daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Made Gaussian equation more readable.

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Roger Labbe 2015-07-22 11:56:32 -07:00
parent 566f43e32f
commit e7954be704
1 changed files with 3 additions and 2 deletions
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5
05-Multivariate-Gaussians.ipynb
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@ -357,7 +357,8 @@
"source": [
"Now, without explanation, here is the multivariate normal distribution in $n$ dimensions.\n",
"\n",
"$$f(\\mathbf{x},\\, \\mu,\\,\\Sigma) = \\frac{1}{(2\\pi)^{\\frac{n}{2}}|\\Sigma|^{\\frac{1}{2}}}\\, \\exp \\Big [{ -\\frac{1}{2}(\\mathbf{x}-\\mu)^\\mathsf{T}\\Sigma^{-1}(\\mathbf{x}-\\mu) \\Big ]}\n",
"$$\n",
"f(\\mathbf{x},\\, \\mu,\\,\\Sigma) = \\frac{1}{\\sqrt{(2\\pi)^n|\\Sigma|}}\\, \\exp \\Big [{ -\\frac{1}{2}(\\mathbf{x}-\\mu)^\\mathsf{T}\\Sigma^{-1}(\\mathbf{x}-\\mu) \\Big ]}\n",
"$$\n",
"\n",
"I urge you to not try to remember this equation. We will program it in a Python function and then call it if we need to compute a specific value. Plus, the Kalman filter equations compute this for us automatically; we never have to explicitly compute it. However, note that it has the same form as the univariate normal distribution. It uses matrices instead of scalar values, and the root of $\\pi$ is scaled by $n$. If you set n=1 then it turns into the univarate equation. Here is the univariate equation for reference:\n",
@ -1427,7 +1428,7 @@
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