# Dynamic Programming

Dynamic Programming (DP) is a general algorithm for finding the optimal solutions to a limited class of exponential problems in linear time. It is commonly used for aligning two sequences that may have slightly inconsistent time bases - this is known as Dynamic Time Warping (DTW) and was an early technique used in speech recognition; it is now widely applied in analogous problems in matching nucleotide sequences in bioinformatics.

The core algorithm is a implemented via nested loops, which typically defeat automatic vectorization. Thus, naive implementations in high-level languages such as Python and Matlab tend to be very inefficient. The code below shows 500x speedup for the core DP operation by replacing the Python with C code.

First, we set up the problem by loading two soundfiles and calculating their spectrograms. We then calculate the cosine similarity between each spectrum (columns of the spectrogram) as a local cost matrix, then use DP to find the lowest-cost path through the cost matrix, giving the optimal DTW alignment.

In [3]:
%matplotlib inline

In [7]:
cd /Users/dpwe/projects/dtw/dp_python

/Users/dpwe/projects/dtw/dp_python

In [57]:
import numpy as np
import matplotlib.pyplot as plt
import librosa  # soundfile access library - see https://github.com/bmcfee/librosa
import dpcore   # the current library

In [93]:
# Mirror matlab example from http://www.ee.columbia.edu/ln/rosa/matlab/dtw/
# Two different speakers saying "Cottage cheese with chives is delicious"
# Calculate their short-time Fourier transforms
D1C = librosa.stft(d1, n_fft=512, hop_length=128)
D2C = librosa.stft(d2, n_fft=512, hop_length=128)
# We'll use the magnitudes to calculate similarity (ignore phase)
D1 = np.abs(D1C)
D2 = np.abs(D2C)

In [94]:
# Plot the spectrograms.  You see a similar sequence of sounds, but different timing details
plt.subplot(2,1,1)
librosa.display.specshow(20*np.log10(D1), sr=sr, hop_length=128, y_axis="linear", x_axis="time", cmap='jet')
plt.subplot(2,1,2)
librosa.display.specshow(20*np.log10(D2), sr=sr, hop_length=128, y_axis="linear", x_axis="time", cmap='jet')

Out[94]:
<matplotlib.image.AxesImage at 0x10c9a25d0>

In [95]:
# Cosine similarity matrix (slow one-liner)
# Each cell SM[i,j] is the cosine similarity between D1[:,i] and D2[:,j]
# i.e. the inner product of the two unit-normalized feature vectors
SM = np.array([[np.sum(a*b)/np.sqrt(np.sum(a**2)*np.sum(b**2)) for b in D2.T] for a in D1.T])

In [96]:
# Find the minimum-cost path.  We use 1-SM so that cosine similarity == 1 -> cost = 0
# penalty is the additional cost incurred by non-diagonal steps (promotes diagonality)
localcost = np.array(1.0-SM, order='C', dtype=float)
p, q, C, phi = dpcore.dp(localcost, penalty=0.1)
# p and q are vectors giving the row and column indices along the best path
# C returns the full minimum-cost matrix, and phi is the full traceback matrix

In [97]:
# Plot the best path on top of local similarity matrix
fig, ax = plt.subplots()
ax.imshow(SM, interpolation='none', cmap='binary')
ax.hold(True)
ax.plot(q,p,'-r')
ax.hold(False)
ax.axis([0, np.size(D2, axis=1), 0, np.size(D1, axis=1)])

Out[97]:
[0, 354, 0, 435]

In [98]:
# We can use the best-aligned columns to plot aligned spectrograms
# Notice how the features in the two spectrograms are now lined up horizontally
plt.subplot(2,1,1)
librosa.display.specshow(20*np.log10(D1[:, p]), sr=sr, hop_length=128, y_axis="linear", x_axis="time", cmap='jet')
plt.subplot(2,1,2)
librosa.display.specshow(20*np.log10(D2[:, q]), sr=sr, hop_length=128, y_axis="linear", x_axis="time", cmap='jet')

Out[98]:
<matplotlib.image.AxesImage at 0x10dcaaed0>

In [90]:
# We can compare the timings of the C external implementation
# to the pure Python version of the core DP routine
# with the use_extension argument:
%timeit C, phi = dpcore.dpcore(localcost, 0.1, use_extension=True)
%timeit C, phi = dpcore.dpcore(localcost, 0.1, use_extension=False)

100 loops, best of 3: 2.82 ms per loop
1 loops, best of 3: 1.77 s per loop

In [101]:
# Just to check, they do give the same result (within numerical precision)
Cext, phiext = dpcore.dpcore(localcost, 0.1, use_extension=True)
Cpyt, phipyt = dpcore.dpcore(localcost, 0.1, use_extension=False)
print np.max(np.abs(Cext-Cpyt))

5.68434188608e-14

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