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Kloppenburg Ernst (CR/PJ-AI-R4) 2017-11-15 17:43:57 +01:00
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06-Multivariate-Kalman-Filters.ipynb
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@ -1354,7 +1354,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Keeping track of all of these variables is burdomsome, so FilterPy also implements the filter with the class `KalmanFilter`. I will use the class in the rest of this book, but I wanted you to see the procedural form of these functions since I know some of you are not fans of object oriented programming."
"Keeping track of all of these variables is burdensome, so FilterPy also implements the filter with the class `KalmanFilter`. I will use the class in the rest of this book, but I wanted you to see the procedural form of these functions since I know some of you are not fans of object oriented programming."
]
},
{
@ -1363,14 +1363,14 @@
"source": [
"## Implementing the Kalman Filter\n",
"\n",
"I've given you all of the code for the filter, but now let's collect it in one place. First we construct an `KalmanFilter` object. We have to specify the number of variables in the state with the `dim_x` parameter, and the number of measurements with `dim_z`. We have two random variables in the state and one measurement, so we write:\n",
"I've given you all of the code for the filter, but now let's collect it in one place. First we construct a `KalmanFilter` object. We have to specify the number of variables in the state with the `dim_x` parameter, and the number of measurements with `dim_z`. We have two random variables in the state and one measurement, so we write:\n",
"\n",
"```python\n",
"from filterpy.kalman import KalmanFilter\n",
"dog_filter = KalmanFilter(dim_x=2, dim_z=1)\n",
"```\n",
"\n",
"This creates an obect with default values for all the Kalman filter matrices:"
"This creates an object with default values for all the Kalman filter matrices:"
]
},
{
@ -1585,7 +1585,7 @@
"\n",
"The covariance matrix $\\mathbf P$ tells us the *theoretical* performance of the filter *assuming* everything we tell it is true. Recall that the standard deviation is the square root of the variance, and that approximately 68% of a Gaussian distribution occurs within one standard deviation. If at least 68% of the filter output is within one standard deviation the filter may be performing well. In the top chart I have displayed the one standard deviation as the yellow shaded area between the two dotted lines. To my eye it looks like perhaps the filter is slightly exceeding that bounds, so the filter probably needs some tuning.\n",
"\n",
"In the univariate chapter we filtered very noisy signals with much simpler code than the code above. However, realize that right now we are working with a very simple example - an object moving through 1-D space and one sensor. That is about the limit of what we can compute with the code in the last chapter. In contrast, we can implement very complicated, multidimensional filter with this code merely by altering our assignments to the filter's variables. Perhaps we want to track 100 dimensions in financial models. Or we have an aircraft with a GPS, INS, TACAN, radar altimeter, baro altimeter, and airspeed indicator, and we want to integrate all those sensors into a model that predicts position, velocity, and accelerations in 3D space. We can do that with the code in this chapter.\n",
"In the univariate chapter we filtered very noisy signals with much simpler code than the code above. However, realize that right now we are working with a very simple example - an object moving through 1-D space and one sensor. That is about the limit of what we can compute with the code in the last chapter. In contrast, we can implement very complicated, multidimensional filters with this code merely by altering our assignments to the filter's variables. Perhaps we want to track 100 dimensions in financial models. Or we have an aircraft with a GPS, INS, TACAN, radar altimeter, baro altimeter, and airspeed indicator, and we want to integrate all those sensors into a model that predicts position, velocity, and acceleration in 3D space. We can do that with the code in this chapter.\n",
"\n",
"I want you to get a better feel for how the Gaussians change over time, so here is a 3D plot showing the Gaussians every 7th epoch (time step). Every 7th separates them enough so can see each one independently. The first Gaussian at $t=0$ is to the left."
]
@ -1658,7 +1658,7 @@
"\n",
"As a reminder, the linear equation $\\mathbf{Ax} = \\mathbf B$ represents a system of equations, where $\\mathbf{A}$ holds the coefficients of the state $\\mathbf x$. Performing the multiplication $\\mathbf{Ax}$ computes the right hand side values for that set of equations.\n",
"\n",
"If $\\mathbf F$ contains the state transition for a given time step, then the product $\\mathbf{Fx}$ computes the state after that transition. Easy! Likewise, $\\mathbf B$ is the control function, $\\mathbf u$ is the control input, so $\\mathbf{Bu}$ computes the contribution of the controls to the state after the transition. Thus, the prior$\\mathbf{\\bar x}$ is computed as the sum of $\\mathbf{Fx}$ and $\\mathbf{Bu}$.\n",
"If $\\mathbf F$ contains the state transition for a given time step, then the product $\\mathbf{Fx}$ computes the state after that transition. Easy! Likewise, $\\mathbf B$ is the control function, $\\mathbf u$ is the control input, so $\\mathbf{Bu}$ computes the contribution of the controls to the state after the transition. Thus, the prior $\\mathbf{\\bar x}$ is computed as the sum of $\\mathbf{Fx}$ and $\\mathbf{Bu}$.\n",
"\n",
"The equivalent univariate equation is\n",
"\n",
@ -2698,7 +2698,7 @@
"source": [
"### Solution\n",
"\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",
"The x-axis is for position, and y-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",
@ -2907,7 +2907,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"This output is fantastic! Two things are very apparent to me in this chart. First, the RTS smoother's output is much smoother than the KF output. Second, it is almost always more accurate than the KF output (we will examine this claim in detail in the **Smoothing** chapter). The improvement in the velocity, which is an hidden variable, is even more dramatic:"
"This output is fantastic! Two things are very apparent to me in this chart. First, the RTS smoother's output is much smoother than the KF output. Second, it is almost always more accurate than the KF output (we will examine this claim in detail in the **Smoothing** chapter). The improvement in the velocity, which is a hidden variable, is even more dramatic:"
]
},
{
@ -3026,7 +3026,7 @@
"\n",
"The Kalman filter is a mathematical model of the world. The output is only as accurate as that model. To make the math tractable we had to make some assumptions. We assume that the sensors and motion model have Gaussian noise. We assume that everything is linear. If that is true, the Kalman filter is *optimal* in a least squares sense. This means that there is no way to make a better estimate than what the filter gives us. However, these assumption are almost never true, and hence the model is necessarily limited, and a working filter is rarely optimal.\n",
"\n",
"In later chapters we will deal with the problem of nonlinearity. For now I want you to understand that designing the matrices of a linear filter is an experimental procedure more than a mathematical one. Use math to establish the initial values, but then you need to experiment. If there is a lot of unaccounted noise in the world (wind, etc) you may have to make $\\mathbf Q$ larger. If you make it too large the filter fails to respond quickly to changes. In the **Adaptive Filters** chapter you will learn some alternative techniques that allow you to change the filter design in real time in response to the inputs and performance, but for now you need to find one set of values that works for the conditions your filter will encounter. Noise matrices for an acrobatic plane might be different if the pilot is a student than if the pilot is an expert as the dynamics will be quite different."
"In later chapters we will deal with the problem of nonlinearity. For now I want you to understand that designing the matrices of a linear filter is an experimental procedure more than a mathematical one. Use math to establish the initial values, but then you need to experiment. If there is a lot of unaccounted noise in the world (wind, etc.) you may have to make $\\mathbf Q$ larger. If you make it too large the filter fails to respond quickly to changes. In the **Adaptive Filters** chapter you will learn some alternative techniques that allow you to change the filter design in real time in response to the inputs and performance, but for now you need to find one set of values that works for the conditions your filter will encounter. Noise matrices for an acrobatic plane might be different if the pilot is a student than if the pilot is an expert as the dynamics will be quite different."
]
},
{