|
| 1 | +""" |
| 2 | +===================== |
| 3 | +Time Series Histogram |
| 4 | +===================== |
| 5 | +
|
| 6 | +This example demonstrates how to efficiently visualize large numbers of time |
| 7 | +series in a way that could potentially reveal hidden substructure and patterns |
| 8 | +that are not immediately obvious, and display them in a visually appealing way. |
| 9 | +
|
| 10 | +In this example, we generate multiple sinusoidal "signal" series that are |
| 11 | +buried under a larger number of random walk "noise/background" series. For an |
| 12 | +unbiased Gaussian random walk with standard deviation of σ, the RMS deviation |
| 13 | +from the origin after n steps is σ*sqrt(n). So in order to keep the sinusoids |
| 14 | +visible on the same scale as the random walks, we scale the amplitude by the |
| 15 | +random walk RMS. In addition, we also introduce a small random offset ``phi`` |
| 16 | +to shift the sines left/right, and some additive random noise to shift |
| 17 | +individual data points up/down to make the signal a bit more "realistic" (you |
| 18 | +wouldn't expect a perfect sine wave to appear in your data). |
| 19 | +
|
| 20 | +The first plot shows the typical way of visualizing multiple time series by |
| 21 | +overlaying them on top of each other with ``plt.plot`` and a small value of |
| 22 | +``alpha``. The second and third plots show how to reinterpret the data as a 2d |
| 23 | +histogram, with optional interpolation between data points, by using |
| 24 | +``np.histogram2d`` and ``plt.pcolormesh``. |
| 25 | +""" |
| 26 | +from copy import copy |
| 27 | +import time |
| 28 | + |
| 29 | +import numpy as np |
| 30 | +import numpy.matlib |
| 31 | +import matplotlib.pyplot as plt |
| 32 | +from matplotlib.colors import LogNorm |
| 33 | + |
| 34 | +fig, axes = plt.subplots(nrows=3, figsize=(6, 8), constrained_layout=True) |
| 35 | + |
| 36 | +# Make some data; a 1D random walk + small fraction of sine waves |
| 37 | +num_series = 1000 |
| 38 | +num_points = 100 |
| 39 | +SNR = 0.10 # Signal to Noise Ratio |
| 40 | +x = np.linspace(0, 4 * np.pi, num_points) |
| 41 | +# Generate unbiased Gaussian random walks |
| 42 | +Y = np.cumsum(np.random.randn(num_series, num_points), axis=-1) |
| 43 | +# Generate sinusoidal signals |
| 44 | +num_signal = int(round(SNR * num_series)) |
| 45 | +phi = (np.pi / 8) * np.random.randn(num_signal, 1) # small random offest |
| 46 | +Y[-num_signal:] = ( |
| 47 | + np.sqrt(np.arange(num_points))[None, :] # random walk RMS scaling factor |
| 48 | + * (np.sin(x[None, :] - phi) |
| 49 | + + 0.05 * np.random.randn(num_signal, num_points)) # small random noise |
| 50 | +) |
| 51 | + |
| 52 | + |
| 53 | +# Plot series using `plot` and a small value of `alpha`. With this view it is |
| 54 | +# very difficult to observe the sinusoidal behavior because of how many |
| 55 | +# overlapping series there are. It also takes a bit of time to run because so |
| 56 | +# many individual artists need to be generated. |
| 57 | +tic = time.time() |
| 58 | +axes[0].plot(x, Y.T, color="C0", alpha=0.1) |
| 59 | +toc = time.time() |
| 60 | +axes[0].set_title("Line plot with alpha") |
| 61 | +print(f"{toc-tic:.3f} sec. elapsed") |
| 62 | + |
| 63 | + |
| 64 | +# Now we will convert the multiple time series into a histogram. Not only will |
| 65 | +# the hidden signal be more visible, but it is also a much quicker procedure. |
| 66 | +tic = time.time() |
| 67 | +# Linearly interpolate between the points in each time series |
| 68 | +num_fine = 800 |
| 69 | +x_fine = np.linspace(x.min(), x.max(), num_fine) |
| 70 | +y_fine = np.empty((num_series, num_fine), dtype=float) |
| 71 | +for i in range(num_series): |
| 72 | + y_fine[i, :] = np.interp(x_fine, x, Y[i, :]) |
| 73 | +y_fine = y_fine.flatten() |
| 74 | +x_fine = np.matlib.repmat(x_fine, num_series, 1).flatten() |
| 75 | + |
| 76 | + |
| 77 | +# Plot (x, y) points in 2d histogram with log colorscale |
| 78 | +# It is pretty evident that there is some kind of structure under the noise |
| 79 | +# You can tune vmax to make signal more visible |
| 80 | +cmap = copy(plt.cm.plasma) |
| 81 | +cmap.set_bad(cmap(0)) |
| 82 | +h, xedges, yedges = np.histogram2d(x_fine, y_fine, bins=[400, 100]) |
| 83 | +pcm = axes[1].pcolormesh(xedges, yedges, h.T, cmap=cmap, |
| 84 | + norm=LogNorm(vmax=1.5e2), rasterized=True) |
| 85 | +fig.colorbar(pcm, ax=axes[1], label="# points", pad=0) |
| 86 | +axes[1].set_title("2d histogram and log color scale") |
| 87 | + |
| 88 | +# Same data but on linear color scale |
| 89 | +pcm = axes[2].pcolormesh(xedges, yedges, h.T, cmap<
5DBC
span class=pl-c1>=cmap, |
| 90 | + vmax=1.5e2, rasterized=True) |
| 91 | +fig.colorbar(pcm, ax=axes[2], label="# points", pad=0) |
| 92 | +axes[2].set_title("2d histogram and linear color scale") |
| 93 | + |
| 94 | +toc = time.time() |
| 95 | +print(f"{toc-tic:.3f} sec. elapsed") |
| 96 | +plt.show() |
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