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Added equation for relationship of h to covariance for g-h filter.

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Roger Labbe 2014-05-26 11:51:11 -07:00
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Kalman_Filters.ipynb
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"source": [
"### Relationship to the g-h Filter\n",
"\n",
"In the first chapter I stated that the Kalman filter is a form of g-h filter. However, we have been reasoning about the probability of Gaussians, and not used any of the reasoning or equations of the first chapter. A trivial amount of algebra will reveal the relationship, so let's do that now.\n",
"In the first chapter I stated that the Kalman filter is a form of g-h filter. However, we have been reasoning about the probability of Gaussians, and not used any of the reasoning or equations of the first chapter. A trivial amount of algebra will reveal the relationship, so let's do that now. It's not particularly illuminating algebra, so feel free to skip to the bottom to see the final equation that relates *g* to the variances.\n",
"\n",
"The equation for our estimate is:\n",
"\n",
@ -1236,15 +1236,20 @@
"\\begin{align*}\n",
"\\mu_{x'}&=x+ \\frac{a}{a+b}(y-x)\\\\\n",
"&= x + \\frac{\\sigma^2_{x'}}{b}(y-x) \\\\\n",
"&= x + g(y-x)\\blacksquare\n",
"&= x + g_n(y-x)\\\\\n",
"\\blacksquare\n",
"\\end{align*}\n",
"$$\n",
"\n",
"where\n",
"\n",
"$$g = \\frac{\\sigma^2_{x'}}{\\sigma^2_{y}}$$\n",
"$$g_n = \\frac{\\sigma^2_{x'}}{\\sigma^2_{y}}$$\n",
"\n",
"The end result is multipying the residual of the two measurements by a constant and adding to our previous value, which is the *g* equation for the g-h filter. *g* is the variance of the new estimate divided by the variance of the measurement. Of course in this case g is not truly a constant, as it varies with each time step as the variance changes, but it is truly the same formula. We can also derive the formula for *h* in the same way."
"The end result is multipying the residual of the two measurements by a constant and adding to our previous value, which is the *g* equation for the g-h filter. *g* is the variance of the new estimate divided by the variance of the measurement. Of course in this case g is not truly a constant, as it varies with each time step as the variance changes, but it is truly the same formula. We can also derive the formula for *h* in the same way but I don't find this a particularly interesting derivation. The end result is\n",
"\n",
"$$h_n = \\frac{COV (x,\\dot{x})}{\\sigma^2_{y}}$$\n",
"\n",
"The takeaway point is that *g* and *h* are specified fully by the variance and covariances of the measurement and preditions at time *n*. In other words, we are just picking a point between the measurement and prediction by a scale factor determined by the quality of each of those two inputs. That is all the Kalman filter is. "
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