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Vaibhavdixit02/JuliaCon.ipynb

Created July 15, 2018 11:00
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JuliaCon.ipynb
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"source": [
"## Introduction"
]
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{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
"using OrdinaryDiffEq, ParameterizedFunctions, DiffEqBayes, Plots, RecursiveArrayTools"
]
},
{
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"Parameter Estimation in one line can be described as obtaining parameters for mathematical models by data fitting. \n",
"\n",
"Let's discuss what that implies specifically in context of our interest here, which are Differential Equation Models. The general structure of a differential equation model can be described as \n",
"$$\\mathbf{\\dot{x}}(\\textit{t}) = f(\\textbf{x}(\\textit{t}), \\textbf{u}, \\textit{t} \\hspace{0.1cm}|\\hspace{0.1cm} \\theta)$$\n",
"\n",
"Here, $t$ denotes the independent variable, usually time, $\\theta$ is a p-dimensional vector of unknown parameters, $x = x(t,\\theta)$ is an n-dimensional state variable vector depending on $t$ and $\\theta$, and thus the function $f(t,x,θ)$ maps $R×R^n×R^p$ into $R^n$.\n",
"\n",
"We are interested in obtaining this vector $\\theta$ with the help of data describing the trajectory of the solution of our equation with some noise, to give you a more intuitive understanding of how different parameter values affect the solutions let's take a look at FitzHugh-Nagumo model which is often used as a generic model for excitable media such as a neuron."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
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"source": [
"fitz = @ode_def_nohes FitzhughNagumo begin\n",
" dV = c*(V - V^3/3 + R)\n",
" dR = -(1/c)*(V - a - b*R)\n",
"end a b c\n",
"u0 = [-1.0;1.0]\n",
"tspan = (0.0,20.0)\n",
"p = [0.2,0.2,3.0]\n",
"prob = ODEProblem(fitz,u0,tspan,p)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### With $\\theta$ as [0.2,0.2,3.0]"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
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},
"execution_count": 7,
"metadata": {},
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}
],
"source": [
"sol = solve(prob,Tsit5())\n",
"plot(sol)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### With $\\theta$ as [0.5,0.5,5.0]"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
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],
"source": [
"p = [1.0,1.0,5.0]\n",
"prob = ODEProblem(fitz,u0,tspan,p)\n",
"sol = solve(prob,Tsit5())\n",
"plot(sol)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's suppose you know that the model that you want to use is FitzHugh-Nagumo, you have some data points with some noise as depicted below."
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
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"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t = collect(linspace(1,20,20))\n",
"rand_sol = VectorOfArray([(sol(t[i]) + .1randn(2)) for i in 1:length(t)])\n",
"scatter(t,rand_sol[1,:],color=:red)\n",
"scatter!(t,rand_sol[2,:],color=:green)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So our problem now becomes pretty much this, what values of $\\theta$ can we get such that the model adequately describes this exact phenomena given that we are sure that the model is correct, this part of the scientific modelling pipeline is parameter estimation."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's look at this problem from a Bayesian Perspective."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"ename": "LoadError",
"evalue": "\u001b[91msyntax: extra token \"inverse\" after end of expression\u001b[39m",
"output_type": "error",
"traceback": [
"\u001b[91msyntax: extra token \"inverse\" after end of expression\u001b[39m",
""
]
}
],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
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