{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Simulations" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this tutorial, we explore\n", "\n", "- The mathematical representations of the simulations\n", "- Plots showing each simulation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Mathematical Equations" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Simulations for the power curves were generated using the following equations:\n", "\n", "- Linear$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$:\n", "\n", "$$X \\sim {\\mathcal{U} \\left( -1, 1 \\right)}^p,$$\n", "\n", "$$Y = w ^T X + \\kappa \\epsilon.$$\n", "\n", "- Exponential$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$:\n", "\n", "$$X \\sim {\\mathcal{U} \\left( 0, 3 \\right)}^p,$$\n", "\n", "$$Y = \\exp \\left( w ^T X \\right) + 10 \\kappa \\epsilon.$$\n", "\n", "- Cubic$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$:\n", "\n", "$$X \\sim {\\mathcal{U} \\left( -1, 1 \\right)}^p,$$\n", "\n", "$$Y = 128 {\\left( w ^T X - \\frac{1}{3} \\right)}^3 + 48 {\\left( w ^T X - \\frac{1}{3} \\right)}^2 - 12 \\left( w ^T X - \\frac{1}{3} \\right) + 80 \\kappa \\epsilon.$$\n", "\n", "- Joint Normal$\\left ( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}^p$: Let $\\rho = 1/2 p$, $I_p$ be the identity matrix of size $p \\times p$, $J_p$ be the matrix of ones of size $p \\times p$, and \n", "$\\Sigma =\n", "\\begin{bmatrix}\n", " I_p & \\rho J_p \\\\\n", " \\rho J_p & \\left(1 + 0.5 \\kappa \\right) I_p\\\\\n", "\\end{bmatrix}$.\n", "Then,\n", "\n", "$$\\left( X, Y \\right) \\sim \\mathcal{N} \\left( 0, \\Sigma \\right).$$\n", "\n", "- Step Function$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$:\n", "\n", "$$X \\sim {\\mathcal{U} \\left( -1, 1 \\right)}^p,$$\n", "\n", "$$Y = \\mathcal{I} \\left( w ^T X > 0 \\right) + \\epsilon,$$\n", "\n", "where $\\mathcal{I}$ is the indicator function, that is $\\mathcal{I} \\left( z \\right)$ is unity whenever $z$ is true, and $0$ otherwise.\n", "\n", "- Quadratic$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$:\n", "\n", "$$X \\sim {\\mathcal{U} \\left( -1, 1 \\right)}^p,$$\n", "\n", "$$Y = {\\left( w ^T X \\right)}^2 + 0.5 \\kappa \\epsilon.$$\n", "\n", "- W-Shape$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$: For $U \\sim {\\mathcal{U} \\left( -1, 1 \\right)}^p$,\n", "\n", "$$X \\sim {\\mathcal{U} \\left( -1, 1 \\right)}^p,$$\n", "\n", "$$Y = 4 \\left[ {\\left( {\\left( w ^T X \\right)}^2 - \\frac{1}{2} \\right)}^2 + \\frac{w ^T U}{500} \\right] + 0.5 \\kappa \\epsilon.$$\n", "\n", "- Spiral$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$: For $U \\sim \\mathcal{U} \\left( 0, 5 \\right)$, $\\epsilon \\sim \\mathcal{N} \\left( 0, 1 \\right)$,\n", "\n", "$$X_{\\left| d \\right|} = U \\sin \\left(\\pi U \\right) \\cos^d \\left(\\pi U \\right)\\ \\mathrm{for}\\ d = 1, ..., p - 1,$$\n", "\n", "$$X_{\\left| p \\right|} = U \\cos^p \\left(\\pi U \\right),$$\n", "\n", "$$Y = U \\sin \\left( \\pi U \\right) + 0.4 p \\epsilon.$$\n", "\n", "- Uncorrelated Bernoulli$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$: For $U \\sim \\mathcal{B} \\left( 0.5 \\right)$, $\\epsilon_1 \\sim \\mathcal{N} \\left( 0, I_p \\right)$, $\\epsilon_2 \\sim \\mathcal{N} \\left( 0, 1 \\right)$,\n", "\n", "$$X \\sim {\\mathcal{B} \\left( 0.5 \\right)}^p + 0.5 \\epsilon_1,$$\n", "\n", "$$Y = \\left( 2 U - 1 \\right) w ^T X + 0.5 \\epsilon_2.$$\n", "\n", "- Logarithmic$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}^p$: For $\\epsilon \\sim \\mathcal{N} \\left( 0, I_p \\right)$,\n", "\n", "$$X \\sim \\mathcal{N} \\left( 0, I_p \\right),$$\n", "\n", "$$Y_{\\left| d \\right|} = 2 \\log_2 \\left( \\left| X_{\\left| d \\right|} \\right| \\right) + 3 \\kappa \\epsilon_{\\left| d \\right|}\\ \\mathrm{for}\\ d = 1, ..., p.$$\n", "\n", "- Fourth\\ Root$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$:\n", "\n", "$$X \\sim {\\mathcal{U} \\left( -1, 1 \\right)}^p,$$\n", "\n", "$$Y = {\\left| w ^T X \\right|}^{1/4} + \\frac{\\kappa}{4} \\epsilon.$$\n", "\n", "- Sine\\ Period 4$\\pi$ $\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}^p$: For $U \\sim \\mathcal{U} \\left( -1, 1 \\right)$, $V \\sim {\\mathcal{N} \\left( 0, 1 \\right)}^p$, $\\theta = 4 \\pi$,\n", "\n", "$$X_{\\left| d \\right|} = U + 0.02 p V_{\\left| d \\right|}\\ \\mathrm{for}\\ d = 1, ..., p,$$\n", "\n", "$$Y=\\sin(\\theta X)+\\kappa \\epsilon.$$\n", "\n", "- Sine\\ Period 16$\\pi$ $\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}^p$: Same as above except $\\theta = 16 \\pi$ and the noise on $Y$ is changed to $0.5 \\kappa \\epsilon$.\n", "\n", "- Square$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}^p$: For $U \\sim \\mathcal{U} \\left( -1, 1 \\right)$, $V \\sim \\mathcal{U} \\left( -1, 1 \\right)$, $\\epsilon \\sim {\\mathcal{N} \\left( 0, 1 \\right)}^p$, $\\theta = -\\frac{\\pi}{8}$,\n", "\n", "$$X_{\\left| d \\right|} = U \\cos \\left( \\theta \\right) + V \\sin \\left( \\theta \\right) + 0.05 p \\epsilon_{\\left| d \\right|},$$\n", "\n", "$$Y_{\\left| d \\right|} = -U \\sin \\left( \\theta \\right) + V \\cos \\left( \\theta \\right).$$\n", "\n", "- Diamond$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}^p$: Same as above except $\\theta = \\pi/4$.\n", "\n", "- Two\\ Parabolas$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$: For $\\epsilon \\sim \\mathcal{U} \\left( 0, 1 \\right)$, $U \\sim \\mathcal{B} \\left( 0.5 \\right)$,\n", "\n", "$$X \\sim {\\mathcal{U} \\left( -1, 1 \\right)}^p,$$\n", "\n", "$$Y = \\left( {\\left( w ^T X \\right)}^2 + 2 \\kappa \\epsilon \\right) \\cdot \\left(U - \\frac{1}{2} \\right).$$\n", "\n", "- Circle$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$: For $U \\sim {\\mathcal{U} \\left( -1, 1 \\right)}^p$, $\\epsilon \\sim \\mathcal{N} \\left( 0, I_p \\right)$, $r = 1$,\n", "\n", "$$X_{\\left| d \\right|} = r \\left( \\sin \\left( \\pi U_{\\left| d + 1 \\right|} \\right) \\prod \\limits_{j = 1}^d \\cos \\left( \\pi U_{\\left| j \\right|} \\right) + 0.4 \\epsilon_{\\left| d \\right|} \\right)\\ \\mathrm{for}\\ d = 1, ..., p-1,$$\n", "\n", "$$X_{\\left| d \\right|} = r \\left( \\prod \\limits_{j = 1}^p \\cos \\left(\\pi U_{\\left| j \\right|} \\right) + 0.4 \\epsilon_{\\left| p \\right|} \\right),$$\n", "\n", "$$Y_{\\left| d \\right|} = \\sin \\left(\\pi U_{\\left| 1 \\right|} \\right).$$\n", "\n", "- Ellipse$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}^p$: Same as above except $r = 5$.\n", "\n", "- Multiplicative Noise$\\left( x, y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}^p$: $u \\sim \\mathcal{N} \\left( 0, I_p \\right)$,\n", "\n", "$$x \\sim \\mathcal{N} \\left( 0, I_p \\right),$$\n", "\n", "$$y_{\\left| d \\right|} = u_{\\left| d \\right|} x_{\\left| d \\right|}\\ \\mathrm{for}\\ d = 1, ..., p.$$\n", "\n", "- Multimodal Independence$\\left( X, Y \\right) \\in \\mathbb{R}^p \\times \\mathbb{R}$: For $U \\sim \\mathcal{N} \\left( 0, I_p \\right)$, $V \\sim \\mathcal{N} \\left( 0, I_p \\right)$, $U' \\sim {\\mathcal{B} \\left( 0.5 \\right)}^p$, $V' \\sim {\\mathcal{B} \\left( 0.5 \\right)}^p$,\n", "\n", "$$X = U/3 + 2U' - 1,$$\n", "\n", "$$Y = V/3 + 2V' - 1.$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Plots" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's import some useful packages and create a function that plots our simulated 1D data, to ensure consistency in these examples, we set the seed:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt; plt.style.use('classic')\n", "import seaborn as sns; sns.set(style=\"white\")\n", "\n", "from mgcpy.benchmarks.simulations import *\n", "\n", "np.random.seed(12345678)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "def plot_sims(sim_name, sim_func):\n", " \"\"\"\n", " Plots all of the simulations\n", " \"\"\"\n", " if sim_name == 'Sine (16$\\pi$)':\n", " x, y = sim_func(num_samp=1000, num_dim=1, noise=0, period=16*np.pi)\n", " elif sim_name == 'Ellipse':\n", " x, y = sim_func(num_samp=1000, num_dim=1, noise=0, radius=5)\n", " elif sim_name == 'Diamond':\n", " x, y = sim_func(num_samp=1000, num_dim=1, noise=0, period=-np.pi/4)\n", " elif sim_name == 'Multiplicative Noise' or sim_name == 'Multimodal Independence':\n", " x, y = sim_func(num_samp=1000, num_dim=1)\n", " else:\n", " x, y = sim_func(num_samp=1000, num_dim=1, noise=0)\n", " \n", " # Normalize\n", " x = x / np.max(x)\n", " y = y / np.max(y)\n", " \n", " fig = plt.figure(figsize=(8,8))\n", " fig.suptitle(sim_name + \" Simulation\", fontsize=17)\n", " ax = sns.scatterplot(x=x[:,0], y=y[:,0])\n", " ax.set_xlabel('Simulated X', fontsize=15)\n", " ax.set_ylabel('Simulated Y', fontsize=15)\n", " plt.axis('equal')\n", " plt.xticks(fontsize=15)\n", " plt.yticks(fontsize=15)\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Simultions are randomly generated with an $x$ which is $(n \\times d)$ and $y$ which is $(n \\times 1)$ that have 2 required parameters: num_samp or the number of samples, and num_dim or the number of dimensions. Optional parameters can be set based on the documentation. Visualizations of are shown below with and without the noise. Here are all the simulations:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "scrolled": false }, "outputs": [ { "data": { "image/png": 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\n", 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