5 Best Ways to Get the Least Squares Fit of Laguerre Series to Data in Python

💡 Problem Formulation: In computational analysis and data science, obtaining an optimal fit for a given set of data points is a common challenge. Specifically, fitting data with a Laguerre series polynomial using the least squares method in Python involves finding the coefficient values that minimize the squared error between the polynomial and the data points. This article demonstrates various methods for achieving this in Python with example inputs being discrete data points and the desired output being the optimal set of coefficients for the Laguerre polynomial that represents the data.

Method 1: Using NumPy’s Polynomial Package

This method employs the polynomial module from NumPy, a fundamental package for numerical computing in Python. The module provides a convenient Laguerre class that includes methods for fitting data using least squares. Function numpy.polynomial.Laguerre.fit is used for obtaining the coefficients.

Here’s an example:

import numpy as np
x = np.linspace(0, 10, 50)
y = np.exp(-x) * np.cos(x)
coefs = np.polynomial.Laguerre.fit(x, y, deg=5).convert().coef
print(coefs)

Output:

[ 0.9998472, -0.5004101, 0.11980448, -0.01493033, 0.00101191, -0.00002937]

This code snippet fits a 5th-degree Laguerre polynomial to the synthetic data generated by the damped cosine function. The np.polynomial.Laguerre.fit function is used to compute the approximation and the method convert() turns the series into a polynomial that produces a list of coefficients.

Method 2: Using Scipy’s Optimize Package

The scipy.optimize package offers general optimization tools that can be adapted for curve fitting tasks. The least squares optimization function curve_fit can fit any user-defined function, including Laguerre polynomials, to data by adjusting the function’s parameters to minimize the least squares error.

Here’s an example:

from scipy.optimize import curve_fit
from scipy.special import eval_genlaguerre
import numpy as np

# Generate sample data
x = np.linspace(0, 10, 50)
y = np.exp(-x) * np.cos(x)

# Define the Laguerre polynomial as a fit function
def laguerre_fit(x, *coefs):
    return sum(c * eval_genlaguerre(i, 0, x) for i, c in enumerate(coefs))

# Perform the curve fitting
popt, _ = curve_fit(laguerre_fit, x, y, p0=np.ones(6))

print(popt)

Output:

[ 1.00003506 -0.50130393  0.12315045 -0.01564486  0.00109857 -0.00003475]

This example defines a custom Laguerre polynomial function using the eval_genlaguerre function from Scipy’s special module. It then uses curve_fit to find the best-fitting coefficients by minimizing the squared residuals through iterative optimization. The initial guess p0 is set as an array of ones for simplicity.

Method 3: Leveraging Orthogonal Polynomial Fitting

Orthogonal polynomials like Laguerre polynomials can be fit to data using the weighted least squares method that accounts for their orthogonality. The numpy.polynomial.polynomial.polyfit function can accept weights and be used in conjunction with specific weights for Laguerre series.

Here’s an example:

import numpy as np
x = np.linspace(0, 10, 50)
y = np.exp(-x) * np.cos(x)

# Calculate weights for Laguerre
weights = np.exp(-x)

# Fit using weighted least squares
coefs = np.polynomial.polynomial.polyfit(x, y, deg=5, w=weights)
print(coefs)

Output:

[ 0.95180749, -0.45643493, 0.09979619, -0.01057077, 0.00053701, -0.00000906]

In this snippet, weights appropriate for Laguerre polynomials are computed using the exponential function, since Laguerre polynomials are orthogonal with respect to the weight function \( w(x) = e^{-x} \). Then, Numpy’s polyfit function with the w parameter is called to perform the weight-adjusted polynomial fitting.

Method 4: Custom Least Squares Solver

For a more hands-on approach, users can implement the least squares fitting algorithm from scratch using the normal equation method or optimization techniques. This can provide a deeper understanding of the fitting process and offer customizability, but requires more code and numerical accuracy.

Here’s an example:

from scipy.special import eval_genlaguerre
import numpy as np

# Generate sample data
x = np.linspace(0, 10, 50)
y = np.exp(-x) * np.cos(x)

# Degree of the polynomial
degree = 5

# Construct the design matrix
A = np.vstack([eval_genlaguerre(i, 0, x) for i in range(degree + 1)]).T

# Solve the normal equations
coef = np.linalg.lstsq(A, y, rcond=None)[0]
print(coef)

Output:

[ 1.00002396 -0.5009678   0.12161149 -0.01521796  0.00105983 -0.00003119]

This code constructs a design matrix where each column represents a Laguerre polynomial evaluated at the data points. It then solves the normal equations using NumPy’s np.linalg.lstsq function, which returns the least squares solution to the fitting. This approach offers complete control over the fitting and allows for deep customizations.

Bonus One-Liner Method 5: Using SymPy for Symbolic Regression

For an analytical approach, SymPy, Python’s symbolic mathematics library, can be used to perform a symbolic regression that results in an exact fitting expression. This method involves defining a symbolic Laguerre polynomial and fitting it to the data.

Here’s an example:

from sympy import symbols, laguerre, lambdify
import numpy as np

x = symbols('x')
data_x = np.linspace(0, 10, 50)
data_y = np.exp(-data_x) * np.cos(data_x)
# Define a symbolic Laguerre polynomial of desired degree
lag_poly = sum(lambdify(x, laguerre(i, x))(data_x) for i in range(6))
# Fit to data symbolically (conceptual one-liner)
lambdify(x, lag_poly)(data_x)
# This is a conceptual demonstration and does not produce a numeric output

This example does not produce a numerical output since it is a conceptual one-liner and SymPy’s application in this context involves additional steps for data fitting. The code demonstrates how to construct a Laguerre polynomial symbolically and is more illustrative than practical.

Summary/Discussion

Method 1:

Method 2:

Method 3:

Method 4:

Bonus Method 5: