============ Bounded KDEs ============ In many situations, we may have a set of samples which are bounded by a given domain. For this case, the standard `scipy.stats.gaussian_kde `_ will not accurately capture the true shape of the distribution. For cases like this, we require a bounded KDE. There are a number of ways to take into account the bounded nature of the distribution. The common methods include: - Reflective: Making the boundaries reflective, i.e. the derivative is zero at the boundary. - Transform: Where the values are transformed to new coordinates in which the PDF does not have a boundary and looks close to Gaussian. This makes it easier for `scipy.stats.gaussian_kde` to represent the distribution. `pesummary` handles bounded KDEs through the `pesummary.core.plots.bounded_1d_kde` module. Below is an example which shows a distribution which is bounded in the domain `0 < x < 1`. We show how each method handles the boundary: .. code-block:: python >>> from pesummary.core.plots.bounded_1d_kde import bounded_1d_kde >>> import numpy as np >>> from scipy import stats >>> import matplotlib.pyplot as plt >>> rands = np.random.random_sample(10000) >>> transf = 0.5 + np.arcsin(2. * rands - 1.) / np.pi >>> fig = plt.figure() >>> plt.hist(transf, bins=50, density=True, histtype="step") >>> xsmooth = np.linspace(0., 1., 100) >>> unbounded_kde = stats.gaussian_kde(transf) >>> reflection = bounded_1d_kde(transf, xlow=0., xhigh=1., method="Reflection") >>> transform = bounded_1d_kde(transf, xlow=0., xhigh=1., method="Transform", smooth=2) >>> plt.plot(xsmooth, unbounded_kde(xsmooth), label="gaussiankde") >>> plt.plot(xsmooth, reflection(xsmooth), label="reflection") >>> plt.plot(xsmooth, transform(xsmooth), label="transform") >>> plt.ylabel("Probability Density") >>> plt.legend() >>> plt.show() .. image:: ./examples/bounded_kde.png Of course, different techniques for handling the boundaries are useful in different situations. Clearly, the `transform` method is best for the example above. Below we show an example where the `reflection` method is best: .. code-block:: python >>> xsmooth = np.linspace(0., 1., 100) >>> pts = np.random.uniform(0, 1, 10000) >>> fig = plt.figure() >>> plt.hist(pts, bins=50, density=True, histtype="step") >>> unbounded_kde = stats.gaussian_kde(pts) >>> reflection = bounded_1d_kde(pts, xlow=0., xhigh=1., method="Reflection") >>> transform = bounded_1d_kde(pts, xlow=0., xhigh=1., method="Transform", smooth=6) >>> plt.plot(xsmooth, unbounded_kde(xsmooth), label="gaussiankde") >>> plt.plot(xsmooth, reflection(xsmooth), label="reflection") >>> plt.plot(*transform(xsmooth), label="transform") >>> plt.ylabel("Probability Density") >>> plt.legend() >>> plt.show() .. image:: ./examples/bounded_kde_uniform.png