Spatially Varying Autoregressive Processes

SVAR(p) Aline A. Nobre PROCC-FIOCRUZ

Spatially Varying Autoregressive Processes

EBEB X Março 2010 Outline Motivation

Aline A. Nobre

A Spatio-temporal model Priors

PROCC-FIOCRUZ

An illustrative example Discussion

Bruno Sanso´ AMS-UCSC

Alexandra M. Schmidt DME-UFRJ

10o Encontro Brasileiro de Estat´ıstica Bayesiana Angra dos Reis, RJ - Março, 2010

Outline of the talk

SVAR(p) Aline A. Nobre PROCC-FIOCRUZ EBEB X Março 2010

I

Motivation

Outline Motivation

I

A Spatio-temporal model

A Spatio-temporal model Priors An illustrative example

I

Priors

I

An illustrative example

I

Discussion

Discussion

Motivation


Temperature Anomalies at two different locations

Outline Motivation A Spatio-temporal model Priors An illustrative example Discussion

I

Are there periodic cycles in the data?

I

Can we detect a trend in the level or the amplitude of the signal?

Motivation




I


I


Motivation




I


I


Motivation

SVAR(p) Aline A. Nobre PROCC-FIOCRUZ EBEB X Março 2010 Outline Motivation

I

The data in the previous plots are not real data.

I

They were simulated from AR(3) processes.

A Spatio-temporal model Priors An illustrative example Discussion

I

Both processes are stationary and so they have no trends or cycles.

Motivation


I


I



I


Motivation


I


I



I


Motivation


I

Autoregressive processes provide great flexibility in spite of their simple formulation.


I

Positive real roots provide some persistence and complex roots provide quasi periodicities.


I

This features are likely to be present in long environmental time series with high frequency data.

I

When roots are outside the region of stationarity, the process shows an explosive behavior.

Motivation


I



I



I


I


Motivation


I



I



I


I


Motivation


I



I



I


I


Motivation

SVAR(p) Aline A. Nobre PROCC-FIOCRUZ EBEB X Março 2010 Outline

I

How can you make sure that the coeficients are in the stationarity region?

Motivation A Spatio-temporal model Priors

I

How can you make sure that spatial interpolations are not explosive?

Goal :To build spatio-temporal models that satisfy those conditions.


Motivation

SVAR(p) Aline A. Nobre PROCC-FIOCRUZ EBEB X Março 2010 Outline

I

How can you make sure that the coeficients are in the stationarity region?

Motivation A Spatio-temporal model Priors

I

How can you make sure that spatial interpolations are not explosive?

Goal :To build spatio-temporal models that satisfy those conditions.




Let s denote location, then for a space-time process, we have xt (s) =

p X

Outline

φi (s)xt−i (s) + t (s),

i=1

or using the backwards operator B p Y

EBEB X Março 2010

Motivation A Spatio-temporal model Priors An illustrative example Discussion

(1 − Gi (s)B)xt (s) = t (s).

i=1

Gi are the roots of the characteristic polynomial. The process is stationary when |Gi (s)| < 1 ∀i, ∀s. We obtain time stationarity and spatial coherence by assuming appropriate priors on Gi (s).



Let s denote location, then for a space-time process, we have xt (s) =

p X

EBEB X Março 2010 Outline

φi (s)xt−i (s) + t (s),

i=1

or using the backwards operator B p Y (1 − Gi (s)B)xt (s) = t (s). i=1

Gi are the roots of the characteristic polynomial. The process is stationary when |Gi (s)| < 1 ∀i, ∀s. We obtain time stationarity and spatial coherence by assuming appropriate priors on Gi (s).

Motivation A Spatio-temporal model Priors An illustrative example Discussion



Denote the vector of observations at time t as yt . We assume that yt can be decomposed into a time-varying mean plus a spatially correlated autoregressive vector xt .

EBEB X Março 2010 Outline Motivation A Spatio-temporal model Priors

yt =

F 0t θt p

xt =

X

+ Kxt

(1)

Discussion

Φj xt−j + t ,

t ∼ Nn (0, τ 2 In )

(2)

j=1

θt = Gθt−1 + wt ,


wt ∼ Nk (0, W t )

where t = p + 1, . . . , T, Σ = KK0 and Φj = diag(φj (s1 ), . . . , φj (sn ))

(3)

Collapsing Equations (1) and (2) we obtain that   p X y = F 0t θt + K  Φj xt−j + t 


t

EBEB X Março 2010

j=1

= F 0t θt +

p X

Φj (yt−j − F 0t−j θt−j ) + vt

j=1

Motivation A Spatio-temporal model

and thus

Priors

p

y∗t

Outline

=

X


Φj y∗t−j

+ vt ,

2

vt ∼ Nn (0, τ Σ)

j=1

where y∗t = yt − F 0t θt . Thus, y∗t is a multivariate AR(p) with spatially varying coefficients, Φj , j = 1, . . . , p and spatially correlated variance, Σ. We refer to this model as SVAR(p).

Discussion



t

EBEB X Março 2010

j=1

= F 0t θt +

p X


j=1


and thus

Priors

p

y∗t

Outline

=

X


Φj y∗t−j

+ vt ,

2

vt ∼ Nn (0, τ Σ)

j=1


Discussion



t

EBEB X Março 2010

j=1

= F 0t θt +

p X


j=1


and thus

Priors

p

y∗t

Outline

=

X


Φj y∗t−j

+ vt ,

2

vt ∼ Nn (0, τ Σ)

j=1


Discussion



t

EBEB X Março 2010

j=1

= F 0t θt +

p X


j=1


and thus

Priors

p

y∗t

Outline

=

X


Φj y∗t−j

+ vt ,

2

vt ∼ Nn (0, τ Σ)

j=1


Discussion


For any location s we write p Y

(1 −

Gi (s)B)y∗t (s)

EBEB X Março 2010

= vt (s).

i=1

Outline Motivation

Suppose that p = R + 2C, ⇓ R ⇒ real roots (aj (s)); C ⇒ complex roots (rj (s)e±iωj (s) ) Then the AR(p) model at location s can written as R R+C Y Y (1−aj (s)B) (1−2rj (s) cos(ωj (s))B+r2j (s)B2 )y∗t (s) = vt (s). j=1

j=R+1



For any location s we write p Y

(1 −

Gi (s)B)y∗t (s)

EBEB X Março 2010

= vt (s).

i=1

Outline Motivation

Suppose that p = R + 2C, ⇓ R ⇒ real roots (aj (s)); C ⇒ complex roots (rj (s)e±iωj (s) ) Then the AR(p) model at location s can written as R R+C Y Y (1−aj (s)B) (1−2rj (s) cos(ωj (s))B+r2j (s)B2 )y∗t (s) = vt (s). j=1

j=R+1


Prior processes for the real roots


The prior for aj (s) is given by

Outline Motivation

aj (s) = 2

M X

k(s − um )zaj (um ) − 1 , j = 1, . . . , R ,

m=1

A Spatio-temporal model Priors An illustrative example

where za1 (um ) ∼ Beta(α1 , β1 ), m = 1, . . . , M, and zaj (um )|zaj−1 (um ) ∼ Beta(αj , βj )I[0,zaj−1 (um )] (zaj (um )) m = 1, . . . , M and j = 2, . . . , R. Thus −1 < aj (s) < 1, ∀s, ∀j.

Discussion

for

Prior processes for the complex roots


I

I

To obtain a stationary process at each location s, 0 < rj (s) < 1 and −π < ωj (s) < π, ∀s, ∀j. Let φ1j (s) = 2rj (s) cos ωj (s) and φ2j (s) = −rj (s)2 .

Outline Motivation A Spatio-temporal model Priors An illustrative example

I

The stationarity conditions translate into the restrictions φ1j (s) ∈ (l1j (s), l2j (s)) and φ2j (s) ∈ (−1, 0) where p p l1j (s) = −2 −φ2j (s) and l2j (s) = 2 cos(4π/t) −φ2j (s).

Discussion

Prior processes for the complex roots


We assume that φ2j (s) =

M X

k(s − um )zφj (um ) − 1

m=1

Outline Motivation A Spatio-temporal model

and

Priors An illustrative example

φ1j (s) = (l2j (s) − l1j (s))

M X

k(s − um )zφj (um ) + l1j (s)

m=1

where j = 1, . . . , C. Then, we have, zφ1 (um ) ∼ Beta(α1 , β1 ) and zφj (um )|zφj−1 (um ) ∼ Beta(αj , βj )I[0,zφ (um )] (zφj (um )) for j−1

m = 1, . . . , M and j = 2, . . . , R.

Discussion

An illustrative example I

I

The measurements correspond to a 10 × 4 (longitude × latitude) grid with a spatial resolution of 0.5◦ . The data were recorded every 8 days, from July 2000 to May 2005 (T=240).

SVAR(p) Aline A. Nobre PROCC-FIOCRUZ EBEB X Março 2010 Outline Motivation A Spatio-temporal model Priors An illustrative example Discussion

Figure: Satellite sea surface temperature data for a region in the Pacific Ocean off the coast of California

Preliminary analysis


I

Multivariate dynamic linear model (DLM);

I

Time-varying mean structure comprising a baseline and a seasonal component with an annual cycle

I

Covariance matrix based on an exponential correlation function.

Figure: Partial autocorrelation function of the residuals after fitting the multivariate DLM model.


Preliminary analysis


I I

I

Multivariate dynamic linear model (DLM); Time-varying mean structure comprising a baseline and a seasonal component with an annual cycle Covariance matrix based on an exponential correlation function.

EBEB X Março 2010 Outline Motivation A Spatio-temporal model Priors An illustrative example Discussion

Figure: Partial autocorrelation function of the residuals after fitting the multivariate DLM model.

Fitting SVAR(p)


Model M0 M1 M2 M3 M4 M5

Autoregressive structure p p p p p p

= = = = = =

0 1 (R 2 (R 2 (C 3 (R 3 (R

= = = = =

1) 2) 1) 3) 1 and C = 1)

Predictive likelihood 8.70e-34 3.06e-22 1.36e-21 2.96e-23 3.01e-21 4.53e-22

P 5446.4 3962.3 3814.2 4123.2 3765.4 3973.8

EPD G 5813.5 6352.3 6360.6 5977.1 6184.1 6106.5

D 8353.2 7138.5 6994.5 7111.7 6857.4 7027.1

Table: Predictive likelihood and EPD criteria for each fitted model.


Fitting SVAR(p)


Figure: Posterior mean and limits of the 95% credible intervals of βt (baseline) and αt (amplitude).

Fitting SVAR(p)


Figure: Posterior mean and limits of the 95% credible intervals of Φ1 (s), Φ2 (s) and Φ3 (s) based on model M4.

Fitting SVAR(p)


Figure: Original data at 0.5◦ spatial resolution: Temperatures for 8 points in time.

Fitting SVAR(p)


Figure: Spatial interpolation at 0.25◦ spatial resolution: Posterior mean of the estimated temperatures for 8 points in time.

Discussion


I

Autoregressive processes are included to capture persistent trends or cycles even after removing the mean structure.

EBEB X Março 2010 Outline Motivation A Spatio-temporal model

I

I

Imposing priors on the roots of the characteristic polynomial guarantees time stationarity at all locations, avoiding explosive behaviors. We illustrate our method with an analysis of satellite data of ocean temperatures that this is a pertinent example, as data from remote sensors are likely to be contaminated by noise that is not white.

Priors An illustrative example Discussion

Discussion


I



I

I



Discussion


I



I

I

