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23 votes
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Splines vs Gaussian Process Regression

I agree with @j__'s answer. However, I would like to highlight the fact that splines are just a special case of Gaussian Process regression/kriging. If you take a certain type of kernel in Gaussian ...
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17 votes
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How Does Kriging Interpolation work?

This answer consists of an introductory section I wrote recently for a paper describing a (modest) spatio-temporal extension of "Universal Kriging" (UK), which itself is a modest generalization of "...
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15 votes
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Ordinary kriging example step by step?

Apart from this answer, there are also some nice additional answers to a similar question on gis.stackexchange.com First I'll describe ordinary kriging with three points mathematically. Assume we have ...
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9 votes

Estimating probability of attack in Ukraine, given count data

This is not an answer, but rather a side comment: Keep in mind that the new attacks are not independent of the previous ones. Historical data is not necessarily relevant for the future. It is probably ...
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7 votes
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Is a function describable by a Gaussian process smooth?

Not all choices of kernel function yield a smooth function. The exponential kernel $K(x_i, x_j) = \exp\left(-\gamma d(x_i, x_j)\right)$ for $\gamma > 0$ and $d$ a valid distance is the covariance ...
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6 votes

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

$$ \newcommand{\E}{\mathbb{E}} \DeclareMathOperator{\cov}{Cov} \newcommand{\zhat}{\widehat{Z}} $$ I'll try and answer the title of your question, about spatial independence of the residuals. For ...
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6 votes

Splines vs Gaussian Process Regression

It is a very interesting question: The equivalent between Gaussian processes and smoothing splines has been shown in Kimeldorf and Wahba 1970. The generalization of this correspondence in the case of ...
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6 votes

Estimating probability of attack in Ukraine, given count data

Does anyone know what kind of model I would use for something like this? ... I was just wondering if anyone know some common approaches. Two approaches you may want to look into: "Self exciting ...
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5 votes
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Gaussian process regression: leave-one-out prediction

In the general noisy or ``signal $+$ noise'' framework $y_i = f(\mathbf{x}_i) + \epsilon_i$, several observations can be done at the same location $\mathbf{x}_i$, so the notations $Y(\mathbf{x}_i)$ ...
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5 votes
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what is the difference between Bayesian optimization and kriging?

I believe you mean Gaussian processes rather than Bayesian optimisation. Bayesian optimisation is the use of Gaussian processes for global optimisation. Essentially you use the mean and variance of ...
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5 votes

what is the difference between Bayesian optimization and kriging?

I debated answering this because I have not used kriging in probably twelve years. They are also closely related. It is somewhat of a bridge between the extremes. Still, that kriging is the BLUP ...
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4 votes
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Is the spherical covariance function not positive definite for d > 3?

There are at least two conventional meanings of "spherical covariance function." Based on the conclusion you want to draw, we can infer you are referring to the family of functions $G:\...
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3 votes

What is the nugget effect?

The nugget effect is like the random noise. It's just the small scale variability that you can't estimate with your large scale variability model. The nugget effect is made of the measurement error ...
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3 votes
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Gstat: Modelled semivariogram values not matching plotted model using the variogramLine function

You are working with an anisotropic variogram model, but did not inform variogramLine in which direction you want to look. From the documentation of ...
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3 votes
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Computing Issues with Kriging

It looks like what is happening is that 3 prediction locations are identical to observation locations ((508,592), (760,592), (508,364)), so their prediction error is zero. Also, the call to ...
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3 votes
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Can I replace the distribution in Gaussian Process Regression with a different regression?

Disclaimer: This is an unfinished answer which still needs some (mathematical / research / googling) work. Feel free to downvote if there is somebody with more experience concerning Weibull processes.....
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2 votes

Is it valid to use a model-variogram fit not on the full range of lag distance?

Ordinary kriging (OK) uses least squares methods to make predictions of values at unsampled locations. The model variogram provides information about the correlations among all locations, both sampled ...
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2 votes

Splines vs Gaussian Process Regression

I agree with @xeon's comment also GPR puts a probability distribution over infinite number of possible functions and the mean function (which is spline like) is only the MAP estimate but you also have ...
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2 votes

Geographic regression

I prefer autoregressive models to kriging, because I feel there is more of an economic rationale behind their use. Others are free to disagree! Autoregressive models may be difficult to use/...
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2 votes

Why is positive definiteness necessary for kriging?

The terminology about positive definite matrices is very inconsistent. Some authors use "positive definite" to mean all-positive eigenvalues and "positive semidefinite" to mean all-nonnegative ...
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2 votes

Is kriging suitable for high dimensional regression problems?

I completely agree with David Kozak's answer, but I would like to bring some additional points: 11 inputs is not too high for Gaussian process regression (GPR)/Kriging. I have already dealt with ...
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2 votes

Is kriging suitable for high dimensional regression problems?

You might look into Gaussian Process Regression... Kriging is the name used for Gaussian Processes in Spatial Statistics, and is the domain in which they found the most use until relatively recently. ...
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2 votes
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What is the nugget effect?

In the context of estimating a variogram, a nugget allows for the variogram to assume a non-zero value for two observations having a distance of zero. The implication also is that the correlation ...
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2 votes
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Cokriging and collocated cokriging data requirements

Co-kriging is often used, as you mentioned, when we have a 'secondary' source of data. The main idea is that the abundant data is a good guess of the primary data, but they're not the same so you ...
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  • 2,385
2 votes

Who first proposed Bayesian optimisation with Gaussian processes?

Kushner (1964) (H. J. Kushner. A new method of locating the maximum of an arbitrary multipeak curve in the presence of noise.J. Basic Engineering, 86:97–106,1964.) is probably the first one with ...
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  • 284
2 votes
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What is the difference between a non-zero nugget and a noise term in Kriging/GPR?

Random noise and nugget effect are indeed quite similar to some extent. The difference between the two appears when there are repeated observations (i.e., several observations at the same location), ...
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1 vote

How do you interpret this variogram?

This is known as the hole effect in the literature. The oscillations correspond to cyclic patterns in your domain. It is very likely that you have "blobs" of high values and low values. The size of ...
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  • 233
1 vote
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What is the difference between accounting for anisotropy and trend removal when performing Kriging?

I will try to address your concerns one by one: Anisotropy might be the reason of the trend. In that case, there is no need to remove trend if I account for anisotropy in my semi-variogram. On the ...
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  • 233
1 vote

Kriging variance results

In ordinary kriging, the variance does not depend on the measurements but only on their location. So it is merely a measure of how far you are from a measurement location. It is not higher in areas ...
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1 vote

Determining covariance of irregularly spaced spatial data

If not all the samples are from the same location is equivalent to almost all the samples are from the same location or put differently that $2000$ and $2010$ spatial supports broadly ...
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