"
]
}
],
"prompt_number": 39
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There are two lessons to be learned here. First, use the *h* term to respond to changes in velocity that you are not modelling. But, far more importantly, there is a tradeoff here between responding quickly and accurately to changes in behavior and producing ideal output for when the system is in a steady state that you have. If the train never changes velocity we would make *h* extremely small to avoid having the filtered estimate unduly affected by the noise in the measurement. But in an interesting problem there is almost always changes in state, and we want to react to them quickly. The more quickly we react to it, the more we are affected by the noise in the sensors. \n",
"\n",
"I could go on, but my aim is not to develop g-h filter theory here so much as to build insight into how combining measurements and predictions lead to a filtered solution, so I will stop here. Do understand that there is extensive literature on choosing *g* and *h* for problems such as this, and that there are optimal ways of choosing them to achieve various goals. In the subsequent chapters we will learn how the Kalman filter solves this problem in the same basic manner, but with far more sophisticated mathematics. "
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"g-h Filters with FilterPy"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Final Thoughts"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"*author's note*: the following few paragraphs really belong in the not yet written least squares chapter.\n",
"\n",
"Near the beginning of the chapter I used `numpy.polyfit()` to fit a straight line to the weight measurements. It fits a n-th degree polynomial to the data using a 'least squared fit'. How does this differ from the g-h filter?\n",
"\n",
"Well, it depends. We will eventually learn that the Kalman filter is optimal from a least squared fit perspective under certain conditions. However, `polyfit()` fits a polynomial to the data, not an arbitrary curve, by minimizing the value of this formula:\n",
"$$E = \\sum_{j=0}^k |p(x_j) - y_j|^2$$\n",
"\n",
"I assumed that my weight gain was constant at 1 lb/day, and so when I tried to fit a polynomial of $n=1$, which is a line, the result very closely matched the actual weight gain. But, of course, no one consistently only gains or loses weight. We fluctuate. Using 'polyfit()' for a longer series of data would yield poor results. In contrast, the g-h filter reacts to changes in the rate - the $h$ term controls how quickly the filter reacts to these changes. If we gain weight, hold steady for awhile, then lose weight, the filter will track that change automatically. 'polyfit()' would not be able to do that unless the gain and loss could be well represented by a polynomial.\n",
"\n",
"Another advantage of this form of filter, even if the data fits a *n*-degree polynomial, is that it is *recursive*. That is, we can compute the estimate for this time period knowing nothing more than the estimate and rate from the last time period. In contrast, if you dig into the implementation for `polyfit()` you will see that it needs all of the data before it can produce an answer. Therefore algorithms like `polyfit()` are not well suited for real-time data filtering. In the 60's when the Kalman filter was developed computers were very slow and had extremely limited memory. They were utterly unable to store, for example, thousands of readings from an aircraft's inertial navigation system, nor could they process all of that data in the short period of time needed to provide accurate and up-to-date navigation information. \n",
"\n",
"\n",
"Up until the mid 20th century various forms of Least Squares Estimation was used for this type of filtering. For example, for NASA's Apollo program had a ground network for tracking the Command and Service Model (CSM) and the Lunar Module (LM). They took measurements over many minutes, batched the data together, and slowly computed an answer. In 1960 Stanley Schmidt at NASA Ames recognized the utility of Rudolf Kalman's seminal paper and invited him to Ames. Schmidt applied Kalman's work to the onboard navigation systems on the CSM and LM, and called it the \"Kalman filter\".[1] Soon after, the world moved to this faster, recursive filter.\n",
"\n",
"The Kalman filter only needs to store the last estimate and a few related parameters, and requires only a relatively small number of computations to generate the next estimate. Today we have so much memory and processing power that this advantage is somewhat less important, but at the time the Kalman filter was a major breakthrough not just because of the mathematical properties, but because it could (barely) run on the hardware of the day. \n",
"\n",
"This subject is much deeper than this short discussion suggests. We will consider these topics many more times throughout the book."
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Summary"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"I encourage you to experiment with this filter to develop your understanding of how it reacts. It shouldn't take too many attempts to come to the realization that ad-hoc choices for $g$ and $h$ do not perform very well. A particular choice might perform well in one situation, but very poorly in another. Even when you understand the effect of $g$ and $h$ it can be difficult to choose proper values. In fact, it is extremely unlikely that you will choose values for $g$ and $h$ that is optimal for any given problem. Filters are *designed*, not selected *ad hoc*. \n",
"\n",
"In some ways I do not want to end the chapter here, as there is a significant amount that we can say about selecting $g$ and $h$. But the g-h filter in this form is not the purpose of this book. Designing the Kalman filter requires you to specify a number of parameters - indirectly they do relate to choosing $g$ and $h$, but you will never refer to them directly when designing Kalman filters. Furthermore, $g$ and $h$ will vary at every time step in a very non-obvious manner. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There is another feature of these filters we have barely touched upon - Bayesian statistics. You will note that the term 'Bayesian' is in the title of this book; this is not a coincidence! For the time being we will leave $g$ and $h$ behind, largely unexplored, and develop a very powerful form of probabilistic reasoning about filtering. Yet suddenly this same g-h filter algorithm will appear, this time with a formal mathematical edifice that allows us to create filters from multiple sensors, to accurately estimate the amount of error in our solution, and to control robots."
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"References"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"* [1] NASA Kalman Filtering Presentation http://nescacademy.nasa.gov/review/downloadfile.php?file=FundamentalsofKalmanFiltering_Presentation.pdf&id=199&distr=Public\n",
"\n",
"\n"
]
}
],
"metadata": {}
}
]
}