Data_Assimilation.ipynb

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    "For a system with a $D$ dimensional state variable $\\vec{x}$ (for example, $D=4$ for a NaKL neuron), the conditional probability distribution $P$ of the system's path $𝐗$ on the measurement path $𝐘$, at time $t = n+1$, is:\n",
    "\n",
    "$$P\n",
    "\\left(\n",
    "𝐗\\color{gray}{(n+1)}\n",
    "| \n",
    "𝐘\\color{gray}{(n+1)}\n",
    "\\right) \n",
    "= ℐ.ℳ.P(\n",
    "𝐗\\color{gray}{(n)}\n",
    "|\n",
    "𝐘\\color{gray}{(n)}\n",
    ")$$\n",
    "\n",
    "where\n",
    "\n",
    "𝐗: $\\{\n",
    "\\vec{x}(t_0),\\ \n",
    "\\vec{x}(t_1 = t_0 + \\Delta t),\\ \n",
    "...,\\ \n",
    "\\vec{x}(t_{f} = t_0 + N\\Delta t) \n",
    "\\}$, and thus dimensions $D.(n+1)$,\n",
    "\n",
    "𝐘: $\\{\n",
    "\\vec{y}(\\tau_0),\\ \n",
    "\\vec{y}(\\tau_1),\\ \n",
    "...,\\ \n",
    "\\vec{y}(\\tau_{f} = t_0 + kN\\Delta t) \n",
    "\\}$, and thus dimensions $L.(n/k+1)$,\n",
    "\n",
    "ℐ: represents information transfer, and\n",
    "\n",
    "ℳ: represents the model.\n",
    "\n",
    "Here, the subscripts denote time indices.\n",
    "\n",
    "Notation:\n",
    "\n",
    "$𝐗(n+1) = \\{\\vec{x}(n+1), 𝐗(n)\\}$\n",
    "\n",
    "\n",
    "$𝐘(n+1) = \\{\\vec{y}(n+1), 𝐘(n)\\}$"
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    "$$\\begin{align}\n",
    "P\\left(\n",
    "𝐗\\color{gray}{(n+1)} | 𝐘\\color{gray}{(n+1)}\\right) \n",
    "&= \\frac{\n",
    "P\\left( 𝐗\\color{gray}{(n+1)},\\ 𝐘\\color{gray}{(n+1)}\\right)}\n",
    "{P\\left(𝐘\\color{gray}{(n+1)}\\right)} \\\\\n",
    "&= \\frac{\n",
    "P\\left(\n",
    "\\vec{x}\\color{gray}{(n+1)},\\ 𝐗\\color{gray}{(n)},\\ \n",
    "\\vec{y}\\color{gray}{(n+1)},\\ 𝐘\\color{gray}{(n)}\\right)}\n",
    "{P\\left(\n",
    "\\vec{y}\\color{gray}{(n+1)}, 𝐘\\color{gray}{(n)}\\right)} \\\\\n",
    "&= \\frac{\n",
    "P\\left(\n",
    "\\vec{x}\\color{gray}{(n+1)},\\ 𝐗\\color{gray}{(n)},\\ \n",
    "\\vec{y}\\color{gray}{(n+1)} \\color{red}{ | 𝐘(n)}\\right)}\n",
    "{P\\left(\n",
    "\\vec{y}\\color{gray}{(n+1)} \\color{green}{| 𝐘(n)}\\right)}\n",
    ". \\frac{\\color{green}{P(𝐘(n))}}{\\color{red}{P(𝐘(n))}}\\\\\n",
    "&= \\frac{\n",
    "P\\left(\n",
    "\\vec{x}\\color{gray}{(n+1)},\\ 𝐗\\color{gray}{(n)},\\ \n",
    "\\vec{y}\\color{gray}{(n+1)} | 𝐘\\color{gray}{(n)}\\right)}\n",
    "{P\\left(\n",
    "\\vec{y}\\color{gray}{(n+1)} | 𝐘\\color{gray}{(n)}\\right)}\n",
    ". \\color{orange}{\n",
    "\\frac\n",
    "{P\\left(\\vec{x}(n+1) | 𝐗(n),\\ 𝐘(n)\\right)}\n",
    "{P\\left(\\vec{x}(n+1) | 𝐗(n),\\ 𝐘(n)\\right)}\n",
    "} \\\\\n",
    "&= \\frac{\n",
    "P\\left(\n",
    "\\vec{x}\\color{gray}{(n+1)},\\ 𝐗\\color{gray}{(n)},\\ \n",
    "\\vec{y}\\color{gray}{(n+1)} | 𝐘\\color{gray}{(n)}\\right)}\n",
    "{P\\left(\n",
    "\\vec{y}\\color{gray}{(n+1)} | 𝐘\\color{gray}{(n)}\\right)}\n",
    ". \\frac\n",
    "{P\\left(\\vec{x}\\color{gray}{(n+1)} | 𝐗\\color{gray}{(n)},\\ 𝐘\\color{gray}{(n)}\\right)}\n",
    "{\n",
    "\\color{purple}{\n",
    "P\\left(\\vec{x}(n+1),\\ 𝐗(n),\\ 𝐘(n)\\right)}\n",
    "}\n",
    ". \\color{purple}{P\\left(𝐗(n),\\ 𝐘(n)\\right)} \\\\\n",
    "&= \\frac{\n",
    "P\\left(\n",
    "\\vec{x}\\color{gray}{(n+1)},\\ 𝐗\\color{gray}{(n)},\\ \n",
    "\\vec{y}\\color{gray}{(n+1)} | 𝐘\\color{gray}{(n)}\\right)}\n",
    "{P\\left(\n",
    "\\vec{y}\\color{gray}{(n+1)} | 𝐘\\color{gray}{(n)}\\right)}\n",
    ". \\frac\n",
    "{P\\left(\\vec{x}\\color{gray}{(n+1)} | 𝐗\\color{gray}{(n)},\\ 𝐘\\color{gray}{(n)}\\right)}\n",
    "{\n",
    "\\color{purple}{\n",
    "P\\left(\\vec{x}(n+1),\\ 𝐗(n),\\ 𝐘(n)\\right)}\n",
    "}\n",
    ". \\frac\n",
    "{\\color{purple}{P\\left(𝐗(n),\\ 𝐘(n)\\right)}}\n",
    "{\\color{red}{P\\left(𝐘(n)\\right)}}\n",
    ". \\color{red}{P\\left(𝐘(n)\\right)} \\\\\n",
    "&= \\frac{\n",
    "P\\left(\n",
    "\\vec{x}\\color{gray}{(n+1)},\\ 𝐗\\color{gray}{(n)},\\ \n",
    "\\vec{y}\\color{gray}{(n+1)} | 𝐘\\color{gray}{(n)}\\right)}\n",
    "{P\\left(\n",
    "\\vec{y}\\color{gray}{(n+1)} | 𝐘\\color{gray}{(n)}\\right)}\n",
    ". \\frac\n",
    "{P\\left(\\vec{x}\\color{gray}{(n+1)} | 𝐗\\color{gray}{(n)},\\ 𝐘\\color{gray}{(n)}\\right)}\n",
    "{\n",
    "\\color{purple}{\n",
    "P\\left(\\vec{x}(n+1),\\ 𝐗(n),\\ 𝐘(n)\\right)}\n",
    "}\n",
    ".\\color{purple}{P\\left(𝐗(n) \\color{red}{|  𝐘(n)}\\right)}\n",
    ". \\color{red}{P\\left(𝐘(n)\\right)}\n",
    "\\end{align}$$"
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