daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Fixed section headings, added math

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Added section in math chapter for Bayesian statistics, and improved
on the discussion of system dynamics.

The PDF generator cannot deal with the markdown for section headings
(using hash marks), so I needed to fix chapters that use them.

Appendix B still uses them, but for now that is just a list of
notes for myself, so I left that alone.

This contains the new PDF generated from these changes.
This commit is contained in:
Roger Labbe
2014-12-20 14:20:23 -08:00
parent 4d632bdc2d
commit c865b57a74
4 changed files with 326 additions and 77 deletions
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91
14_Smoothing/Smoothing.ipynb
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@@ -267,12 +267,18 @@
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Introduction"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Introduction\n",
"\n",
"It has probably struck you by now that the performance of the Kalman filter is not optimal when you consider future data. For example, suppose we are tracking an aircraft, and the latest measurement is far from the current track, like so (I'll only consider 1 dimension for simplicity):\n",
"\n",
" 10.1 10.2 9.8 10.1 10.2 10.3 10.1 9.9 10.2 10.0 9.9 12.4"
@@ -381,10 +387,21 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Since this type of smoothing requires knowing data from \"the future\", there are some applications for the Kalman filter where these observations are not helpful. For example, if we are using a Kalman filter as our navigation filter for an aircraft we have little interest in where we have been. While we could use a smoother to create a smooth history for the plane, we probably are not interested in it. However if we can afford a bit of latency some smoothers only require a few measurements into the future prode better results. And, of course any problem where we can batch collect the data and then run the Kalman filter on the data will be able to take maximum advantage of this type of algorithm.\n",
"\n",
"## Types of Smoothers\n",
"\n",
"Since this type of smoothing requires knowing data from \"the future\", there are some applications for the Kalman filter where these observations are not helpful. For example, if we are using a Kalman filter as our navigation filter for an aircraft we have little interest in where we have been. While we could use a smoother to create a smooth history for the plane, we probably are not interested in it. However if we can afford a bit of latency some smoothers only require a few measurements into the future prode better results. And, of course any problem where we can batch collect the data and then run the Kalman filter on the data will be able to take maximum advantage of this type of algorithm."
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Types of Smoothers"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There are three broad classes of Kalman smoothers that produce better tracking in these situations.\n",
"\n",
"* Fixed Point Smoothing\n",
@@ -405,23 +422,51 @@
"\n",
"Fixed lag smoothing only requires you to store a window of data, and processing requirements are modest because only that window is processed for each new measurement. The drawback is that the filter's output always lags the input, and the smoothing is not as pronounced as is possible with fixed interval smoothing.\n",
"\n",
"Fixed interval smoothing produces the most smoothed output at the cost of having to be batch processed. Most algorithms use some sort of forwards/backwards algorithm that is only twice as slow as a recursive Kalamn filter. \n",
"\n",
"## Fixed Point Smoothing\n",
"\n",
"not done\n",
"\n",
"## Fixed Lag Smoothing\n",
"\n",
"not done"
"Fixed interval smoothing produces the most smoothed output at the cost of having to be batch processed. Most algorithms use some sort of forwards/backwards algorithm that is only twice as slow as a recursive Kalamn filter. "
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Fixed Point Smoothing"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"not done"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Fixed Lag Smoothing"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"not done"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Fixed Interval Smoothing"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Fixed Interval Smoothing\n",
"\n",
"There are several fixed lag smoothers available in the literature. I have chosen to implement the smoother invented by Rauch, Tung, and Striebel because of its ease of implementation and efficiency of computation. This smoother is commonly known as an RTS smoother, and that is what we will call it\n",
"\n",
"Derivation of the RTS smoother runs to several pages of densely packed math, and to be honest I have never read it through. I'm certainly not going to inflict it on you. I don't think anyone but thesis writers really need to understand the derivation. Instead I will briefly present the algorithm, equations, and then move directly to implementation and demonstration of the smoother.\n",
@@ -548,15 +593,21 @@
],
"prompt_number": 9
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"References"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## References\n",
"\n",
"[1] H. Rauch, F. Tung, and C. Striebel. \"Maximum likelihood estimates of linear dynamic systems,\" *AIAA Journal*, **3**(8), pp. 1445-1450 (August 1965).\n",
"\n",
"http://arc.aiaa.org/doi/abs/10.2514/3.3166\n"
"http://arc.aiaa.org/doi/abs/10.2514/3.3166"
]
}
],