Whittaker-Eilers Smoothing and Interpolation

The Whittaker-Eilers smoother is the perfect smoother. It offers extremely quick, efficient smoothing with built-in interpolation via weights on each measurement. This package provides a sparse-matrix implementation for additional speed and memory efficiency and can handle both equally and unequally spaced measurements. This package was originally written in Rust so additional examples, tests, and benchmarks are also available in addition to it being super speedy. The API is almost identical.

pip install whittaker-eilers

Usage

To start smoothing and interpolating data, create a reusable WhittakerSmoother class. You'll only need to recreate this class if the length or sampling rate of your data changes.

Equally spaced data

This is the fastest smoothing option. It smooths equally spaced y measurements using two tunable parameters, lambda (2e4) and the smoother order (2). The larger the lambda, the smoother the data.

from whittaker_eilers import WhittakerSmoother

data_to_smooth = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]

whittaker_smoother = WhittakerSmoother(lmbda=2e4, order=2, data_length = len(data_to_smooth))

smoothed_data = whittaker_smoother.smooth(data_to_smooth)

print("Smoothed data: {}".format(smoothed_data))

Non-equally spaced data

If you wish to smooth unequally spaced data, you need to provide an x_input with the sample times/positions.

from whittaker_eilers import WhittakerSmoother

x_input = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]
data_to_smooth = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]

whittaker_smoother = WhittakerSmoother(
    lmbda=2e4, order=2, data_length=len(data_to_smooth), x_input=x_input
)

smoothed_data = whittaker_smoother.smooth(data_to_smooth)

print("Smoothed non-equally spaced data: {}".format(smoothed_data))

Weighted data & Interpolation

Each measurement can then be weighted to trust some measurements more than others. Setting weights to 0 for measurements will lead to interpolation.

from whittaker_eilers import WhittakerSmoother

x_input = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]
data_to_smooth = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]
weights = [1.0] * len(x_input)
weights[5] = 0.0

whittaker_smoother = WhittakerSmoother(
    lmbda=2e4,
    order=2,
    data_length=len(data_to_smooth),
    x_input=x_input,
    weights=weights,
)

smoothed_data = whittaker_smoother.smooth(data_to_smooth)

print("Smoothed and interpolated weighted data: {}".format(smoothed_data))

Smoothing with cross validation

With this package, you can also calculate the cross validation error alongside the smoothed series. This shouldn't really be used in production where speed is necessary though!

from whittaker_eilers import WhittakerSmoother

x_input = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]
data_to_smooth = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]

whittaker_smoother = WhittakerSmoother(
    lmbda=2e4, order=2, data_length=len(data_to_smooth), x_input=x_input
)

smoothed_data_with_cross_validation = whittaker_smoother.smooth_and_cross_validate(
    data_to_smooth
)

print(
    "Error :{}".format(smoothed_data_with_cross_validation.get_cross_validation_error())
)
print("Smoothed :{}".format(smoothed_data_with_cross_validation.get_smoothed()))

Automatic smoothing

Smoothing data requires a choice of Lambda. This can be done using visual inspection or by finding the lambda which results in the lowest cross validation error. The smooth_optimal function runs the smoother for a variety of lambdas and returns the results with the ability to retrieve the optimal one.

from whittaker_eilers import WhittakerSmoother

x_input = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]
data_to_smooth = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]

whittaker_smoother = WhittakerSmoother(
    lmbda=2e4, order=2, data_length=len(data_to_smooth), x_input=x_input
)

optimal_smooth = whittaker_smoother.smooth_optimal(data_to_smooth)

print("Optimal lambda: {}".format(optimal_smooth.get_optimal().get_lambda()))

print("Optimally smoothed data: {}".format(optimal_smooth.get_optimal().get_smoothed()))

Parallel Smoothing

With this package you're able to smooth multiple data series in parallel. While y, the data to be smoothed can vary, x, weights, and the other smoothing parameters cannot be varied across the different data series.

y = [[1.2, 4.5, 6.5, 3.2, 1.23, 23.23, 231.23]] * 10

whittaker_smoother = WhittakerSmoother(lmbda=20, order=2, data_length=len(y[0]))

result = whittaker_smoother.smooth_parallel(y)

print("Smoothed data: {}".format(result))

You can use these methods in combination with each other for instance, interpolating measurements without providing an x input. For more advanced examples of usage take a look at the examples, tests, and benches in the Github repository. Here's an image of some smoothed data from an example:

Other methods

from whittaker_eilers import WhittakerSmoother
x_input = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]
data_to_smooth = [1.1, 1.9, 3.1, 3.91, 5.0, 6.02, 7.01, 7.7, 9.0, 10.0]
weights = [1.0] * len(x_input)
weights[5] = 0.0

whittaker_smoother = WhittakerSmoother(
    lmbda=2e4,
    order=2,
    data_length=len(data_to_smooth),
    x_input=x_input,
    weights=weights,
)

whittaker_smoother.get_order()
whittaker_smoother.get_lambda()
whittaker_smoother.get_data_length()
whittaker_smoother.update_weights([0.5] * len(x_input))
whittaker_smoother.update_order(3)
whittaker_smoother.update_lambda(4321.0)

Further Reading

If you'd like to see a more detailed run through of the library, check out this Medium post. Within it, I run through examples and benchmarks against other smoothing methods.

Future Features

• Scatter plot smoothing
• Generic typing

References

The algorithm implemented here mirrors a 2003 implementation by Paul H. C. Eilers in Matlab. I've included scripts and data from the original paper in the tests for this package. The original paper and code can be found here:

A Perfect Smoother Paul H. C. Eilers Analytical Chemistry 2003 75 (14), 3631-3636 DOI: 10.1021/ac034173t