daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Merge pull request #198 from slayoo/slayoo-patch-5

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is the ... elements -> are the ... elements
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Roger Labbe 2018-01-10 09:21:23 -08:00 committed by GitHub
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06-Multivariate-Kalman-Filters.ipynb
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@ -2296,7 +2296,7 @@
"\n",
"The prediction $\\bar x$ was computed as $x + \\bar x \\Delta t$. If the prediction was perfect then the residual will be $y=0$ (ignoring noise in the measurement) and the velocity estimate will be unchanged. On the other hand, if the velocity estimate was very bad then the prediction will be very bad, and the residual will be large: $y >> 0$. In this case we update the velocity estimate with $yK_{\\dot x}$. $K_{\\dot x}$ is proportional to $COV(x,\\dot x)$. Therefore the velocity is updated by the error in the position times the value proportional to the covariance between the position and velocity. The higher the correlation the larger the correction. \n",
"\n",
"To bring this full circle, $COV(x,\\dot x)$ is the off-diagonal elements of $\\mathbf P$. Recall that those values were computed with $\\mathbf{FPF}^\\mathsf T$. So the covariance of position and velocity is computed during the predict step. The Kalman gain for the velocity is proportional to this covariance, and we adjust the velocity estimate based on how inaccurate it was during the last epoch times a value proportional to this covariance. \n",
"To bring this full circle, $COV(x,\\dot x)$ are the off-diagonal elements of $\\mathbf P$. Recall that those values were computed with $\\mathbf{FPF}^\\mathsf T$. So the covariance of position and velocity is computed during the predict step. The Kalman gain for the velocity is proportional to this covariance, and we adjust the velocity estimate based on how inaccurate it was during the last epoch times a value proportional to this covariance. \n",
"\n",
"In summary, these linear algebra equations may be unfamiliar to you, but computation is actually very simple. It is essentially the same computation that we performed in the g-h filter. Our constants are different in this chapter because we take the noise in the process model and sensors into account, but the math is the same."
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