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@@ -5,4 +5,7 @@ build | |
| dist | ||
| htmlcov | ||
| uravu.egg-info | ||
| **/.DS_Store | ||
| **/.DS_Store | ||
| **/Untitled.ipynb | ||
| coverage_html_report | ||
| paper | ||
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| @@ -0,0 +1,14 @@ | ||
| language: python | ||
| python: | ||
| - '3.6' | ||
| - 3.6-dev | ||
| - '3.7' | ||
| - 3.7-dev | ||
| - '3.8' | ||
| - 3.8-dev | ||
| install: | ||
| - pip install -r requirements.txt | ||
| script: | ||
| - coverage run --source=uravu setup.py test | ||
| after_success: | ||
| - coveralls |
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| environment: | ||
| matrix: | ||
|
|
||
| # For Python versions available on Appveyor, see | ||
| # http://www.appveyor.com/docs/installed-software#python | ||
| # The list here is complete (excluding Python 2.6, which | ||
| # isn't covered by this document) at the time of writing. | ||
|
|
||
| - PYTHON: "C:\\Python36-x64" | ||
| MINICONDA: "C:\\Miniconda" | ||
| - PYTHON: "C:\\Python37-x64" | ||
| MINICONDA: "C:\\Miniconda" | ||
| - PYTHON: "C:\\Python38-x64" | ||
| MINICONDA: "C:\\Miniconda" | ||
|
|
||
| install: | ||
| # We need wheel installed to build wheels | ||
| - "SET PATH=%MINICONDA%;%MINICONDA%\\Scripts;%PATH%" | ||
| - "%PYTHON%\\python.exe -m pip install -U twine wheel setuptools" | ||
| - "%PYTHON%\\python.exe -m pip install -r requirements.txt" | ||
| - "set PATH=%PATH%;C:\"%PYTHON%\\Scripts" | ||
|
|
||
| build: off | ||
|
|
||
| test_script: | ||
| # Put your test command here. | ||
| # If you don't need to build C extensions on 64-bit Python 3.3 or 3.4, | ||
| # you can remove "build.cmd" from the front of the command, as it's | ||
| # only needed to support those cases. | ||
| # Note that you must use the environment variable %PYTHON% to refer to | ||
| # the interpreter you're using - Appveyor does not do anything special | ||
| # to put the Python version you want to use on PATH. | ||
| - "%PYTHON%\\python.exe setup.py test" |
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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,219 @@ | ||
| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Maximum likelihood\n", | ||
| "\n", | ||
| "In Bayesian modelling, the **likelihood**, $L$, is the name given to the measure of the goodness of fit between the model, with some given parameters, and the data. \n", | ||
| "When the maximum likelihood, $\\hat{L}$, is found, the most likely statistical model has been found for the given data. \n", | ||
| "\n", | ||
| "In this tutorial we will see how `uravu` can be used to maximize the likelihood of a model for some dataset.\n", | ||
| "\n", | ||
| "In `uravu`, the likelihood is calculated as follows, \n", | ||
| "\n", | ||
| "$$ \\ln L = -0.5 \\sum_{i=1}^n \\bigg[ \\frac{(y_i - m_i) ^2}{s_i^2} + \\ln(2 \\pi s_i^2) \\bigg], $$\n", | ||
| "\n", | ||
| "where, $y$ is the data ordinate, $m$ is the model ordinate, and $s$ is, \n", | ||
| "\n", | ||
| "$$ s_i^2 = \\delta y_i^2 + f^2 m_i^2 $$,\n", | ||
| "\n", | ||
| "where $\\delta y$ is the uncertainty in $y$ and $f$ is the fraction of unaccounted for uncertainty (more about this can be found in the [Unaccounted uncertainty](./unaccounted_uncertainty.html) tutorial.)\n", | ||
| "`uravu` is able to maximize this function with the help of the [`scipy.optimize.minimize()`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html) function (we minimize the negative of the likelihood).\n", | ||
| "\n", | ||
| "Before we maximise the likelihood, is it necessary to create some *synthetic* data to analyse. " | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "import numpy as np\n", | ||
| "import matplotlib.pyplot as plt" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "x = np.linspace(0, 10, 20)\n", | ||
| "y = np.exp(0.5 * x) * 4\n", | ||
| "y += y * np.random.randn(20) * 0.1\n", | ||
| "dy = y * 0.2" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "plt.errorbar(x, y, dy, marker='o', ls='')" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "The data plotted above may be modelled with the following relationship, \n", | ||
| "\n", | ||
| "$$ y = a\\exp(bx), $$\n", | ||
| "\n", | ||
| "where $a$ and $b$ are the variables of interest in the modelling process. \n", | ||
| "We want to find the values of these variables, which maximises the likelihood. \n", | ||
| "\n", | ||
| "First, we must write a function do describe the model." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "def my_model(x, a, b):\n", | ||
| " \"\"\"\n", | ||
| " A function to describe the model under investgation.\n", | ||
| " \n", | ||
| " Args:\n", | ||
| " x (array_like): Abscissa data.\n", | ||
| " a (float): The pre-exponential factor.\n", | ||
| " b (float): The x-multiplicative factor.\n", | ||
| " \n", | ||
| " Returns\n", | ||
| " y (array_like): Ordinate data.\n", | ||
| " \"\"\"\n", | ||
| " return a * np.exp(b * x)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "With our model defined, we can construct a `Relationship` object. " | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "from uravu.relationship import Relationship" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "modeller = Relationship(my_model, x, y, dy)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "print(modeller)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "The `Relationship` object gives us access to a few exciting Bayesian modelling methods.\n", | ||
| "However, this tutorial is focused on maximising the likelihood, this is achieved with the `max_likelihood()` class method. " | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "modeller.max_likelihood()" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "print(modeller.variables)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "We can see that the variables have changed to be close to the values into in the data synthesis. \n", | ||
| "\n", | ||
| "Let's inspect the model visually. \n", | ||
| "This can be achieved easily with the `plotting` module in `uravu`." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "from uravu import plotting" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "ax = plotting.plot_relationship(modeller)\n", | ||
| "plt.show()" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "Above, we can see that the orange line of maximum likelihood agrees well with the data. " | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Python 3", | ||
| "language": "python", | ||
| "name": "python3" | ||
| }, | ||
| "language_info": { | ||
| "codemirror_mode": { | ||
| "name": "ipython", | ||
| "version": 3 | ||
| }, | ||
| "file_extension": ".py", | ||
| "mimetype": "text/x-python", | ||
| "name": "python", | ||
| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython3", | ||
| "version": "3.7.3" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 4 | ||
| } |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
|
|
@@ -8,4 +8,6 @@ pint | |
| uncertainties | ||
| nbsphinx | ||
| jupyter-sphinx | ||
| sphinx_rtd_theme | ||
| sphinx_rtd_theme | ||
| coverage | ||
| coveralls | ||
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