[PPT] - On Statistical Inference of Spatio-Temporal Random Fields Yoshihiro PowerPoint Presentation

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On Statistical Inference of Spatio-Temporal Random Fields

Yoshihiro Yajima and Yasumasa Matsuda University of Tokyo and Tohoku University

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Outline

Model The Frameworks of Asymptotics A Test Statistic Theoretical Results Future Studies

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Model

Weakly Stationary Random Field (Continuous Parameter Case)

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Model

Examples

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Model

The Spectral Representation

where is the transpose and

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Model

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Stationary Random Fields

If the spectral distribution function is absolutely continuous, where is the spectral density function. Hereafter we assume that the spectral distribution function is absolutely continuous.

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The Frameworks of Asymptotics

The Three Frameworks (1)Increasing Domain Asymptotics The Equidistant Sampling Points(Rectangular Lattice): The Sample Size:

cf. Dahlhaus and Künsch(1987), Biometrika, 74, 877-882.

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The Frameworks of Asymptotics

(2)Fixed Domain(Infill) Asymptotics The Sampling Points: The Sample Size: .

cf. Stein(1995)J.Amer.Statist.Assoc.,, 90, 1277-1288.

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The Frameworks of Asymptotics

(3)Mixed Asymptotics (a)Hall and Patil(1994). Probab. Th. Rel. Fields, 99, 399-424. where and is uniformly distributed on .

RemarkThe speed of divergence of

relative to that of is important to develope asymptotic theory.

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The Frameworks of Asymptotics

(b)Karr(1986). Adv. Appl. Probab., 18, 406-422. We consider mixed asymptotics.

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Our Sampling Scheme

Assumption 1

where with a density function with a compact support in . and diverge to as .

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Our Sampling Scheme

Figure 1: Mixed Asymptotics

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A Statistical Hypothesis

Testing a simple hypothesis

for some where is a family of spectral density functions.

Testing a composite hypothesis

:(A parametric class)

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A Test Statistic

Preparation

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A Test Statistic

Estimation of where is a kernel function on and is a bandwidth.

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A Test Statistic

Test Statistic(Simple Hypothesis) where is a symmetric compact set with .

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A Test Statistic

The lag window is a continuous even function such that

Remark

is a bias term caused by irregularly sampling.

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The Assumptions

Assumption 2. is a stationary Gaussian random field with mean 0. Assumption 3. (a) as and (b) as . Assumption 4 is bounded, bounded away from 0 and twice partially differentiable such that

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The Assumptions

Assumption 5 (a) has a compact support in and there exists . (b) has a compact support in and there exists for .

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A Test Statistic

Assumption 6. is twice continuously differentiable and and as . Under Assumption 1,2,3(a),4,5(a) and 6，if the null hypothesis is true, Matsuda,Y. and Yajima,Y.(2009). J.Roy.Statist.Soc., B, 71, 191- 217.

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A Test Statistic

Test Statistic(Composite Hypothesis) Let be the true parameter if the null hypothesis is true. We estimate it by the Whittle estimator.

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The Original Idea

Paparoditis, E.(2000).Scand. J. Statist., 27, 143-176. (Equidistant Observations)

The Test Statistic(Simple Hypothesis)

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The Original Idea

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The Original Idea

Theorem 1(Paparoditis, E.(2000))

Under some assmuption on , , and , if the null hypothesis is true, as

Remark. An advantage of this test statistic is that it is scale

invariant and its asymptotic mean and variance are simple and depend only on .

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The Original Idea

The Test Statistic(Composite Hypothesis)

where

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The Original Idea

Theorem 2(Paparoditis(2000))

Under some assumptions, if the null hypothesis is true, as Hence has the same limiting distribution as .

Theorem 3(Paparoditis(2000))(Consistency)

If the null hypothesis is not true, diverges to as in probability.

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Theoretical Results

Assumption 7. as .

Theorem 4(Simple Hypothesis)

(i)Under Assumption 1,2,3(a),4,5(a),6 and 7，if the null hypothesis is true, as

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Theoretical Results

Assumption 6’. as .

Theorem 4(continued)

(ii)Under Assumption 1,2,3(a),4,5(a),6’ and 7，if the null hypothesis is true, as

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Theoretical Results

Theorem 5(Simple Hypothesis)

Under the same assumptions as Theorem 4.(ii) , if the null hypothesis is true, as

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Theoretical Results

Assumption 8. The set of parameters is a compact subset of . is a positive twice continuously differentiable function in . If ,

n a subset of

with a positive Lesbegue measure.

Theorem 6(Composite Hypothesis)

Under the same assumptions as Theorem 4.(ii) and Assumption 8, if the null hypothesis is true and is an inner point of , as

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Theoretical Results

Assumption 9. is bounded in .

Theorem 7(Consistency)

Under the same assumptions as Theorem 6 and As- sumption 9, if the null hypothesis is not true, diverges to as .

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Future Studies

1.Other Test Statistics

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Future Studies

1.Other Test Statistics(continued) Examples

cf. Yajima Y. and Matsuda,Y.(2009). Ann.Statist., 37, 3529-

3554. (

Time Series Case)

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Future Studies

2.Random Fields with Stationary Increments Definition(Intrinsic Stationary Random Fields)

A random field satisfies that for any fixed , the random field is stationary where Put

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Future Studies

The Spectral Representation of Variogram

Under some conditions, a variogram is expressed by where is positive and satisfies

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Future Studies

An Example

Istas,J.(2007). Statist. Inf. Stoch. Proc., 10, 97-106.

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Future Studies

Estimation of . If samples were continuously observed, we would be able to estimate f( ) by

cf. Solo,V.(1992). SIAM J. Appl. Math., 52, 270-291.

In our sampling scheme,

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Future Studies

3.Bootstrap

Does the grid-based block bootstrap proposed by Lahiri and Zhu work for our test statistic?

cf. Lahiri,S.N. and Zhu,J.(2006). Resampling methods for

spatial regression models under a class of stochastic de-

signs. Ann.Statist., 34, 1774-1813.

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Future Studies

4.Empirical Analysis An Example

The Land Price Data(yen ) of Kanto Area 5573 sampling points(10m mesh data,100km 100km) The Ministry of Land, Infrastructure and Transportation

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An Empirical Data

Figure 2: All of the Sampling Points

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Model for the data

Mat´ ern class( )

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Model for the data

Remark(1)

, (2)If , depends only on (Isotropic model). (3)Implication. First rotate the axes by . Then the contour on the

plane is the ellipsoid given

by

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Model for the Data

The Spectral Density Function

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