# Tag Info

24

"Spatial autocorrelation" means various things to various people. An overarching concept, though, is that a phenomenon observed at locations $\mathbf{z}$ may depend in some definite way on (a) covariates, (b) location, and (c) its values at nearby locations. (Where the technical definitions vary lie in the kind of data being considered, what "definite way" ...

17

There are two things that will impact the smoothness of the plot, the bandwidth used for your kernel density estimate and the breaks you assign colors to in the plot. In my experience, for exploratory analysis I just adjust the bandwidth until I get a useful plot. Demonstration below. library(spatstat) set.seed(3) X <- rpoispp(10) par(mfrow = c(2,2)) ...

12

The main issue in any statistical model is the assumptions that underlay any inference procedure. In the sort of model you describe, the residuals are assumed independent. If they have some spatial dependence and this is not modelled in the sytematic part of the model, the residuals from that model will also exhibit spatial dependence, or in other words they ...

8

An outlier detector for your irregular ("vector") point data is available in GRASS as v.outlier. An overview of spatial outlier detection methods appears in a 2004 paper by Cheng and Li. An older method, specialized for topographic data, relies on "drainage enforcement" (making the water flow downhill continuously without accumulating in sinks). That can ...

8

One of the problems with multivariate data is deciding on, and then interpreting, a suitable metric for calculating distances, hence clever but somewhat hard-to-explain concepts such as Mahalanobis distance. But in this case surely the choice is obvious - Euclidean distance. I'd suggest a simple heuristic algorithm something like: Calculate the ...

8

1) You can model spatial correlation with the nlme library; there are several possible models you might choose. See pages 260-266 of Pinheiro/Bates. A good first step is to make a variogram to see how the correlation depends on distance. library(nlme) m0 <- gls(response ~ level, data = layout) plot(Variogram(m0, form=~x+y)) Here the sample ...

7

From ten points you are going to have 45 (10*(10-1)/2) points in your variogram cloud from the distances between each pair of points. Once the system has binned that, or even without binning, its going to be dominated by noise, I reckon. Get a plot of the variogram cloud to see what I mean. If autokrige can't fit a nice smooth variogram then it will do what ...

6

As the Encyclopedia of GIS states, the conditional autoregressive model (CAR) is appropriate for situations with first order dependency or relatively local spatial autocorrelation, and simultaneous autoregressive model (SAR) is more suitable where there are second order dependency or a more global spatial autocorrelation. This is made clear by the fact ...

6

It seems to me that you have enough data to model the dependence on space-time and meteorological influences of both the bias of forecast errors (i.e. tendency to systematically over-/underestimate [first moment]) and their variance [second moment]. For exploration of the bias, I'd just do a lot of scatterplots, heatmaps or hexbin plots. For exploration of ...

6

These terms probably do not have a universally accepted technical definition, but their meanings are reasonably clear: they refer to second order and first order variation of a spatial process, respectively. Let's take them by order after first introducing some standard concepts. A spatial process or spatial stochastic process can be thought of as a ...

6

There's an extensive literature on this area in spatial ecology; but it comes down to what kind of assumptions you're willing to make. A very common method for estimating spatial distributions based on abundance data is a maximum entropy approach, take a look at the software and publications There's a variety of other algorithms designed for this purpose ...

6

Ok. These books seem to be general books on spatial statistics, not restricted to particular area: Bivand et al - Applied Spatial Data Analysis with R - This book was recommended in some presentation at an ecological conference. Banerjee et. al - Hierarchical Modeling and Analysis for Spatial Data. This one I just found randomly I think, don't know nothing ...

6

Suppose $U$, $V$, and $W$ are independent and have a standard normal distribution. Then by definition of the chi-squared distribution (as a sum of squares of iid normal variables), the sum of squares $U^2 + V^2 + W^2$ has a chi-squared distribution with three degrees of freedom. We find (via tables or calculation) that its upper 95th percentile is $7.815$. ...

6

Variography is part statistics, part science, and partly a practical art. Entire books (or major parts thereof) have been written about it, beginning with Journel & Huijbregts' Mining Geostatistics in 1978, so it will not be possible to do justice to this question in the space of one Web page. Let's just examine the issues briefly. What a variogram ...

5

You can use this module of the pysal python library for the spatial data analysis methods I discuss below. Your description of how each person's attitude is influenced by the attitudes of the people surrounding her can be represented by a spatial autoregressive model (SAR) (also see my simple SAR explanation from this SE answer 2). The simplest approach is ...

5

We (A colleague and I) finally wrote a paper on that one. To summarized things we proposed two solution to quantify and give a statistical summary of the (spatio-temporal) propagation of errors along Denmark and along look ahead times. In the first one we compute the correlation between all pairs of wind farms and for all pairs of look ahead times (this ...

5

Non-spatial model My House Value is a function of my home Gardening Investment. SAR model My House Value is a function of the House Values of my neighbours. CAR model My House Value is a function of the Gardening Investment of my neighbours.

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One option may be to split the original data into two subsets: one that will be used in interpolating values and one that will be used to validate the interpolation results. The error is then estimated by comparing interpolated values at the validation point locations with the actual validation point values. Note that the appropriateness of this approach is ...

5

In short, identifying anisotropy is hopeless with these sparse data. The two parameters in question, psiA and psiR, describe anisotropy (the angle and ratio, respectively, of a "geometric anisotropy": consult GSLIB or Journel & Huijbregts for details, because the geoR documentation in Diggle & Ribeiro Jr is indeed inadequate concerning anisotropy). ...

5

Moran's I is a diagnostic statistic that can be used to detect spatial autocorrelation in the residuals of a regression, given that you have a weight matrix $\mathbf{w}$, with entries $w_{ij}$ representing distances between observation (residuals) $X_i$ and $X_j$. You can think of it as a spatially-weighted measure of correlation. Significance of the ...

5

Mantel test and Moran's I refer to very different concepts. The reason for using Moran's I is the question of spatial autocorrelation: correlation of a variable with itself through space. One uses Moran's I when wants to know to which extent the occurrence of an event in an areal unit makes more likely or unlikely the occurrence of an event in a ...

4

One model for this situation is to view $Z$ as a realization of a stationary 2D stochastic process. The limiting behavior at zero (distance) of its empirical variogram or correlogram provides information about its smoothness: if the limiting correlation is less than one, the process is not even (mean square) continuous. Otherwise (Theorem) A ...

4

I have recently come across this method that was displayed in Johnson and Wichern. Let the data points that you want to test for bivariate normality be designated as $\{ x_{i} \}$. Next, compute the sample covariance matrix and deisgnate it as $S$. For each observed point calculate $d_{j}^{2} = (x_{j} - \bar{x})^{T} S^{-1} (x_{j} - \bar{x})$. Order the ...

4

Your idea about the "rasters" is not very clearly stated, but you might have a look at the paper by Borcard and Legendre (1994) and their later works on spatial eigenvector-based analyses to see if one of the approaches will fit to your problem. Borcard, D., Legendre, P., (1994) Environmental control and spatial structure in ecological communities: an ...

4

There are many ways you can characterize homogeneity, so there could be many answers to your question. One of the most intuitive ways I have seen it displayed is in a book chapter, "Spatial Analysis of Regional Income Inequality" by Sergio Rey in the book Spatially Integrated Social Science (PDF). The approach Rey takes in that chapter is to visualize the ...

4

1) What is your spatial explaining variable? Looks like the x*y plane would be a poor model for the spatial effect. i=c(1,3,5,7,8,11,14,15,16,17,18,22,23,25,28,30,31,32,35,36,39,39,41,42) l=rep(NA,42)[i];l[i]=level r=rep(NA,42)[i];r[i]=response image(t(matrix(-l,6)));title("treatment") image(t(matrix(-r,6)));title("response") 2) Seeing as how the ...

4

One thing you could use is a distance measure from a central point, ${\bf c}=(c_{1},c_{2})$, such as the sample mean of the points $(\overline{x}, \overline{y})$, or perhaps the centroid of the observed points. Then a measure of dispersion would be the average distance from that central point: $$\frac{1}{n} \sum_{i=1}^{n} || {\bf z}_{i} - {\bf c} ||$$ ...

4

The work of Gary King, in particular his book "A Solution to the Ecological Inference Problem" (the first two chapters are available here), would be of interest (as well as the accompanying software he uses for ecological inference). King shows in his book how the estimates of regression models using aggregate data can be improved by examining the potential ...

4

A point process is a collection of random variables that are positions in some space (like locations on a plane). A marked point process is a point process in which some additional features are measured at each point. For your situation, the locations of the points are by design rather than random, and so while you could call it a marked point process, the ...

4

This is actually an extremely sophisticated problem and a tough ask from your lecturer! In terms of how you organise your data, a 1070 x 10 rectangle is fine. For example, in R: > conflict.data <- data.frame( + confl = sample(0:1, 1070, replace=T), + country = factor(rep(1:107,10)), + period = factor(rep(1:10, rep(107,10))), + landdeg = ...

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