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Merge pull request #175 from tv3141/Ch01_formatting_and_typos

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Ch01: fix formatting and typos
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Roger Labbe 2018-01-01 11:43:09 -08:00 committed by GitHub
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01-g-h-filter.ipynb
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@ -798,7 +798,7 @@
"weights = [158.0, 164.2, 160.3, 159.9, 162.1, 164.6, \n",
" 169.6, 167.4, 166.4, 171.0, 171.2, 172.6]\n",
"\n",
"time_step = 1.0 # day\n",
"time_step = 1.0 # day\n",
"scale_factor = 4.0/10\n",
"\n",
"def predict_using_gain_guess(weight, gain_rate, do_print=True, sim_rate=0): \n",
@ -824,6 +824,7 @@
"\n",
" # plot results\n",
" gh.plot_gh_results(weights, estimates, predictions, sim_rate)\n",
"\n",
"initial_guess = 160.\n",
"predict_using_gain_guess(weight=initial_guess, gain_rate=1) "
]
@ -2482,7 +2483,7 @@
"source": [
"import time\n",
"with interactive_plot():\n",
" for x in range(2,6):\n",
" for x in range(2, 6):\n",
" plt.plot([0, 1], [1, x])\n",
" plt.gcf().canvas.draw()\n",
" time.sleep(0.5)"
@ -3337,7 +3338,7 @@
],
"source": [
"weight = 160. # initial guess\n",
"gain_rate = -1.0 # initial guess\n",
"gain_rate = -1.0 # initial guess\n",
"\n",
"time_step = 1.\n",
"weight_scale = 4./10\n",
@ -3373,7 +3374,7 @@
"```python\n",
"gain_rate = gain_rate\n",
"``` \n",
"This obviously has no effect, and can be removed. I wrote this to emphasize that in the prediction step you need to predict next value for all variables, both `weight` and `gain_rate`. In this case we are assuming that the gain does not vary, but when we generalize this algorithm we will remove that assumption. "
"This obviously has no effect, and can be removed. I wrote this to emphasize that in the prediction step you need to predict the next value for all variables, both `weight` and `gain_rate`. In this case we are assuming that the gain does not vary, but when we generalize this algorithm we will remove that assumption. "
]
},
{
@ -3957,15 +3958,15 @@
" x_est = x0\n",
" results = []\n",
" for z in data:\n",
" #prediction step\n",
" # prediction step\n",
" x_pred = x_est + (dx*dt)\n",
" dx = dx\n",
"\n",
" # update step\n",
" residual = z - x_pred\n",
" dx = dx + h * (residual) / dt\n",
" x_est = x_pred + g * residual \n",
" results.append(x_est) \n",
" dx = dx + h * (residual) / dt\n",
" x_est = x_pred + g * residual\n",
" results.append(x_est)\n",
" return np.array(results)\n",
"\n",
"book_plots.plot_track([0, 11], [160, 172], label='Actual weight')\n",
@ -3977,7 +3978,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Choice of g and h"
"## Choice of $g$ and $h$"
]
},
{
@ -4283,14 +4284,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise: Varying g"
"## Exercise: Varying $g$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's look at the effect of varying g. Before you perform this exercise, recall that g is the scale factor for choosing between the measurement and prediction. What do you think the effect of a large value of g will be? A small value? \n",
"Now let's look at the effect of varying $g$. Before you perform this exercise, recall that $g$ is the scale factor for choosing between the measurement and prediction. What do you think the effect of a large value of $g$ will be? A small value?\n",
"\n",
"Now, let the `noise_factor=50` and `dx=5`. Plot the results of $g = 0.1\\mbox{, } 0.4,\\mbox{ and } 0.8$."
]
@ -4344,7 +4345,7 @@
" book_plots.plot_filter(data2, label='g=0.4', marker='v', c='C1')\n",
" book_plots.plot_filter(data3, label='g=0.8', c='C2')\n",
" plt.legend(loc=4)\n",
" book_plots.set_limits([20,40], [50, 250])"
" book_plots.set_limits([20, 40], [50, 250])"
]
},
{
@ -4393,7 +4394,7 @@
"source": [
"Here we can see the effects of ignoring the signal. We not only filter out noise, but legitimate changes in the signal as well. \n",
"\n",
"Maybe we need a 'Goldilocks' filter, where is not too large, not too small, but just right? Well, not exactly. As alluded to earlier, different filters choose g and h in different ways depending on the mathematical properties of the problem. For example, the Benedict-Bordner filter was invented to minimize the transient error in this example, where $\\dot{x}$ makes a step jump. We will not discuss this filter in this book, but here are two plots chosen with different allowable pairs of g and h. This filter design minimizes transient errors for step jumps in $\\dot{x}$ at the cost of not being optimal for other types of changes in $\\dot{x}$."
"Maybe we need a 'Goldilocks' filter, where $g$ is not too large, not too small, but just right? Well, not exactly. As alluded to earlier, different filters choose $g$ and $h$ in different ways depending on the mathematical properties of the problem. For example, the Benedict-Bordner filter was invented to minimize the transient error in this example, where $\\dot{x}$ makes a step jump. We will not discuss this filter in this book, but here are two plots chosen with different allowable pairs of $g$ and $h$. This filter design minimizes transient errors for step jumps in $\\dot{x}$ at the cost of not being optimal for other types of changes in $\\dot{x}$."
]
},
{
@ -4429,14 +4430,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Varying h"
"## Varying $h$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's leave g unchanged and investigate the effect of modifying h. We know that h affects how much we favor the measurement of $\\dot{x}$ vs our prediction. But what does this *mean*? If our signal is changing a lot (quickly relative to the time step of our filter), then a large $h$ will cause us to react to those transient changes rapidly. A smaller $h$ will cause us to react more slowly.\n",
"Now let's leave $g$ unchanged and investigate the effect of modifying $h$. We know that $h$ affects how much we favor the measurement of $\\dot{x}$ vs our prediction. But what does this *mean*? If our signal is changing a lot (quickly relative to the time step of our filter), then a large $h$ will cause us to react to those transient changes rapidly. A smaller $h$ will cause us to react more slowly.\n",
"\n",
"We will look at three examples. We have a noiseless measurement that slowly goes from 0 to 1 in 50 steps. Our first filter uses a nearly correct initial value for $\\dot{x}$ and a small $h$. You can see from the output that the filter output is very close to the signal. The second filter uses the very incorrect guess of $\\dot{x}=2$. Here we see the filter 'ringing' until it settles down and finds the signal. The third filter uses the same conditions but it now sets $h=0.5$. If you look at the amplitude of the ringing you can see that it is much smaller than in the second chart, but the frequency is greater. It also settles down a bit quicker than the second filter, though not by much."
]