//////////////////////////////////////
// Plot the spectrum
fig, ax = plt.subplots()
img = librosa.display.specshow(librosa.amplitude_to_db(S_full, ref=np.max),
y_axis="log", x_axis="time", sr=sr, ax=ax)
fig.colorbar(img, ax=ax);
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// As you can see, there are periods of silence and
// non-silence throughout this recording.
//
// As a first step, we can plot the root-mean-square (RMS) curve
rms = librosa.feature.rms(y=y)[0]
times = librosa.frames_to_time(np.arange(len(rms)))
fig, ax = plt.subplots()
ax.plot(times, rms)
ax.axhline(0.02, color="r", alpha=0.5)
ax.set(xlabel="Time", ylabel="RMS");
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// The red line at 0.02 indicates a reasonable threshold for silence detection.
// However, the RMS curve occasionally dips below the threshold momentarily,
// and we would prefer the detector to not count these brief dips as silence.
// This is where the Viterbi algorithm comes in handy!
//////////////////////////////////////////////////////////////////////////////////////////////////////////
// As a first step, we will convert the raw RMS score
// into a likelihood (probability) by logistic mapping
//
// :math:`P[V=1 | x] = \frac{\exp(x - \tau)}{1 + \exp(x - \tau)}`
//
// where :math:`x` denotes the RMS value and :math:`\tau=0.02` is our threshold.
// The variable :math:`V` indicates whether the signal is non-silent (1) or silent (0).
//
// We"ll normalize the RMS by its standard deviation to expand the
// range of the probability vector
r_normalized = (rms - 0.02) / np.std(rms)
p = np.exp(r_normalized) / (1 + np.exp(r_normalized))
////////////////////////////////////////////////////////////////////////////////////////////
// We can plot the probability curve over time:
fig, ax = plt.subplots()
ax.plot(times, p, label="P[V=1|x]")
ax.axhline(0.5, color="r", alpha=0.5, label="Descision threshold")
ax.set(xlabel="Time")
ax.legend();
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// which looks much like the first plot, but with the decision threshold
// shifted to 0.5. A simple silence detector would classify each frame
// independently of its neighbors, which would result in the following plot:
plt.figure(figsize=(12, 6))
fig, ax = plt.subplots(nrows=2, sharex=True)
librosa.display.specshow(librosa.amplitude_to_db(S_full, ref=np.max),
y_axis="log", x_axis="time", sr=sr, ax=ax[0])
ax[0].label_outer()
ax[1].step(times, p>=0.5, label="Non-silent")
ax[1].set(ylim=[0, 1.05])
ax[1].legend()
//////////////////////////////////////////////////////////////////////////////////////////////
// We can do better using the Viterbi algorithm.
// We"ll use state 0 to indicate silent, and 1 to indicate non-silent.
// We"ll assume that a silent frame is equally likely to be followed
// by silence or non-silence, but that non-silence is slightly
// more likely to be followed by non-silence.
// This is accomplished by building a self-loop transition matrix,
// where `transition[i, j]` is the probability of moving from state
// `i` to state `j` in the next frame.
transition = librosa.sequence.transition_loop(2, [0.5, 0.6])
print(transition)
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// Our `p` variable only indicates the probability of non-silence,
// so we need to also compute the probability of silence as its complement.
full_p = np.vstack([1 - p, p])
print(full_p)
////////////////////////////////////////////////////////////////////////
// Now, we"re ready to decode!
// We"ll use `viterbi_discriminative` here, since the inputs are
// state likelihoods conditional on data (in our case, data is rms).
states = librosa.sequence.viterbi_discriminative(full_p, transition)
// sphinx_gallery_thumbnail_number = 5
fig, ax = plt.subplots(nrows=2, sharex=True)
librosa.display.specshow(librosa.amplitude_to_db(S_full, ref=np.max),
y_axis="log", x_axis="time", sr=sr, ax=ax[0])
ax[0].label_outer()
ax[1].step(times, p>=0.5, label="Frame-wise")
ax[1].step(times, states, linestyle="--", color="orange", label="Viterbi")
ax[1].set(ylim=[0, 1.05])
ax[1].legend()
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
