Emely

Emely is a lightweight Python package for maximum likelihood estimation (MLE)–based parameter fitting.
It provides a scipy.optimize.curve_fit-like interface built on top of scipy.optimize.minimize, with support for Poisson, Normal, Laplace, Folded Normal, Rayleigh, and Rice noise models.

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Features

• Parameter estimation using MLE for Poisson, Normal, Laplace, Folded Normal, Rayleigh, and Rice noise
• Parameter error estimation using the Fisher information matrix
• Model selection criteria: Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for comparing models
• emely.curve_fit provides a scipy.optimize.curve_fit-like interface
• The underlying BaseMLE classes (NormalMLE, PoissonMLE, LaplaceMLE, FoldedNormalMLE, RayleighMLE, RiceMLE) provide a modern object-oriented API

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Installation

Install from source using pip (requires Python 3.8+):

pip install .

For development, install in editable mode:

pip install -e .

Dependencies are managed via pyproject.toml.

Quick Start

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import poisson

from emely import curve_fit

# define plot style
blue = "#648fff"
red = "#dc267f"
black = "#000000"
plt.style.use("seaborn-v0_8")

# define the model
def gaussian(x, a, mu, sigma):
    return a / np.sqrt(2 * np.pi * sigma**2) * np.exp(-((x - mu) ** 2) / (2 * sigma**2))

# create the Gaussian data with Poisson noise
p = (100, 5, 1)
x_data = np.linspace(0, 10, 51)
y_data = poisson.rvs(gaussian(x_data, *p))

# fit using MLE for Poisson noise
p0 = (50, 10, 5)
p_opt, p_cov = curve_fit(
    gaussian,
    x_data,
    y_data,
    p0=p0,
    noise="poisson",  # "normal", "poisson", "laplace", "folded-normal", "rayleigh", "rice"
)

# show the fit
plt.figure(figsize=(6, 2))

plt.scatter(x_data, y_data, label="Measurement", edgecolor=black, facecolor=blue)
plt.plot(x_data, gaussian(x_data, *p_opt), label="Fit", color=red)

plt.xlabel("x")
plt.ylabel("y")
plt.legend()

Why MLE for parameter estimation?

Maximum likelihood estimation (MLE) can correctly model different noise distributions, leading to more accurate and unbiased parameter estimates as compared to least-squares fitting, which is only optimal for normally-distributed noise.

Examples

See the provided notebooks in the examples/ folder for detailed usage:

• example_1.ipynb: Fitting a 1D Gaussian signal with Poisson noise
• example_2.ipynb: Fitting a 1D Gaussian signal with Laplace noise
• example_3.ipynb: Fitting a 2D dynamic Gaussian signal with folded normal-noise, comparing folded-normal and normal MLE estimators

For more details, see the examples README.

License

MIT License © 2025