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346 lines
13 KiB
Plaintext
346 lines
13 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "closed-glenn",
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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"import torch\n",
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"from torch.autograd.functional import jacobian\n",
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"import itertools"
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]
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},
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{
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"cell_type": "markdown",
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"id": "naval-ivory",
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"metadata": {},
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"source": [
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"# Setup Functions\n",
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"## General CompositionFunctions"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "italian-enforcement",
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"metadata": {},
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"outputs": [],
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"source": [
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"### Set up functions to compose functions \n",
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"# These functions will \n",
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"# - compose two functions together\n",
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"# - compose a function to itself n times.\n",
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"\n",
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"def compose(f,g):\n",
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" return lambda *args: f(g(*args))\n",
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"\n",
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"def compose_recursive_functions(fn,n):\n",
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" #takes a function fn and composes it with itself n times\n",
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" #Returns each of the iterations of the functions.\n",
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" \n",
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" #Set base conditions\n",
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" out_func = None\n",
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" out_func_list =[]\n",
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"\n",
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" #build the compositions of functions\n",
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" for f in itertools.repeat(fn, n):\n",
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" #if first iteration\n",
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" if out_func == None:\n",
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" out_func = f\n",
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" else:\n",
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" out_func = compose(f,out_func)\n",
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"\n",
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" #append the ou\n",
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" out_func_list.append(out_func)\n",
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" \n",
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" return out_func_list"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fancy-tucson",
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"metadata": {},
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"source": [
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"## Setup functions related to the problem"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"id": "outside-arrangement",
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"metadata": {},
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"outputs": [],
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"source": [
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"### Background functions\n",
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"\n",
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"\n",
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"def survival(stock, debris):\n",
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" #Gompertz distribution for simplicity\n",
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" #commonly used with saturation\n",
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" #TODO: ACTUALLY DERIVE A SURVIVAL FUNCTION. THIS IS JUST A PLACEHOLDER. PROBABLY SHOULD BE AN EXPONENTIAL DISTRIBUTION\n",
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" \n",
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" #eta = 1.0/(SCALING@stock)\n",
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" #b = 1/debris\n",
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" #return 1 - ( b*eta*torch.exp(eta+b*stock-eta*torch.exp(b*stock)))\n",
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" return 1 - torch.exp(-SCALING * stock-debris)\n",
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"\n",
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"def test_launch(stock, debris):\n",
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" return torch.ones(5, requires_grad=True)\n",
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"\n",
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"def laws_of_motion(stock, debris, launches):\n",
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" \n",
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" new_stock = stock*survival(stock,debris) + launches#(stock,debris) #TODO: Launches will become a function (neural network)\n",
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" \n",
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" #TODO: Currently Ignoring autocatalysis\n",
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" new_debris = (1-DELTA)*debris + LAUNCH_DEBRIS_RATE * launches.sum() + COLLISION_DEBRIS_RATE*(1-survival(stock,debris)) @ stock\n",
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" \n",
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" return (new_stock, new_debris)\n",
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"\n",
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"#This is not a good specification of the profit function, but it will work for now.\n",
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"def profit(x):\n",
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" return UTIL_WEIGHTS @ x"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"id": "romance-generation",
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"metadata": {},
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"outputs": [],
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"source": [
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"def single_transition(item_to_iterate, laws_motion, profit,launch, stocks, debris ):\n",
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" #TODO: change launch from a direct tensor, to a function.\n",
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" \n",
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" #Calculate the inverse\n",
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" bA = BETA * jacobian(laws_motion, (stocks,debris,launch))[0][0]\n",
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" #TODO: figure out some diagnostics for this section\n",
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" \n",
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" \n",
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" return bA.inverse() @ (item_to_iterate - jacobian(profit,stocks))\n",
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"\n",
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"# This function wraps the single transition and handles updating dates etc.\n",
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"def transition_wrapper(data_in):\n",
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" #unpack states and functions\n",
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" stocks, debris,profit, launch, laws_motion, item_to_transition = data_in\n",
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" \n",
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" #Calculate new states\n",
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" new_stocks, new_debris = laws_motion(stocks,debris,launch)\n",
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" \n",
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" #WARNING: RECURSION: You may break your head...\n",
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" #This gets the transition of the value function derivatives over time.\n",
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" transitioned = single_transition(\n",
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" item_to_transition, #item to iterate, i.e. the derivatives of the value function\n",
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" #functions\n",
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" laws_motion, \n",
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" profit, \n",
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" launch, #TODO: reimplement with launch as a function\n",
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" stocks, debris #states\n",
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" )\n",
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" \n",
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" #collects the data back together for return, including the updated state variables\n",
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" data_out = new_stocks, new_debris, profit, launch, laws_motion, transitioned\n",
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" \n",
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" return data_out"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fluid-parks",
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"metadata": {},
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"source": [
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"# Actual calculations"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"id": "changing-january",
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"metadata": {},
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"outputs": [],
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"source": [
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"#set states\n",
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"stocks = torch.ones(5)\n",
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"#Last one is different\n",
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"stocks[-1] = 0.5\n",
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"#now add the tracking requirement in place\n",
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"stocks.requires_grad_()\n",
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"\n",
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"#Setup Debris\n",
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"debris = torch.tensor([2.2],requires_grad=True)\n",
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"\n",
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"#CHANGE LATER: Launch is currently a value, should be a function (i.e. neural network)\n",
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"launch = torch.ones(5, requires_grad=True)\n",
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"\n",
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"#compose the functions together.\n",
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"base_data = (stocks,debris, profit, launch, laws_of_motion, torch.ones(5, requires_grad=True))\n",
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"\n",
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"#Parameters\n",
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"SCALING = torch.ones(5)\n",
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"DELTA = 0.9 \n",
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"LAUNCH_DEBRIS_RATE = 0.005\n",
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"COLLISION_DEBRIS_RATE = 0.0007\n",
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"UTIL_WEIGHTS = torch.tensor([1,-0.2,0,0,0])\n",
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"BETA = 0.95"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 19,
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"id": "dominant-boost",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"tensor([0.0000, 1.2632, 1.0526, 1.0526, 1.0892], grad_fn=<MvBackward>)\n",
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"tensor([-0.9519, 1.3928, 1.0020, 1.0020, 1.0575], grad_fn=<MvBackward>)\n",
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"tensor([-1.8565, 1.5150, 0.9530, 0.9530, 0.9882], grad_fn=<MvBackward>)\n",
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"tensor([-2.8103, 1.6872, 0.9376, 0.9376, 0.9474], grad_fn=<MvBackward>)\n",
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"tensor([-3.8626, 1.9131, 0.9505, 0.9505, 0.9408], grad_fn=<MvBackward>)\n",
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"tensor([-5.0235, 2.1830, 0.9819, 0.9819, 0.9598], grad_fn=<MvBackward>)\n",
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"tensor([-6.2860, 2.4869, 1.0247, 1.0247, 0.9951], grad_fn=<MvBackward>)\n",
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"tensor([-7.6403, 2.8175, 1.0746, 1.0746, 1.0403], grad_fn=<MvBackward>)\n",
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"tensor([-9.0802, 3.1712, 1.1293, 1.1293, 1.0918], grad_fn=<MvBackward>)\n",
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"tensor([-10.6035, 3.5462, 1.1879, 1.1879, 1.1477],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-12.2108, 3.9422, 1.2501, 1.2501, 1.2075],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-13.9045, 4.3597, 1.3157, 1.3157, 1.2708],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-15.6882, 4.7995, 1.3849, 1.3849, 1.3375],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-17.5662, 5.2625, 1.4577, 1.4577, 1.4079],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-19.5432, 5.7500, 1.5345, 1.5345, 1.4820],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-21.6244, 6.2631, 1.6152, 1.6152, 1.5600],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-23.8151, 6.8033, 1.7002, 1.7002, 1.6421],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-26.1212, 7.3719, 1.7897, 1.7897, 1.7285],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-28.5486, 7.9704, 1.8839, 1.8839, 1.8195],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-31.1038, 8.6004, 1.9831, 1.9831, 1.9152],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-33.7934, 9.2636, 2.0874, 2.0874, 2.0160],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-36.6247, 9.9617, 2.1973, 2.1973, 2.1221],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-39.6049, 10.6965, 2.3129, 2.3129, 2.2338],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-42.7420, 11.4700, 2.4347, 2.4347, 2.3514],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-46.0442, 12.2842, 2.5628, 2.5628, 2.4751],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-49.5202, 13.1413, 2.6977, 2.6977, 2.6054],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-53.1792, 14.0435, 2.8397, 2.8397, 2.7425],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-57.0307, 14.9931, 2.9891, 2.9891, 2.8869],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-61.0850, 15.9928, 3.1465, 3.1465, 3.0388],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-65.3526, 17.0450, 3.3121, 3.3121, 3.1988],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-69.8449, 18.1526, 3.4864, 3.4864, 3.3671],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-74.5736, 19.3186, 3.6699, 3.6699, 3.5443],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-79.5511, 20.5459, 3.8630, 3.8630, 3.7309],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-84.7907, 21.8378, 4.0664, 4.0664, 3.9272],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-90.3060, 23.1976, 4.2804, 4.2804, 4.1339],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-96.1115, 24.6291, 4.5057, 4.5057, 4.3515],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-102.2227, 26.1359, 4.7428, 4.7428, 4.5805],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-108.6555, 27.7220, 4.9924, 4.9924, 4.8216],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-115.4268, 29.3916, 5.2552, 5.2552, 5.0754],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-122.5545, 31.1490, 5.5318, 5.5318, 5.3425],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-130.0574, 32.9990, 5.8229, 5.8229, 5.6237],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-137.9552, 34.9463, 6.1294, 6.1294, 5.9197],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-146.2686, 36.9961, 6.4520, 6.4520, 6.2313],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-155.0196, 39.1538, 6.7916, 6.7916, 6.5592],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-164.2312, 41.4251, 7.1490, 7.1490, 6.9044],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-173.9275, 43.8158, 7.5253, 7.5253, 7.2678],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-184.1343, 46.3325, 7.9213, 7.9213, 7.6503],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-194.8782, 48.9815, 8.3382, 8.3382, 8.0530],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-206.1876, 51.7701, 8.7771, 8.7771, 8.4768],\n",
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" grad_fn=<MvBackward>)\n",
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"tensor([-218.0922, 54.7053, 9.2391, 9.2391, 8.9230],\n",
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" grad_fn=<MvBackward>)\n"
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]
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}
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],
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"source": [
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"#calculate results for first 5 iterations\n",
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"for f in compose_recursive_functions(transition_wrapper,50):\n",
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" result = f(base_data)\n",
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" print(result[5])"
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]
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},
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{
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"cell_type": "markdown",
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"id": "unnecessary-architect",
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"metadata": {},
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"source": [
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"Note how this fails on the last few iterations.\n",
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"I need to get better model functions (profit, laws_of_motion, etc) together to test this out.\n",
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"Alternatively, I can check for areas where the determinant of $A$ is zero, possibly by doing some sort of grid search?\n",
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"\n",
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"Maybe with a standard RBC model?\n",
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"\n",
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"Also, maybe I can create a `Model` class that upon construction will capture the necesary constants, functions, etc."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "varying-organization",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.8"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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