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Orbits/Code/TransitionDerivatives.ipynb

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{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "committed-cincinnati",
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"import torch\n",
"from torch.autograd.functional import jacobian"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "accepting-valentine",
"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": "reserved-parallel",
"metadata": {},
"outputs": [],
"source": [
"launch = torch.ones(5, requires_grad=True)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "expected-consensus",
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"debris = torch.tensor([2.2],requires_grad=True)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "cross-drain",
"metadata": {},
"outputs": [],
"source": [
"#Parameters\n",
"SCALING = torch.ones(5)\n",
"DELTA = 0.9 \n",
"launch_debris_rate = 0.005\n",
"collision_debris_rate = 0.0007"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "charitable-frost",
"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",
" \n",
" new_stock = stock*survival(stock,debris) + launches\n",
" \n",
" #TODO: Currently 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": "nonprofit-vintage",
"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([[4.2202e-05, 4.2202e-05, 4.2202e-05, 4.2202e-05, 1.2820e-05]]),\n",
" tensor([[0.0997]]),\n",
" tensor([[0.0050, 0.0050, 0.0050, 0.0050, 0.0050]])))"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#Examples\n",
"survival(stocks,debris)\n",
"\n",
"#Get the derivatives seperately\n",
"jacobian(survival, (stocks,debris))\n",
"\n",
"#Get the derivatives as a single result\n",
"torch.cat(jacobian(survival, (stocks,debris)), axis=1)\n",
"\n",
"#Testing state updates\n",
"g(stocks, debris, launch)\n",
"\n",
"#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": "future-greenhouse",
"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": 8,
"id": "higher-brook",
"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\n",
"\n",
"w = torch.zeros(5)\n",
"w[0]=1"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "approximate-motivation",
"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)\n",
"\n",
"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))\n",
"\n",
"#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": 10,
"id": "prescribed-unemployment",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([ 0.0372, -1.2487, 0.0372, 0.0372, 0.0164])"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"single_transition(w,g,util,stocks,debris,launch,beta)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "ordinary-admission",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"tensor([ 0.0372, -1.2487, 0.0372, 0.0372, 0.0164])\n",
"tensor([-1.0135, -2.3671, 0.0391, 0.0391, 0.0172])\n",
"tensor([-2.1195, -3.5443, 0.0412, 0.0412, 0.0181])\n"
]
}
],
"source": [
"r1 = recurse_transitions(w, g, util, stocks, debris, launch, beta, 1)\n",
"r2 = recurse_transitions(w, g, util, stocks, debris, launch, beta, 2)\n",
"r3 = recurse_transitions(w, g, util, stocks, debris, launch, beta, 3)\n",
"\n",
"print(r1)\n",
"print(r2)\n",
"print(r3)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "committed-porcelain",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(tensor([1.8600, 1.8600, 1.8600, 1.8600, 1.4401], grad_fn=<AddBackward0>),\n",
" tensor([0.2454], grad_fn=<AddBackward0>),\n",
" tensor([2.8600, 2.8600, 2.8600, 2.8600, 2.4401], grad_fn=<AddBackward0>),\n",
" tensor([0.0495], grad_fn=<AddBackward0>))"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"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)\n",
"stocks_1,debris_1,stocks_2,debris_2"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "touched-judge",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"tensor([ 0.0372, -1.2487, 0.0372, 0.0372, 0.0164])\n",
"tensor([-1.0135, -2.3671, 0.0391, 0.0391, 0.0172])\n",
"tensor([-2.1195, -3.5443, 0.0412, 0.0412, 0.0181])\n"
]
}
],
"source": [
"#Recurse 1\n",
"sin1 = single_transition(w,g,util,stocks,debris,launch,beta)\n",
"#Recurse 2\n",
"sin2 = single_transition(sin1 ,g,util,stocks_1,debris_1,launch,beta)\n",
"#recurse 3\n",
"sin3 = single_transition(sin2,g,util,stocks_2,debris_2,launch,beta)\n",
"\n",
"print(sin1)\n",
"print(sin2)\n",
"print(sin3)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "cooked-absence",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-2.1195, -3.5443, 0.0412, 0.0412, 0.0181])"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"recurse_transitions(sin2, g, util, stocks_2, debris_2, launch, beta, 1)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "accepting-grade",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-2.1195, -3.5443, 0.0412, 0.0412, 0.0181])"
]
},
"execution_count": 15,
"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": "indoor-papua",
"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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