<SYSTEM_TASK:>
Given the following text description, write Python code to implement the functionality described below step by step
<END_TASK>
<USER_TASK:>
Description:
 Let's see if we can get Aaron's delay network to recognize two different patterns that are multidimensional.  We want patterns where you need information from both dimensions, so let's do phase-shifted sign waves.
Step1:  Now let's create a network that represents a rolling window in time (Aaron's "delay network").  The process determines what sort of pattern the network will be optimized for -- here we just go with white noise of a maximum of 3Hz.  theta determines how big the rolling window is -- here we use 0.5 seconds.
Step2:  Now we need to create the training data for decoding out of the rolling window.  Our patterns are larger than the rolling window, so to create our training data we will take our patterns, shift them, and cut them down to the right size.  In order to then give that to nengo, we also need to project from the window's space to the internal representation space (using the inv_basis).
The target array is the desired output value for each of the slices of the pattern in eval_points.  We'll use 1 for pattern1 and -1 for pattern2.
Step3:  Now we can create a connection optimized to do this decoding
Step4:  Let's try feeding in those two patterns and see what the response is
Step5:  Yay it works!  Still a pretty hight rmse, though.  Can we improve that?
One possible problem is that the data used for the output connection is assuming that the delay network does a perfect job.  What if we instead train it based on the real data coming out of the delay network?  This is basically just making our eval_points a bit more realistic.
Step6:  That's some improvement!  What else can we do?
The pool Ensemble is pretty high-dimensional, and for some high-dimensional functions, it's useful to get a better encoder distribution.  So, let's use feedback alignment to find better encoders
Step7:  That looks like it should improve things!  How well does it do?
<ASSISTANT_TASK:>
Python Code:
s_pattern = 4000                # number of data points in the pattern
t = np.arange(s_pattern)*0.001  # time points for the elements in the pattern
D = 2
pattern1 = np.vstack([np.sin(t*np.pi), np.cos(t*np.pi)]).T
pattern2 = np.vstack([np.sin(t*np.pi), -np.sin(t*np.pi)]).T
plt.subplot(1, 2, 1)
plt.plot(t, pattern1)
plt.title('pattern 1')
plt.subplot(1, 2, 2)
plt.plot(t, pattern2)
plt.title('pattern 2')
plt.show()
Explanation: Let's see if we can get Aaron's delay network to recognize two different patterns that are multidimensional.  We want patterns where you need information from both dimensions, so let's do phase-shifted sign waves.
End of explanation
net = nengo.Network()
with net:
    process = nengo.processes.WhiteSignal(period=100., high=3., y0=0)
    
    rw = []
    for i in range(D):
        rw.append(nengolib.networks.RollingWindow(theta=0.5, n_neurons=3000, process=process, neuron_type=nengo.LIFRate()))
Explanation: Now let's create a network that represents a rolling window in time (Aaron's "delay network").  The process determines what sort of pattern the network will be optimized for -- here we just go with white noise of a maximum of 3Hz.  theta determines how big the rolling window is -- here we use 0.5 seconds.
End of explanation
s_window = 500
t_window = np.linspace(0, 1, s_window)
inv_basis = rw[0].inverse_basis(t_window)
    
eval_points = []
target = []
for i in range(s_pattern):
    eval_points.append(np.dot(inv_basis, np.roll(pattern1, i)[:s_window]).T)
    eval_points.append(np.dot(inv_basis, np.roll(pattern2, i)[:s_window]).T)
    target.append([1])
    target.append([-1])
eval_points = np.array(eval_points).reshape(len(eval_points), -1)
Explanation: Now we need to create the training data for decoding out of the rolling window.  Our patterns are larger than the rolling window, so to create our training data we will take our patterns, shift them, and cut them down to the right size.  In order to then give that to nengo, we also need to project from the window's space to the internal representation space (using the inv_basis).
The target array is the desired output value for each of the slices of the pattern in eval_points.  We'll use 1 for pattern1 and -1 for pattern2.
End of explanation
with net:
    
    pool = nengo.Ensemble(n_neurons=3000, dimensions=eval_points.shape[1],
                          neuron_type=nengo.LIFRate(), seed=1)
    start = 0
    for r in rw:
        nengo.Connection(r.state, pool[start:start+r.state.size_out])
        start += r.state.size_out
        
    result = nengo.Node(None, size_in=1)
    dec_conn = nengo.Connection(pool, result,
                     eval_points=eval_points, scale_eval_points=False,
                     function=target, synapse=0.1)
Explanation: Now we can create a connection optimized to do this decoding
End of explanation
model = nengo.Network()
model.networks.append(net)
with model:
    offsets = [-np.pi/2, np.pi]
    def stim_func(t):
        offset = offsets[int(t/5) % len(offsets)]
        return np.sin(t*np.pi), np.sin(t*np.pi+offset)
    ideal_results = [1.0, -1.0]
    def ideal_func(t):
        return ideal_results[int(t/5) % len(ideal_results)]
    ideal_result = nengo.Node(ideal_func)
    
    
    
    stim = nengo.Node(stim_func)
    for i in range(D):
        nengo.Connection(stim[i], rw[i].input, synapse=None)
    
    p_result = nengo.Probe(result)
    p_stim = nengo.Probe(stim)
    p_pool = nengo.Probe(pool)
    p_ideal = nengo.Probe(ideal_result)
    
sim = nengo.Simulator(model)
sim.run(10)
plt.plot(sim.trange(), sim.data[p_stim], label='input')
plt.plot(sim.trange(), sim.data[p_result], label='output (rmse:%1.3f)' % np.sqrt(np.mean((sim.data[p_ideal]-sim.data[p_result])**2)))
plt.legend(loc='best')
Explanation: Let's try feeding in those two patterns and see what the response is
End of explanation
model = nengo.Network()
model.networks.append(net)
with model:
    offsets = [-np.pi/2, np.pi]
    def stim_func(t):
        offset = offsets[int(t/5) % len(offsets)]
        return np.sin(t*np.pi), np.sin(t*np.pi+offset)
    
    stim = nengo.Node(stim_func)
    for i in range(D):
        nengo.Connection(stim[i], rw[i].input, synapse=None)
        
    result2 = nengo.Node(None, size_in=1)
    nengo.Connection(pool, result2, eval_points=sim.data[p_pool], function=sim.data[p_ideal], scale_eval_points=False)
    
    p_result = nengo.Probe(result)
    p_result2 = nengo.Probe(result2)
    p_stim = nengo.Probe(stim)
    
sim2 = nengo.Simulator(model)
sim2.run(10)
plt.plot(sim2.trange(), sim2.data[p_stim], label='input')
plt.plot(sim2.trange(), sim2.data[p_result], label='output (rmse:%1.3f)' % np.sqrt(np.mean((sim.data[p_ideal]-sim2.data[p_result])**2)))
plt.plot(sim2.trange(), sim2.data[p_result2], label='trained (rmse:%1.3f)' % np.sqrt(np.mean((sim.data[p_ideal]-sim2.data[p_result2])**2)))
plt.legend(loc='best')
Explanation: Yay it works!  Still a pretty hight rmse, though.  Can we improve that?
One possible problem is that the data used for the output connection is assuming that the delay network does a perfect job.  What if we instead train it based on the real data coming out of the delay network?  This is basically just making our eval_points a bit more realistic.
End of explanation
model = nengo.Network()
model.networks.append(net)
with model:
    result2 = nengo.Node(None, size_in=1)
    conn = nengo.Connection(pool, result2, eval_points=sim.data[p_pool], function=sim.data[p_ideal], scale_eval_points=False)
    
import nengo_encoder_learning
error = nengo_encoder_learning.improve(conn, learning_rate=1e-2, steps=10)
plt.plot(error)
Explanation: That's some improvement!  What else can we do?
The pool Ensemble is pretty high-dimensional, and for some high-dimensional functions, it's useful to get a better encoder distribution.  So, let's use feedback alignment to find better encoders:
End of explanation
model = nengo.Network()
model.networks.append(net)
with model:
    offsets = [-np.pi/2, np.pi]
    def stim_func(t):
        offset = offsets[int(t/5) % len(offsets)]
        return np.sin(t*np.pi), np.sin(t*np.pi+offset)
    
    stim = nengo.Node(stim_func)
    for i in range(D):
        nengo.Connection(stim[i], rw[i].input, synapse=None)
        
    result2 = nengo.Node(None, size_in=1)
    nengo.Connection(pool, result2, eval_points=sim.data[p_pool], function=sim.data[p_ideal], scale_eval_points=False)
    
    p_result = nengo.Probe(result)
    p_result2 = nengo.Probe(result2)
    p_stim = nengo.Probe(stim)
    
sim2 = nengo.Simulator(model)
sim2.run(10)
plt.plot(sim2.trange(), sim2.data[p_stim], label='input')
plt.plot(sim2.trange(), sim2.data[p_result], label='output (rmse:%1.3f)' % np.sqrt(np.mean((sim.data[p_ideal]-sim2.data[p_result])**2)))
plt.plot(sim2.trange(), sim2.data[p_result2], label='trained (rmse:%1.3f)' % np.sqrt(np.mean((sim.data[p_ideal]-sim2.data[p_result2])**2)))
plt.legend(loc='best')
Explanation: That looks like it should improve things!  How well does it do?
End of explanation
<END_TASK>