Natural Splines

Natural Splines#

Splines are flexible functions that can be used to fit rating curves. In fact, the multi-segment power law is a form of linear spline, but other types of spline can be used as well. One alternative is the natural spline. They have the advantage of being faster to fit, but their form is less constrained than the segmented power law. As a result, natural splines may produce strange results, particularly with small datasets.

%load_ext autoreload
%autoreload 2

from ratingcurve import data
from ratingcurve.ratings import SplineRating

data.list()

['chalk artificial',
 'co channel',
 'green channel',
 'provo natural',
 '3-segment simulated',
 'mahurangi artificial',
 'nordura',
 'skajalfandafljot',
 'isere']

Green River#

In practice splines can work quite well, particularly for simpler ratings. Here is an example showing a natural spline fit to the Green River dataset.

df = data.load('green channel')

spline = SplineRating(df=8)

# converges much faster than the power law
trace = spline.fit(q=df['q'],
                   h=df['stage'],
                   q_sigma=df['q_sigma'],
                   n=70_000)

spline.plot()

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Finished [100%]: Average Loss = -39.236





Sampling: [model_q, sigma, w]





Sampling: [model_q]





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Sampling: [model_q]





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../_images/edf986f254dd4698eb0b281991ad90b21fa16f37a8d37b3ae5ffe4515a4ec13f.png









Simulated Rating#






sim_df = data.load('3-segment simulated')

# subsample the simulated rating curve
n_sample = 30
df = sim_df.sample(n_sample, random_state=12345)

ax = sim_df.plot(x='q', y='stage', color='gray', ls='-', legend=False)
df.plot.scatter(x='q', y='stage', marker='o', color='blue', ax=ax)
ax.set_xlabel("Discharge (cfs)")
ax.set_ylabel("Stage (ft)")









Text(0, 0.5, 'Stage (ft)')




../_images/fad8188929d4cdb129aad43b5b8e13991335d7ce5c774d95ccaf6404e756a6bd.png









spline = SplineRating(df=10)













trace = spline.fit(q=df['q'],
                   h=df['stage'],
                   n=100_000)

spline.plot()









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Finished [100%]: Average Loss = -2.7793





Sampling: [model_q, sigma, w]





Sampling: [model_q]





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Sampling: [model_q]





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../_images/91ae2f627abc24b4c10edeef2d4e2ce28f0ec772da5ccc70628cfaa9298b3c9e.png







Excercise#


Splines can give unexpectedly poor results.
Experiment with the following:


    

  1. Try changing the random state: sim_df.sample(n=30, random_state=771)
    2. 

  2. Rerun with n=200. The fit has improved, but the uncertainty is unreasonably high.
    3. 

  3. Rerun with more degrees of freedon (df=30) in SplineRating(), and observe how the uncertainty changes.
    4. 







%load_ext watermark
%watermark -n -u -v -iv -w -p pytensor,xarray









Last updated: Tue, 01 Sep 2026

Python implementation: CPython
Python version       : 3.12.14
IPython version      : 9.17.1

pytensor: 3.3.0
xarray  : 2026.7.0

ratingcurve: 0.1.dev1+ga5a0fe6ba

Watermark: 2.6.0