DOCSLIB.ORG

A Note on Kriging and Gaussian Processes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Note on Kriging and Gaussian Processes A NOTE ON KRIGING AND GAUSSIAN PROCESSES Mohammad Shekaramiz, Todd K. Moon, and Jacob H. Gunther Electrical and Computer Engineering Department and Information Dynamics Laboratory Utah State University [email protected], [email protected], [email protected] 1. INTRODUCTION TO GAUSSIAN PROCESSES [1–4] • Training data: input-output pairs D = f(xi; yi)ji = 1; 2; :::; ng • Each input is a vector xi of dimension d • Each target is a real-valued scalar yi = f(xi) for some unknown function f, possibly corrupted by noise • Collect inputs into matrix X 2 Rd×n and the correspond- ing outputs into vector y 2 Rn, resulting in the input- output training set D := fX; yg • Assumption: There exit an underlying process for the un- known function f Figure 1: Example showing a graphical representation of two • Objective: Predict the output y? at the input test x? i.e., input-output training data and one test data [1]. y? = f(x?) using the training data set D • Restate the Objective: Infer distribution over function f given the training data i.e., p(fjD = fX; yg) • How does it work? GP assumes that p(f(x1); :::; f(xn)) is jointly Gaussian with some mean µ(X) and covariance • Then use this information to make predictions for y? corre- Σ(X) given by Σij = κ(xi; xj ), where k is a P.D. kernel sponding to new (test) input data X? i.e., computing function i.e., p f(x1); :::; f(xn) ∼ N µ(X); Σ(X) Z p y?jX?;D = fX; yg) = p(y?jf; X? p(fjD)df • Key idea: If xi and xj are by the kernel similar, we expect that the output of the function at those points to be similar, as well • Remark: Need direct representation for function f via p(fjD) rather than the parametric description p(θjD) • Graphical illustration of a Gaussian Process: As an ex- ample, a Gaussian Process for 2 training data points and 1 • In other words, we want to find a way to perform Bayesian testing point is illustrated in Fig. 1, which is a mixture of inference over “functions” directed and undirected graphical model describing • Solution: “Gaussian Process” Q p(y; fjX) = N fjm(X);K(X) i p yijf(xi) 1.1. Definition of Gaussian Process Modeling 1.2. GPs for Regression • GP is a way of defining distribution over functions • Let the prior on the regression function be a GP, denoted by It defines a prior over function f and then based on Bayesian inference convert it into a posterior once some data is ob- f(x) ∼ GPm(x); κ(x; x0); served • Gaussian Process is a stochastic process whose realizations consist of a collection random variables (r.v.s) associated where m(x) = E[f(x)] is the mean function κ(x; x0) = with every point in range of either time or space E[(f(x)−m(x))(f(x0)−m(x0))T ] is the covariance func- In such a process each r.v. has a normal distribution tion • A GP is completely specified by a mean function and a pos- • For any finite set of n data points, this process defines a itive definite (p.d.) covariance function joint Gaussian distribution This property facilitates model fitting as only the first- and second-order moments of the process are required to be p(fjX) = N (fjµ;K) specified. where Kij = κ(xi; xj ) and µ = [m(x1); :::; m(xn)] 1.2.1. Prediction Using GP Modeling for Noise-Free Ob- servations • In this case, we expect the GP to predict f(x) for a value at location x that has already seen, to return the answer f(x) with no uncertainty • In other words, GP-regression model should act as an inter- polator • The joint distribution between the training and test data has the following form f µ KK? ∼ N ; T ; f? µ? K? K?? Figure 2: Finding z(u∗) based on spatial interpolation where f contains the set of output training data and f? are the predictions that are after at the input test data points 1.2.3. Prediction Using Semi-Parametric GP Modeling • Inference • Semi-parametric GP combines a parametric model with a non-parametric model p(f jX ; X; f) = N (f jµ ; Σ ); ? ? ? ? ? • It uses a linear model for the mean of the process where f(x) = βT φ(x) + r(x); where r(x) ∼ GP(0; κ(x; x0)) models the residuals and µ = µ(X ) + KT K−1f − µ(X) ? ? ? assuming that β ∼ N(b;B): Σ = K − KT K−1K ? ?? ? ? f(x) ∼ GPφ(x)T b; κ(x; x0) + φ(x)T Bφ(x0) • Posterior predictive density • Prediction: µ? can be used as the prediction for f? p(f?jX?; X; y) ∼ N (µf? ; Σf? ); 1.2.2. Prediction Using GP Modeling for Noisy Observa- where tions 8µ = ΦT β¯ + KT K−1(y − ΦT β¯) > f? ? ? y > T −1 T −1 −1 T −1 <Σf = K?? − K? Ky K? + R (B + ΦKy Φ ) R • Suppose the ith input-output training data is (Xi; yi), where ? ¯ −1 T −1 −1 −1 −1 yi = f(xi) + " and >β = (ΦKy Φ + B ) (ΦKy y + B b) > −1 :R = Φ? − ΦKy K?: 2 " ∼ N (0; σy); 8i = 1; :::; n: 2. KRIGING [5–8] • This model does not necessarily perform interpolation, but it must come close to the observed data • Illustrating example: Fig. (2) • Define • Need a model for spatial dependency to interpolate at un- 2 sampled locations Ky =: cov(yjX) = K + σyIN ; where K is the covariance matrix of the training data 2.1. Kriging • Joint density of the observed and the test data • Spatial interpolation method based on statistical model • Common types of Kriging y Ky K? ∼ N 0; T ; Simple Kriging (SK) f? K? K?? Ordinary Kriging (OK) where zero-mean case is considered for simplicity Kriging with a trend (KT) • Posterior predictive density Factorial Kriging (FK) Co-Kriging p(f?jX?; X; f) = N (f?jµ?; Σ?); • Kriging is based on GP modeling • Problem Description: where Estimating the value of a continuous attribute Z at any un- T −1 sampled location u∗ using only the available training data- µ? = K K y ? y pairs D = f(uα; z(uα))jα = 1; :::; n(α)g over the study T −1 Σ? = K?? − K? Ky K? area 2.2. Kriging Structure b) Ordinary Kriging (OK): Accounts for local fluc- tuations of the mean by limiting the domain of stationarity • Kriging estimators are based on linear regression estimation to local neighborhood W (u) of the R.V. Z(u∗) m(u0) = m; unknown 8u0 2 W (u) n(α) c) Kriging with a trend model(KT): Unknown local ∗ X Z (u∗) − µ(Z(u∗)) = λα(u)[Z(uα) − µ(uα)]; mean m(u0) smoothly varies within each local neighbor- α=1 hood W (u), hence over the entire under-study area Trend component is modeled as a linear combination where of functions fk(u) of the coordinates Z(uα): Random variable at point u K z(uα): Realization of Z(uα) at point u X m(u0) = ak(u0)fk(u0); ak(u0) ≈ ak; 8u0 2 W (u) λα(u): Kriging weight assigned to data z(uα) to compute k=0 z(u∗) µ(Z(uα)): Expected value of R.V. Z(uα) Remark: Functions fk(u0) may be directly suggested by the physics of the problem • Remark: The number of data, n(u), involved in the esti- Example: A linear trend model mation may change from one location to another. In prac- tice, only the n(u) data closest to the location u∗ are re- m(u) = m(ux; uy) = a0 + a1ux + a2uy tained. 2.4. Simple Kriging (SK) Model • Kriging interprets the unknown value z(u∗) and data val- ues z(uα) as realizations of R.V, Z(u∗) and Z(uα), re- • Model the trend component m(u) as a known stationary spectively mean i.e., µ(Z(u))) = m(u) = m • Goal: Minimizing the estimation error i.e., error variance 2 σE (u) under unbiasedness assumption of the estimator 2 ∗ minλ σE (u∗) = Var(Z (u∗) − Z(u∗)) ∗ s.t. Ef(Z (u∗) − Z(u∗))g = 0 2 ∗ minλ σE (u∗) = Var(Z (u∗) − Z(u∗)) • In Simple Kriging the constraint condition is already satis- s.t. Ef(Z∗(u ) − Z(u ))g = 0 ∗ ∗ fied i.e., SK estimator is unbiased ∗ • Decomposing Z(u), Based on semi-parametric GP EfZSK (u∗) − Z(u∗)g = m − m = 0 Z(u) := R(u) + µ(Z(u)); • Unconstraint minimization problem 2 ∗ where R(u) is the residual component and µ(Z(u)) is the minλ σE (u∗) = Var(Z (u∗) − Z(u∗)) trend component • Estimator error Z∗(u ) − Z(u ) can be viewed as a linear • Modeling the components ∗ ∗ combination of residual variables R(uα) and R.V. R(u) 1) Residual component: Modeled as a stationary GP with zero mean as n(α) ∗ X Z (u ) − Z(u ) = λ (u)(Z(u ) − m) − R(u) ∗ ∗ α α R(u) ∼ GP 0;CR(h) α=1 In other words, R(uα) := Z(uα) − m n(α) EfR(u)g = 0 ∗ X R (u) := λ (u)(Z(u ) − m) CovR(u);R(u + h) = EfR(u)R(u + h)g α α α=1 ∗ ∗ 2) Trend component: Z (u∗) − Z(u∗) := R (u) − R(u) EfZ(u)g = µ(Z(u)) := m(u) • Variance error n(u) n(u) 2 X X 2.3. Common Kriging Types σE (u) = λα(u)λβ (u)CR(uα − uβ ) + CR(0) α=1 β=1 • Here, we present three types of Kriging according to the n(u) trend component model X −2 λα(u)CR(uα − u); a) Simple Kriging (SK): Mean m(u) is known and α=1 constant throughout the study area where m(u) = m; known 8u 2 funderstudy regiong CR(h) := EfR(u)R(u + h)g • Error variance is a quadratic function of Kriging weights 2 σE (u) = Q(λα(u)); 8α = 1; :::; n(α) • Optimal weights to minimize error variance 1 @σ2 (u) E = 0; 8α = 1; :::; n(u) 2 @λα(u) n(u) n(u) X X λα(u)CR(uα − uβ ) − λα(u)CR(uα − u) = 0; α=1 α=1 8α = 1; ::; n(α) Figure 3: Semi-variogram • Stationarity of the mean CR(h) = Ef(R(u) − µ(R(u)))(R(u + h) − µ(R(u + h)))g 3.
Load more

Recommended publications
  • Spatial Diversification, Dividend Policy, and Credit Scoring in Real
    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2006 Spatial diversification, dividend policy, and credit scoring in real estate Darren Keith Hayunga Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Finance and Financial Management Commons Recommended Citation Hayunga, Darren Keith, "Spatial diversification, dividend policy, and credit scoring in real estate" (2006). LSU Doctoral Dissertations. 2954. https://digitalcommons.lsu.edu/gradschool_dissertations/2954 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. SPATIAL DIVERSIFICATION, DIVIDEND POLICY, AND CREDIT SCORING IN REAL ESTATE A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Interdepartmental Program in Business Administration (Finance) by Darren K. Hayunga B.S., Western Illinois University, 1988 M.B.A., The College of William and Mary, 1997 August 2006 Table of Contents ABSTRACT....................................................................................................................................iv
    [Show full text]
  • Stationary Processes and Their Statistical Properties
    Stationary Processes and Their Statistical Properties Brian Borchers March 29, 2001 1 Stationary processes A discrete time stochastic process is a sequence of random variables Z1, Z2, :::. In practice we will typically analyze a single realization z1, z2, :::, zn of the stochastic process and attempt to esimate the statistical properties of the stochastic process from the realization. We will also consider the problem of predicting zn+1 from the previous elements of the sequence. We will begin by focusing on the very important class of stationary stochas- tic processes. A stochastic process is strictly stationary if its statistical prop- erties are unaffected by shifting the stochastic process in time. In particular, this means that if we take a subsequence Zk+1, :::, Zk+m, then the joint distribution of the m random variables will be the same no matter what k is. Stationarity requires that the mean of the stochastic process be a constant. E[Zk] = µ. and that the variance is constant 2 V ar[Zk] = σZ : Also, stationarity requires that the covariance of two elements separated by a distance m is constant. That is, Cov(Zk;Zk+m) is constant. This covariance is called the autocovariance at lag m, and we will use the notation γm. Since Cov(Zk;Zk+m) = Cov(Zk+m;Zk), we need only find γm for m 0. The ≥ correlation of Zk and Zk+m is the autocorrelation at lag m. We will use the notation ρm for the autocorrelation. It is easy to show that γk ρk = : γ0 1 2 The autocovariance and autocorrelation ma- trices The covariance matrix for the random variables Z1, :::, Zn is called an auto- covariance matrix.
    [Show full text]
  • Stochastic Process - Introduction
    Stochastic Process - Introduction • Stochastic processes are processes that proceed randomly in time. • Rather than consider fixed random variables X, Y, etc. or even sequences of i.i.d random variables, we consider sequences X0, X1, X2, …. Where Xt represent some random quantity at time t. • In general, the value Xt might depend on the quantity Xt-1 at time t-1, or even the value Xs for other times s < t. • Example: simple random walk . week 2 1 Stochastic Process - Definition • A stochastic process is a family of time indexed random variables Xt where t belongs to an index set. Formal notation, { t : ∈ ItX } where I is an index set that is a subset of R. • Examples of index sets: 1) I = (-∞, ∞) or I = [0, ∞]. In this case Xt is a continuous time stochastic process. 2) I = {0, ±1, ±2, ….} or I = {0, 1, 2, …}. In this case Xt is a discrete time stochastic process. • We use uppercase letter {Xt } to describe the process. A time series, {xt } is a realization or sample function from a certain process. • We use information from a time series to estimate parameters and properties of process {Xt }. week 2 2 Probability Distribution of a Process • For any stochastic process with index set I, its probability distribution function is uniquely determined by its finite dimensional distributions. •The k dimensional distribution function of a process is defined by FXX x,..., ( 1 x,..., k ) = P( Xt ≤ 1 ,..., xt≤ X k) x t1 tk 1 k for anyt 1 ,..., t k ∈ I and any real numbers x1, …, xk .
    [Show full text]
  • Autocovariance Function Estimation Via Penalized Regression
    Autocovariance Function Estimation via Penalized Regression Lina Liao, Cheolwoo Park Department of Statistics, University of Georgia, Athens, GA 30602, USA Jan Hannig Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA Kee-Hoon Kang Department of Statistics, Hankuk University of Foreign Studies, Yongin, 449-791, Korea Abstract The work revisits the autocovariance function estimation, a fundamental problem in statistical inference for time series. We convert the function estimation problem into constrained penalized regression with a generalized penalty that provides us with flexible and accurate estimation, and study the asymptotic properties of the proposed estimator. In case of a nonzero mean time series, we apply a penalized regression technique to a differenced time series, which does not require a separate detrending procedure. In penalized regression, selection of tuning parameters is critical and we propose four different data-driven criteria to determine them. A simulation study shows effectiveness of the tuning parameter selection and that the proposed approach is superior to three existing methods. We also briefly discuss the extension of the proposed approach to interval-valued time series. Some key words: Autocovariance function; Differenced time series; Regularization; Time series; Tuning parameter selection. 1 1 Introduction Let fYt; t 2 T g be a stochastic process such that V ar(Yt) < 1, for all t 2 T . The autoco- variance function of fYtg is given as γ(s; t) = Cov(Ys;Yt) for all s; t 2 T . In this work, we consider a regularly sampled time series f(i; Yi); i = 1; ··· ; ng. Its model can be written as Yi = g(i) + i; i = 1; : : : ; n; (1.1) where g is a smooth deterministic function and the error is assumed to be a weakly stationary 2 process with E(i) = 0, V ar(i) = σ and Cov(i; j) = γ(ji − jj) for all i; j = 1; : : : ; n: The estimation of the autocovariance function γ (or autocorrelation function) is crucial to determine the degree of serial correlation in a time series.
    [Show full text]
  • An Introduction to Spatial Autocorrelation and Kriging
    An Introduction to Spatial Autocorrelation and Kriging Matt Robinson and Sebastian Dietrich RenR 690 – Spring 2016 Tobler and Spatial Relationships • Tobler’s 1st Law of Geography: “Everything is related to everything else, but near things are more related than distant things.”1 • Simple but powerful concept • Patterns exist across space • Forms basic foundation for concepts related to spatial dependency Waldo R. Tobler (1) Tobler W., (1970) "A computer movie simulating urban growth in the Detroit region". Economic Geography, 46(2): 234-240. Spatial Autocorrelation (SAC) – What is it? • Autocorrelation: A variable is correlated with itself (literally!) • Spatial Autocorrelation: Values of a random variable, at paired points, are more or less similar as a function of the distance between them 2 Closer Points more similar = Positive Autocorrelation Closer Points less similar = Negative Autocorrelation (2) Legendre P. Spatial Autocorrelation: Trouble or New Paradigm? Ecology. 1993 Sep;74(6):1659–1673. What causes Spatial Autocorrelation? (1) Artifact of Experimental Design (sample sites not random) Parent Plant (2) Interaction of variables across space (see below) Univariate case – response variable is correlated with itself Eg. Plant abundance higher (clustered) close to other plants (seeds fall and germinate close to parent). Multivariate case – interactions of response and predictor variables due to inherent properties of the variables Eg. Location of seed germination function of wind and preferred soil conditions Mechanisms underlying patterns will depend on study system!! Why is it important? Presence of SAC can be good or bad (depends on your objectives) Good: If SAC exists, it may allow reliable estimation at nearby, non-sampled sites (interpolation).
    [Show full text]
  • Spatial Autocorrelation: Covariance and Semivariance Semivariance
    Spatial Autocorrelation: Covariance and Semivariancence Lily Housese ­P eters GEOG 593 November 10, 2009 Quantitative Terrain Descriptorsrs Covariance and Semivariogram areare numericc methods used to describe the character of the terrain (ex. Terrain roughness, irregularity) Terrain character has important implications for: 1. the type of sampling strategy chosen 2. estimating DTM accuracy (after sampling and reconstruction) Spatial Autocorrelationon The First Law of Geography ““ Everything is related to everything else, but near things are moo re related than distant things.” (Waldo Tobler) The value of a variable at one point in space is related to the value of that same variable in a nearby location Ex. Moranan ’s I, Gearyary ’s C, LISA Positive Spatial Autocorrelation (Neighbors are similar) Negative Spatial Autocorrelation (Neighbors are dissimilar) R(d) = correlation coefficient of all the points with horizontal interval (d) Covariance The degree of similarity between pairs of surface points The value of similarity is an indicator of the complexity of the terrain surface Smaller the similarity = more complex the terrain surface V = Variance calculated from all N points Cov (d) = Covariance of all points with horizontal interval d Z i = Height of point i M = average height of all points Z i+d = elevation of the point with an interval of d from i Semivariancee Expresses the degree of relationship between points on a surface Equal to half the variance of the differences between all possible points spaced a constant distance apart
    [Show full text]
  • Statistical Analysis of NSI Data Chapter 2
    Statistical and geostatistical analysis of the National Soil Inventory SP0124 CONTENTS Chapter 2 Statistical Methods........................................................................................2 2.1 Statistical Notation and Summary.......................................................................2 2.1.1 Variables .......................................................................................................2 2.1.2 Notation........................................................................................................2 2.2 Descriptive statistics ............................................................................................2 2.3 Transformations ...................................................................................................3 2.3.1 Logarithmic transformation. .........................................................................3 2.3.2 Square root transform...................................................................................3 2.4 Exploratory data analysis and display.................................................................3 2.4.1 Histograms....................................................................................................3 2.4.2 Box-plots.......................................................................................................3 2.4.3 Spatial aspects...............................................................................................4 2.5 Ordination............................................................................................................4
    [Show full text]
  • Augmenting Geostatistics with Matrix Factorization: a Case Study for House Price Estimation
    International Journal of Geo-Information Article Augmenting Geostatistics with Matrix Factorization: A Case Study for House Price Estimation Aisha Sikder ∗ and Andreas Züfle Department of Geography and Geoinformation Science, George Mason University, Fairfax, VA 22030, USA; azufl[email protected] * Correspondence: [email protected] Received: 14 January 2020; Accepted: 22 April 2020; Published: 28 April 2020 Abstract: Singular value decomposition (SVD) is ubiquitously used in recommendation systems to estimate and predict values based on latent features obtained through matrix factorization. But, oblivious of location information, SVD has limitations in predicting variables that have strong spatial autocorrelation, such as housing prices which strongly depend on spatial properties such as the neighborhood and school districts. In this work, we build an algorithm that integrates the latent feature learning capabilities of truncated SVD with kriging, which is called SVD-Regression Kriging (SVD-RK). In doing so, we address the problem of modeling and predicting spatially autocorrelated data for recommender engines using real estate housing prices by integrating spatial statistics. We also show that SVD-RK outperforms purely latent features based solutions as well as purely spatial approaches like Geographically Weighted Regression (GWR). Our proposed algorithm, SVD-RK, integrates the results of truncated SVD as an independent variable into a regression kriging approach. We show experimentally, that latent house price patterns learned using SVD are able to improve house price predictions of ordinary kriging in areas where house prices fluctuate locally. For areas where house prices are strongly spatially autocorrelated, evident by a house pricing variogram showing that the data can be mostly explained by spatial information only, we propose to feed the results of SVD into a geographically weighted regression model to outperform the orginary kriging approach.
    [Show full text]
  • Covariances of ARMA Models
    Statistics 910, #9 1 Covariances of ARMA Processes Overview 1. Review ARMA models: causality and invertibility 2. AR covariance functions 3. MA and ARMA covariance functions 4. Partial autocorrelation function 5. Discussion Review of ARMA processes ARMA process A stationary solution fXtg (or if its mean is not zero, fXt − µg) of the linear difference equation Xt − φ1Xt−1 − · · · − φpXt−p = wt + θ1wt−1 + ··· + θqwt−q φ(B)Xt = θ(B)wt (1) 2 where wt denotes white noise, wt ∼ WN(0; σ ). Definition 3.5 adds the • identifiability condition that the polynomials φ(z) and θ(z) have no zeros in common and the • normalization condition that φ(0) = θ(0) = 1. Causal process A stationary process fXtg is said to be causal if there ex- ists a summable sequence (some require square summable (`2), others want more and require absolute summability (`1)) sequence f jg such that fXtg has the one-sided moving average representation 1 X Xt = jwt−j = (B)wt : (2) j=0 Proposition 3.1 states that a stationary ARMA process fXtg is causal if and only if (iff) the zeros of the autoregressive polynomial Statistics 910, #9 2 φ(z) lie outside the unit circle (i.e., φ(z) 6= 0 for jzj ≤ 1). Since φ(0) = 1, φ(z) > 0 for jzj ≤ 1. (The unit circle in the complex plane consists of those x 2 C for which jzj = 1; the closed unit disc includes the interior of the unit circle as well as the circle; the open unit disc consists of jzj < 1.) If the zeros of φ(z) lie outside the unit circle, then we can invert each Qp of the factors (1 − B=zj) that make up φ(B) = j=1(1 − B=zj) one at a time (as when back-substituting in the derivation of the AR(1) representation).
    [Show full text]
  • Covariance Functions
    C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006, ISBN 026218253X. c 2006 Massachusetts Institute of Technology. www.GaussianProcess.org/gpml Chapter 4 Covariance Functions We have seen that a covariance function is the crucial ingredient in a Gaussian process predictor, as it encodes our assumptions about the function which we wish to learn. From a slightly different viewpoint it is clear that in supervised learning the notion of similarity between data points is crucial; it is a basic similarity assumption that points with inputs x which are close are likely to have similar target values y, and thus training points that are near to a test point should be informative about the prediction at that point. Under the Gaussian process view it is the covariance function that defines nearness or similarity. An arbitrary function of input pairs x and x0 will not, in general, be a valid valid covariance covariance function.1 The purpose of this chapter is to give examples of some functions commonly-used covariance functions and to examine their properties. Section 4.1 defines a number of basic terms relating to covariance functions. Section 4.2 gives examples of stationary, dot-product, and other non-stationary covariance functions, and also gives some ways to make new ones from old. Section 4.3 introduces the important topic of eigenfunction analysis of covariance functions, and states Mercer’s theorem which allows us to express the covariance function (under certain conditions) in terms of its eigenfunctions and eigenvalues. The covariance functions given in section 4.2 are valid when the input domain X is a subset of RD.
    [Show full text]
  • Geostatistical Approach for Spatial Interpolation of Meteorological Data
    Anais da Academia Brasileira de Ciências (2016) 88(4): 2121-2136 (Annals of the Brazilian Academy of Sciences) Printed version ISSN 0001-3765 / Online version ISSN 1678-2690 http://dx.doi.org/10.1590/0001-3765201620150103 www.scielo.br/aabc Geostatistical Approach for Spatial Interpolation of Meteorological Data DERYA OZTURK1 and FATMAGUL KILIC2 1Department of Geomatics Engineering, Ondokuz Mayis University, Kurupelit Campus, 55139, Samsun, Turkey 2Department of Geomatics Engineering, Yildiz Technical University, Davutpasa Campus, 34220, Istanbul, Turkey Manuscript received on February 2, 2015; accepted for publication on March 1, 2016 ABSTRACT Meteorological data are used in many studies, especially in planning, disaster management, water resources management, hydrology, agriculture and environment. Analyzing changes in meteorological variables is very important to understand a climate system and minimize the adverse effects of the climate changes. One of the main issues in meteorological analysis is the interpolation of spatial data. In recent years, with the developments in Geographical Information System (GIS) technology, the statistical methods have been integrated with GIS and geostatistical methods have constituted a strong alternative to deterministic methods in the interpolation and analysis of the spatial data. In this study; spatial distribution of precipitation and temperature of the Aegean Region in Turkey for years 1975, 1980, 1985, 1990, 1995, 2000, 2005 and 2010 were obtained by the Ordinary Kriging method which is one of the geostatistical interpolation methods, the changes realized in 5-year periods were determined and the results were statistically examined using cell and multivariate statistics. The results of this study show that it is necessary to pay attention to climate change in the precipitation regime of the Aegean Region.
    [Show full text]
  • Kriging Models for Linear Networks and Non‐Euclidean Distances
    Received: 13 November 2017 | Accepted: 30 December 2017 DOI: 10.1111/2041-210X.12979 RESEARCH ARTICLE Kriging models for linear networks and non-Euclidean distances: Cautions and solutions Jay M. Ver Hoef Marine Mammal Laboratory, NOAA Fisheries Alaska Fisheries Science Center, Abstract Seattle, WA, USA 1. There are now many examples where ecological researchers used non-Euclidean Correspondence distance metrics in geostatistical models that were designed for Euclidean dis- Jay M. Ver Hoef tance, such as those used for kriging. This can lead to problems where predictions Email: [email protected] have negative variance estimates. Technically, this occurs because the spatial co- Handling Editor: Robert B. O’Hara variance matrix, which depends on the geostatistical models, is not guaranteed to be positive definite when non-Euclidean distance metrics are used. These are not permissible models, and should be avoided. 2. I give a quick review of kriging and illustrate the problem with several simulated examples, including locations on a circle, locations on a linear dichotomous net- work (such as might be used for streams), and locations on a linear trail or road network. I re-examine the linear network distance models from Ladle, Avgar, Wheatley, and Boyce (2017b, Methods in Ecology and Evolution, 8, 329) and show that they are not guaranteed to have a positive definite covariance matrix. 3. I introduce the reduced-rank method, also called a predictive-process model, for creating valid spatial covariance matrices with non-Euclidean distance metrics. It has an additional advantage of fast computation for large datasets. 4. I re-analysed the data of Ladle et al.
    [Show full text]