Generaded PDF.

15_Adaptive_Filtering/Adaptive_Filtering.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:070b3521e43c9acc2c9fe7593b6fbc1b8894cf90bdcba38da2ba06db8b53984b"
+  "signature": "sha256:3d0003a848712ccf6ba0dae09f5d3f04279d82fdc23791a1936c2ab60833a690"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -1270,7 +1270,7 @@
      "source": [
       "Fading memory filters are not normally classified as an adaptive filter since they do not adapt to the the input, but they do provide good performance with manuevering targets. They also have the benefit of having a very simple computational form for first, second, and third order kinematic filters (e.g. the filters we are using in this chapter). This simple form does not require the Ricatti equations to compute the gain of the Kalman filter, which drastically reduces the amount of computation. However, there is also a form that works with the standard Kalman filter. I will focus on the latter in this chapter since our focus is more on adaptive filters. Both forms of the fading memory filter are implemented in `FilterPy`.\n",
       "\n",
-      "The Kalman filter is recursive, but it incorporates all of the previous measurements into the current computation of the filter gain. If the target behavior is consistant with the process model than this allows the Kalman filter to find the optimal estimate for every measurement. Consider a ball in flight - we can clearly estimate the position of the ball at time t better if we take into account all the previous measurement. If we only used some of the measurements we would be less certain about the current position, and thus more influenced by the noise in the measurement. If this is still not clear, consider the worst case. Suppose we forget all but the last measurement and estimates. We would then have no confidence in the position and trajectory of the ball, and would little choice but to weight the current measurement heavily. If the measurement is noisy, the estimate is noisy. We see this effect every time a Kalman filter is initialized. The early estimates are noisy, but then they settle down as more measurements are acquired.\n",
+      "The Kalman filter is recursive, but it incorporates all of the previous measurements into the current computation of the filter gain. If the target behavior is consistant with the process model than this allows the Kalman filter to find the optimal estimate for every measurement. Consider a ball in flight - we can clearly estimate the position of the ball at time t better if we take into account all the previous measurement. If we only used some of the measurements we would be less certain about the current position, and thus more influenced by the noise in the measurement. If this is still not clear, consider the worst case. Suppose we forget all but the last measurement and estimates. We would then have no confidence in the position and trajectory of the ball, and would have little choice but to weight the current measurement heavily. If the measurement is noisy, the estimate is noisy. We see this effect every time a Kalman filter is initialized. The early estimates are noisy, but then they settle down as more measurements are acquired.\n",
       "\n",
       "However, if the target is manuevering it is not always behaving like the process model predicts. In this case remembering all of the past measurements and estimates is a liability. We can see this in all of the charts above. The target initiates a turn, and the Kalman filter continues to project movement in a straight line. This is because the filter has built a history of the target's movement, and incorrectly 'feels' confident that the target is moving in a straight line at a given heading and velocity. \n",
       "\n",