Minor rewordings.

06-Multivariate-Kalman-Filters.ipynb

@@ -1747,7 +1747,7 @@
     "\n",
     "$\\mathbf{P} = (\\mathbf{I}-\\mathbf{KH})\\mathbf{\\bar{P}}$\n",
     "\n",
-    "$\\mathbf{I}$ is the identity matrix, and is the way we represent $1$ in multiple dimensions. $\\mathbf{H}$ is our measurement function, and is a constant.  So, simplified, this is simply $\\mathbf{P} = (1-c\\mathbf{K})\\mathbf{P}$. $\\mathbf{K}$ is our ratio of how much prediction vs measurement we use. So, if $\\mathbf{K}$ is large then $(1-\\mathbf{cK})$ is small, and $\\mathbf{P}$ will be made smaller than it was. If $\\mathbf{K}$ is small, then $(1-\\mathbf{cK})$ is large, and $\\mathbf{P}$ will be relatively larger. So we adjust the size of our uncertainty by some factor of the Kalman gain.\n",
+    "$\\mathbf{I}$ is the identity matrix, and is the way we represent $1$ in multiple dimensions. $\\mathbf{H}$ is our measurement function, and is a constant.  We can think of the equation as $\\mathbf{P} = (1-c\\mathbf{K})\\mathbf{P}$. $\\mathbf{K}$ is our ratio of how much prediction vs measurement we use. If $\\mathbf{K}$ is large then $(1-\\mathbf{cK})$ is small, and $\\mathbf{P}$ will be made smaller than it was. If $\\mathbf{K}$ is small, then $(1-\\mathbf{cK})$ is large, and $\\mathbf{P}$ will be relatively larger. This means that we adjust the size of our uncertainty by some factor of the Kalman gain.\n",
     "\n",
     "This equation can be numerically unstable and I don't use it in FilterPy. Later I'll share more complicated but numerically stable forms of this equation."
    ]
@@ -1789,7 +1789,7 @@
     "\\\\\\end{aligned}\n",
     "$$\n",
     "\n",
-    "This notation uses the Bayesian $a\\mid b$ notation, which means $a$ given the evidence of $b$. The hat means estimate. So, $\\hat{\\mathbf{x}}_{k\\mid k}$ means the estimate of the state $\\mathbf{x}$ at step $k$ (the first k) given the evidence from step $k$ (the second k). The posterior, in other words. $\\hat{\\mathbf{x}}_{k\\mid k-1}$ means the estimate for the state $\\mathbf{x}$ at step $k$ given the estimate from step $k - 1$. The prior, in other words. \n",
+    "This notation uses the Bayesian $a\\mid b$ notation, which means $a$ given the evidence of $b$. The hat means estimate. Thus $\\hat{\\mathbf{x}}_{k\\mid k}$ means the estimate of the state $\\mathbf{x}$ at step $k$ (the first k) given the evidence from step $k$ (the second k). The posterior, in other words. $\\hat{\\mathbf{x}}_{k\\mid k-1}$ means the estimate for the state $\\mathbf{x}$ at step $k$ given the estimate from step $k - 1$. The prior, in other words. \n",
     "\n",
     "This notation, copied from [Wikipedia](https://en.wikipedia.org/wiki/Kalman_filter#Details) [[1]](#[wiki_article]), allows a mathematician to express himself exactly. In formal publications presenting new results this precision is necessary. As a programmer I find it fairly unreadable. I am used to thinking about variables changing state as a program runs, and do not use a different variable name for each new computation. There is no agreed upon format in the literature, so each author makes different choices. I find it challenging to switch quickly between books and papers, and so have adopted my admittedly less precise notation. Mathematicians may write scathing emails to me, but I hope programmers and students will rejoice at my simplified notation.\n",
     "\n",
@@ -2047,7 +2047,7 @@
     "\n",
     "Let's remind ourselves of what the term *process uncertainty* means. Consider the problem of tracking a ball. We can accurately model its behavior in a vacuum with math, but with wind, varying air density, temperature, and an spinning ball with an imperfect surface our model will diverge from reality. \n",
     "\n",
-    "In the first case we set `Q_var=20 m^2`, which is quite large. In physical terms this is telling the filter \"I don't trust my motion prediction step\" as we are saying that the variance in the velocity is 20. Strictly speaking, we are telling the filter there is a lot of external noise that we are not modeling with $\\small{\\mathbf{F}}$, but the upshot of that is to not trust the motion prediction step. So the filter will be computing velocity ($\\dot{x}$), but then mostly ignoring it because we are telling the filter that the computation is extremely suspect. Therefore the filter has nothing to trust but the measurements, and thus it follows the measurements closely. \n",
+    "In the first case we set `Q_var=20 m^2`, which is quite large. In physical terms this is telling the filter \"I don't trust my motion prediction step\" as we are saying that the variance in the velocity is 20. Strictly speaking, we are telling the filter there is a lot of external noise that we are not modeling with $\\small{\\mathbf{F}}$, but the upshot of that is to not trust the motion prediction step. The filter will be computing velocity ($\\dot{x}$), but then mostly ignoring it because we are telling the filter that the computation is extremely suspect. Therefore the filter has nothing to trust but the measurements, and thus it follows the measurements closely. \n",
     "\n",
     "In the second case we set `Q_var=0.02 m^2`, which is quite small. In physical terms we are telling the filter \"trust the prediction, it is really good!\". More strictly this actually says there is very small amounts of process noise (variance 0.02 $m^2$), so the process model is very accurate. So the filter ends up ignoring some of the measurement as it jumps up and down, because the variation in the measurement does not match our trustworthy velocity prediction."
    ]
@@ -2433,7 +2433,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The x-axis is for position, and x-axis is velocity. An ellipse that is vertical, or nearly so, says there is no correlation between position and velocity, and an ellipse that is diagonal says that there is a lot of correlation. Phrased that way, the results sound unlikely. The tilt of the ellipse changes, but the correlation shouldn't be changing over time. But this is a measure of the *output of the filter*, not a description of the actual, physical world. When $\\mathbf{R}$ is very large we are telling the filter that there is a lot of noise in the measurements. In that case the Kalman gain $\\mathbf{K}$ is set to favor the prediction over the measurement, and the prediction comes from the velocity state variable. So, there is a large correlation between $x$ and $\\dot{x}$. Conversely, if $\\mathbf{R}$ is small, we are telling the filter that the measurement is very trustworthy, and $\\mathbf{K}$ is set to favor the measurement over the prediction. Why would the filter want to use the prediction if the measurement is nearly perfect? If the filter is not using much from the prediction there will be very little correlation reported. \n",
+    "The x-axis is for position, and x-axis is velocity. An ellipse that is vertical, or nearly so, says there is no correlation between position and velocity, and an ellipse that is diagonal says that there is a lot of correlation. Phrased that way, the results sound unlikely. The tilt of the ellipse changes, but the correlation shouldn't be changing over time. But this is a measure of the *output of the filter*, not a description of the actual, physical world. When $\\mathbf{R}$ is very large we are telling the filter that there is a lot of noise in the measurements. In that case the Kalman gain $\\mathbf{K}$ is set to favor the prediction over the measurement, and the prediction comes from the velocity state variable. Thus there is a large correlation between $x$ and $\\dot{x}$. Conversely, if $\\mathbf{R}$ is small, we are telling the filter that the measurement is very trustworthy, and $\\mathbf{K}$ is set to favor the measurement over the prediction. Why would the filter want to use the prediction if the measurement is nearly perfect? If the filter is not using much from the prediction there will be very little correlation reported. \n",
     "\n",
     "**This is a critical point to understand!**. The Kalman filter is a mathematical model for a real world system. A report of little correlation *does not mean* there is no correlation in the physical system, just that there was no *linear* correlation in the mathematical model. It's a report of how much measurement vs prediction was incorporated into the model.  \n",
     "\n",