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Orbits/Code/TransitionDerivatives.ipynb

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{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "innovative-filename",
"metadata": {},
"outputs": [],
"source": [
"import torch\n",
"from torch.autograd.functional import jacobian"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "filled-question",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([1.0000, 1.0000, 1.0000, 1.0000, 0.5000], requires_grad=True)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#set states\n",
"stocks = torch.ones(5)\n",
"#Last one is different\n",
"stocks[-1] = 0.5\n",
"#now add the tracking requirement in place\n",
"stocks.requires_grad_()"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "middle-lease",
"metadata": {},
"outputs": [],
"source": [
"launch = torch.ones(5, requires_grad=True)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "regular-grounds",
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"debris = torch.tensor([2.2],requires_grad=True)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "satisfied-hawaiian",
"metadata": {},
"outputs": [],
"source": [
"scaling = torch.ones(5)\n",
"delta = 0.9 \n",
"launch_debris_rate = 0.05\n",
"collision_debris_rate = 0.07"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "saved-corporation",
"metadata": {},
"outputs": [],
"source": [
"def survival(stock, debris):\n",
" #Gompertz distribution for simplicity\n",
" #commonly used with saturation\n",
" #TODO: ACTUALLY DERIVE A SURVIVAL FUNCTION. THIS IS JUST A PLACEHOLDER. PROBABLY SHOULD BE AN EXPONENTIAL DISTRIBUTION\n",
" eta = 1.0/(scaling@stock)\n",
" b = 1/debris\n",
" \n",
" return 1 - ( b*eta*torch.exp(eta+b*stock-eta*torch.exp(b*stock)))\n",
"\n",
"\n",
"def g(stock, debris, launches):\n",
" new_stock = stock*survival(stock,debris) + launches\n",
" \n",
" #Ignoring autocatalysis\n",
" new_debris = (1-delta)*debris + launch_debris_rate * launches.sum() + collision_debris_rate*(1-survival(stock,debris)) @ stock\n",
" \n",
" return (new_stock, new_debris)\n"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "nominated-visitor",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([0.8600, 0.8600, 0.8600, 0.8600, 0.8802], grad_fn=<RsubBackward1>)"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"survival(stocks,debris)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "clean-panama",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(tensor([[-0.0142, 0.0271, 0.0271, 0.0271, 0.0271],\n",
" [ 0.0271, -0.0142, 0.0271, 0.0271, 0.0271],\n",
" [ 0.0271, 0.0271, -0.0142, 0.0271, 0.0271],\n",
" [ 0.0271, 0.0271, 0.0271, -0.0142, 0.0271],\n",
" [ 0.0251, 0.0251, 0.0251, 0.0251, -0.0142]]),\n",
" tensor([[0.0825],\n",
" [0.0825],\n",
" [0.0825],\n",
" [0.0825],\n",
" [0.0634]]))"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#Get the derivatives seperately\n",
"jacobian(survival, (stocks,debris))"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "cardiovascular-music",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([[-0.0142, 0.0271, 0.0271, 0.0271, 0.0271, 0.0825],\n",
" [ 0.0271, -0.0142, 0.0271, 0.0271, 0.0271, 0.0825],\n",
" [ 0.0271, 0.0271, -0.0142, 0.0271, 0.0271, 0.0825],\n",
" [ 0.0271, 0.0271, 0.0271, -0.0142, 0.0271, 0.0825],\n",
" [ 0.0251, 0.0251, 0.0251, 0.0251, -0.0142, 0.0634]])"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#Get the derivatives as a single result\n",
"torch.cat(jacobian(survival, (stocks,debris)), axis=1)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "funky-illness",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(tensor([1.8600, 1.8600, 1.8600, 1.8600, 1.4401], grad_fn=<AddBackward0>),\n",
" tensor([0.5134], grad_fn=<AddBackward0>))"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#Testing state updates\n",
"g(stocks, debris, launch)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "charged-dairy",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((tensor([[0.8457, 0.0271, 0.0271, 0.0271, 0.0271],\n",
" [0.0271, 0.8457, 0.0271, 0.0271, 0.0271],\n",
" [0.0271, 0.0271, 0.8457, 0.0271, 0.0271],\n",
" [0.0271, 0.0271, 0.0271, 0.8457, 0.0271],\n",
" [0.0126, 0.0126, 0.0126, 0.0126, 0.8731]]),\n",
" tensor([[0.0825],\n",
" [0.0825],\n",
" [0.0825],\n",
" [0.0825],\n",
" [0.0317]]),\n",
" tensor([[1., 0., 0., 0., 0.],\n",
" [0., 1., 0., 0., 0.],\n",
" [0., 0., 1., 0., 0.],\n",
" [0., 0., 0., 1., 0.],\n",
" [0., 0., 0., 0., 1.]])),\n",
" (tensor([[0.0042, 0.0042, 0.0042, 0.0042, 0.0013]]),\n",
" tensor([[0.0747]]),\n",
" tensor([[0.0500, 0.0500, 0.0500, 0.0500, 0.0500]])))"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#Note the two tuples of jacobians: the first is for stock evolution, the second is for debris evolution\n",
"jacobian(g, (stocks,debris,launch))"
]
},
{
"cell_type": "markdown",
"id": "marked-flower",
"metadata": {},
"source": [
"## Next step: Construct the intertemporal-transition function(s)\n",
" - Note: There are a couple of different ways to do this\n",
" - Just a single period transition function, manually iterated\n",
" - A recursive function that creates a $p$ period iterated function\n",
" - A recursive function that returns a list of functions iterated from 1 to $p$ periods\n",
"\n",
"I am planning on doing the latter, as each version is needed."
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "outdoor-action",
"metadata": {},
"outputs": [],
"source": [
"#setup\n",
"beta = torch.tensor([0.95])\n",
"util_weights = torch.tensor([1.0,1.0,0,0,0])\n",
"sigma = 0.5"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "unavailable-hawaii",
"metadata": {},
"outputs": [],
"source": [
"#This is not a good specification of the profit function, but it will work for now.\n",
"def util(x):\n",
" return (util_weights@x)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "composite-cooperative",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor(2., grad_fn=<DotBackward>)"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"util(stocks)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "activated-advancement",
"metadata": {},
"outputs": [],
"source": [
"w = torch.zeros(5)\n",
"w[0]=1"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "wireless-concord",
"metadata": {},
"outputs": [],
"source": [
"def single_transition(w, g, util, stocks, debris, launch, beta):\n",
" #TODO: change launch from a direct tensor, to a function.\n",
" bA = beta * jacobian(g, (stocks,debris,launch))[0][0]\n",
" return bA.inverse() @ (w - jacobian(util,stocks))"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "detected-cooking",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([ 0.0372, -1.2487, 0.0372, 0.0372, 0.0164])"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"single_transition(w,g,util,stocks,debris,launch,beta)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "sharp-pound",
"metadata": {},
"outputs": [],
"source": [
"#need to create recursive transitions\n",
"def recurse_transitions(w, g, util, stocks, debris, launch, beta, iterations):\n",
" #This is of type two from the discussion above\n",
" if iterations <= 1:\n",
" return single_transition(w, g, util, stocks, debris, launch, beta)\n",
" else: \n",
" #Get recursive results\n",
" curse = recurse_transitions(w,g,util,stocks,debris,launch,beta,iterations-1)\n",
" \n",
" #Get updated stocks and debris\n",
" stocks_iterated,debris_iterated = g(stocks, debris, launch) \n",
" \n",
" #Return the updated values\n",
" return single_transition(curse, g, util, stocks_iterated, debris_iterated, launch, beta)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "radical-reason",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([ 0.0372, -1.2487, 0.0372, 0.0372, 0.0164])"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"recurse_transitions(w, g, util, stocks, debris, launch, beta, 1)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "intelligent-angle",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-0.4740, -1.0278, -0.0434, -0.0434, -0.0575])"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"recurse_transitions(w, g, util, stocks, debris, launch, beta, 2)"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "jewish-lemon",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-0.7046, -0.9430, -0.0884, -0.0884, -0.1102])"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"recurse_transitions(w, g, util, stocks, debris, launch, beta, 3)"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "scenic-literature",
"metadata": {},
"outputs": [],
"source": [
"#TODO: manually check single_transition\n",
"stocks_1,debris_1 = g(stocks, debris, launch) \n",
"stocks_2,debris_2 = g(stocks_1, debris_1, launch)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "moving-slave",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([ 0.0372, -1.2487, 0.0372, 0.0372, 0.0164])"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#Recurse 1\n",
"sin1 = single_transition(w,g,util,stocks,debris,launch,beta)\n",
"sin1"
]
},
{
"cell_type": "code",
"execution_count": 31,
"id": "corrected-radar",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-0.4740, -1.0278, -0.0434, -0.0434, -0.0575])"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#Recurse 2\n",
"sin2 = single_transition(sin1 \n",
" ,g\n",
" ,util\n",
" ,stocks_1\n",
" ,debris_1\n",
" ,launch,beta)\n",
"sin2"
]
},
{
"cell_type": "code",
"execution_count": 36,
"id": "handy-pencil",
"metadata": {
"tags": []
},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-1.5516, -2.1345, -0.0456, -0.0456, -0.1302])"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#recurse 3\n",
"single_transition(sin2,g,util,stocks_2,debris_2,launch,beta)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"id": "attempted-affairs",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-1.5516, -2.1345, -0.0456, -0.0456, -0.1302])"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"recurse_transitions(sin2, g, util, stocks_2, debris_2, launch, beta, 1)"
]
},
{
"cell_type": "code",
"execution_count": 39,
"id": "honest-diana",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-0.7046, -0.9430, -0.0884, -0.0884, -0.1102])"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#I think this highlights the error.\n",
"recurse_transitions(recurse_transitions(sin1, g, util, stocks_1, debris_1, launch, beta, 1), g, util, stocks_1, debris_1, launch, beta, 1)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "animated-nudist",
"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": "3.8.8"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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