snippets.md

Using Weights to Specify Dependence

12 = [(Yi1 - Y-bar1)*(Yi2 - Y-bar2)] / [(Yi1 - Y-bar1)2 * (Yi2 - Y-bar2)2]1/2 display(YouTubeVideo("MQACCcfTpXc"))

• Measured using a semivariogram *
  • Look at pairs of observations separated by a distance $h$
  • Gives us a lot of data — every observation has $n-1$ possible pairs!
  • Illustrates the spatial structure of a variable

^\* We usually just call it a variogram (there is a technical difference between the two but for clarity we'll use the term 'variogram' generally)

Calculating Semivariance

• For each pair we know the distance between points
  • We measure the similarity of the point pairs (semivariance)
  • We plot the semivariance for a range of distances

^\* In practice there is an important distinction between a *theoretical* and an *empirical* variogram — an empirical variogram is used to *fit* a theoretical variogram

• $N(h)$ denotes the set of pairs of observations separated by distance $h$
• $|N(h)|$ is the number of pairs in the set, and
• $h$ is usually an approximate distance implemented using a certain tolerance

$$ \hat\lambda(h) = {1 \over {|N(h)|}} \sum_{(i,j)\in N(h)} (x_i - x_j)^2 $$

The $\in$ under the summation denotes all $i$,$j$ pairs within the set $N(h)$

Meuse Soil Data

• The Maas river bank soil pollution data (Limburg, The Netherlands)
  • Sampled along the Dutch bank of the river Maas (Meuse) north of Maastricht
  • Data are those used in Burrough and McDonnell (1998, pp. 309–311)
• The version we use here is a subset of the data provided with the gstat and sp R packages

Variogram Parts

• Range
  • Distance at which point pairs stop being similar
  • In our meause example
    • Nearby samples have similar levels of zinc
    • Means samples separated by less than ... will be similar
    • Beyond 'range' samples are "independent"
• Sill
  • Background level of variance. Sort of a baseline for study region
• Nugget
  • Small scale discontinuity... error?

Loading/Plotting Data

Grab the data from online

meuse = pd.read_csv("https://raw.githubusercontent.com/filipkral/meuse/master/meuse.txt")

Quick plot using simple pandas plotting...

ax = meuse.plot.scatter('x', 'y', c='zinc', s=100).set_axis_off()

You can see the top of the data frame with:

meuse.head()