{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Linear Algebra and Linear Regression\n", "### [Neil D. Lawrence](http://inverseprobability.com), University of Sheffield\n", "### 2015-10-13\n", "\n", "**Abstract**: In this session we combine the objective function perspective and the\n", "probabilistic perspective on *linear regression*. We motivate the\n", "importance of *linear algebra* by showing how much faster we can\n", "complete a linear regression using linear algebra.\n", "\n", "$$\n", "\\newcommand{\\tk}[1]{}\n", "%\\newcommand{\\tk}[1]{\\textbf{TK}: #1}\n", "\\newcommand{\\Amatrix}{\\mathbf{A}}\n", "\\newcommand{\\KL}[2]{\\text{KL}\\left( #1\\,\\|\\,#2 \\right)}\n", "\\newcommand{\\Kaast}{\\kernelMatrix_{\\mathbf{ \\ast}\\mathbf{ \\ast}}}\n", "\\newcommand{\\Kastu}{\\kernelMatrix_{\\mathbf{ \\ast} \\inducingVector}}\n", "\\newcommand{\\Kff}{\\kernelMatrix_{\\mappingFunctionVector \\mappingFunctionVector}}\n", "\\newcommand{\\Kfu}{\\kernelMatrix_{\\mappingFunctionVector \\inducingVector}}\n", "\\newcommand{\\Kuast}{\\kernelMatrix_{\\inducingVector \\bf\\ast}}\n", "\\newcommand{\\Kuf}{\\kernelMatrix_{\\inducingVector \\mappingFunctionVector}}\n", "\\newcommand{\\Kuu}{\\kernelMatrix_{\\inducingVector \\inducingVector}}\n", "\\newcommand{\\Kuui}{\\Kuu^{-1}}\n", "\\newcommand{\\Qaast}{\\mathbf{Q}_{\\bf \\ast \\ast}}\n", "\\newcommand{\\Qastf}{\\mathbf{Q}_{\\ast \\mappingFunction}}\n", "\\newcommand{\\Qfast}{\\mathbf{Q}_{\\mappingFunctionVector \\bf \\ast}}\n", "\\newcommand{\\Qff}{\\mathbf{Q}_{\\mappingFunctionVector \\mappingFunctionVector}}\n", "\\newcommand{\\aMatrix}{\\mathbf{A}}\n", "\\newcommand{\\aScalar}{a}\n", "\\newcommand{\\aVector}{\\mathbf{a}}\n", "\\newcommand{\\acceleration}{a}\n", "\\newcommand{\\bMatrix}{\\mathbf{B}}\n", "\\newcommand{\\bScalar}{b}\n", "\\newcommand{\\bVector}{\\mathbf{b}}\n", "\\newcommand{\\basisFunc}{\\phi}\n", "\\newcommand{\\basisFuncVector}{\\boldsymbol{ \\basisFunc}}\n", "\\newcommand{\\basisFunction}{\\phi}\n", "\\newcommand{\\basisLocation}{\\mu}\n", "\\newcommand{\\basisMatrix}{\\boldsymbol{ \\Phi}}\n", "\\newcommand{\\basisScalar}{\\basisFunction}\n", "\\newcommand{\\basisVector}{\\boldsymbol{ \\basisFunction}}\n", "\\newcommand{\\activationFunction}{\\phi}\n", "\\newcommand{\\activationMatrix}{\\boldsymbol{ \\Phi}}\n", "\\newcommand{\\activationScalar}{\\basisFunction}\n", "\\newcommand{\\activationVector}{\\boldsymbol{ \\basisFunction}}\n", "\\newcommand{\\bigO}{\\mathcal{O}}\n", "\\newcommand{\\binomProb}{\\pi}\n", "\\newcommand{\\cMatrix}{\\mathbf{C}}\n", "\\newcommand{\\cbasisMatrix}{\\hat{\\boldsymbol{ \\Phi}}}\n", "\\newcommand{\\cdataMatrix}{\\hat{\\dataMatrix}}\n", "\\newcommand{\\cdataScalar}{\\hat{\\dataScalar}}\n", "\\newcommand{\\cdataVector}{\\hat{\\dataVector}}\n", "\\newcommand{\\centeredKernelMatrix}{\\mathbf{ \\MakeUppercase{\\centeredKernelScalar}}}\n", "\\newcommand{\\centeredKernelScalar}{b}\n", "\\newcommand{\\centeredKernelVector}{\\centeredKernelScalar}\n", "\\newcommand{\\centeringMatrix}{\\mathbf{H}}\n", "\\newcommand{\\chiSquaredDist}[2]{\\chi_{#1}^{2}\\left(#2\\right)}\n", "\\newcommand{\\chiSquaredSamp}[1]{\\chi_{#1}^{2}}\n", "\\newcommand{\\conditionalCovariance}{\\boldsymbol{ \\Sigma}}\n", "\\newcommand{\\coregionalizationMatrix}{\\mathbf{B}}\n", "\\newcommand{\\coregionalizationScalar}{b}\n", "\\newcommand{\\coregionalizationVector}{\\mathbf{ \\coregionalizationScalar}}\n", "\\newcommand{\\covDist}[2]{\\text{cov}_{#2}\\left(#1\\right)}\n", "\\newcommand{\\covSamp}[1]{\\text{cov}\\left(#1\\right)}\n", "\\newcommand{\\covarianceScalar}{c}\n", "\\newcommand{\\covarianceVector}{\\mathbf{ \\covarianceScalar}}\n", "\\newcommand{\\covarianceMatrix}{\\mathbf{C}}\n", "\\newcommand{\\covarianceMatrixTwo}{\\boldsymbol{ \\Sigma}}\n", "\\newcommand{\\croupierScalar}{s}\n", "\\newcommand{\\croupierVector}{\\mathbf{ \\croupierScalar}}\n", "\\newcommand{\\croupierMatrix}{\\mathbf{ \\MakeUppercase{\\croupierScalar}}}\n", "\\newcommand{\\dataDim}{p}\n", "\\newcommand{\\dataIndex}{i}\n", "\\newcommand{\\dataIndexTwo}{j}\n", "\\newcommand{\\dataMatrix}{\\mathbf{Y}}\n", "\\newcommand{\\dataScalar}{y}\n", "\\newcommand{\\dataSet}{\\mathcal{D}}\n", "\\newcommand{\\dataStd}{\\sigma}\n", "\\newcommand{\\dataVector}{\\mathbf{ \\dataScalar}}\n", "\\newcommand{\\decayRate}{d}\n", "\\newcommand{\\degreeMatrix}{\\mathbf{ \\MakeUppercase{\\degreeScalar}}}\n", "\\newcommand{\\degreeScalar}{d}\n", "\\newcommand{\\degreeVector}{\\mathbf{ \\degreeScalar}}\n", "% Already defined by latex\n", "%\\newcommand{\\det}[1]{\\left|#1\\right|}\n", "\\newcommand{\\diag}[1]{\\text{diag}\\left(#1\\right)}\n", "\\newcommand{\\diagonalMatrix}{\\mathbf{D}}\n", "\\newcommand{\\diff}[2]{\\frac{\\text{d}#1}{\\text{d}#2}}\n", "\\newcommand{\\diffTwo}[2]{\\frac{\\text{d}^2#1}{\\text{d}#2^2}}\n", "\\newcommand{\\displacement}{x}\n", "\\newcommand{\\displacementVector}{\\textbf{\\displacement}}\n", "\\newcommand{\\distanceMatrix}{\\mathbf{ \\MakeUppercase{\\distanceScalar}}}\n", "\\newcommand{\\distanceScalar}{d}\n", "\\newcommand{\\distanceVector}{\\mathbf{ \\distanceScalar}}\n", "\\newcommand{\\eigenvaltwo}{\\ell}\n", "\\newcommand{\\eigenvaltwoMatrix}{\\mathbf{L}}\n", "\\newcommand{\\eigenvaltwoVector}{\\mathbf{l}}\n", "\\newcommand{\\eigenvalue}{\\lambda}\n", "\\newcommand{\\eigenvalueMatrix}{\\boldsymbol{ \\Lambda}}\n", "\\newcommand{\\eigenvalueVector}{\\boldsymbol{ \\lambda}}\n", "\\newcommand{\\eigenvector}{\\mathbf{ \\eigenvectorScalar}}\n", "\\newcommand{\\eigenvectorMatrix}{\\mathbf{U}}\n", "\\newcommand{\\eigenvectorScalar}{u}\n", "\\newcommand{\\eigenvectwo}{\\mathbf{v}}\n", "\\newcommand{\\eigenvectwoMatrix}{\\mathbf{V}}\n", "\\newcommand{\\eigenvectwoScalar}{v}\n", "\\newcommand{\\entropy}[1]{\\mathcal{H}\\left(#1\\right)}\n", "\\newcommand{\\errorFunction}{E}\n", "\\newcommand{\\expDist}[2]{\\left<#1\\right>_{#2}}\n", "\\newcommand{\\expSamp}[1]{\\left<#1\\right>}\n", "\\newcommand{\\expectation}[1]{\\left\\langle #1 \\right\\rangle }\n", "\\newcommand{\\expectationDist}[2]{\\left\\langle #1 \\right\\rangle _{#2}}\n", "\\newcommand{\\expectedDistanceMatrix}{\\mathcal{D}}\n", "\\newcommand{\\eye}{\\mathbf{I}}\n", "\\newcommand{\\fantasyDim}{r}\n", "\\newcommand{\\fantasyMatrix}{\\mathbf{ \\MakeUppercase{\\fantasyScalar}}}\n", "\\newcommand{\\fantasyScalar}{z}\n", "\\newcommand{\\fantasyVector}{\\mathbf{ \\fantasyScalar}}\n", "\\newcommand{\\featureStd}{\\varsigma}\n", "\\newcommand{\\gammaCdf}[3]{\\mathcal{GAMMA CDF}\\left(#1|#2,#3\\right)}\n", "\\newcommand{\\gammaDist}[3]{\\mathcal{G}\\left(#1|#2,#3\\right)}\n", "\\newcommand{\\gammaSamp}[2]{\\mathcal{G}\\left(#1,#2\\right)}\n", "\\newcommand{\\gaussianDist}[3]{\\mathcal{N}\\left(#1|#2,#3\\right)}\n", "\\newcommand{\\gaussianSamp}[2]{\\mathcal{N}\\left(#1,#2\\right)}\n", "\\newcommand{\\given}{|}\n", "\\newcommand{\\half}{\\frac{1}{2}}\n", "\\newcommand{\\heaviside}{H}\n", "\\newcommand{\\hiddenMatrix}{\\mathbf{ \\MakeUppercase{\\hiddenScalar}}}\n", "\\newcommand{\\hiddenScalar}{h}\n", "\\newcommand{\\hiddenVector}{\\mathbf{ \\hiddenScalar}}\n", "\\newcommand{\\identityMatrix}{\\eye}\n", "\\newcommand{\\inducingInputScalar}{z}\n", "\\newcommand{\\inducingInputVector}{\\mathbf{ \\inducingInputScalar}}\n", "\\newcommand{\\inducingInputMatrix}{\\mathbf{Z}}\n", "\\newcommand{\\inducingScalar}{u}\n", "\\newcommand{\\inducingVector}{\\mathbf{ \\inducingScalar}}\n", "\\newcommand{\\inducingMatrix}{\\mathbf{U}}\n", "\\newcommand{\\inlineDiff}[2]{\\text{d}#1/\\text{d}#2}\n", "\\newcommand{\\inputDim}{q}\n", "\\newcommand{\\inputMatrix}{\\mathbf{X}}\n", "\\newcommand{\\inputScalar}{x}\n", "\\newcommand{\\inputSpace}{\\mathcal{X}}\n", "\\newcommand{\\inputVals}{\\inputVector}\n", "\\newcommand{\\inputVector}{\\mathbf{ \\inputScalar}}\n", "\\newcommand{\\iterNum}{k}\n", "\\newcommand{\\kernel}{\\kernelScalar}\n", "\\newcommand{\\kernelMatrix}{\\mathbf{K}}\n", "\\newcommand{\\kernelScalar}{k}\n", "\\newcommand{\\kernelVector}{\\mathbf{ \\kernelScalar}}\n", "\\newcommand{\\kff}{\\kernelScalar_{\\mappingFunction \\mappingFunction}}\n", "\\newcommand{\\kfu}{\\kernelVector_{\\mappingFunction \\inducingScalar}}\n", "\\newcommand{\\kuf}{\\kernelVector_{\\inducingScalar \\mappingFunction}}\n", "\\newcommand{\\kuu}{\\kernelVector_{\\inducingScalar \\inducingScalar}}\n", "\\newcommand{\\lagrangeMultiplier}{\\lambda}\n", "\\newcommand{\\lagrangeMultiplierMatrix}{\\boldsymbol{ \\Lambda}}\n", "\\newcommand{\\lagrangian}{L}\n", "\\newcommand{\\laplacianFactor}{\\mathbf{ \\MakeUppercase{\\laplacianFactorScalar}}}\n", "\\newcommand{\\laplacianFactorScalar}{m}\n", "\\newcommand{\\laplacianFactorVector}{\\mathbf{ \\laplacianFactorScalar}}\n", "\\newcommand{\\laplacianMatrix}{\\mathbf{L}}\n", "\\newcommand{\\laplacianScalar}{\\ell}\n", "\\newcommand{\\laplacianVector}{\\mathbf{ \\ell}}\n", "\\newcommand{\\latentDim}{q}\n", "\\newcommand{\\latentDistanceMatrix}{\\boldsymbol{ \\Delta}}\n", "\\newcommand{\\latentDistanceScalar}{\\delta}\n", "\\newcommand{\\latentDistanceVector}{\\boldsymbol{ \\delta}}\n", "\\newcommand{\\latentForce}{f}\n", "\\newcommand{\\latentFunction}{u}\n", "\\newcommand{\\latentFunctionVector}{\\mathbf{ \\latentFunction}}\n", "\\newcommand{\\latentFunctionMatrix}{\\mathbf{ \\MakeUppercase{\\latentFunction}}}\n", "\\newcommand{\\latentIndex}{j}\n", "\\newcommand{\\latentScalar}{z}\n", "\\newcommand{\\latentVector}{\\mathbf{ \\latentScalar}}\n", "\\newcommand{\\latentMatrix}{\\mathbf{Z}}\n", "\\newcommand{\\learnRate}{\\eta}\n", "\\newcommand{\\lengthScale}{\\ell}\n", "\\newcommand{\\rbfWidth}{\\ell}\n", "\\newcommand{\\likelihoodBound}{\\mathcal{L}}\n", "\\newcommand{\\likelihoodFunction}{L}\n", "\\newcommand{\\locationScalar}{\\mu}\n", "\\newcommand{\\locationVector}{\\boldsymbol{ \\locationScalar}}\n", "\\newcommand{\\locationMatrix}{\\mathbf{M}}\n", "\\newcommand{\\variance}[1]{\\text{var}\\left( #1 \\right)}\n", "\\newcommand{\\mappingFunction}{f}\n", "\\newcommand{\\mappingFunctionMatrix}{\\mathbf{F}}\n", "\\newcommand{\\mappingFunctionTwo}{g}\n", "\\newcommand{\\mappingFunctionTwoMatrix}{\\mathbf{G}}\n", "\\newcommand{\\mappingFunctionTwoVector}{\\mathbf{ \\mappingFunctionTwo}}\n", "\\newcommand{\\mappingFunctionVector}{\\mathbf{ \\mappingFunction}}\n", "\\newcommand{\\scaleScalar}{s}\n", "\\newcommand{\\mappingScalar}{w}\n", "\\newcommand{\\mappingVector}{\\mathbf{ \\mappingScalar}}\n", "\\newcommand{\\mappingMatrix}{\\mathbf{W}}\n", "\\newcommand{\\mappingScalarTwo}{v}\n", "\\newcommand{\\mappingVectorTwo}{\\mathbf{ \\mappingScalarTwo}}\n", "\\newcommand{\\mappingMatrixTwo}{\\mathbf{V}}\n", "\\newcommand{\\maxIters}{K}\n", "\\newcommand{\\meanMatrix}{\\mathbf{M}}\n", "\\newcommand{\\meanScalar}{\\mu}\n", "\\newcommand{\\meanTwoMatrix}{\\mathbf{M}}\n", "\\newcommand{\\meanTwoScalar}{m}\n", "\\newcommand{\\meanTwoVector}{\\mathbf{ \\meanTwoScalar}}\n", "\\newcommand{\\meanVector}{\\boldsymbol{ \\meanScalar}}\n", "\\newcommand{\\mrnaConcentration}{m}\n", "\\newcommand{\\naturalFrequency}{\\omega}\n", "\\newcommand{\\neighborhood}[1]{\\mathcal{N}\\left( #1 \\right)}\n", "\\newcommand{\\neilurl}{http://inverseprobability.com/}\n", "\\newcommand{\\noiseMatrix}{\\boldsymbol{ E}}\n", "\\newcommand{\\noiseScalar}{\\epsilon}\n", "\\newcommand{\\noiseVector}{\\boldsymbol{ \\epsilon}}\n", "\\newcommand{\\norm}[1]{\\left\\Vert #1 \\right\\Vert}\n", "\\newcommand{\\normalizedLaplacianMatrix}{\\hat{\\mathbf{L}}}\n", "\\newcommand{\\normalizedLaplacianScalar}{\\hat{\\ell}}\n", "\\newcommand{\\normalizedLaplacianVector}{\\hat{\\mathbf{ \\ell}}}\n", "\\newcommand{\\numActive}{m}\n", "\\newcommand{\\numBasisFunc}{m}\n", "\\newcommand{\\numComponents}{m}\n", "\\newcommand{\\numComps}{K}\n", "\\newcommand{\\numData}{n}\n", "\\newcommand{\\numFeatures}{K}\n", "\\newcommand{\\numHidden}{h}\n", "\\newcommand{\\numInducing}{m}\n", "\\newcommand{\\numLayers}{\\ell}\n", "\\newcommand{\\numNeighbors}{K}\n", "\\newcommand{\\numSequences}{s}\n", "\\newcommand{\\numSuccess}{s}\n", "\\newcommand{\\numTasks}{m}\n", "\\newcommand{\\numTime}{T}\n", "\\newcommand{\\numTrials}{S}\n", "\\newcommand{\\outputIndex}{j}\n", "\\newcommand{\\paramVector}{\\boldsymbol{ \\theta}}\n", "\\newcommand{\\parameterMatrix}{\\boldsymbol{ \\Theta}}\n", "\\newcommand{\\parameterScalar}{\\theta}\n", "\\newcommand{\\parameterVector}{\\boldsymbol{ \\parameterScalar}}\n", "\\newcommand{\\partDiff}[2]{\\frac{\\partial#1}{\\partial#2}}\n", "\\newcommand{\\precisionScalar}{j}\n", "\\newcommand{\\precisionVector}{\\mathbf{ \\precisionScalar}}\n", "\\newcommand{\\precisionMatrix}{\\mathbf{J}}\n", "\\newcommand{\\pseudotargetScalar}{\\widetilde{y}}\n", "\\newcommand{\\pseudotargetVector}{\\mathbf{ \\pseudotargetScalar}}\n", "\\newcommand{\\pseudotargetMatrix}{\\mathbf{ \\widetilde{Y}}}\n", "\\newcommand{\\rank}[1]{\\text{rank}\\left(#1\\right)}\n", "\\newcommand{\\rayleighDist}[2]{\\mathcal{R}\\left(#1|#2\\right)}\n", "\\newcommand{\\rayleighSamp}[1]{\\mathcal{R}\\left(#1\\right)}\n", "\\newcommand{\\responsibility}{r}\n", "\\newcommand{\\rotationScalar}{r}\n", "\\newcommand{\\rotationVector}{\\mathbf{ \\rotationScalar}}\n", "\\newcommand{\\rotationMatrix}{\\mathbf{R}}\n", "\\newcommand{\\sampleCovScalar}{s}\n", "\\newcommand{\\sampleCovVector}{\\mathbf{ \\sampleCovScalar}}\n", "\\newcommand{\\sampleCovMatrix}{\\mathbf{s}}\n", "\\newcommand{\\scalarProduct}[2]{\\left\\langle{#1},{#2}\\right\\rangle}\n", "\\newcommand{\\sign}[1]{\\text{sign}\\left(#1\\right)}\n", "\\newcommand{\\sigmoid}[1]{\\sigma\\left(#1\\right)}\n", "\\newcommand{\\singularvalue}{\\ell}\n", "\\newcommand{\\singularvalueMatrix}{\\mathbf{L}}\n", "\\newcommand{\\singularvalueVector}{\\mathbf{l}}\n", "\\newcommand{\\sorth}{\\mathbf{u}}\n", "\\newcommand{\\spar}{\\lambda}\n", "\\newcommand{\\trace}[1]{\\text{tr}\\left(#1\\right)}\n", "\\newcommand{\\BasalRate}{B}\n", "\\newcommand{\\DampingCoefficient}{C}\n", "\\newcommand{\\DecayRate}{D}\n", "\\newcommand{\\Displacement}{X}\n", "\\newcommand{\\LatentForce}{F}\n", "\\newcommand{\\Mass}{M}\n", "\\newcommand{\\Sensitivity}{S}\n", "\\newcommand{\\basalRate}{b}\n", "\\newcommand{\\dampingCoefficient}{c}\n", "\\newcommand{\\mass}{m}\n", "\\newcommand{\\sensitivity}{s}\n", "\\newcommand{\\springScalar}{\\kappa}\n", "\\newcommand{\\springVector}{\\boldsymbol{ \\kappa}}\n", "\\newcommand{\\springMatrix}{\\boldsymbol{ \\mathcal{K}}}\n", "\\newcommand{\\tfConcentration}{p}\n", "\\newcommand{\\tfDecayRate}{\\delta}\n", "\\newcommand{\\tfMrnaConcentration}{f}\n", "\\newcommand{\\tfVector}{\\mathbf{ \\tfConcentration}}\n", "\\newcommand{\\velocity}{v}\n", "\\newcommand{\\sufficientStatsScalar}{g}\n", "\\newcommand{\\sufficientStatsVector}{\\mathbf{ \\sufficientStatsScalar}}\n", "\\newcommand{\\sufficientStatsMatrix}{\\mathbf{G}}\n", "\\newcommand{\\switchScalar}{s}\n", "\\newcommand{\\switchVector}{\\mathbf{ \\switchScalar}}\n", "\\newcommand{\\switchMatrix}{\\mathbf{S}}\n", "\\newcommand{\\tr}[1]{\\text{tr}\\left(#1\\right)}\n", "\\newcommand{\\loneNorm}[1]{\\left\\Vert #1 \\right\\Vert_1}\n", "\\newcommand{\\ltwoNorm}[1]{\\left\\Vert #1 \\right\\Vert_2}\n", "\\newcommand{\\onenorm}[1]{\\left\\vert#1\\right\\vert_1}\n", "\\newcommand{\\twonorm}[1]{\\left\\Vert #1 \\right\\Vert}\n", "\\newcommand{\\vScalar}{v}\n", "\\newcommand{\\vVector}{\\mathbf{v}}\n", "\\newcommand{\\vMatrix}{\\mathbf{V}}\n", "\\newcommand{\\varianceDist}[2]{\\text{var}_{#2}\\left( #1 \\right)}\n", "% Already defined by latex\n", "%\\newcommand{\\vec}{#1:}\n", "\\newcommand{\\vecb}[1]{\\left(#1\\right):}\n", "\\newcommand{\\weightScalar}{w}\n", "\\newcommand{\\weightVector}{\\mathbf{ \\weightScalar}}\n", "\\newcommand{\\weightMatrix}{\\mathbf{W}}\n", "\\newcommand{\\weightedAdjacencyMatrix}{\\mathbf{A}}\n", "\\newcommand{\\weightedAdjacencyScalar}{a}\n", "\\newcommand{\\weightedAdjacencyVector}{\\mathbf{ \\weightedAdjacencyScalar}}\n", "\\newcommand{\\onesVector}{\\mathbf{1}}\n", "\\newcommand{\\zerosVector}{\\mathbf{0}}\n", "$$\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "## Review\n", "\n", "- Last time: Looked at objective functions for movie recommendation.\n", "- Minimized sum of squares objective by steepest descent and\n", " stochastic gradients.\n", "- This time: explore least squares for regression.\n", "\n", "## Regression Examples \\[edit\\]\n", "\n", "Regression involves predicting a real value, $\\dataScalar_i$, given an\n", "input vector, $\\inputVector_i$. For example, the Tecator data involves\n", "predicting the quality of meat given spectral measurements. Or in\n", "radiocarbon dating, the C14 calibration curve maps from radiocarbon age\n", "to age measured through a back-trace of tree rings. Regression has also\n", "been used to predict the quality of board game moves given expert rated\n", "training data.\n", "\n", "## Olympic 100m Data\n", "\n", "- Gold medal times for Olympic 100 m runners since 1896.\n", " \n", "
\n", "- Gold medal times for Olympic Marathon since 1896.\n", "- Marathons before 1924 didn't have a standardised distance.\n", "- Present results using pace per km.\n", "- In 1904 Marathon was badly organised leading to very slow times.\n", "\n", " | \n", "\n",
"\n",
"Image from Wikimedia Commons | \n",
"