Merge branch 'master' of https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python

01_g-h_filter.ipynb

@@ -985,10 +985,10 @@
     "\n",
     "**Let me repeat the key points as they are so important**. If you do not understand these you will not understand the rest of the book. If you do understand them, then the rest of the book will unfold naturally for you as mathematically elaborations to various 'what if' questions we will ask about *g* and *h*.\n",
     "\n",
-    "* Multiple data points are more accurate than one data point, so throw nothing away no matter how inaccurate it is\n",
-    "* Always choose a number part way between two data points to create a more accurate estimate\n",
-    "* Predict the next measurement and rate of change based on the current estimate and how much we think it will change\n",
-    "* The new estimate is then chosen as part way between the prediction and next measurement\n",
+    "* Multiple data points are more accurate than one data point, so throw nothing away no matter how inaccurate it is.\n",
+    "* Always choose a number part way between two data points to create a more accurate estimate.\n",
+    "* Predict the next measurement and rate of change based on the current estimate and how much we think it will change.\n",
+    "* The new estimate is then chosen as part way between the prediction and next measurement.\n",
     "\n",
     "Let's look at a visual depiction of the algorithm."
    ]
@@ -1487,7 +1487,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "It is clear that as $g$ is larger we more closely follow the measurement instead of the prediction. When $g=0.9$ we follow the signal almost exactly, and reject almost none of the noise. One might naively conclude that $g$ should always be very small to maximize noise rejection. However, that means that we are mostly ignoring the measurements in favor of our prediction. What happens when the signal changes not due to noise, but an actual state change? Let's look. I will create data that has $\\dot{x}=1$ for 9 steps before changing to $\\dot{x}=0$. "
+    "It is clear that as $g$ is larger we more closely follow the measurement instead of the prediction. When $g=0.9$ we follow the signal almost exactly, and reject almost none of the noise. One might naively conclude that $g$ should always be very small to maximize noise rejection. However, that means that we are mostly ignoring the measurements in favor of our prediction. What happens when the signal changes not due to noise, but an actual state change? Let's have a look. I will create data that has $\\dot{x}=1$ for 9 steps before changing to $\\dot{x}=0$. "
    ]
   },
   {
@@ -1934,9 +1934,9 @@
    "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 modeling. But, far more importantly, there is a trade off 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",
+    "There are two lessons to be learned here. First, use the *h* term to respond to changes in velocity that you are not modeling. But, far more importantly, there is a trade off 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 are almost always changes in state, and we want to react to them quickly. The more quickly we react to them, 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. "
+    "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 leads 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. "
    ]
   },
   {
@@ -1990,12 +1990,7 @@
     "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": [
+    "\n",
     "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."
    ]
   },

02_Discrete_Bayes.ipynb

@@ -1659,7 +1659,7 @@
     "\n",
     "If you are not familiar with this notation, let's review. $P(A)$ means the probability of event $A$. If $A$ is the event of a fair coin landing heads, then $P(A) = 0.5$.\n",
     "\n",
-    "$P(A|B)$ is called a *conditional probability*. That is, it represents the probability of $A$ happening *if* $B$ happened. For example, it is more likely to rain today if it also rained yesterday. We'd write that as $P(rain_{yesterday}|rain_{today})$.\n",
+    "$P(A|B)$ is called a *conditional probability*. That is, it represents the probability of $A$ happening *if* $B$ happened. For example, it is more likely to rain today if it also rained yesterday. We'd write that as $P(rain_{today}|rain_{yesterday})$.\n",
     "\n",
     "In Bayesian statistics $P(A)$ is the *prior*, and $P(A|B)$ is the *posterior*. To see why, let's rewrite the equation in terms of our problem. We will use $x_i$ for the position at *i*, and $Z$ for the measurement. Hence, we want to know $P(x_i|Z)$, that is, the probability of the dog being at $x_i$ given the measurement $Z$. \n",
     "\n",