# Rolling Median vs Rolling Mean:

**URL:** <https://discourse.julialang.org/t/rolling-median-vs-rolling-mean/119227>\
**Category:** General Usage\
**Tags:** plotting, statistics\
**Created:** [September 9, 2024, 8:59pm UTC](https://discourse.julialang.org/t/rolling-median-vs-rolling-mean/119227 "2024-09-09T20:59:00Z")\
**Posts on this page:** 7\
**Page:** 1

<div class="post-metadata">

**Author:** ![GeodeticR](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/geodeticr/32/218661_2.png) [@GeodeticR](https://discourse.julialang.org/u/GeodeticR)\
**Post date:** [September 9, 2024, 8:59pm UTC](https://discourse.julialang.org/t/rolling-median-vs-rolling-mean/119227/1 "2024-09-09T20:59:00Z")

</div>

Hi all;

I have a questions that I am hoping someone smarter than me (not a difficult thing to be) can help me out with.

I am smoothing out some GPS/GNSS data and I am wondering why at many of the “excursions” the absolute median value is larger than the average. I, at least intuitively, would expect to see the opposite.

Here is the code I used for the rolling mean:

```julia
function moving_window(data, bandwidth)
    
    result = []
    moving_sum = sum(data[1:bandwidth])
    push!(result, round(10 .* (moving_sum / bandwidth); digits = 5))

    for i in (bandwidth + 1):length(data)

        moving_sum += (data[i] - data[i - bandwidth])
        push!(result, round(10 .* (moving_sum / bandwidth); digits = 5))

    end

    return result
end

```

and I tried two different packages for a rolling median `RollingFunction.jl` and `FastRunningMedian.jl`, both of which produced the same outputs.

Here is what I wrote for those:

```julia
window7 = 7;
window15 = 15;
window31 = 31;

# didn't provide expected result - trying different package to compare:
#=
w7_result = round.(rollmedian((10 .* agmt_df.value), window7); digits = 5)
w15_result = round.(rollmedian((10 .* agmt_df.value), window15); digits = 5)
w31_result = round.(rollmedian((10 .* agmt_df.value), window31); digits =5)
=#

w7_result = round.(running_median((10 .* agmt_df.value), window7, :none); digits = 5)
w15_result = round.(running_median((10 .* agmt_df.value), window15, :none); digits = 5)
w31_result = round.(running_median((10 .* agmt_df.value), window31, :none); digits = 5)

```

Here is a comparison of the plots with various window sizes:

![30_comparison](https://global.discourse-cdn.com/julialang/original/3X/2/1/21d7ad1728ec805a3f83c7f8edc53f2b9f4552e7.png)  
 ![15_day_comparison](https://global.discourse-cdn.com/julialang/original/3X/6/1/61c6553a2e17e4f14af530dc8405b6f144ee0a36.png)  
 ![7_comparison](https://global.discourse-cdn.com/julialang/original/3X/9/3/9343927e200c8f2f9551c71217152f6033fd5662.png)

Any insights would be greatly appreciated!

Thank you 🙂  
-Rob

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<div class="post-metadata">

**Author:** ![devel-chm](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/devel-chm/32/3572_2.png) [@devel-chm](https://discourse.julialang.org/u/devel-chm)\
**Post date:** [September 9, 2024, 9:11pm UTC](https://discourse.julialang.org/t/rolling-median-vs-rolling-mean/119227/2 "2024-09-09T21:11:17Z")

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Try debugging this with a test data set that is small enough to be  
computed by hand and check that things match.

---

<div class="post-metadata">

**Author:** ![Satvik](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/satvik/32/20486_2.png) [@Satvik](https://discourse.julialang.org/u/Satvik)\
**Post date:** [September 9, 2024, 9:29pm UTC](https://discourse.julialang.org/t/rolling-median-vs-rolling-mean/119227/3 "2024-09-09T21:29:21Z")

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> [@GeodeticR](#):
>
> I am smoothing out some GPS/GNSS data and I am wondering why at many of the “excursions” the absolute median value is larger than the average. I, at least intuitively, would expect to see the opposite.

Can you say more about why you’d expect this – is that a property of the data?

A median can be larger than the mean in many cases, e.g. `[1, 1, 0, -2, -2]`. It seems like most of the points you see where the median is larger are ones where there’s a sudden drop in the average, and the median updates more slowly.

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<div class="post-metadata">

**Author:** ![GeodeticR](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/geodeticr/32/218661_2.png) [@GeodeticR](https://discourse.julialang.org/u/GeodeticR)\
**Post date:** [September 9, 2024, 9:42pm UTC](https://discourse.julialang.org/t/rolling-median-vs-rolling-mean/119227/4 "2024-09-09T21:42:27Z")

</div>

It’s not a property of the data by any means. I guess I am used to seeing that “smoother” trend when using a median vs a mean, and it just didn’t seem to be the case here.

It makes more sense that my assumption/expectation was wrong, rather than two separate packages being incorrect and producing the same result.

But, here is the data, I’m using plotted behind the mean and medians, just so you can see what the distribution looks like:

![30_median_agmt](https://global.discourse-cdn.com/julialang/original/3X/0/e/0e862b1be27bce26ffe0af20f4868c18ff2752cd.png)  
 ![30_day_mean](https://global.discourse-cdn.com/julialang/original/3X/4/c/4c6ca931615025ca1ba6594a8a0387e8c87d7a84.png)

I’m thinking I should instead remove the extreme outliers (maybe a density based algorithm), and then try the rolling mean and medians again. 🤔

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<div class="post-metadata">

**Author:** ![dlakelan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dlakelan/32/8491_2.png) [@dlakelan](https://discourse.julialang.org/u/dlakelan)\
**Post date:** [September 9, 2024, 9:55pm UTC](https://discourse.julialang.org/t/rolling-median-vs-rolling-mean/119227/5 "2024-09-09T21:55:18Z")

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Does this fine-scale variation have meaning for you, or are you smoothing way too little here for what you’re actually interested in?

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<div class="post-metadata">

**Author:** ![GeodeticR](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/geodeticr/32/218661_2.png) [@GeodeticR](https://discourse.julialang.org/u/GeodeticR)\
**Post date:** [September 9, 2024, 10:00pm UTC](https://discourse.julialang.org/t/rolling-median-vs-rolling-mean/119227/6 "2024-09-09T22:00:51Z")

</div>

It’s a balance of not wanting to over-smooth and completely remove the anomaly I’m researching. Since I have various lengths of data for different GPS stations, I need to choose a window that makes sense. For this station in particular, I could probably get away with a window of 50-75 days, but, there are some stations that don’t have 1800+ days of data.

This is also precursory examination of the data before I make an animation. This smoothing is to reduce the flicker of the animation to better evaluate the propagation of the anomaly across the US.

---

<div class="post-metadata">

**Author:** ![dlakelan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dlakelan/32/8491_2.png) [@dlakelan](https://discourse.julialang.org/u/dlakelan)\
**Post date:** [September 9, 2024, 10:23pm UTC](https://discourse.julialang.org/t/rolling-median-vs-rolling-mean/119227/7 "2024-09-09T22:23:11Z")

</div>

For this kind of stuff I’ve used ApproxFun Chebyshev polynomials in Turing models with priors that impose smoothness to extremely good effect.

```julia
using ApproxFun, Turing

@model smooth(x,y, t0, tend)
   # overall size of coefficients on the scale of about +- 5, but this is just the base 
   # prior, we modify below
   c ~ MvNormal(fill(0.0,10),fill(5.0,10)) 
   e ~ Gamma(5.0,5.0/4.0) # sd of error around 5

   f = Fun(Chebyshev(t0 .. tend), c)
   fpm = Derivative() * f
   fpmpm = Derivative() * fpm
   ifpmsq = Integral() * (fpm^2)
   ifpmpmsq = Integral() * (fpmpm^2)

   # modify coefficient priors in a dependent way, 
   # encourage low mean square derivative behavior
   Turing.@addlogprob!(Exponential(5.0^2),(ifpmsq(t0)-ifpmsq(tend))/(tend-t0))

   # encourage low mean square 2nd deriv behavior
   Turing.@addlogprob!(Exponential(2.0^2), (ifpmpmsq(t0) - ifpmpmsq(tend))/(tend-t0))

   preds = f.(x)
   y ~ MvNormal(preds,e)
   return(preds)
end

```

You can tune the priors in terms of the mean squared derivative and 2nd deriv behavior or add additional conditions to your prior knowledge. The prior is expressed in terms of the change per unit time, so is independent of both the number of data points and the span of the data (but not independent of units unless you first non-dimensionalize the data)