fix a few typos, word omissions

07-Kalman-Filter-Math.ipynb

@@ -364,7 +364,7 @@
     "\n",
     "$\\mathbf w$ may strike you as a poor choice for the name, but you will soon see that the Kalman filter assumes *white* noise.\n",
     "\n",
-    "Finally, we need to consider any inputs into the system. We assume an input $\\mathbf u$, and that there exists a linear model that defines how that input changes the system. For example, pressing the accelerator in your car makes it accelerate, and gravity causes balls to fall. Both are contol inputs. We will need a matrix $\\mathbf B$ to convert $u$ into the effect on the system. We add that into our equation:\n",
+    "Finally, we need to consider any inputs into the system. We assume an input $\\mathbf u$, and that there exists a linear model that defines how that input changes the system. For example, pressing the accelerator in your car makes it accelerate, and gravity causes balls to fall. Both are control inputs. We will need a matrix $\\mathbf B$ to convert $u$ into the effect on the system. We add that into our equation:\n",
     "\n",
     "$$ \\dot{\\mathbf x} = \\mathbf{Ax} + \\mathbf{Bu} + \\mathbf{w}$$\n",
     "\n",
@@ -557,7 +557,7 @@
     "\n",
     "That series is found by doing a Taylor series expansion of $e^{\\mathbf At}$, which I will not cover here.\n",
     "\n",
-    "Let's use this to find the solution to Newton's equations. Using $v$ as an substitution for $\\dot x$, and assuming constant velocity we get the linear matrix-vector form \n",
+    "Let's use this to find the solution to Newton's equations. Using $v$ as a substitution for $\\dot x$, and assuming constant velocity we get the linear matrix-vector form \n",
     "\n",
     "$$\\begin{bmatrix}\\dot x \\\\ \\dot v\\end{bmatrix} =\\begin{bmatrix}0&1\\\\0&0\\end{bmatrix} \\begin{bmatrix}x \\\\ v\\end{bmatrix}$$\n",
     "\n",
@@ -754,7 +754,7 @@
    "source": [
     "## Design of the Process Noise Matrix\n",
     "\n",
-    "In general the design of the $\\mathbf Q$ matrix is among the most difficult aspects of Kalman filter design. This is due to several factors. First, the math requires a good foundation in signal theory. Second, we are trying to model the noise in something for which we have little information. Consider trying to model the process noise for a thrown baseball. We can model it as a sphere moving through the air, but that leaves many unknown factors - the wind, ball rotation and spin decay, the coefficient of drag of a ball with stitches, the effects of wind and air density, and so on. We develop the equations for an exact mathematical solution for a given process model, but since the process model is incomplete the result for $\\mathbf Q$ will also be incomplete. This has a lot of ramifications for the behavior of the Kalman filter. If $\\mathbf Q$ is too small then the filter will be overconfident in its prediction model and will diverge from the actual solution. If $\\mathbf Q$ is too large than the filter will be unduly influenced by the noise in the measurements and perform sub-optimally. In practice we spend a lot of time running simulations and evaluating collected data to try to select an appropriate value for $\\mathbf Q$. But let's start by looking at the math.\n",
+    "In general the design of the $\\mathbf Q$ matrix is among the most difficult aspects of Kalman filter design. This is due to several factors. First, the math requires a good foundation in signal theory. Second, we are trying to model the noise in something for which we have little information. Consider trying to model the process noise for a thrown baseball. We can model it as a sphere moving through the air, but that leaves many unknown factors - ball rotation and spin decay, the coefficient of drag of a ball with stitches, the effects of wind and air density, and so on. We develop the equations for an exact mathematical solution for a given process model, but since the process model is incomplete the result for $\\mathbf Q$ will also be incomplete. This has a lot of ramifications for the behavior of the Kalman filter. If $\\mathbf Q$ is too small then the filter will be overconfident in its prediction model and will diverge from the actual solution. If $\\mathbf Q$ is too large than the filter will be unduly influenced by the noise in the measurements and perform sub-optimally. In practice we spend a lot of time running simulations and evaluating collected data to try to select an appropriate value for $\\mathbf Q$. But let's start by looking at the math.\n",
     "\n",
     "\n",
     "Let's assume a kinematic system - some system that can be modeled using Newton's equations of motion. We can make a few different assumptions about this process. \n",
@@ -1658,9 +1658,9 @@
     "\n",
     "The details of the mathematics for this computation varies based on the problem. The **Discrete Bayes** and **Univariate Kalman Filter** chapters gave two different formulations which you should have been able to reason through. The univariate Kalman filter assumes that for a scalar state both the noise and process are linear model are affected by zero-mean, uncorrelated Gaussian noise. \n",
     "\n",
-    "The Multivariate Kalman filter make the same assumption but for states and measurements that are vectors, not scalars. Dr. Kalman was able to prove that if these assumptions hold true then the Kalman filter is *optimal* in a least squares sense. Colloquially this means there is no way to derive more information from the noise. In the remainder of the book I will present filters that relax the constraints on linearity and Gaussian noise.\n",
+    "The Multivariate Kalman filter make the same assumption but for states and measurements that are vectors, not scalars. Dr. Kalman was able to prove that if these assumptions hold true then the Kalman filter is *optimal* in a least squares sense. Colloquially this means there is no way to derive more information from the noisy measurements. In the remainder of the book I will present filters that relax the constraints on linearity and Gaussian noise.\n",
     "\n",
-    "Before I go on, a few more words about statistical inversion. As Calvetti and Somersalo write in *Introduction to Bayesian Scientific Computing*, \"we adopt the Bayesian point of view: *randomness simply means lack of information*.\"[3] Our state parametize physical phenomena that we could in principle measure or compute: velocity, air drag, and so on. We lack enough information to compute or measure their value, so we opt to consider them as random variables. Strictly speaking they are not random, thus this is a subjective position. \n",
+    "Before I go on, a few more words about statistical inversion. As Calvetti and Somersalo write in *Introduction to Bayesian Scientific Computing*, \"we adopt the Bayesian point of view: *randomness simply means lack of information*.\"[3] Our state parameterizes physical phenomena that we could in principle measure or compute: velocity, air drag, and so on. We lack enough information to compute or measure their value, so we opt to consider them as random variables. Strictly speaking they are not random, thus this is a subjective position. \n",
     "\n",
     "They devote a full chapter to this topic. I can spare a paragraph. Bayesian filters are possible because we ascribe statistical properties to unknown parameters. In the case of the Kalman filter we have closed-form solutions to find an optimal estimate. Other filters, such as the discrete Bayes filter or the particle filter which we cover in a later chapter, model the probability in a more ad-hoc, non-optimal manner. The power of our technique comes from treating lack of information as a random variable, describing that random variable as a probability distribution, and then using Bayes Theorem to solve the statistical inference problem."
    ]