# 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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### Distribution of the approximation error in Gaussian Process Regression (finite data setting)

I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...
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### What am I not understanding about semivariogram and Normal Score Transformation?

I have generated this two dimensional random field: This is done following this page. In particular, I have selected t=23 as dataframe and I have changed some parameters. As you can noticed, I have ...
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### Generate a syntetic log-normal two dimensional random field

I would like to test some functions that I wrote related to the kriging applied to rain data. In order to do that, I would like to generate a synthetic log-normal 2D random field. The idea is to ...
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### How to constrain ordinary kriging weights to sum to 1 in R

I need to perform ordinary kriging on a dataset and I understand that I need the weights to sum to 1, I just don't understand how to set that up properly. For example, my covariance function is C(h) =...
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### Kriging - Derivation of equations [duplicate]

While discussing about a problem, I told her about Kriging. My daughter asked me for the derivation of equations. Can you help me with derivation of Kriging Equations for Simple Kriging? (that can be ...
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### Estimating probability of attack in Ukraine, given count data

I was looking at some attack count data in Ukraine for different days. The data is gathered from the ACCLED dataset, and there is a picture below. The picture shows individual attacks, but I can apply ...
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### Is there a Sequential Gaussian Simulation that uses Ordinary Kriging?

As far as I have read, Sequential Gaussian Simulation always uses Simple Kriging. Is there any chance that it uses Ordinary Kriging?
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### Kriging with positivity constraint using the STK toolbox

I work in the aerospace world, specifically doing calibration and characterization of detectors and space instruments (cubesats, space spectrographic satellites etc.) We always need to interpolate our ...
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### What is the difference between a non-zero nugget and a noise term in Kriging/GPR?

With some Gaussian Process Regression/Kriging models, it's possible to specify both a non-zero nugget, and a noise term. For example, in Scikit-learn's GPR model, there is an ...
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### Multicollinearity in regression kriging

I am doing regression kriging for a certain pollutant as a dependent variable, and for independent variables I'm using 4 variables that add to 100% for all observations. Because of this when I use ...
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### Analytical Leave-one-out prediction variance for Kriging

I make extensive use of Kriging (Gaussian Process regression) methods in my work especially using the leave-one-out error calculation that you can get from the Gram matrix. Background: To compute the ...
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### gstat kriging vs. mgcv Gaussian process smooth

In Simon Wood's Generalized Additive Models: An Introduction with R, he states that the Gaussian process smooth (i.e., Gaussian process regression) is analogous to kriging. This is also stated on ...
1 vote
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### How can concentrated (profile) log marginal likelihood be used to optimize the mean and scale(outputscale) parameters in Gaussian Process Regression?

The log marginal likelihood which is used in Gaussian Process Regression comes from a Multivariate Normal pdf Gaussian Processes for Machine Learning, p.19, eqn. 2.30, Surrogates, Chapter 5, eqn. 5.4 \...
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### Who first proposed Bayesian optimisation with Gaussian processes?

From what I understand, the 'standard' approach to Bayesian Optimisation uses a Gaussian process for the prior (as opposed to more recent proposals like TPE or Bayesian Optimisation with random ...
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### Interpreting a Distance & Time 3D Variogram for Variogram modeling

I am trying to understand some concepts of variograms. I have made several variogram models in R and am trying to understand exactly what they mean. My data is ...
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### Variogram fit in R not converging

I have taken a shapefile from Open NYC Data and performed the following method. My end goal is to predict Taxi trip_duration at various points across the city of ...
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### Cokriging and collocated cokriging data requirements

In this wiki article and elsewhere in educational materials/papers, I have seen people refer to the idea that secondary data, if used (appropriately) in cokriging or collocated cokriging, is usually ...
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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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### “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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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