Say we have discrete CDF values (percentiles from the field for one metric) such as

percentile 1 => 1.4

…

percentile 10 => 10.3

…

percentile 50 => 50.3

…

percentile 80 => 70.3

…

percentile 100 => 90.3

I am construction a CDF using linear interpolation such as

```
percentile_values = [
[0.0, 0.01, 0.1,
1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0,
10.0, 20.0, 30.0, 40.0, 50.0, 60.0, 70.0, 80.0,
90.0, 91.0, 92.0, 93.0, 94.0, 95.0, 96.0, 97.0, 98.0, 99.0,
99.9, 99.99, 100.0]
using Interpolations
Xs = [...] # same length as percentile_values
nodes = (percentile_values,)
itp = interpolate(nodes, Xs, Gridded(Linear()))
```

Questions:

- Can i generate PDF from this interpolated function using differentiation ? If so can you help me with sample code ?
- I have attached standalone sample code that describes my current process ( code uses quantiles [0,1] instead of percentiles [0,100] just to adhere to CDF rules), can you suggest more robust process to convert information from CDF to PDF ?
- Can you provide suggestions an comments on the current method used by me using
`Spline1D`

?

Please refer sample code

```
using Dierckx
using Interpolations
using ImageFiltering
import Plots
import StatsPlots
using Plots.PlotMeasures
Plots.gr()
Plots.theme(:ggplot2)
# sample input
q_values = [ 0.0, 0.0001, 0.001,
0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09,
0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9,
0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99,
0.999, 0.9999, 1.0 ]
feature_values = [ 9.0, 12.0, 14.0,
23.0, 28.0, 31.0, 33.0, 36.0, 38.0, 39.0, 41.0, 43.0,
45.0, 58.0, 70.0, 80.0, 87.0, 91.0, 94.0, 95.0, 97.0,
97.0, 97.0, 97.0, 97.0, 97.0, 97.0, 98.0, 98.0, 98.0,
99.0, 99.0, 100.0 ]
# TODO task: Generate PDF from discrete quantiles (also called discrete CDF)
p1 = Plots.scatter(feature_values, q_values, markersize=1.75, color="black", label="33 quantile data points")
Plots.plot!(p1, feature_values, q_values, label="method.0) simple CDF",
title = "input discrete CDF of feature value", color=:green, linewidth=2,
ylabel="Cumulative Density [or Quantile]",
xlim=(-5,105), xlabel="Feature value ∈ [0,100]",
border=nothing
)
# using Spline1D
# interpolate function mapping quantiles -> feature values
spl = Spline1D(q_values, feature_values, k=1, bc="extrapolate")
sample_cdf_q_values1 = [rand() for p in 1:25000]
pdf_of_feature_values1 = [evaluate(spl, p) for p in sample_cdf_q_values1]
p2 = Plots.density(pdf_of_feature_values1,
title = "generated PDF of feature value", color=:green, linewidth=2, label="method.0) using Spline1D",
ylabel="Density",
xlim=(-5,105), xlabel="Feature value ∈ [0,100]",
border=nothing
)
# using methods from Interpolations
# https://discourse.julialang.org/t/interpolations-jl-discrete-cdf-to-pdf/60124/8?u=bicepjai
# smoothing to get a more reasonable-looking PDF
smoothed_feature_values = imfilter(feature_values, ImageFiltering.Kernel.gaussian((1,)));
sample_feature_values = 0.01:0.01:100.0 # just a range for plotting
# you can change SteffenMonotonicInterpolation to some other monotonic algorithm from Interpolations.jl
itp_cdf = extrapolate(interpolate(smoothed_feature_values, q_values, SteffenMonotonicInterpolation()), Flat());
Plots.plot!(p1, sample_feature_values, itp_cdf.(sample_feature_values),
color=:orange, linewidth=1, label="imfilter SteffenMonotonicInterpolation CDF")
itp_pdf(x) = Interpolations.gradient1(itp_cdf, x); # this is the PDF generate
Plots.plot!(p2, sample_feature_values, itp_pdf.(sample_feature_values),
color=:orange, linewidth=1.5, label="method.1) from Interpolations.gradient1 PDF")
# using new methods update the plots with different colors
# so that its easier for comparision
l = @Plots.layout([ a{0.5w} b{0.5w} ])
Plots.plot(p1, p2, layout=l,
legend=:topleft,
top_margin=5mm, bottom_margin=5mm, left_margin=5mm,
dpi=200, size=(1000,500), fmt = :png
)
```

References:

update: sample code and plots