Kernel Density Estimation

The problem, from first principles

01-histogram-bins showed that a histogram's look depends on where its bins start and how wide they are, and its blocky bars are a poor picture of a smooth quantity. Kernel density estimation gives a smooth curve instead. Place a small bump, a kernel, on every data point and add them up: where data is dense the bumps pile up into a high ridge, and where it is sparse the curve is low. One number, the bandwidth, sets how wide each bump is, and it plays the role the bin width played, trading smoothness against detail. This question builds the estimator, a standard rule for choosing the bandwidth and a check that the result is a valid density.

From theory to code

Implement gaussian_kde(x, grid, bandwidth), which evaluates the density estimate at a set of points, then silverman_bandwidth(x), a rule-of-thumb bandwidth, then integrate_density(density, grid), which uses the trapezoid rule to check that the curve has total area one. The signatures and docstrings are already in the editor.

Constraints

• x is a 1D float array of at least two values and grid is a 1D float array of the points at which to evaluate the density. bandwidth is a positive float.
• gaussian_kde returns an array the shape of grid where entry j is mean_i( N(grid[j] - x[i]; 0, bandwidth) ), the average of Gaussian bumps of standard deviation bandwidth centred on every data point. It raises ValueError if bandwidth <= 0.
• silverman_bandwidth returns 0.9 * min(std, IQR / 1.34) * n ** (-1/5) as a float, where std is the sample standard deviation with ddof=1, IQR is the interquartile range from np.percentile and n = len(x). If the minimum of the two spread measures is zero, it uses whichever is nonzero, and 1.0 if both are zero.
• integrate_density(density, grid) returns the trapezoid-rule area under the curve as a float, for a strictly increasing grid. Do not call np.trapz or np.trapezoid.
• Inputs must not be modified.

Hints