4. Plotting

At the core of SciKit-GStat is a set of classes, that can be used interactively to perform variogram analysis. One important aspect of this analysis is a rich collection of plotting functions. These are directly available as class methods of the Variogram, DirectionalVariogram and SpaceTimeVariogram method. With version 0.3.3, SciKit-GStat implements two different plotting backend: matplotlib and plotly. Generally speaking, matplotlib is great for creating publication ready figures in a variety of formats, including vector-graphic PDF files. Plotly, on the other hand, will translate the figure into their Javascript library and open a webbrowser with an interactive plot. This way you can obtain the same figure either for your publication as PDF, or as a HTML object that can be injected into a project report website.

With the newly introduced skgstat.plotting backend, you can easily read and change the backend with a single convenient function. The default backend is matplotlib. Please be aware, that plotly is only a soft dependency, meaning you need to take care of the installation yourself, to keep SciKit-GStat’s dependency list shorter.

The data used to create the Variogram and DirectionalVariogram is from Mälicke (2021). Here, pancake dataset is used. The spatio-temporal data is derived from Fersch et al. (2020). From that data publication, the wireless sensor network data is used. The originaly published 15 minutes intervals soil temperature data at 20 cm depth was taken for all 55 stations and aggregated to mean hourly values. To further decrease the data size, only every 6th data point is used here. Estimating the full data set will take approx. 120GB RAM and processing took about 30 minutes. The results for the thinned data sample are very comparable.

Both data samples can either be obtained by the orignial publications, or from the SciKit-GStat documentation. Both samples are published under Creative Commons BY 4.0 license. Please cite the original publications if you use the data, and not SciKit-GStat.

References

Fersch, Benjamin, et al. “A dense network of cosmic-ray neutron sensors for soil moisture observation in a pre-alpine headwater catchment in Germany.” Earth System Science Data Discussions 2020 (2020): 1-35.

Mälicke, M.: SciKit-GStat 1.0: A SciPy flavoured geostatistical variogram estimation toolbox written in Python, Geosci. Model Dev. Discuss. [preprint], https://doi.org/10.5194/gmd-2021-174, in review, 2021.

import skgstat as skg
from skgstat.plotting import backend
import numpy as np
import json
import warnings
import matplotlib.pyplot as plt
from plotly.subplots import make_subplots
warnings.filterwarnings('ignore')

4.1 Load data

Load a pancake sample from the data directory.

c, v = skg.data.pancake(N=300, seed=42).get('sample')

Load a artificial random field, generated from a Gaussian covariance function, with a 2x larger range in x-axis direction:

ac, av = skg.data.aniso(N=300, seed=42).get('sample')

Load the TERENO soil temperature data from Fersch et al. (2020):

with open('./data/tereno_fendt/tereno.json', 'r') as js:
    data_obj = json.load(js)

coords = np.array(data_obj['coordinates'])
vals = np.array(data_obj['values'])
print(data_obj['description'])

Out:

Data derived from Fersch et al. (2020) https://doi.org/10.5194/essd-2020-48. Published under CC BY 4.0.
 It is  From the WSN product, the T_a in 20cm depth is extracted

Estimate a variogram, with a few more lag classes, as there are enough observation points available.

V = skg.Variogram(c,v, n_lags=25)
print(V)

Out:

spherical Variogram
-------------------
Estimator:         matheron

Effective Range:   303.59

Sill:              1465.45

Nugget:            0.00

Estimate the directional variogram with a few more lag classes and an azimuth of 90°. The tolerance is set rather low to illustrate the graphs better (fewer point connections.):

DV = skg.DirectionalVariogram(ac, av, n_lags=20, azimuth=40., tolerance=15.0)
print(DV)

Out:

spherical Variogram
-------------------
Estimator:         matheron

Effective Range:   324.51

Sill:              1407.98

Nugget:            0.00

Estimate the spatio-temporal variogram with a product-sum model. Only every 6th hour is taken into account to decrease the memory footprint. If you use the full dataset, you need ^120 GiB RAM. The marginal variograms are kept as they are.

STV = skg.SpaceTimeVariogram(coords, vals[:,::6], x_lags=20, t_lags=20, model='product-sum')
print(STV)

Out:

<skgstat.SpaceTimeVariogram.SpaceTimeVariogram object at 0x7fbbaf3195d0>

4.2 Backend

You can switch to plotly as a plotting backend by calling the plotting.backend function and passing the name of the backend. Note that plotly is only a soft dependency and will not automatically be installed along with SciKit-GStat. You can install it like:

pip install plotly

Note that in a Jupyter environment you might want to use the plotly.offline environment to embed the needed Javascript into the notebook. In these cases you have to catch the Figure object and use the iplot function from the offline submodule.

4.3 Variogram

4.3.1 Variogram.plot

The Variogram.plot is the main plotting function in SciKit-GStat. Before you use the variogram for further geostatistical methods, like kriging, or further analysis, make sure, that a suitable model was found and fitted to the experimental data. Further, you have to make sure that the statistical foundation of this estimation is sound, the lag classes are well designed and backed by a suiatable amount of data. Otherwise, any other geostatistical analysis or method will have to fail, no matter how nice the results might look like.

# from skgstat.plotting import backend
backend('plotly')

Plotly

fig = V.plot(show=False)
fig

A useful argument for plot is the ax, this takes a matplotlib.AxesSubplot for the 'matplotlib' backend and a plotly.Figure for the 'plotly' backend. You need to supply the correct amount of subplots (two). For convenience, the histogram in the upper subplot can be disabled.

fig = make_subplots(rows=1, cols=1)
fig.update_layout(
    width=800,
    height=200,
    template='seaborn',
    showlegend=False,
    margin=dict(l=0, r=0, b=0, t=0)
)

V.plot(axes=fig, hist=False, show=False)
fig

The Variogram.plot functions is customizable and takes a lot of arguments. However, the same interface is used as for the matplotlib version of that function. Many matplotlib arguments are mapped to the corresponding plotly arguments. Beyond that, you can either try common plotly arguments, or update the figure afterwards:

fig = V.plot(show=False)

fig.update_layout(
    legend=dict(x=0.05, y=1.1, xanchor='left', yanchor='top', orientation='h'),
    template='plotly_dark',
    annotations=[dict(
        text="AWESOME",
        xref="paper",
        yref="paper",
        x=0.5,
        y=0.5,
        font=dict(color="white", size=100),
        textangle=-30,
        opacity=.3
    )]
)
fig

Matplotlib

backend('matplotlib')

fig = V.plot()

With matplotlib, you can set any matplotlib.AxesSubplot as axes to plot on other figures. You can send two axes, for the variogram and the histogram, or only one and disable the histogram plotting.

fig, ax = plt.subplots(2,2, figsize=(8, 4))

fig = V.plot(axes=ax.flatten()[1], hist=False)

4.3.2 Variogram.scattergram

You can plot a scattergram of all point pairs formed by the class. The pairs can be grouped by the lag classes, they were formed in. This way you can analyze how the two values of the point pait (head and tail) scatter and if this follows a pattern (i.e. anisotropy). It is recommended to use the 'plotly' backend, as you can click on the legend entries to hide a specific class, or double-click to show only the selected lag class. This makes it much easier to inspect the classes.

Plotly

backend('plotly')
fig = V.scattergram(show=False)
fig

It is, however possible to re-create the plot that was used up to SciKit-GStat version 0.3.0 with only one color. This is still the default for the 'matplotlib' backend.

fig = V.scattergram(single_color=True, show=False)
fig

Matplotlib

backend(‘matplotlib’)

fig = V.scattergram()

Out:

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4.3.3 Variogram.location_trend

Another useful helper plot is the location_trend. This will plot the observation values related to their coordinate position, for each coordinate dimension separatedly. With the 'plotly' backend, each dimension will appear as a coloured group in a single plot. By double-clicking the legend, you can inspect each group separately.

The 'plotly' backend will automatically switch the used plot type from a ordinary scatter-plot to a WebGL backed scatter-plot, if there are more than 5000 observations. This will add some startup-overhead for the plot to appear, but the interactivity actions (like pan, zoom) are speed up by magnitudes.

Plotly

backend('plotly')
fig = V.location_trend(show=False)
fig

Since version 0.3.5 the location_trend <skgstat.Variogram.location_trend>`# function accepts a ``add_trend_line`() parameter, that defaults to False. If set to true, the class will fit linear models to each of the point clouds and output a trend line. It will also calculate the R², which you can use to either accept the input data as trend free or not (a high R² indicates a linear trend and hence you should decline using the input data).

fig = V.location_trend(add_trend_line=True, show=False)
fig

# Matplotlib
# """"""""""
#
# There is a difference between the ``'matplotlib'`` and ``'plotly'`` backend in
# this plotting function. As Plotly utilizes the legend by default to show and
# hide traces on the plot, the user can conveniently switch between the
# coordinate dimensions.
# In Matplotlib, the figures are not interactive by default and therefore
# SciKit-GStat will create one subplot for each coordinate dimension.
backend('matplotlib')
fig = V.location_trend()

4.3.4 distance_difference plot

The final utility plot presented here is a scatter-plot that relates all pairwise-differences in value to the spatial distance of the respective point pairs. This can already be considered to be a variogram. For convenience, the plotting method will mark all upper lag class edges in the plot. This can already give you an idea, if the number of lag classes is chosen wisely, or if you need to adjust. To estimate valid, expressive variograms, this is maybe the most important preparation step. If your lag classes do not represent your data well, you will never find a useful variogram.

Plotly

backend('plotly')
fig = V.distance_difference_plot(show=False)
fig

You might also consider to adapt the maximum lag distance using this plot, to exclude distances that are not well backed by data. Alternatively, the binning method can be changed. Or both

Vcopy = V.clone()
Vcopy.bin_func = 'uniform'

fig = Vcopy.distance_difference_plot(show=False)
fig


# Matplotlib
# """"""""""
backend('matplotlib')
fig = V.distance_difference_plot()

4.4 Directional Variogram

4.4.1 pair_field

The DirectionalVariogram class is inheriting from Variogram. Therefore all plotting method shown above are available for directional variograms, as well. Additionally, there is one more plotting method, DirectionalVariogram.pair_field. This function will plot all coordinate locations and draw a line between all point pairs, that were not masked by the directional mask array and will, thus, be used for variogram estimation. By default, the method will draw all lines for all point pairs and you will see nothing on the plot. But there is also the possibility to draw these lines only for a subset of the coordinate locations.

# Matplotlib
# """"""""""
backend('matplotlib')
fig = DV.pair_field()

Obviously, one can see the azimuth (40°) and narrow tolerance (15°) settings in the cone-like shapes of the connection lines, but the whole plot is not really instructive or helpful like this. Using the points keyword, you can show the lines only for a given set of coordinate locations. You have to pass a list of coordinate indices. With add_points=True, the seleceted points will be highlighted in red.

fig = DV.pair_field(points=[0, 42, 104, 242], add_points=True)

Plotly

Note: It is not recommended to plot the full pair_field with all points using plotly. Due to the implementation, that makes the plot really, really slow for rendering.

backend('plotly')
fig = DV.pair_field(points=[0,42, 104, 242], add_points=True, show=False)
fig

4.5 ST Variogram

The SpaceTimeVariogram does not inherit from the Variogram has two properties, SpaceTimeVariogram.XMarginal and SpaceTimeVariogram.TMarginal, which are both instances of Variogram for the spatial and temporal marginal variogram. These instances in turn, have all plotting methods available, in addition to the plotting methods of SpaceTimeVariogram itself.

# 4.5.1 `plot(kind='scatter') <skgstat.SpaceTimeVariogram.plot>`
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# The scatter plot can be used to inspect the experimental variogram data on a
# spatial and temporal axis, with the fitted spatio-temporal model fitted to the data.
#
# Plotly
# """"""
backend('plotly')
fig = STV.plot(kind='scatter', show=False)
fig

The method can also remove the model from the plot. This can be helpful in case the experimental data should be analyzed. Then, the model plane might be disturbing.

fig = STV.plot(kind='scatter', no_model=True, show=False)
fig

And finally, the experimental point data can be connected to a surface grid, to emphasize an apparent structure more easily in a 3D plot. This can be done by switching to kind='surf'.

fig = STV.plot(kind='surf', show=False)
fig


# Matplotlib
# """"""""""
backend('matplotlib')
fig = STV.plot(kind='surf')

4.5.2 contour

3D plots are great for data exploration, especially if they are interactive. For publications, 3D plots are not that helpful. Additionally, it can be quite tricky sometimes to find a good angle to focus on the main message of a 3D plot. Hence, there are more plotting modes. They can either be used by setting kind='contour' or kind='contourf'. Alternatively, these two plotting types also have their own method. In both cases, the experimental semi-variance is plotted on a two dimensional plane. The spatial dimension can be found on the x-axis and the temporal dimension on the y-axis. The semi-variance itself is shown as a contour plot, that can either only plot the lines ('contour') or filled areas for each contour ('contourf').

Plotly

backend('plotly')
fig = STV.contour(show=False)
fig

fig = STV.contourf(show=False)
fig

Matplotlib

The matplotlib versions of the contour plots are not that sophisticated, but the returned figure can be adjusted to your needs.

backend('matplotlib')
fig = STV.plot(kind='contour')

fig = STV.plot(kind='contourf')

4.5.3 :func`marginals <skgstat.SpaceTimeVariogram.marginals>`

A very important step for the estimation of spatio-temporal variogram models, is the estimation of marginal models. While the marginal models are implemented as Variogram instances and can be changed and plotted like any other Variogram instance, it can come very handy to plot the marginal models side-by-side.

This can be done with the :func`marginals <skgstat.SpaceTimeVariogram.marginals>` method.

backend('plotly')
fig = STV.marginals(show=False)
fig

backend('matplotlib')
fig = STV.marginals()

Additionally, the separated spatial and temporal models can be plotted into each sub-plot:

fig = STV.marginals(include_model=True)

backend('plotly')
fig = STV.marginals(include_model=True, show=False)
fig

previous

2 - Semi-variance Estimators

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7. Maximum Likelihood fit