.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "tutorials/weights.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_tutorials_weights.py: Using Weights ============= One of the advantages of using a Green's functions approach to interpolation is that we can easily weight the data to give each point more or less influence over the results. This is a good way to not let data points with large uncertainties bias the interpolation or the data decimation. .. GENERATED FROM PYTHON SOURCE LINES 16-27 .. code-block:: default import cartopy.crs as ccrs import matplotlib.pyplot as plt import numpy as np import pyproj # The weights vary a lot so it's better to plot them using a logarithmic color # scale from matplotlib.colors import LogNorm import verde as vd .. GENERATED FROM PYTHON SOURCE LINES 28-31 We'll use some sample GPS vertical ground velocity which has some variable uncertainties associated with each data point. The data are loaded as a pandas.DataFrame: .. GENERATED FROM PYTHON SOURCE LINES 31-34 .. code-block:: default data = vd.datasets.fetch_california_gps() print(data.head()) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none latitude longitude height ... std_north std_east std_up 0 34.116409 242.906804 762.11978 ... 0.0002 0.00037 0.00053 1 34.116409 242.906804 762.10883 ... 0.0002 0.00037 0.00053 2 34.116409 242.906805 762.09364 ... 0.0002 0.00037 0.00053 3 34.116409 242.906805 762.09073 ... 0.0002 0.00037 0.00053 4 34.116409 242.906805 762.07699 ... 0.0002 0.00037 0.00053 [5 rows x 9 columns] .. GENERATED FROM PYTHON SOURCE LINES 35-38 Let's plot our data using Cartopy to see what the vertical velocities and their uncertainties look like. We'll make a function for this so we can reuse it later on. .. GENERATED FROM PYTHON SOURCE LINES 38-79 .. code-block:: default def plot_data(coordinates, velocity, weights, title_data, title_weights): "Make two maps of our data, one with the data and one with the weights" fig, axes = plt.subplots( 1, 2, figsize=(9.5, 7), subplot_kw=dict(projection=ccrs.Mercator()) ) crs = ccrs.PlateCarree() ax = axes[0] ax.set_title(title_data) maxabs = vd.maxabs(velocity) pc = ax.scatter( *coordinates, c=velocity, s=30, cmap="seismic", vmin=-maxabs, vmax=maxabs, transform=crs, ) plt.colorbar(pc, ax=ax, orientation="horizontal", pad=0.05).set_label("m/yr") vd.datasets.setup_california_gps_map(ax) ax = axes[1] ax.set_title(title_weights) pc = ax.scatter( *coordinates, c=weights, s=30, cmap="magma", transform=crs, norm=LogNorm() ) plt.colorbar(pc, ax=ax, orientation="horizontal", pad=0.05) vd.datasets.setup_california_gps_map(ax) plt.show() # Plot the data and the uncertainties plot_data( (data.longitude, data.latitude), data.velocity_up, data.std_up, "Vertical GPS velocity", "Uncertainty (m/yr)", ) .. image-sg:: /tutorials/images/sphx_glr_weights_001.png :alt: Vertical GPS velocity, Uncertainty (m/yr) :srcset: /tutorials/images/sphx_glr_weights_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 80-93 Weights in data decimation -------------------------- :class:`~verde.BlockReduce` can't output weights for each data point because it doesn't know which reduction operation it's using. If you want to do a weighted interpolation, like :class:`verde.Spline`, :class:`~verde.BlockReduce` won't propagate the weights to the interpolation function. If your data are relatively smooth, you can use :class:`verde.BlockMean` instead to decimated data and produce weights. It can calculate different kinds of weights, depending on configuration options and what you give it as input. Let's explore all of the possibilities. .. GENERATED FROM PYTHON SOURCE LINES 93-96 .. code-block:: default mean = vd.BlockMean(spacing=15 / 60) print(mean) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none BlockMean(spacing=0.25) .. GENERATED FROM PYTHON SOURCE LINES 97-118 Option 1: No input weights ++++++++++++++++++++++++++ In this case, we'll get a standard mean and the output weights will be 1 over the variance of the data in each block: .. math:: \bar{d} = \dfrac{\sum\limits_{i=1}^N d_i}{N} \: , \qquad \sigma^2 = \dfrac{\sum\limits_{i=1}^N (d_i - \bar{d})^2}{N} \: , \qquad w = \dfrac{1}{\sigma^2} in which :math:`N` is the number of data points in the block, :math:`d_i` are the data values in the block, and the output values for the block are the mean data :math:`\bar{d}` and the weight :math:`w`. Notice that data points that are more uncertain don't necessarily have smaller weights. Instead, the blocks that contain data with sharper variations end up having smaller weights, like the data points in the south. .. GENERATED FROM PYTHON SOURCE LINES 118-130 .. code-block:: default coordinates, velocity, weights = mean.filter( coordinates=(data.longitude, data.latitude), data=data.velocity_up ) plot_data( coordinates, velocity, weights, "Mean vertical GPS velocity", "Weights based on data variance", ) .. image-sg:: /tutorials/images/sphx_glr_weights_002.png :alt: Mean vertical GPS velocity, Weights based on data variance :srcset: /tutorials/images/sphx_glr_weights_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 131-154 Option 2: Input weights are not related to the uncertainty of the data ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ This is the case when data weights are chosen by the user, not based on the measurement uncertainty. For example, when you need to give less importance to a portion of the data and no uncertainties are available. The mean will be weighted and the output weights will be 1 over the weighted variance of the data in each block: .. math:: \bar{d}^* = \dfrac{\sum\limits_{i=1}^N w_i d_i}{\sum\limits_{i=1}^N w_i} \: , \qquad \sigma^2_w = \dfrac{\sum\limits_{i=1}^N w_i(d_i - \bar{d}*)^2}{ \sum\limits_{i=1}^N w_i} \: , \qquad w = \dfrac{1}{\sigma^2_w} in which :math:`w_i` are the input weights in the block. The output will be similar to the one above but points with larger initial weights will have a smaller influence on the mean and also on the output weights. .. GENERATED FROM PYTHON SOURCE LINES 154-173 .. code-block:: default # We'll use 1 over the squared data uncertainty as our input weights. data["weights"] = 1 / data.std_up**2 # By default, BlockMean assumes that weights are not related to uncertainties coordinates, velocity, weights = mean.filter( coordinates=(data.longitude, data.latitude), data=data.velocity_up, weights=data.weights, ) plot_data( coordinates, velocity, weights, "Weighted mean vertical GPS velocity", "Weights based on weighted data variance", ) .. image-sg:: /tutorials/images/sphx_glr_weights_003.png :alt: Weighted mean vertical GPS velocity, Weights based on weighted data variance :srcset: /tutorials/images/sphx_glr_weights_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 174-200 Option 3: Input weights are 1 over the data uncertainty squared +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ If input weights are 1 over the data uncertainty squared, we can use uncertainty propagation to calculate the uncertainty of the weighted mean and use it to define our output weights. Use option ``uncertainty=True`` to tell :class:`~verde.BlockMean` to calculate weights based on the propagated uncertainty of the data. The output weights will be 1 over the propagated uncertainty squared. In this case, the **input weights must not be normalized**. This is preferable if you know the uncertainty of the data. .. math:: w_i = \dfrac{1}{\sigma_i^2} \: , \qquad \sigma_{\bar{d}^*}^2 = \dfrac{1}{\sum\limits_{i=1}^N w_i} \: , \qquad w = \dfrac{1}{\sigma_{\bar{d}^*}^2} in which :math:`\sigma_i` are the input data uncertainties in the block and :math:`\sigma_{\bar{d}^*}` is the propagated uncertainty of the weighted mean in the block. Notice that in this case the output weights reflect the input data uncertainties. Less weight is given to the data points that had larger uncertainties from the start. .. GENERATED FROM PYTHON SOURCE LINES 200-218 .. code-block:: default # Configure BlockMean to assume that the input weights are 1/uncertainty**2 mean = vd.BlockMean(spacing=15 / 60, uncertainty=True) coordinates, velocity, weights = mean.filter( coordinates=(data.longitude, data.latitude), data=data.velocity_up, weights=data.weights, ) plot_data( coordinates, velocity, weights, "Weighted mean vertical GPS velocity", "Weights based on data uncertainty", ) .. image-sg:: /tutorials/images/sphx_glr_weights_004.png :alt: Weighted mean vertical GPS velocity, Weights based on data uncertainty :srcset: /tutorials/images/sphx_glr_weights_004.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 219-235 .. note:: Output weights are always normalized to the ]0, 1] range. See :func:`verde.variance_to_weights`. Interpolation with weights -------------------------- The Green's functions based interpolation classes in Verde, like :class:`~verde.Spline`, can take input weights if you want to give less importance to some data points. In our case, the points with larger uncertainties shouldn't have the same influence in our gridded solution as the points with lower uncertainties. Let's setup a projection to grid our geographic data using the Cartesian spline gridder. .. GENERATED FROM PYTHON SOURCE LINES 236-243 .. code-block:: default projection = pyproj.Proj(proj="merc", lat_ts=data.latitude.mean()) proj_coords = projection(data.longitude.values, data.latitude.values) region = vd.get_region(coordinates) spacing = 5 / 60 .. GENERATED FROM PYTHON SOURCE LINES 244-250 Now we can grid our data using a weighted spline. We'll use the block mean results with uncertainty based weights. Note that the weighted spline solution will only work on a non-exact interpolation. So we'll need to use some damping regularization or not use the data locations for the point forces. Here, we'll apply a bit of damping. .. GENERATED FROM PYTHON SOURCE LINES 250-267 .. code-block:: default spline = vd.Chain( [ # Convert the spacing to meters because Spline is a Cartesian gridder ("mean", vd.BlockMean(spacing=spacing * 111e3, uncertainty=True)), ("spline", vd.Spline(damping=1e-10)), ] ).fit(proj_coords, data.velocity_up, data.weights) grid = spline.grid( region=region, spacing=spacing, projection=projection, dims=["latitude", "longitude"], data_names="velocity", ) # Avoid showing interpolation outside of the convex hull of the data points. grid = vd.convexhull_mask(coordinates, grid=grid, projection=projection) .. GENERATED FROM PYTHON SOURCE LINES 268-269 Calculate an unweighted spline as well for comparison. .. GENERATED FROM PYTHON SOURCE LINES 269-286 .. code-block:: default spline_unweighted = vd.Chain( [ ("mean", vd.BlockReduce(np.mean, spacing=spacing * 111e3)), ("spline", vd.Spline()), ] ).fit(proj_coords, data.velocity_up) grid_unweighted = spline_unweighted.grid( region=region, spacing=spacing, projection=projection, dims=["latitude", "longitude"], data_names="velocity", ) grid_unweighted = vd.convexhull_mask( coordinates, grid=grid_unweighted, projection=projection ) .. GENERATED FROM PYTHON SOURCE LINES 287-288 Finally, plot the weighted and unweighted grids side by side. .. GENERATED FROM PYTHON SOURCE LINES 288-324 .. code-block:: default fig, axes = plt.subplots( 1, 2, figsize=(9.5, 7), subplot_kw=dict(projection=ccrs.Mercator()) ) crs = ccrs.PlateCarree() ax = axes[0] ax.set_title("Spline interpolation with weights") maxabs = vd.maxabs(data.velocity_up) pc = grid.velocity.plot.pcolormesh( ax=ax, cmap="seismic", vmin=-maxabs, vmax=maxabs, transform=crs, add_colorbar=False, add_labels=False, ) plt.colorbar(pc, ax=ax, orientation="horizontal", pad=0.05).set_label("m/yr") ax.plot(data.longitude, data.latitude, ".k", markersize=0.1, transform=crs) ax.coastlines() vd.datasets.setup_california_gps_map(ax) ax = axes[1] ax.set_title("Spline interpolation without weights") pc = grid_unweighted.velocity.plot.pcolormesh( ax=ax, cmap="seismic", vmin=-maxabs, vmax=maxabs, transform=crs, add_colorbar=False, add_labels=False, ) plt.colorbar(pc, ax=ax, orientation="horizontal", pad=0.05).set_label("m/yr") ax.plot(data.longitude, data.latitude, ".k", markersize=0.1, transform=crs) ax.coastlines() vd.datasets.setup_california_gps_map(ax) plt.show() .. image-sg:: /tutorials/images/sphx_glr_weights_005.png :alt: Spline interpolation with weights, Spline interpolation without weights :srcset: /tutorials/images/sphx_glr_weights_005.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 4.052 seconds) .. _sphx_glr_download_tutorials_weights.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: weights.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: weights.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_