# Get smooth CDF from OnlineStats's Quantile

**URL:** <https://discourse.julialang.org/t/get-smooth-cdf-from-onlinestatss-quantile/113886>\
**Category:** Statistics\
**Created:** [May 6, 2024, 8:50am UTC](https://discourse.julialang.org/t/get-smooth-cdf-from-onlinestatss-quantile/113886 "2024-05-06T08:50:13Z")\
**Posts on this page:** 1\
**Showing post:** 4

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**Author:** ![disberd](https://avatars.discourse-cdn.com/v4/letter/d/8edcca/32.png) [@disberd](https://discourse.julialang.org/u/disberd)\
**Post date:** [May 6, 2024, 1:27pm UTC](https://discourse.julialang.org/t/get-smooth-cdf-from-onlinestatss-quantile/113886/4 "2024-05-06T13:27:42Z")

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Thanks @tbeason, I actually found the `Ash` function from this other somehow related post:

> [@On-the-flight kernel density estimation?](https://discourse.julialang.org/t/on-the-flight-kernel-density-estimation/95817):
>
> I have the following type of problem: imagine a Monte-Carlo simulation generating some data, and I want to estimate the density function for that simulation. I cannot store the data for each Monte-Carlos step, which would be prohibitive, but I want to do something better than simply constructing a histogram. To provide a concrete example: In theory a nice way to obtain the density would be: julia\> using Plots, KernelDensity julia\> data = randn(10^4); julia\> k = kde(data); julia\> plot(k.x,…

I understand that the AverageShiftedHistogram is somehow something that does similarly to the kernel density you mentioned, so using that and integrating its output was indeed the first thing I tried as mentioned in the original post.

Do you think this approach is actually different from what you are suggesting?

I was actually trying out the different options in a Pluto notebook:

> **Notebook Code**
>
> ```julia
> # ╔═╡ 21f03ec8-4e76-4e7c-8cac-3e67d9e79e44
> begin
> using OnlineStats
> using PlutoPlotly
> using Statistics
> end
> 
> # ╔═╡ dff0f3f2-6706-4f60-a607-1fcff9b3b314
> begin
> db2lin(x) = 10.0^(x/10)
> q = Quantile()
> o = OrderStats(100)
> a = db2lin.(randn(10^5) .* 2) # Let's create some lognormal variable
> # a = randn(10^4)
> fit!(q, a)
> fit!(o, a)
> end
> 
> # ╔═╡ d78b749b-a739-4fcf-a5c2-1c7d41fc11f0
> let
> d1 = let
> # We just build an Average Shifted Histogram for the smoothed pdf of the histogram used by Quantile
> ash = Ash(q.eh, 1)
> x, y = value(ash) # This extracts x and y for the smoothed pdf
> y = cumsum(y)
> # eltype to convert the step (which is TwicePrecision) to the actual type of the elements of y
> y *= eltype(y)(x.step) # We have to normalize by the step on x to have the sum to 1
> scatter(;x,y, name = "ASH")
> end
> d2 = let
> y = range(0, 1; length = 101)
> x = map(y) do y
> value(q, y)
> end
> scatter(;x, y, name = "Quantile", line_dash = :dash)
> end
> d3 = let
> y = range(0, 1; length = 101)
> x = map(y) do y
> quantile(o, y)
> end
> scatter(;x, y, name = "OrderStats", line_dash = :dot)
> end
> plot([d1, d2, d3], Layout(;
> template = "none",
> uirevision = 1,
> xaxis = attr(;
> title = "X"
> ),
> yaxis = attr(;
> title = "P{x <= X}"
> )
> ))
> end
> 
> ```

I see that Ash works for smoothing but the actual curve from OrderStats is “closer” to the Quantile curve (see below example zoom of the CDF plot of the notebook):  
 ![image](https://global.discourse-cdn.com/julialang/original/3X/7/0/70693326f8da1943f1e2a1976ba88156b47fb1bd.png)

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