Model Selection

In Evaluating Performance, we saw how to check the performance of an interpolator using cross-validation. We found that the default parameters for verde.Spline are not good for predicting our sample air temperature data. Now, let’s see how we can tune the Spline to improve the cross-validation performance.

Once again, we’ll start by importing the required packages and loading our sample data.

import cartopy.crs as ccrs
import matplotlib.pyplot as plt
import numpy as np
import pyproj

import verde as vd

data = vd.datasets.fetch_texas_wind()

# Use Mercator projection because Spline is a Cartesian gridder
projection = pyproj.Proj(proj="merc", lat_ts=data.latitude.mean())
proj_coords = projection(data.longitude.values, data.latitude.values)

region = vd.get_region((data.longitude, data.latitude))
# The desired grid spacing in degrees
spacing = 15 / 60

Before we begin tuning, let’s reiterate what the results were with the default parameters.

spline_default = vd.Spline()
score_default = np.mean(
    vd.cross_val_score(spline_default, proj_coords, data.air_temperature_c)
)
spline_default.fit(proj_coords, data.air_temperature_c)
print("R² with defaults:", score_default)

/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
R² with defaults: 0.7960368857076603

Tuning

Spline has the damping regularization parameter which smooths the solution and provides a least-squares fit to the data instead of an exact fit at the observation points. This is often desirable to mitigate data errors and provide better results when points are widely spaced. Would changing the default values give us a better score?

We can answer this question by changing the values in our spline and re-evaluating the model score repeatedly for different values of this parameter. Let’s test the following values:

dampings = [None, 1e-4, 1e-3, 1e-2]

Now we can loop over each value and collect the scores for each parameter choice.

spline = vd.Spline()
scores = []
for damping in dampings:
    spline.set_params(damping=damping)
    score = np.mean(vd.cross_val_score(spline, proj_coords, data.air_temperature_c))
    scores.append(score)
print(scores)

/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
[0.7960368857076603, 0.8447749626358183, 0.8382287942980768, 0.8409658539038999]

The largest score will yield the best parameter combination.

best = np.argmax(scores)
print("Best score:", scores[best])
print("Score with defaults:", score_default)
print("Best damping:", dampings[best])

Best score: 0.8447749626358183
Score with defaults: 0.7960368857076603
Best damping: 0.0001

That is a nice improvement over our previous score!

This type of tuning is important and should always be performed when using a new gridder or a new dataset. However, the above implementation requires a lot of coding. Fortunately, Verde provides convenience classes that perform the cross-validation and tuning automatically when fitting a dataset.

Cross-validated gridders

The verde.SplineCV class provides a cross-validated version of verde.Spline. It has almost the same interface but does all of the above automatically when fitting a dataset. The only difference is that you must provide a list of damping values to try instead of only a single value:

spline = vd.SplineCV(dampings=dampings)

Calling fit will run a grid search over all parameter values to find the one that maximizes the cross-validation score.

spline.fit(proj_coords, data.air_temperature_c)

/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/spline.py:245: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  spline = Spline(**params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/spline.py:245: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  spline = Spline(**params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/spline.py:245: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  spline = Spline(**params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/spline.py:245: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  spline = Spline(**params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/spline.py:261: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  self.spline_ = Spline(**parameter_sets[best])

SplineCV(dampings=[None, 0.0001, 0.001, 0.01], mindists=[0])

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The estimated best damping, as well as the cross-validation scores, are stored in class attributes:

print("Highest score:", spline.scores_.max())
print("Best damping:", spline.damping_)

Highest score: 0.8447749626358183
Best damping: 0.0001

The cross-validated gridder can be used like any other gridder (including in verde.Chain and verde.Vector):

grid = spline.grid(
    region=region,
    spacing=spacing,
    projection=projection,
    dims=["latitude", "longitude"],
    data_names="temperature",
)
print(grid)

<xarray.Dataset> Size: 18kB
Dimensions:      (latitude: 43, longitude: 51)
Coordinates:
  * longitude    (longitude) float64 408B -106.4 -106.1 -105.9 ... -94.06 -93.8
  * latitude     (latitude) float64 344B 25.91 26.16 26.41 ... 35.91 36.16 36.41
Data variables:
    temperature  (latitude, longitude) float64 18kB 24.7 24.56 ... 7.542 7.639
Attributes:
    metadata:  Generated by SplineCV(dampings=[None, 0.0001, 0.001, 0.01], mi...

Like verde.cross_val_score, SplineCV can also run the grid search in parallel using Dask by specifying the delayed attribute:

spline = vd.SplineCV(dampings=dampings, delayed=True)

Unlike verde.cross_val_score, calling fit does not result in dask.delayed objects. The full grid search is executed and the optimal parameters are found immediately.

spline.fit(proj_coords, data.air_temperature_c)

print("Best damping:", spline.damping_)

/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/spline.py:245: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  spline = Spline(**params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/sklearn/base.py:125: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  new_object = klass(**new_object_params)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/model_selection.py:784: FutureWarning: The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly.
  score = estimator.score(*test_data)
/usr/share/miniconda/envs/test/lib/python3.12/site-packages/verde/spline.py:261: FutureWarning: The mindist parameter of verde.Spline is no longer required and will be removed in Verde 2.0.0. Use the default value to obtain the future behavior.
  self.spline_ = Spline(**parameter_sets[best])
Best damping: 0.0001

The one caveat is the that the scores_ attribute will be a list of dask.delayed objects instead because the scores are only computed as intermediate values in the scheduled computations.

print("Delayed scores:", spline.scores_)

Delayed scores: [Delayed('mean-0990024f-8a10-405c-aa70-01ea059c0b8f'), Delayed('mean-47c2a56a-24de-4321-9d5c-80f2bb3ff7c2'), Delayed('mean-1b83ec9a-4a9d-4eef-bba8-55c46f28abcc'), Delayed('mean-eb31f18f-696b-41e5-bde4-0422fa20f145')]

Calling dask.compute on the scores will calculate their values but will unfortunately run the entire grid search again. So using delayed=True is not recommended if you need the scores of each parameter combination.

The importance of tuning

To see the difference that tuning has on the results, we can make a grid with the best configuration and see how it compares to the default result.

grid_default = spline_default.grid(
    region=region,
    spacing=spacing,
    projection=projection,
    dims=["latitude", "longitude"],
    data_names="temperature",
)

Let’s plot our grids side-by-side:

mask = vd.distance_mask(
    (data.longitude, data.latitude),
    maxdist=3 * spacing * 111e3,
    coordinates=vd.grid_coordinates(region, spacing=spacing),
    projection=projection,
)

grid = grid.where(mask)
grid_default = grid_default.where(mask)

plt.figure(figsize=(14, 8))
for i, title, grd in zip(range(2), ["Defaults", "Tuned"], [grid_default, grid]):
    ax = plt.subplot(1, 2, i + 1, projection=ccrs.Mercator())
    ax.set_title(title)
    pc = grd.temperature.plot.pcolormesh(
        ax=ax,
        cmap="plasma",
        transform=ccrs.PlateCarree(),
        vmin=data.air_temperature_c.min(),
        vmax=data.air_temperature_c.max(),
        add_colorbar=False,
        add_labels=False,
    )
    plt.colorbar(pc, orientation="horizontal", aspect=50, pad=0.05).set_label("C")
    ax.plot(
        data.longitude, data.latitude, ".k", markersize=1, transform=ccrs.PlateCarree()
    )
    vd.datasets.setup_texas_wind_map(ax)
plt.show()

Notice that, for sparse data like these, smoother models tend to be better predictors. This is a sign that you should probably not trust many of the short wavelength features that we get from the defaults.