12+12t} ∼ WN(0, σ2

sequentially assemble in time and create a single random process

{η∗t , t = 0,±1,±2, . . .},

we do not get white noise, it is recalled that:

Eη∗t η∗t+h = 0 only where h that are multiples of 12

Eη∗t ηt+h 6= 0 may occur for any other h,

therefore model the process ηt as a general ARMA(pq) process

Φ(B)η∗t = Θ(B)εt , εt ∼WN(0, σ2ε).

Stationary SARMA models

General stationary seasonal mixed SARMA model:

Φ(B)π(BL)Yt = Θ(B)Ψ(BL)εt ∼ SARMA(p, q)× (P,Q)L

Φ(B) = 1− ϕ1B − · · · − ϕpBp

π(BL) = 1− π1BL − · · · − πPBPL

Θ(B) = 1 + θ1B + · · ·+ θqBp

Ψ(BL) = 1 + ψ1BL + · · ·+ ψQBQL

MA homogeneous seasonal models

Yt = Θ(B)Ψ(BL)εt ∼ SARMA(0, q)× (0,Q)L.

AR homogeneous seasonal models

Φ(B)π(BL)Yt = εt ∼ SARMA(p, 0)× (P, 0)L

SARMA model as a special type of ARMA model

Consider a simple example SARMA(1, 0)× (1, 0)12 model:

Φ(B)π(B12)Yt = εt

(1− ϕ1B)(1− π1B12)Yt = εt

(1− ϕ1B − π1B12 + ϕ1π1B13)Yt = εt

Yt − ϕ1Yt−1 − π1Yt−12 + ϕ1π1Yt−13 = εt

We see that it is a special case of AR(13) model in which:

10 coefficients are zero,three remaining non-zero coefficients were created on two parameters:.

Relationship between SARMA and ARMA models

Model SARMA(p, q)× (P,Q)L is actually ARMA(p + PL,QL + q) modelwith additional conditions on AR and MA coefficients.

Pure seasonal homogeneous model with MA parts: Yt = Ψ(B12)εt

SARMA(0, 0)(0, 1)12 : Yt = (1− 0.95B12)εt : Yt = εt − 0.95εt−12, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−60 −40 −20 0 20 40 60

−0.5

fSMA(ω) = σ2ε

∣∣Ψ (e−i12ω)∣∣2

−3 −2 −1 0 1 2 3

Pure seasonal homogeneous model with AR parts: π(B12)Yt = εt

SARMA(0, 0)(0, 1)12 : (1− 0.95B12)Yt = εt : Yt = 0.95Yt−12 + εt , εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−60 −40 −20 0 20 40 60

fSAR(ω) = σ2ε

|π(e−i12ω)|2

−3 −2 −1 0 1 2 3

Pure seasonal homogeneous model with AR parts: π(B12)Yt = εt

SARMA(0, 0)(2, 0)12 : (1− 0.3B12 + 0.1B24)Yt = εt : Yt = 0.3Yt−12 − 0.1Yt−24 + εt , εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−60 −40 −20 0 20 40 60

fSAR(ω) = σ2ε

|π(e−i12ω)|2

−3 −2 −1 0 1 2 3

Seasonal homogeneous model with MA parts: Yt = Θ(B)Ψ(B12)εt

SARMA(0, 1)(0, 1)12 : Yt = (1 + 0.9B)(1− 0.4B12)εt : Yt = εt + 0.9εt−1 − 0.4εt−12 − 0.36εt−13, εt ∼ N(0, 1)

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0 50 100 150 200 250 300−4

−60 −40 −20 0 20 40 60

−0.2

fSMA(ω) = σ2ε

∣∣Θ (e−iω)∣∣2∣∣Ψ (e−i12ω)∣∣2

−3 −2 −1 0 1 2 3

Seasonal homogeneous model with MA parts: Yt = Θ(B)Ψ(B12)εt

SARMA(0, 1)(0, 1)12 : Yt = (1 + 0.9B)(1 + 0.4B12)εt : Yt = εt + 0.9εt−1 + 0.4εt−12 + 0.36εt−13, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−60 −40 −20 0 20 40 60

fSMA(ω) = σ2ε

∣∣Θ (e−iω)∣∣2∣∣Ψ (e−i12ω)∣∣2

−3 −2 −1 0 1 2 3

Seasonal homogeneous model with AR parts: Φ(B)π(B12)Yt = εt

SARMA(1, 0)(1, 0)12 : (1− 0.5B)(1− 0.7B12)Yt = εt : Yt = 0.5Yt−1 + 0.7Yt−12 − 0.35Yt−13 + εt , εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−60 −40 −20 0 20 40 60

fSAR(ω) = σ2ε

|Φ(e−iω)|21

|π(e−i12ω)|2

−3 −2 −1 0 1 2 3

Seasonal homogeneous model with AR parts: Φ(B)π(B12)Yt = εt

SARMA(1, 0)(1, 0)12 : (1− 0.9B)(1− 0.7B12)Yt = εt : Yt = 0.9Yt−1 + 0.7Yt−12 − 0.63Yt−13 + εt , εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−60 −40 −20 0 20 40 60

fSAR(ω) = σ2ε

|Φ(e−iω)|21

|π(e−i12ω)|2

−3 −2 −1 0 1 2 3

Seasonal mixed model: Φ(B)π(B12)Yt = Θ(B)Ψ(B12)εt

SARMA(1, 1)(1, 1)12 : (1− 0.5B)(1− 0.7B12)Yt = (1 + 0.9B)(1− 0.4B12)εtYt = 0.5Yt−1 + 0.7Yt−12 − 0.35Yt−13 + εt + 0.9εt−1 − 0.4εt−12 − 0.36εt−13, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−60 −40 −20 0 20 40 60

fSARMA(ω) = σ2ε

|Ψ(e−i12ω)|2|π(e−i12ω)|2

−3 −2 −1 0 1 2 3

SARMA(1, 1)(1, 1)12 : (1− 0.5B)(1− 0.7B12)Yt = (1 + 0.9B)(1 + 0.4B12)εtYt = 0.5Yt−1 + 0.7Yt−12 − 0.35Yt−13 + εt + 0.9εt−1 + 0.4εt−12 + 0.36εt−13, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−60 −40 −20 0 20 40 60

fSARMA(ω) = σ2ε

|Ψ(e−i12ω)|2|π(e−i12ω)|2

−3 −2 −1 0 1 2 3

SARMA(1, 1)(1, 1)12 : (1− 0.9B)(1− 0.7B12)Yt = (1 + 0.9B)(1− 0.4B12)εtYt = 0.9Yt−1 + 0.7Yt−12 − 0.63Yt−13 + εt + 0.9εt−1 − 0.4εt−12 − 0.36εt−13, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−60 −40 −20 0 20 40 60

fSARMA(ω) = σ2ε

|Ψ(e−i12ω)|2|π(e−i12ω)|2

−3 −2 −1 0 1 2 3

SARMA(1, 1)(1, 1)12 : (1− 0.9B)(1− 0.7B12)Yt = (1 + 0.9B)(1 + 0.4B12)εtYt = 0.9Yt−1 + 0.7Yt−12 − 0.635Yt−13 + εt + 0.9εt−1 + 0.4εt−12 + 0.36εt−13, εt ∼ N(0, 1)

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0 50 100 150 200 250 300

−60 −40 −20 0 20 40 60

fSARMA(ω) = σ2ε

|Ψ(e−i12ω)|2|π(e−i12ω)|2

−3 −2 −1 0 1 2 3

Estimation of moments

Let Y = (Y1, . . . ,Yn)T be a time series observed at equally-spaced timepoints t1, . . . , tn. We consider the problem of using these data to forecastYn+1 at time tn+1.

-�∆

t1 t2 · · ·

-�∆

ti ti+1 · · ·

-�∆

tn−1 tn

Denote by ∆ = ti+1 − ti . Thenti = t1 + (i − 1)∆ for i = 2, . . . , ni = ti−t1

∆ + 1

Without loss of generality, we can therefore assume that ti = i .

Estimation of the second order moments

Suppose we have data Y1, . . . ,Yn from a stationary time series. We canestimate

Empirical Mean Estimator

Y = 1n

n∑t=1

Empirical Autocovariance Function Estimator

Ck = γ(k) = 1n−k

n−k∑t=1

(Yt − Y )(Yt+k − Y ) for k = 0, 1, . . . , n − 1

Empirical Autocorrelation Function Estimator

ρ(k) = γ(k)γ(0)

Example with ACF estimates for AR(2)

AR(2) : (1− 1.5B + 0.75B2)Yt = εt : Yt = 1.5Yt−1 − 0.75Yt−2 + εt , εt ∼ N(0, 1)

0 100 200 300 400 500

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Example with ACF estimates for AR(2)

Empirical estimate for ACFn = 100

0 5 10 15 20 25 30

−0.5

0 5 10 15 20 25 30

−0.5

red . . . theoretical values

Example with ACF estimates for AR(2) (cont.)

0 5 10 15 20 25 30

−0.5

0 5 10 15 20 25 30

−0.5

Example with ACF estimates for AR(2) (cont.)

0 5 10 15 20 25 30

−0.5

0 5 10 15 20 25 30

−0.5

Monte Carlo study for the 1000 replication

Empirical estimate for ACF●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 n = 100●

●●

● ● ●●

●●

● ● ● ●● ● ● ●

Empirical estimate for ACF●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 n = 200●

●●

● ● ●●

●●

● ● ● ●● ● ● ●

Monte Carlo study for the 1000 replication (cont. 1)

Empirical estimate for ACF●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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● ●

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 n = 300●

●●

● ● ● ●● ● ● ●

Empirical estimate for ACF●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 n = 500●

●●

● ● ● ●● ● ● ●

Monte Carlo study for the 1000 replication (cont. 2)

Empirical estimate for ACF●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 n = 1000●

●●

● ● ● ●● ● ● ●

Empirical estimate for ACF●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 n = 5000●

●●

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●● ● ●

LA Pollution-Mortality Study: Spectral Density

LA Pollution-Mortality Study: Total Mortality (weeklydata)

n = 508

1970 1972 1974 1976 1978 1980

HistogramKernel Density EstimateNormal Density

140 160 180 200 220

Assessment of seasonality

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

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0 10 20 30 40 50

LA Pollution-Mortality Study: Total Mortality

Spectral Density

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Angular Frequency

Parametric AR(2) estimateParametric ARMA(2,0) estimateParametric SARMA(3,0)(2,2)[52] estimate

LA Pollution-Mortality Study: Total Mortality●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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Empirical ACFTheoretical ACF for AR(2)Theoretical ACF for ARMA(2,0)Theoretical ACF for SARMA(3,0)(2,2)[52]

0 20 40 60 80 100 120

−0.2

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0 10 20 30 40

AIC criterion for choice of AR order

Optimal order of AR = 2

AIC = n(log σ2ε + 1) + 2(p + 1)

LA Pollution-Mortality Study: Cardiovascular Mortalityn = 508

1970 1972 1974 1976 1978 1980

70 80 90 100 110 120 130

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

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0 10 20 30 40 50

LA Pollution-Mortality Study: Cardiovascular MortalitySpectral Density

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Angular Frequency

LA Pollution-Mortality Study: Cardiovascular Mortality

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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0 20 40 60 80 100 120

−0.5

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0 10 20 30 40

AIC = n(log σ2ε + 1) + 2(p + 1)

LA Pollution-Mortality Study: Temperature (weekly data)n = 508

1970 1972 1974 1976 1978 1980

50 60 70 80 90 100

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

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0 10 20 30 40 50

LA Pollution-Mortality Study: Temperature

Spectral Density

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Angular Frequency

LA Pollution-Mortality Study: Temperature●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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0 20 40 60 80 100 120

−0.5

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● ● ● ● ●●

0 10 20 30 40

AIC = n(log σ2ε + 1) + 2(p + 1)

LA Pollution-Mortality Study: Relative Humidityn = 508

1970 1972 1974 1976 1978 1980

20 40 60 80

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

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0 10 20 30 40 50

LA Pollution-Mortality Study: Relative Humidity

Spectral Density

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Angular Frequency

LA Pollution-Mortality Study: Relative Humidity (weeklydata)

●●

●●●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

0 20 40 60 80 100 120

● ●

●●

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0 10 20 30 40

AIC = n(log σ2ε + 1) + 2(p + 1)

LA Pollution-Mortality Study: Carbon Monoxide (weeklydata)

n = 508

1970 1972 1974 1976 1978 1980

5 10 15 20

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

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0 10 20 30 40 50

LA Pollution-Mortality Study: Carbon Monoxide (weekly data)

Spectral Density

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Angular Frequency

LA Pollution-Mortality Study: Carbon Monoxide (weeklydata)

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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0 20 40 60 80 100 120

−0.5

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● ●

● ●● ●

● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●

0 10 20 30 401.

AIC = n(log σ2ε + 1) + 2(p + 1)

LA Pollution-Mortality Study: Hydrocarbonsn = 508

1970 1972 1974 1976 1978 1980

20 40 60 80 100

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

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● ●

0 10 20 30 40 50

LA Pollution-Mortality Study: HydrocarbonsSpectral Density

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Angular Frequency

LA Pollution-Mortality Study: Hydrocarbons●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●●●

●●

●●●●●●●

●●

●●●●●●●●●●●●

●●

●●●●●●●●●●●●●●●●●●

0 20 40 60 80 100 120

−0.4

−0.2

● ● ● ●●

● ●

●● ●

● ●

●●

● ● ●●

●●

●● ●

●● ● ● ● ●

●●

● ● ●●

0 10 20 30 40

AIC = n(log σ2ε + 1) + 2(p + 1)

LA Pollution-Mortality Study: Nitrogen Dioxiden = 508

1970 1972 1974 1976 1978 1980

5 10 15 20 25

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

● ●

●●

● ●

●●

● ●

●●

● ●

●● ●

●●

● ●

●●

● ●

●●

● ●

●●

● ●

●●

0 10 20 30 40 50

LA Pollution-Mortality Study: Nitrogen Dioxide

Spectral Density

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Angular Frequency

LA Pollution-Mortality Study: Nitrogen Dioxide)●

●●

●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

0 20 40 60 80 100 120

−0.2

● ●

● ●●

0 10 20 30 40

AIC = n(log σ2ε + 1) + 2(p + 1)

LA Pollution-Mortality Study: Nitrogen Dioxiden = 508

1970 1972 1974 1976 1978 1980

5 10 15 20 25

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

● ●

●●

● ●

●●

● ●

●●

● ●

●● ●

●●

● ●

●●

● ●

●●

● ●

●●

● ●

●●

0 10 20 30 40 50

LA Pollution-Mortality Study: Nitrogen Dioxide

Spectral Density

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Angular Frequency

LA Pollution-Mortality Study: Nitrogen Dioxide●

●●

●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

0 20 40 60 80 100 120

−0.2

● ●

● ●●

0 10 20 30 40

AIC = n(log σ2ε + 1) + 2(p + 1)

Literatura

Andel, J. Statisticka analyza casovych rad . Praha. SNTL 1976.

Altman, N. S. (1990): Kernel Smoothing of Data With CorrelatedErrors. Journal of the American Statistical Association. Vol. 85, No.411, pp. 749–759.

Anderson, T.W. The Statistical Analysis of Time Series. John Wiley& Sons Inc. 1971.

Box, G., Jenkins, G. Time series analysis - forecasting and control .Holden-Day 1976.

Brabanter, K., Brabanter, J., Suykens, J. A. K. (2011): KernelRegression in the Presence of Correlated Errors. Journal of MachineLearning Research 12, pp. 1955–1976.

Brillinger, D. R. (1975): Time Series. Data Analysis and Theory. SanFrancisco: Holden–Day.

Brockwell, P.J., Davis, R.A. Time Series: Theory and Methods.Springer–Verlag, New York, 1991.

Literatura (pokracovanı)

Brockwell, P.J., Davis, R.A. Introduction to time series andforecasting . Springer-Verlag, New York, 2002.

Cipra, T. Analyza casovych rad s aplikacemi v ekonomii . SNTL,Praha, 1986.

Doob, J.L. Stochastic processes. New York, Wiley 1953.

Damsleth, E., Spjøtvoll, E. Estimation of Trigonometric Componentsin Time Series, J. Amer. Statistics, Assoc. 77. 1982, pp. 382–387.

Daniels, H.E. Rank correlation and population models. Journals ofthe Royal Statistical Society, B, 12 1950. 171–181.

Doob, J.L. Stochastic processes. New York, Wiley 1953.

Fisher, R.A. Tests of significance in harmonic analysis. Proc. RoyalSoc. A 125, 1929, 54–59.

Forbelska, M. Detekce periodicity v hydrologickych datech. In XIII.letnı skola bometriky, Biometricke metody a modely v soucasne vedea vyzkumu. 1. vyd. Brno: UKZUZ Brno, 1998. s. 173–178.

Forbelska, M. Stochasticke modelovanı jednorozmernych casovychrad. Munipress, 2009.

Geweke, J.F., Meese, R. Estimating Regression Models of Finite butUnknown Order. International Economic Review, 22, 1981. 55–70.

Gichman, I.I., Skorochod, A.V. Teorija slucajnych processov . Moskva.Nauka 1971.

Hamilton, J.D. Time Series Analysis. Princeton University Press.1994.

Hannan, E.J., Quinn, B. G. The Determination of the Order of anAutoregression, Journal of the Royal Statistical Society, Series B, 41,No.2, 1979, 190–195.

Michalek, J., Budıkova, M., Brazdil, R. Metody odhadu trendu casoverady na prıkladu stredoevropskych teplotnıch rad. 1. vyd. Praha :Cesky hydrometeorologicky ustav, 1993, 53 s.

Priestley, M. Spectral analysis and time series. Academic Press 1989.

Stulajter, F. Odhady v nahodnych procesoch. Alfa. Bratislava. 1989.

Vesely, V. Knihovna programu TSA-M pro analyzu casovych rad. Ed.P.Flak. In XIV. letna skola biometriky, Biometricke metody a modelyv podohospodarskej vede, vyskume a vyuke. Nitra: AgenturaSlovenskej akademie podohospodarskych vied, 2000. s. 239–248.

Vesely, V. Uvod do casovych rad. In Proceedings ANALYZADAT’2003/II. Pardubice (Czech Rep.): Trilobyte, Ltd., 2004. od s.7–31.

Zvara, K. Regresnı analyza Praha. Academia. 1989.

spectral density estimation via autoregressive modeling

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