Smooth data fitting in N dimensions

## Project description

Given experimental data, it is often desirable to produce a function whose values match the data to some degree. This package implements a robust approach to data fitting based on the minimization problem (A similar idea is used in for data smoothing in signal processing; see, e.g., section 8.3 in this document.)

Unlike polynomial regression or Gauss-Newton, smoothfit makes no assumptions about the function other than that it is smooth.

The generality of the approach makes it suitable for function whose domain is multidimensional, too.

### Pics or it didn't happen

#### Runge's example ```import matplotlib.pyplot as plt
import numpy
import smoothfit

a = -1.5
b = +1.5

# plot original function
x = numpy.linspace(a, b, 201)
plt.plot(x, 1 / (1 + 25 * x ** 2), "-", color="0.8", label="1 / (1 + 25 * x**2)")

# 21 sample points
x0 = numpy.linspace(-1.0, 1.0, 21)
y0 = 1 / (1 + 25 * x0 ** 2)
plt.plot(x0, y0, "xk")

u = smoothfit.fit1d(x0, y0, a, b, 1000, degree=1, lmbda=1.0e-6)
x = numpy.linspace(a, b, 201)
vals = [u(xx) for xx in x]
plt.plot(x, vals, "-", label="smooth fit")

plt.ylim(-0.1)
plt.grid()
plt.show()
```

Runge's example function is a tough nut for classical polynomial regression.

If there is no noise in the input data, the parameter `lmbda` can be chosen quite small such that all data points are approximated well. Note that there are no oscillations in the output function `u`.

#### Runge's example with noise   `lmbda = 0.001` `lmbda = 0.05` `lmbda = 0.2`
```import matplotlib.pyplot as plt
import numpy
import smoothfit

a = -1.5
b = +1.5

# plot original function
x = numpy.linspace(a, b, 201)
plt.plot(x, 1 / (1 + 25 * x ** 2), "-", color="0.8", label="1 / (1 + 25 * x**2)")

# 21 sample points
numpy.random.seed(0)
n = 51
x0 = numpy.linspace(-1.0, 1.0, n)
y0 = 1 / (1 + 25 * x0 ** 2)
y0 += 1.0e-1 * (2 * numpy.random.rand(n) - 1)
plt.plot(x0, y0, "xk")

lmbda = 5.0e-2
u = smoothfit.fit1d(x0, y0, a, b, 1000, degree=1, lmbda=lmbda)
x = numpy.linspace(a, b, 201)
vals = [u(xx) for xx in x]
plt.plot(x, vals, "-", label="smooth fit")

plt.grid()
plt.show()
```

If the data is noisy, `lmbda` needs to be chosen more carefully. If too small, the approximation tries to resolve all data points, resulting in many small oscillations. If it's chosen too large, no details are resolved, not even those of the underlying data.

#### Few samples ```import numpy
import smoothfit

x0 = numpy.array([0.038, 0.194, 0.425, 0.626, 1.253, 2.500, 3.740])
y0 = numpy.array([0.050, 0.127, 0.094, 0.2122, 0.2729, 0.2665, 0.3317])
u = smoothfit.fit1d(x0, y0, 0, 4, 1000, degree=1, lmbda=1.0)
```

Some noisy example data taken from Wikipedia.

#### A two-dimensional example ```import meshzoo
import numpy
import smoothfit

n = 200
numpy.random.seed(123)
x0 = numpy.random.rand(n, 2) - 0.5
y0 = numpy.cos(numpy.pi * numpy.sqrt(x0.T ** 2 + x0.T ** 2))

# create a triangle mesh for the square
points, cells = meshzoo.rectangle(-1.0, 1.0, -1.0, 1.0, 32, 32)

u = smoothfit.fit2d(x0, y0, points, cells, lmbda=1.0e-4, solver="dense-direct")

# Write the function to a file
from dolfin import XDMFFile
xdmf = XDMFFile("temp.xdmf")
xdmf.write(u)
```

This example approximates a function from R2 to R (without noise in the samples). Note that the absence of noise the data allows us to pick a rather small `lmbda` such that all sample points are approximated well.

### Comparison with other approaches

#### Polynomial fitting/regression The classical approach to data fitting is polynomial regression. Polynomials are chosen because they are very simple, can be evaluated quickly, and can be made to fit any function very closely.

There are, however, some fundamental problems:

This above plot highlights the problem with oscillations.

#### Fourier smoothing One approach to data fitting with smoothing is to create a function with all data points, and simply cut off the high frequencies after Fourier transformation.

This approach is fast, but only works for evenly spaced samples.

```import matplotlib.pyplot as plt
import numpy

numpy.random.seed(0)

# original function
x0 = numpy.linspace(-1.0, 1.0, 1000)
y0 = 1 / (1 + 25 * x0 ** 2)
plt.plot(x0, y0, color="k", alpha=0.2)

# create sample points
n = 51
x1 = numpy.linspace(-1.0, 1.0, n)  # only works if samples are evenly spaced
y1 = 1 / (1 + 25 * x1 ** 2) + 1.0e-1 * (2 * numpy.random.rand(x1.shape) - 1)
plt.plot(x1, y1, "xk")

# Cut off the high frequencies in the transformed space and transform back
X = numpy.fft.rfft(y1)
X[5:] = 0.0
y2 = numpy.fft.irfft(X, n)
#
plt.plot(x1, y2, "-", label="5 lowest frequencies")

plt.grid()
plt.show()
```

### Testing

To run the smoothfit unit tests, check out this repository and type

``````pytest
``````

smoothfit is published under the GPLv3+ license.

## Project details

This version 0.2.0 0.1.1 0.1.0