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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Table of Contents
Smoothing
Step1:  Introduction
The performance of the Kalman filter is not optimal when you consider future data. For example, suppose we are tracking an aircraft, and the latest measurement deviates far from the current track, like so (I'll only consider 1 dimension for simplicity)
Step2:  After a period of near steady state, we have a very large change. Assume the change is past the limit of the aircraft's flight envelope. Nonetheless the Kalman filter incorporates that new measurement into the filter based on the current Kalman gain. It cannot reject the noise because the measurement could reflect the initiation of a turn. Granted it is unlikely that we are turning so abruptly, but it is impossible to say whether 
The aircraft started a turn awhile ago, but the previous measurements were noisy and didn't show the change.
The aircraft is turning, and this measurement is very noisy
The measurement is very noisy and the aircraft has not turned
The aircraft is turning in the opposite direction, and the measurement is extremely noisy
Now, suppose the following measurements are
Step3:  Given these future measurements we can infer that yes, the aircraft initiated a turn. 
On the other hand, suppose these are the following measurements.
Step4:  In this case we are led to conclude that the aircraft did not turn and that the outlying measurement was merely very noisy. 
An Overview of How Smoothers Work
The Kalman filter is a recursive filter with the Markov property - it's estimate at step k is  based only on the estimate from step k-1 and the measurement at step k. But this means that the estimate from step k-1 is based on step k-2, and so on back to the first epoch. Hence, the estimate at step k depends on all of the previous measurements, though to varying degrees. k-1 has the most influence, k-2 has the next most, and so on. 
Smoothing filters incorporate future measurements into the estimate for step k. The measurement from k+1 will have the most effect, k+2 will have less effect, k+3 less yet, and so on. 
This topic is called smoothing, but I think that is a misleading name. I could smooth the data above by passing it through a low pass filter. The result would be smooth, but not necessarily accurate because a low pass filter will remove real variations just as much as it removes noise. In contrast, Kalman smoothers are optimal - they incorporate all available information to make the best estimate that is mathematically achievable.
Types of Smoothers
There are three classes of Kalman smoothers that produce better tracking in these situations.
Fixed-Interval Smoothing
This is a batch processing based filter. This filter waits for all of the data to be collected before making any estimates. For example, you may be a scientist collecting data for an experiment, and don't need to know the result until the experiment is complete. A fixed-interval smoother will collect all the data, then estimate the state at each measurement using all available previous and future measurements. If it is possible for you to run your Kalman filter in batch mode it is always recommended to use one of these filters a it will provide much better results than the recursive forms of the filter from the previous chapters.
Fixed-Lag Smoothing
Fixed-lag smoothers introduce latency into the output. Suppose we choose a lag of 4 steps. The filter will ingest the first 3 measurements but not output a filtered result. Then, when the 4th measurement comes in the filter will produce the output for measurement 1, taking measurements 1 through 4 into account. When the 5th measurement comes in, the filter will produce the result for measurement 2, taking measurements 2 through 5 into account. This is useful when you need recent data but can afford a bit of lag. For example, perhaps you are using machine vision to monitor a manufacturing process. If you can afford a few seconds delay in the estimate a fixed-lag smoother will allow you to produce very accurate and smooth results.
Fixed-Point Smoothing
A fixed-point filter operates as a normal Kalman filter, but also produces an estimate for the state at some fixed time $j$.  Before the time $k$ reaches $j$ the filter operates as a normal filter. Once $k>j$ the filter estimates $x_k$ and then also updates its estimate for $x_j$ using all of the measurements between $j\dots k$. This can be useful to estimate initial paramters for a system, or for producing the best estimate for an event that happened at a specific time. For example, you may have a robot that took a photograph at time $j$. You can use a fixed-point smoother to get the best possible pose information for the camera at time $j$ as the robot continues moving.
Choice of Filters
The choice of these filters depends on your needs and how much memory and processing time you can spare. Fixed-point smoothing requires storage of all measurements, and is very costly to compute because the output is for every time step is recomputed for every measurement. On the other hand, the filter does produce a decent output for the current measurement, so this filter can be used for real time applications.
Fixed-lag smoothing only requires you to store a window of data, and processing requirements are modest because only that window is processed for each new measurement. The drawback is that the filter's output always lags the input, and the smoothing is not as pronounced as is possible with fixed-interval smoothing.
Fixed-interval smoothing produces the most smoothed output at the cost of having to be batch processed. Most algorithms use some sort of forwards/backwards algorithm that is only twice as slow as a recursive Kalman filter. 
Fixed-Interval Smoothing
There are many fixed-lag smoothers available in the literature. I have chosen to implement the smoother invented by Rauch, Tung, and Striebel because of its ease of implementation and efficiency of computation. It is also the smoother I have seen used most often in real applications. This smoother is commonly known as an RTS smoother.
Derivation of the RTS smoother runs to several pages of densely packed math. I'm not going to inflict it on you. Instead I will briefly present the algorithm, equations, and then move directly to implementation and demonstration of the smoother.
The RTS smoother works by first running the Kalman filter in a batch mode, computing the filter output for each step. Given the filter output for each measurement along with the covariance matrix corresponding to each output the RTS runs over the data backwards, incorporating its knowledge of the future into the past measurements. When it reaches the first measurement it is done, and the filtered output incorporates all of the information in a maximally optimal form.
The equations for the RTS smoother are very straightforward and easy to implement. This derivation is for the linear Kalman filter. Similar derivations exist for the EKF and UKF. These steps are performed on the output of the batch processing, going backwards from the most recent in time back to the first estimate. Each iteration incorporates the knowledge of the future into the state estimate. Since the state estimate already incorporates all of the past measurements the result will be that each estimate will contain knowledge of all measurements in the past and future. Here is it very important to distinguish between past, present, and future so I have used subscripts to denote whether the data is from the future or not.
Predict Step
$$\begin{aligned}
\mathbf{P} &= \mathbf{FP}_k\mathbf{F}^\mathsf{T} + \mathbf{Q }
\end{aligned}$$
Update Step
$$\begin{aligned}
\mathbf{K}k &= \mathbf{P}_k\mathbf{F}^\mathsf{T}\mathbf{P}^{-1} \
\mathbf{x}_k &= \mathbf{x}_k + \mathbf{K}_k(\mathbf{x}{k+1} - \mathbf{Fx}k) \
\mathbf{P}_k &= \mathbf{P}_k + \mathbf{K}_k(\mathbf{P}{k+1} - \mathbf{P})\mathbf{K}_k^\mathsf{T}
\end{aligned}$$
As always, the hardest part of the implementation is correctly accounting for the subscripts. A basic implementation without comments or error checking would be
Step5:  I've injected a lot of noise into the signal to allow you to visually distinguish the RTS output from the ideal output. In the graph above we can see that the Kalman filter, drawn as the green dotted line, is reasonably smooth compared to the input, but it still wanders from from the ideal line when several measurements in a row are biased towards one side of the line. In contrast, the RTS output is both extremely smooth and very close to the ideal output.
With a perhaps more reasonable amount of noise we can see that the RTS output nearly lies on the ideal output. The Kalman filter output, while much better, still varies by a far greater amount.
Step6:  However, we must understand that this smoothing is predicated on the system model. We have told the filter that what we are tracking follows a constant velocity model with very low process error. When the filter looks ahead it sees that the future behavior closely matches a constant velocity so it is able to reject most of the noise in the signal. Suppose instead our system has a lot of process noise. For example, if we are tracking a light aircraft in gusty winds its velocity will change often, and the filter will be less able to distinguish between noise and erratic movement due to the wind. We can see this in the next graph.
Step7:  This underscores the fact that these filters are not smoothing the data in colloquial sense of the term. The filter is making an optimal estimate based on previous measurements, future measurements, and what you tell it about the behavior of the system and the noise in the system and measurements.
Let's wrap this up by looking at the velocity estimates of Kalman filter vs the RTS smoother.
Step8:  The improvement in the velocity, which is an hidden variable, is even more dramatic. 
Fixed-Lag Smoothing
The RTS smoother presented above should always be your choice of algorithm if you can run in batch mode because it incorporates all available data into each estimate. Not all problems allow you to do that, but you may still be interested in receiving smoothed values for previous estimates. The number line below illustrates this concept.
Step9:  At step $k$ we can estimate $x_k$ using the normal Kalman filter equations. However, we can make a better estimate for $x_{k-1}$ by using the measurement received for $x_k$. Likewise, we can make a better estimate for $x_{k-2}$ by using the measurements recevied for $x_{k-1}$ and $x_{k}$. We can extend this computation back for an arbitrary $N$ steps.
Derivation for this math is beyond the scope of this book; Dan Simon's Optimal State Estimation [2] has  a very good exposition if you are interested. The essense of the idea is that instead of having a state vector $\mathbf{x}$ we make an augmented state containing
$$\mathbf{x} = \begin{bmatrix}\mathbf{x}k \ \mathbf{x}{k-1} \ \vdots\ \mathbf{x}_{k-N+1}\end{bmatrix}$$
This yields a very large covariance matrix that contains the covariance between states at different steps. FilterPy's class FixedLagSmoother takes care of all of this computation for you, including creation of the augmented matrices. All you need to do is compose it as if you are using the KalmanFilter class and then call smooth(), which implements the predict and update steps of the algorithm. 
Each call of smooth computes the estimate for the current measurement, but it also goes back and adjusts the previous N-1 points as well. The smoothed values are contained in the list FixedLagSmoother.xSmooth. If you use FixedLagSmoother.x you will get the most recent estimate, but it is not smoothed and is no different from a standard Kalman filter output.
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Python Code:
from __future__ import division, print_function
%matplotlib inline
#format the book
import book_format
book_format.set_style()
Explanation: Table of Contents
Smoothing
End of explanation
import matplotlib.pyplot as plt
data = [10.1, 10.2, 9.8, 10.1, 10.2, 10.3, 
        10.1, 9.9, 10.2, 10.0, 9.9, 11.4]
plt.plot(data)
plt.xlabel('time')
plt.ylabel('position');
Explanation: Introduction
The performance of the Kalman filter is not optimal when you consider future data. For example, suppose we are tracking an aircraft, and the latest measurement deviates far from the current track, like so (I'll only consider 1 dimension for simplicity):
End of explanation
data2 = [11.3, 12.1, 13.3, 13.9, 14.5, 15.2]
plt.plot(data + data2);
Explanation: After a period of near steady state, we have a very large change. Assume the change is past the limit of the aircraft's flight envelope. Nonetheless the Kalman filter incorporates that new measurement into the filter based on the current Kalman gain. It cannot reject the noise because the measurement could reflect the initiation of a turn. Granted it is unlikely that we are turning so abruptly, but it is impossible to say whether 
The aircraft started a turn awhile ago, but the previous measurements were noisy and didn't show the change.
The aircraft is turning, and this measurement is very noisy
The measurement is very noisy and the aircraft has not turned
The aircraft is turning in the opposite direction, and the measurement is extremely noisy
Now, suppose the following measurements are:
11.3 12.1 13.3 13.9 14.5 15.2
End of explanation
data3 = [9.8, 10.2, 9.9, 10.1, 10.0, 10.3, 9.9, 10.1]
plt.plot(data + data3);
Explanation: Given these future measurements we can infer that yes, the aircraft initiated a turn. 
On the other hand, suppose these are the following measurements.
End of explanation
import numpy as np
from numpy import random
from numpy.random import randn
import matplotlib.pyplot as plt
from filterpy.kalman import KalmanFilter
import kf_book.book_plots as bp
def plot_rts(noise, Q=0.001, show_velocity=False):
    random.seed(123)
    fk = KalmanFilter(dim_x=2, dim_z=1)
    fk.x = np.array([0., 1.])      # state (x and dx)
    fk.F = np.array([[1., 1.],
                     [0., 1.]])    # state transition matrix
    fk.H = np.array([[1., 0.]])    # Measurement function
    fk.P = 10.                     # covariance matrix
    fk.R = noise                   # state uncertainty
    fk.Q = Q                       # process uncertainty
    # create noisy data
    zs = np.asarray([t + randn()*noise for t in range (40)])
    # filter data with Kalman filter, than run smoother on it
    mu, cov, _, _ = fk.batch_filter(zs)
    M, P, C, _ = fk.rts_smoother(mu, cov)
    # plot data
    if show_velocity:
        index = 1
        print('gu')
    else:
        index = 0
    if not show_velocity:
        bp.plot_measurements(zs, lw=1)
    plt.plot(M[:, index], c='b', label='RTS')
    plt.plot(mu[:, index], c='g', ls='--', label='KF output')
    if not show_velocity:
        N = len(zs)
        plt.plot([0, N], [0, N], 'k', lw=2, label='track') 
    plt.legend(loc=4)
    plt.show()
    
plot_rts(7.)
Explanation: In this case we are led to conclude that the aircraft did not turn and that the outlying measurement was merely very noisy. 
An Overview of How Smoothers Work
The Kalman filter is a recursive filter with the Markov property - it's estimate at step k is  based only on the estimate from step k-1 and the measurement at step k. But this means that the estimate from step k-1 is based on step k-2, and so on back to the first epoch. Hence, the estimate at step k depends on all of the previous measurements, though to varying degrees. k-1 has the most influence, k-2 has the next most, and so on. 
Smoothing filters incorporate future measurements into the estimate for step k. The measurement from k+1 will have the most effect, k+2 will have less effect, k+3 less yet, and so on. 
This topic is called smoothing, but I think that is a misleading name. I could smooth the data above by passing it through a low pass filter. The result would be smooth, but not necessarily accurate because a low pass filter will remove real variations just as much as it removes noise. In contrast, Kalman smoothers are optimal - they incorporate all available information to make the best estimate that is mathematically achievable.
Types of Smoothers
There are three classes of Kalman smoothers that produce better tracking in these situations.
Fixed-Interval Smoothing
This is a batch processing based filter. This filter waits for all of the data to be collected before making any estimates. For example, you may be a scientist collecting data for an experiment, and don't need to know the result until the experiment is complete. A fixed-interval smoother will collect all the data, then estimate the state at each measurement using all available previous and future measurements. If it is possible for you to run your Kalman filter in batch mode it is always recommended to use one of these filters a it will provide much better results than the recursive forms of the filter from the previous chapters.
Fixed-Lag Smoothing
Fixed-lag smoothers introduce latency into the output. Suppose we choose a lag of 4 steps. The filter will ingest the first 3 measurements but not output a filtered result. Then, when the 4th measurement comes in the filter will produce the output for measurement 1, taking measurements 1 through 4 into account. When the 5th measurement comes in, the filter will produce the result for measurement 2, taking measurements 2 through 5 into account. This is useful when you need recent data but can afford a bit of lag. For example, perhaps you are using machine vision to monitor a manufacturing process. If you can afford a few seconds delay in the estimate a fixed-lag smoother will allow you to produce very accurate and smooth results.
Fixed-Point Smoothing
A fixed-point filter operates as a normal Kalman filter, but also produces an estimate for the state at some fixed time $j$.  Before the time $k$ reaches $j$ the filter operates as a normal filter. Once $k>j$ the filter estimates $x_k$ and then also updates its estimate for $x_j$ using all of the measurements between $j\dots k$. This can be useful to estimate initial paramters for a system, or for producing the best estimate for an event that happened at a specific time. For example, you may have a robot that took a photograph at time $j$. You can use a fixed-point smoother to get the best possible pose information for the camera at time $j$ as the robot continues moving.
Choice of Filters
The choice of these filters depends on your needs and how much memory and processing time you can spare. Fixed-point smoothing requires storage of all measurements, and is very costly to compute because the output is for every time step is recomputed for every measurement. On the other hand, the filter does produce a decent output for the current measurement, so this filter can be used for real time applications.
Fixed-lag smoothing only requires you to store a window of data, and processing requirements are modest because only that window is processed for each new measurement. The drawback is that the filter's output always lags the input, and the smoothing is not as pronounced as is possible with fixed-interval smoothing.
Fixed-interval smoothing produces the most smoothed output at the cost of having to be batch processed. Most algorithms use some sort of forwards/backwards algorithm that is only twice as slow as a recursive Kalman filter. 
Fixed-Interval Smoothing
There are many fixed-lag smoothers available in the literature. I have chosen to implement the smoother invented by Rauch, Tung, and Striebel because of its ease of implementation and efficiency of computation. It is also the smoother I have seen used most often in real applications. This smoother is commonly known as an RTS smoother.
Derivation of the RTS smoother runs to several pages of densely packed math. I'm not going to inflict it on you. Instead I will briefly present the algorithm, equations, and then move directly to implementation and demonstration of the smoother.
The RTS smoother works by first running the Kalman filter in a batch mode, computing the filter output for each step. Given the filter output for each measurement along with the covariance matrix corresponding to each output the RTS runs over the data backwards, incorporating its knowledge of the future into the past measurements. When it reaches the first measurement it is done, and the filtered output incorporates all of the information in a maximally optimal form.
The equations for the RTS smoother are very straightforward and easy to implement. This derivation is for the linear Kalman filter. Similar derivations exist for the EKF and UKF. These steps are performed on the output of the batch processing, going backwards from the most recent in time back to the first estimate. Each iteration incorporates the knowledge of the future into the state estimate. Since the state estimate already incorporates all of the past measurements the result will be that each estimate will contain knowledge of all measurements in the past and future. Here is it very important to distinguish between past, present, and future so I have used subscripts to denote whether the data is from the future or not.
Predict Step
$$\begin{aligned}
\mathbf{P} &= \mathbf{FP}_k\mathbf{F}^\mathsf{T} + \mathbf{Q }
\end{aligned}$$
Update Step
$$\begin{aligned}
\mathbf{K}k &= \mathbf{P}_k\mathbf{F}^\mathsf{T}\mathbf{P}^{-1} \
\mathbf{x}_k &= \mathbf{x}_k + \mathbf{K}_k(\mathbf{x}{k+1} - \mathbf{Fx}k) \
\mathbf{P}_k &= \mathbf{P}_k + \mathbf{K}_k(\mathbf{P}{k+1} - \mathbf{P})\mathbf{K}_k^\mathsf{T}
\end{aligned}$$
As always, the hardest part of the implementation is correctly accounting for the subscripts. A basic implementation without comments or error checking would be:
```python
def rts_smoother(Xs, Ps, F, Q):
    n, dim_x, _ = Xs.shape
# smoother gain
K = zeros((n,dim_x, dim_x))
x, P, Pp = Xs.copy(), Ps.copy(), Ps.copy
for k in range(n-2,-1,-1):
    Pp[k] = dot(F, P[k]).dot(F.T) + Q # predicted covariance
    K[k]  = dot(P[k], F.T).dot(inv(Pp[k]))
    x[k] += dot(K[k], x[k+1] - dot(F, x[k]))
    P[k] += dot(K[k], P[k+1] - Pp[k]).dot(K[k].T)
return (x, P, K, Pp)
```
This implementation mirrors the implementation provided in FilterPy. It assumes that the Kalman filter is being run externally in batch mode, and the results of the state and covariances are passed in via the Xs and Ps variable.
Here is an example.
End of explanation
plot_rts(noise=1.)
Explanation: I've injected a lot of noise into the signal to allow you to visually distinguish the RTS output from the ideal output. In the graph above we can see that the Kalman filter, drawn as the green dotted line, is reasonably smooth compared to the input, but it still wanders from from the ideal line when several measurements in a row are biased towards one side of the line. In contrast, the RTS output is both extremely smooth and very close to the ideal output.
With a perhaps more reasonable amount of noise we can see that the RTS output nearly lies on the ideal output. The Kalman filter output, while much better, still varies by a far greater amount.
End of explanation
plot_rts(noise=7., Q=.1)
Explanation: However, we must understand that this smoothing is predicated on the system model. We have told the filter that what we are tracking follows a constant velocity model with very low process error. When the filter looks ahead it sees that the future behavior closely matches a constant velocity so it is able to reject most of the noise in the signal. Suppose instead our system has a lot of process noise. For example, if we are tracking a light aircraft in gusty winds its velocity will change often, and the filter will be less able to distinguish between noise and erratic movement due to the wind. We can see this in the next graph.
End of explanation
plot_rts(7.,show_velocity=True)
Explanation: This underscores the fact that these filters are not smoothing the data in colloquial sense of the term. The filter is making an optimal estimate based on previous measurements, future measurements, and what you tell it about the behavior of the system and the noise in the system and measurements.
Let's wrap this up by looking at the velocity estimates of Kalman filter vs the RTS smoother.
End of explanation
from kf_book.book_plots import figsize
from kf_book.smoothing_internal import *
with figsize(y=2):
    show_fixed_lag_numberline()
Explanation: The improvement in the velocity, which is an hidden variable, is even more dramatic. 
Fixed-Lag Smoothing
The RTS smoother presented above should always be your choice of algorithm if you can run in batch mode because it incorporates all available data into each estimate. Not all problems allow you to do that, but you may still be interested in receiving smoothed values for previous estimates. The number line below illustrates this concept.
End of explanation
from filterpy.kalman import FixedLagSmoother, KalmanFilter
import numpy.random as random
fls = FixedLagSmoother(dim_x=2, dim_z=1, N=8)
fls.x = np.array([0., .5])
fls.F = np.array([[1.,1.],
                  [0.,1.]])
fls.H = np.array([[1.,0.]])
fls.P *= 200
fls.R *= 5.
fls.Q *= 0.001
kf = KalmanFilter(dim_x=2, dim_z=1)
kf.x = np.array([0., .5])
kf.F = np.array([[1.,1.],
                 [0.,1.]])
kf.H = np.array([[1.,0.]])
kf.P *= 200
kf.R *= 5.
kf.Q *= 0.001
N = 4 # size of lag
nom =  np.array([t/2. for t in range (0, 40)])
zs = np.array([t + random.randn()*5.1 for t in nom])
for z in zs:
    fls.smooth(z)
    
kf_x, _, _, _ = kf.batch_filter(zs)
x_smooth = np.array(fls.xSmooth)[:, 0]
fls_res = abs(x_smooth - nom)
kf_res = abs(kf_x[:, 0] - nom)
plt.plot(zs,'o', alpha=0.5, marker='o', label='zs')
plt.plot(x_smooth, label='FLS')
plt.plot(kf_x[:, 0], label='KF', ls='--')
plt.legend(loc=4)
print('standard deviation fixed-lag: {:.3f}'.format(np.mean(fls_res)))
print('standard deviation kalman: {:.3f}'.format(np.mean(kf_res)))
Explanation: At step $k$ we can estimate $x_k$ using the normal Kalman filter equations. However, we can make a better estimate for $x_{k-1}$ by using the measurement received for $x_k$. Likewise, we can make a better estimate for $x_{k-2}$ by using the measurements recevied for $x_{k-1}$ and $x_{k}$. We can extend this computation back for an arbitrary $N$ steps.
Derivation for this math is beyond the scope of this book; Dan Simon's Optimal State Estimation [2] has  a very good exposition if you are interested. The essense of the idea is that instead of having a state vector $\mathbf{x}$ we make an augmented state containing
$$\mathbf{x} = \begin{bmatrix}\mathbf{x}k \ \mathbf{x}{k-1} \ \vdots\ \mathbf{x}_{k-N+1}\end{bmatrix}$$
This yields a very large covariance matrix that contains the covariance between states at different steps. FilterPy's class FixedLagSmoother takes care of all of this computation for you, including creation of the augmented matrices. All you need to do is compose it as if you are using the KalmanFilter class and then call smooth(), which implements the predict and update steps of the algorithm. 
Each call of smooth computes the estimate for the current measurement, but it also goes back and adjusts the previous N-1 points as well. The smoothed values are contained in the list FixedLagSmoother.xSmooth. If you use FixedLagSmoother.x you will get the most recent estimate, but it is not smoothed and is no different from a standard Kalman filter output.
End of explanation
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