Questions tagged [kriging]

Kriging is spatial prediction based on a stochastic model of a spatial random field. Such models and methods can also be used in non-spatial context, and then often known as gaussian processes.

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Differences between Kriging and Gaussian Process Regression

I am having quite difficult time to clearly understand the differences between Kriging and Gaussian Process Regression. Here is what I have understood so far: For simple kriging (mean value known), ...
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Conditional distribution of Ornstein-Uhlenbeck on two fixed points

The conditional distribution of a Ornstein-Uhlenbeck $X(t)$ conditional on $X(0)$ is given by $$ X(t)|X(0) = X(0)e^{-t} + \mu(1 - e^{-t}) $$ This process is usually only defined for $t>0$ (future ...
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(Co)kriging / co-located kriging with heterogenous measurement errors

I have a large set of polygons on a map, some of which contain data on 2 count variables, say ($z_{1}$ and $z_{2}$) that are correlated. In fact, $z_{2}$ most likely causes $z_{1}$ without the ...
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Intuition Behind Correlation Function in Kriging Models

I'm thinking and researching extensively to interpret the parameter $\theta$ (activeness parameter) in Gaussian correlation function in a Kriging model, namely as: $$ K(h;\theta)=exp(-h^2/(2\theta^2)) ...
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Intuitive explanation of Gaussian Process Regression

How would you intuitively explain the idea behind Gaussian Process Regression to someone unfamiliar with stochastic processes? Especially the point where you discuss modeling covariance, choice of ...
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Smoothing methods for geographically aggregated data?

I have some Canadian census data with various statistics defined by dissemination block. These blocks are irregular in shape since their boundaries are based on the road network. I thought it would be ...
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“Jumping” among several interpolation techniques?

I am comparing several interpolation methods using monthly climatic data, through RMSE and a 10-fold cross-validation scheme. What I'm observing is that the performances vary from one month to ...
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151 views

Modeling trends with Gaussian Processes

I am trying to use Gaussian Process Regression to model data that has a clear upward rising trend. Here is a basic plot of the data - The most well-known example of the flexibility of GPs in such ...
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learning time series by machine learning

I am trying to learn a mapping between coordinates x,y,z,d, with d=sqrt(x**2 + y**2 + z**2), and a scalar. Each ...
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When regularizing based on an informative prior, how to give model a little more freedom to partially reject regulariziation

I am new here I hope this question is appropriate. I am modelling a spatial domain, whereby I have repeated measures at n locations. I make a bayesian linear model at each n locations based on about ...
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51 views

Can I replace the distribution in Gaussian Process Regression with a different regression?

Sorry for the confusing title. Let me try to clarify: I have a time series of wind speeds with some missing points here and there. I want to interpolate these points and have tried mainly polynomial ...
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Iterative solution to estimate nugget effect

I have to implement the equation to fit an analytical curve to an empirical covariance function. Until now I used the equation $$ c = ae^{-b \Delta t} $$ with $a$ and $b$ are the parameters I want ...
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177 views

Is the spherical covariance function not positive definite for d > 3?

I read in a textbook (Japanese one) that the spherical covariance function is only valid for dimensions $d = 1,$ $2,$ and $3.$ I have the following questions: Does that mean the spherical covariance ...
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Cokriging variances differ using cross validation

I'm investigating cokriging using various metals in the Meuse dataset but the variances output by R when I predict values at gridded points differ substantially from the variances produced by cross ...
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Why do you need a variogram for Kriging? goldingn/gpe package?

I am using golingn/gpe (github) package, and it does not provide a variogram and instead look at co-variances. Is it possible to do kriging without providing variograms?
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How to handle multiple points at the same location in spatial interpolation?

I am new to the topic of spatial interpolation and would appreciate your opinion on a general question which has arisen. Suppose I have a data set containing rental rates for different apartments in ...
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Poor performance from `gstat::krige` with a noise predictor

I'm new to kriging, and I'm considering replacing a use of inverse-distance weighting (IDW) in a spatial modeling project (implemented with gstat::idw in R) with a ...
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Gaussian process with interval observations

The stochastic process $(X_t)_{t \in T}$ is a Gaussian process if the marginal distribution of $X_{t_1}, \ldots, X_{t_n}$ is a multivariate Gaussian distribution for all $t_1, \ldots, t_n \in T$. Let ...
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Cokriging, zero distance semivariance [gstat]

Trying cokriging with simulated data, I faced a problem that did not seem one in the demo(cokriging) with the meuse dataset: I can't use ...
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261 views

Correct way to compute the error variance of ordinary kriging

I'm learning ordinary kriging and I found some discrepancies in the method of computing error variance among different materials I read, in general there are 2 different formula: Method 1: ...
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173 views

spacetime R: How to handle missing data in a space-time-full data structure for spatio-temporal kriging purposes?

I am using R and the spacetime package. I am having problems using STFDF. I want to use the STF data-structure since I have spacetime data with recurrent observations for fixed spatial coordinates. ...
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Noisy conditional simulation

A conditional random field $Z_C(x)$ is a random field whose realisations $z_C(x)$ always take the same values $z_C(x_a)$ at locations $x_a$. Realisations of $Z_C(x)$ can be produced as follows (...
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Ordinary kriging example step by step?

I have followed tutorials online for spatial kriging with both geoR and gstat (and also ...
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What is the nugget effect?

I don't understand exactly what is meant by the term "nugget effect" in geostatistics. When looking at empirical variograms plotting the variogram $\gamma(h)$ vs. the lag $h$, the nugget is defined as ...
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66 views

Are Kriging's residuals (i.e. $Z-\hat{Z}$) spatially independent?

Suppose data ${Z(s_i ):i=1, ..., n}$ are observed at spatial locations ${s_i :i=1, ..., n}$. To carry out the spatial prediction (predict un unknown $Z(s_0)$ at a known location $s_0$) we can use a ...
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596 views

Is a function describable by a Gaussian process smooth?

I understand that a stochastic process or function is considered a Gaussian process if sampling from it at any point some set of times yields a set of observations that match a Gaussian random ...
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How do you interpret this variogram?

Description: 8000 spatial data points spanned over an entire state 200 bins are used My question: Is the variogram telling something about the nature of the data? Why is it fluctuating? Should I do ...
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What is the statistically correct way of performing the following interpolation?

I have a table with thousands of entries similar to the following: It is desired to determine Property 2 at the unknown locations. Property 1 and Property 2 are spatially correlated. By that I mean ...
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450 views

Spherical vs. Exponential Kriging Covariance Functions

A statistical epidemiologist colleague of mine told me that in comparing spherical vs exponential kriging covariance functions, only the latter (i.e., exponential) function is generally a valid model. ...
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753 views

How do I estimate the prediction interval of back transformed log-normal data from Gaussian process?

I have some data that are clearly positively skewed and follow a log-normal distribution, lets assume the initial data is $Z = exp(Y)$, where $Y \sim N(\mu,\sigma^2)$. A Gaussian process assumes ...
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416 views

What is the difference between accounting for anisotropy and trend removal when performing Kriging?

Without being geostatistician, I read a bit about anisotropy detection, mostly from ArcGIS documentation and the R gstat package tutorial. But still, it is hard to have a confident understanding of ...
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What is 5th order kriging?

For our foray into geostatistics we got data that consists of measurements taken from the soil. The dataset has like concentrations of various different minerals. We were divided into a number of ...
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274 views

Kriging variance results

I'm quite a newb at statistics and interpolation, and I cannot understand how to interpret the error estimation computed by Kriging. For example, I performed kriging on temperature values (Celsius ...
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218 views

Determining covariance of irregularly spaced spatial data

I'm comparing concentration $C$ of a contaminant in the same spatial region at two time point 2000 and 2010 with sample size of $N_{2000}$ = 51 and $N_{2010}$ = 26 (not all the samples are from the ...
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Explain Like I'm Five version for Variograms in R's gstat package

Long story short, I asked a question on StackOverflow about Variograms in the gstat package in R. The person who answered gave me some tips on creating the variogram using the package. My dataset is ...
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what is the difference between Bayesian optimization and kriging?

Both methods use Gaussian process, and kriging uses the Best Linear Unbiased Predictor (BLUP) to predict the mean (this is not seen in Bayesian optimization?). At the bottom line, they also have ...
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Is kriging suitable for high dimensional regression problems?

I would like to point out that I am new to this field, so if I am not clear please forgive me (and correct me). I set up a DoE (Design of Experiment) with 11 inputs and 121 runs. I used a STOA (...
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Performing inference across multiple target variables with Gaussian Processes regression

Suppose I have a design matrix $\mathbf{X}$ with targets $\mathbf{y}_1$, when plotted looks like so: the data is very sparse, and the magnitude of the targets is shown by color bar, where each ...
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494 views

How to implement a Gaussian Markov Random field (GMRF) model in R?

I would like to model a (conditional) GMRF using a linear mixed effects model without having grid Data but only a neighbourhood matrix $W$. My model is given by $$Y=X\beta+ \epsilon$$ and the error ...
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Estimation of time for a specific value of a variable

I have a data set: ...
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Doubts on the methodology used for build and evaluate a Kriging model

I have run 3000 finite element (deterministic) simulations of a physical model, which has 10 design variables and one output variable. I have generated the sampling plan using the LH method. My ...
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Kriging with mean data instead of point data

Kriging is a technique to predict a realisation of a Gaussian process. The values of the realisation are known at a finite subset of points and we would like to optimally predict the values of the ...
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405 views

Why is positive definiteness necessary for kriging?

I understand from wikipedia that a variogram model must be positive definite to be used for kriging: Note that the experimental variogram is an empirical ...
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Is there a way to find optimal sensor node locations in a domain when all data is known?

I have an x-y domain where I know the snow depth everywhere. (i.e. the granularity of the values is such that I can assume I know it everywhere). I want to use this information to inform where to ...
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Universal Kriging

In universal kriging the general model is $$y(x)=h(x)^{t}\boldsymbol\beta + f(x)+\epsilon(x)$$ where $h(x)^{t}$ is a regression function such as $\left [1,x,x^{2} \right ]$ and $\boldsymbol\beta$ are ...
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Universal kriging with non-zero mean vector

Its is known that in ordinary Kriging (Gaussian Process) the mean and variance at any new point is given as $$\\\begin{pmatrix} \mathbf{y}\\y^{*} \end{pmatrix} \sim N(\mathbf{0},\begin{bmatrix} K &...
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245 views

Kriging: fit.method and dX argument in variogram

I would like to krige residuals (from multiple linear regression) of yearly precipitation totals from a 50 years time series. Every year has been regressed individually. The residuals will be added to ...
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Transformation of non-negative data including zeros

at first I want to mention that I am fully aware of this question here as well as the answers. Still, things won't work out as intended (using R). I have a lot of hourly rainfall data that include ...
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Why we krige residuals in regression-kriging?

Wikipedia says: In applied statistics, regression-kriging (RK) is a spatial prediction technique that combines a regression of the dependent variable on auxiliary variables (such as parameters ...
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Why likelihood based geostatistical modelling slower than non likelihood based counterpart

Likelihood based geo-statistics (geoR etc.) are usually slower than non-likelihood based geo-statistics (i.e. those based on just least square fitting, for example <...