Note

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# Time Series Histogram¶

This example demonstrates how to efficiently visualize large numbers of time series in a way that could potentially reveal hidden substructure and patterns that are not immediately obvious, and display them in a visually appealing way.

In this example, we generate multiple sinusoidal "signal" series that are
buried under a larger number of random walk "noise/background" series. For an
unbiased Gaussian random walk with standard deviation of σ, the RMS deviation
from the origin after n steps is σ*sqrt(n). So in order to keep the sinusoids
visible on the same scale as the random walks, we scale the amplitude by the
random walk RMS. In addition, we also introduce a small random offset `phi`

to shift the sines left/right, and some additive random noise to shift
individual data points up/down to make the signal a bit more "realistic" (you
wouldn't expect a perfect sine wave to appear in your data).

The first plot shows the typical way of visualizing multiple time series by
overlaying them on top of each other with `plt.plot`

and a small value of
`alpha`

. The second and third plots show how to reinterpret the data as a 2d
histogram, with optional interpolation between data points, by using
`np.histogram2d`

and `plt.pcolormesh`

.

```
from copy import copy
import time
import numpy as np
import numpy.matlib
import matplotlib.pyplot as plt
from matplotlib.colors import LogNorm
fig, axes = plt.subplots(nrows=3, figsize=(6, 8), constrained_layout=True)
# Make some data; a 1D random walk + small fraction of sine waves
num_series = 1000
num_points = 100
SNR = 0.10 # Signal to Noise Ratio
x = np.linspace(0, 4 * np.pi, num_points)
# Generate unbiased Gaussian random walks
Y = np.cumsum(np.random.randn(num_series, num_points), axis=-1)
# Generate sinusoidal signals
num_signal = int(round(SNR * num_series))
phi = (np.pi / 8) * np.random.randn(num_signal, 1) # small random offset
Y[-num_signal:] = (
np.sqrt(np.arange(num_points))[None, :] # random walk RMS scaling factor
* (np.sin(x[None, :] - phi)
+ 0.05 * np.random.randn(num_signal, num_points)) # small random noise
)
# Plot series using `plot` and a small value of `alpha`. With this view it is
# very difficult to observe the sinusoidal behavior because of how many
# overlapping series there are. It also takes a bit of time to run because so
# many individual artists need to be generated.
tic = time.time()
axes[0].plot(x, Y.T, color="C0", alpha=0.1)
toc = time.time()
axes[0].set_title("Line plot with alpha")
print(f"{toc-tic:.3f} sec. elapsed")
# Now we will convert the multiple time series into a histogram. Not only will
# the hidden signal be more visible, but it is also a much quicker procedure.
tic = time.time()
# Linearly interpolate between the points in each time series
num_fine = 800
x_fine = np.linspace(x.min(), x.max(), num_fine)
y_fine = np.empty((num_series, num_fine), dtype=float)
for i in range(num_series):
y_fine[i, :] = np.interp(x_fine, x, Y[i, :])
y_fine = y_fine.flatten()
x_fine = np.matlib.repmat(x_fine, num_series, 1).flatten()
# Plot (x, y) points in 2d histogram with log colorscale
# It is pretty evident that there is some kind of structure under the noise
# You can tune vmax to make signal more visible
cmap = copy(plt.cm.plasma)
cmap.set_bad(cmap(0))
h, xedges, yedges = np.histogram2d(x_fine, y_fine, bins=[400, 100])
pcm = axes[1].pcolormesh(xedges, yedges, h.T, cmap=cmap,
norm=LogNorm(vmax=1.5e2), rasterized=True)
fig.colorbar(pcm, ax=axes[1], label="# points", pad=0)
axes[1].set_title("2d histogram and log color scale")
# Same data but on linear color scale
pcm = axes[2].pcolormesh(xedges, yedges, h.T, cmap=cmap,
vmax=1.5e2, rasterized=True)
fig.colorbar(pcm, ax=axes[2], label="# points", pad=0)
axes[2].set_title("2d histogram and linear color scale")
toc = time.time()
print(f"{toc-tic:.3f} sec. elapsed")
plt.show()
```

Out:

```
0.240 sec. elapsed
0.094 sec. elapsed
```

References

The use of the following functions, methods, classes and modules is shown in this example:

**Total running time of the script:** ( 0 minutes 3.073 seconds)

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