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vankesteren/adam_julia_regression.ipynb

Created November 25, 2019 09:20
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adam_julia_regression.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Arbitrary functional regression with Adam and Julia\n",
"\n",
"This notebook shows how to perform regression estimation with an arbitrary functional form of the outcome variable in Julia."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using Zygote\n",
"using HTTP: request\n",
"\"https://gist.githubusercontent.com/vankesteren/96207abcd16ecd01a2491bcbec12c73f/raw/1b59af6962a1107db5873eba59054acc3f9a8aac/Adamopt.jl\" |>\n",
" url -> request(\"GET\", url) |> \n",
" res -> String(res.body) |> \n",
" str -> include_string(Main, str)\n",
"using .Adamopt\n",
"using Plots"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"First, we create some input data and we determine a functional form"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"fn (generic function with 1 method)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# data size\n",
"N = 1000\n",
"P = 10\n",
"\n",
"# input data\n",
"X = randn(N, P)\n",
"b = rand(P)\n",
"\n",
"# functional form\n",
"fn(x) = abs(x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then, we generate output data with random normal noise"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
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"source": [
"# output data\n",
"y = fn.(X * b) + randn(N) / 2\n",
"\n",
"# plot\n",
"scatter(X * b, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then we can create a loss function on the basis of the data, the functional form, and parameter estimates"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"mse (generic function with 4 methods)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# MSE for model\n",
"function mse(bhat, X = X, y = y, fn = fn)\n",
" res = y - fn.(X * bhat)\n",
" res'res / length(res)\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using Zygote.jl, we can automatically create a gradient function for this loss and precompile it by running it once"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10-element Array{Float64,1}:\n",
" -0.04484990305012533 \n",
" -0.1510650284785526 \n",
" -0.0978044271507378 \n",
" 0.11279328423456693 \n",
" -0.127243714103672 \n",
" -0.06342430977782527 \n",
" -0.07727450179851326 \n",
" -0.2558884279308273 \n",
" 0.1289947107469941 \n",
" 0.004372948353504059"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# create gradient function\n",
"grad(bhat; X = X, y = y, fn = fn) = gradient(mse, bhat, X, y, fn)[1]\n",
"\n",
"# run gradient function once\n",
"g = grad(zeros(P))\n",
"g"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then, we can use the adam implementation from [this gist](https://gist.github.com/vankesteren/96207abcd16ecd01a2491bcbec12c73f) to optimise the parameter estimates"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Step: 500 | Loss:1.6592273394645338\n",
"Step: 1000 | Loss:1.3405064681652905\n",
"Step: 1500 | Loss:0.6615545793593625\n",
"Step: 2000 | Loss:0.34229229998674476\n",
"Step: 2500 | Loss:0.2463537345002543\n",
"Step: 3000 | Loss:0.22680909644350958\n",
"Step: 3500 | Loss:0.22484339990834676\n",
"Step: 4000 | Loss:0.22472726785251848\n",
"Step: 4500 | Loss:0.22472018794240636\n",
"Step: 5000 | Loss:0.22472014925233963\n"
]
}
],
"source": [
"# optimize with adam\n",
"opt = Adam(randn(P), mse, grad)\n",
"opt.a = 0.0005\n",
"for i = 1:5000\n",
" step!(opt)\n",
" if (i % 500 == 0) \n",
" print(string(\"Step: \", opt.t, \" | Loss:\", opt.loss(opt.theta), \"\\n\"))\n",
" end\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then, we can compare the obtained parameter estimates to the true parameter estimates"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10×2 Array{Float64,2}:\n",
" 0.306539 0.316267 \n",
" 0.00602553 0.000868093\n",
" 0.918709 0.935933 \n",
" 0.0992748 0.105651 \n",
" 0.223801 0.183828 \n",
" 0.44857 0.423375 \n",
" 0.87722 0.87261 \n",
" 0.710273 0.729783 \n",
" 0.214625 0.202979 \n",
" 0.0157282 0.0130304 "
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"hcat(opt.theta, b)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Last, let's look at the residuals"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
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]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"scatter(fn.(X * opt.theta), y - fn.(X * opt.theta), color = \"green\", ylab = \"residual\", xlab = \"prediction\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Looking good!"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.2.0",
"language": "julia",
"name": "julia-1.2"
},
"language_info": {
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"mimetype": "application/julia",
"name": "julia",
"version": "1.2.0"
}
},
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}
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