In brief, stationarity is a condition that shows whether the data has a constant mean and variance in each location. Stationarity is widely used in time series function, nevertheless we also need to know its application in terms of spatial data estimation.
There are 2 important things quoted from one of the Michael Pyrcz lecture courses:
- Stationarity is a decision, not an hypothesis; therefore it cannot be tested. Data may demonstrate that it is inappropriate.
- The stationarity assessment depends on scale. This choice of modeling scale(s) should be based on the specific problem and project needs.
To investigate/assess stationarity in spatial data, practically we can use two ways of visualizations, either by using a trend plot or a variogram plot.
Case Study — Python Codes
let’s say in “X” field , we have a Silver (Ag) distribution as follows.
import pandas as pddf = pd.read_csv(‘contoh.csv’)
df = df[[‘ID’,’XCOO’,’YCOO’,’Ag’]]
ID is the identifier of each sample taken from the field, while XCOO and YCOO are X and Y coordinates respectively. To recognize the variation of our data, we visualize it in ID vs AG plot using
pyplot as figure 2 below:
import matplotlib.pyplot as pltplt.figure()
plt.title(‘Ag Field Data’)
The stationarity of the data could be analyzed using a trend plot, we can simply calculate the intercept and slope of the linear regression equation using
from scipy import stats
slope, intercept, corrcoeff, p_value, stderr = stats.linregress(df.ID,df.Ag)m = slope
c = intercept
By extracting slope and intercept from
linregress we can define and plot our linear function (y = mx + c) as shown at Figure 3.
#linear regression function
df.Y = m * df.ID + cplt.figure()
plt.plot(df.ID, df.Ag, color=’k’) #point plotting
plt.plot(df.ID, df.Y, linewidth=3.5, color=’y’, label=’trend’) #trend plottingplt.title(‘Ag Field Data’)
We can see that the trend shown in Figure 3 is a straight horizontal line which indicates that our data is stationary. But is it really stationary? let’s check the trend by partitioning this data (for example on ID 650–750). By using the same code as above, we can visualize the result as in Figure 4.
Figure 4 shows one of the stationarity characteristics. “The stationarity assessment depends on scale”. According to the trend plots, it can be seen that the second plot from ID 650–750 is not stationary despite the result from the whole data appears to be stationary. This stationarity uncertainty also applies whether we take another scale / partition on our data.
In terms of the stationarity assessment, variogram plot has a different approach compare to trend plot. Variogram is a geostatistical method that can visualize variance trend on spatial (lag from one data to another) perspective . Variogram is a highly recommended way to analyze uncertainty since it can see the pattern of relationships between data points at each lag.
Figure 5 shows the Variogram plot consisting of data point (black point) and modelled / theoretical variogram (black line). We also can see other important variables such as:
- Lag distance: The distance between pairs calculated by experimental variogram. e.g: lag distance can be calculated every 10m, 20m,1000m, etc.
- Semivariance/variance: a parameter that describes the dissimilarity between data. The higher the semivariance / variance, the worse the similarity relationship between the data.
- Sill: Variogram value when it reaches a constant point
- Range: Lag distance when the variogram value reaches sill
- Nugget: the variogram “jump” at the starting point (lag distance is ~ 0)
Stationary data can be indicated from a bounded variogram, while non-stationary data can be indicated from an unbounded variogram, as shown in Figure 6.
Bounded variogram is a variogram that has sill & range, while unbounded variogram is a variogram that has an upward trend and does not have sill & range (unflatted).
Now let’s create variogram using our data with
geostatspy is a python library developed by MichaelPyrcz (@geostatsguy) which contains modules related to geostatistics. You can simply install the library by using
pip install geostatspy .
In order to plot the experimental variogram we will plot data point in every 7000 lag with 3000 lag tolerance with n = 30. The azimuth is 0 degree with 90 degree tolerance as the illustration shown in Figure 7.
lag = 7000
lag_tol = 3000
nlag = 30
azi = 0
azi_tol = 90
bandwidth = 9999lags, gammas, npps = geostats.gamv(df,”XCOO”,”YCOO”,”Ag”,-9999,9999,lag,lag_tol,nlag,azi,azi_tol,bandwidth,isill=1.0)
gammas obtained from
geostats.gamv is the variogram value for each lag that we determined before, while
npps is the number of data pairs at a specified lag. After that we can plot
pyplot as Figure 8 shows.
# plot experimental variogram
plt.scatter(lags,gammas,color = ‘black’,s = npps*0.1,label = ‘Azimuth ‘ +str(azi))
plt.title(‘Ag Distribution Variogram’)
Thenceforth we need to plot our modelled variogram to analyze the stationarity in our data. Basically this step needs a “trial and error” to make sure our model fit with our experimental variogram. In this article we already got our best parameter for our modelled variogram.
Sill: 1 | Range: 110000 | Type: Spherical
geostats.vmodel module we can extract the function of our modelled variogram.
nug = 0.5
nst = 2 #whether a normal score transformation is used on this variogram (two (2) is for not using)
it1 = 1 #variogram type (one (1) is for spherical type)
cc1 = 0.5 #sill minus nugget
azi1 = azi #azimuth
hmaj1 = 110000 #maximum major lag distance measured in this variogram (range)
hmin1 = 110000 #considering not using a minor azimuth (we can ignore this parameter)
#assumption no contribution on minor structure (we can ignore this)
it2 = 1; cc2 = 0; azi2 = 0; hmaj2 = 9999.9; hmin2 = 400
xlag=15000#make model object
vario = gslib.make_variogram(nug,nst,it1,cc1,azi1,hmaj1,hmin1,it2,cc2,azi2,hmaj2,hmin2)#extract the important variable of the modelled variogram
index_maj,h_maj,gam_maj,cov_maj,ro_maj = geostats.vmodel(nlag,xlag,azi,vario)
Then we overlay our modelled variogram into our previous experimental variogram plot as shown in Figure 9.
plt.plot(h_maj,gam_maj,color = ‘black’)
From Figure 9 it can be seen that the variogram modelled fit with the experimental variogram. This variogram is categorized as bounded variogram considering the sill / range which is very clear. It can be decided that our data tends to be stationary.
Why is this Important?
Stationarity analysis is useful to consider further geostatistical estimation methods such as kriging. We know that kriging has various types, such as ordinary kriging, universal kriging, kriging with external drift, and cokriging. The stationarity factor is what really determines which method we are going to use.
For example, if we choose simple kriging method as a spatial interpolation method. This method assumes that our data must be stationary. The estimation result will have an error if our data is non-stationary.
The error margin can be measured by cross validation or reconciliation. Ordinary kriging might be used if the result generates a small error number (we could ignore the non-stationarity). Otherwise, other spatial estimation methods such as universal kriging and kriging with external drift might be considered to accommodate the non-stationarity of the dataset.
Ordinary kriging might be used if the result generates a small error number (we could ignore the non-stationarity). Otherwise, other spatial estimation methods such as universal kriging and kriging with external drift might be considered to accommodate the non-stationarity of the dataset.