I have a site called Sudomemo. I record when I first see a user in the database; and I've exported it in JSON format, located here.


I've done this:

dates = Import["http://www.sudomemo.net/statistics/firstSeenDump.php", "JSON"]

I want to plot, since February, how the user-join rate increases/decreases, over the course of the year.

Thinking about it mathematically: Since the derivative of a function shows the slope of a function, and the second derivative shows the concavity/how quickly the slope is changing, I should graph the second derivative of the timestamps over the year. I think I have the right idea... I'm still learning about derivatives in Calculus :P

The only issue is, I have no idea how to go about doing that!

I'm pretty new to Mathematica, and thought this would make a fantastic chart. Not only that, but I've installed Mathematica 10 on my server, and I'd love to be able to present cool graphs :D some interesting statistics to interested users.

Right now, I've simply used the json_encode() PHP function to produce the timestamps like this:

["2014-01-29 16:49:00","2014-01-29 16:49:00","2014-01-29 16:49:00"...

I can change the array format to something different if that'd be better suited for this.

Would anyone be able to point me in the right direction?

Note: The first hundred values or so are all 2014-01-29 16:49:00 because that's when I first started recording the "first seen" timestamps.

http://www.sudomemo.net/statistics/firstSeenDump.php?nospike doesn't show the ones from January 29th 16:49:00.

  • $\begingroup$ You can cast to DateObject like this: Map[StringSplit[#, {" ", ":", "-"}] & /* Map[ToExpression] /* DateObject] $\endgroup$ Nov 20, 2014 at 0:25

2 Answers 2


You're about to realize that derivatives in the real world are a pain. You have to aggregate data as much as possible and average it a lot until you get a "textbook-quality second-derivative!

dates = DateList /@ Import["http://www.sudomemo.net/statistics/firstSeenDump.php", "JSON"]

Grouping by days and counting:

datesDays = Tally@dates[[All, ;; 3]]; 
datesDays // DateListPlot

By days

Very noisy, as every real world data... and when we differentiate noise we make it stronger.

MapThread[{#1[[1]], (#2[[2]] - #1[[2]])/DayCount[#1[[1]], #2[[1]]]} &
    , {Most@#, Drop[#, 1]}] &@dayCounts // DateListPlot

first derivative

We'd get a similar plot if we differentiated it once more.

Some MovingAverage filtering

datesWeeks = Thread[{datesDays[[;; -7, 1]], MovingAverage[datesDays[[All, 2]], 7]}];
datesMonths = Thread[{datesDays[[;; -30, 1]], MovingAverage[datesDays[[All, 2]], 30]}];
GraphicsGrid@{DateListPlot /@ {%%, %}}

Weeks and Months Average

But even the well-behaved 30-day-average data becomes too noisy under a single differentiation:

MapThread[{#1[[1]], (#2[[2]] - #1[[2]])/DayCount[#1[[1]], #2[[1]]]} & , {Most@#, Drop[#, 1]}] &@datesMonths // DateListPlot[#, PlotRange -> All] &

Differentiated Month

You could average the result before taking another derivative, but remember that every averaging step adds more lag to your system... Old data is just as actionable as noisy data!

Just for kicks, second derivative, with and without averaging

Using the same "methodology" as before:

diffMonthAvg = Thread[{diffMonth[[;; -30, 1]], MovingAverage[diffMonth[[All, 2]], 30]}];

MapThread[{#1[[1]], (#2[[2]] - #1[[2]])/DayCount[#1[[1]], #2[[1]]]} &, {Most@#, Drop[#, 1]}] &@diffMonth;
MapThread[{#1[[1]], (#2[[2]] - #1[[2]])/DayCount[#1[[1]], #2[[1]]]} &, {Most@#, Drop[#, 1]}] &@diffMonthAvg;
GraphicsGrid@{DateListPlot[#, PlotRange -> {All, {-3, 3}}] & /@ {%%, %}}

Second derivative

You can also apply the moving average to the second derivative, but it's late and we all know that squinting hard achieves the same result!

  • $\begingroup$ Oh wow, thanks! :D These are both neat, I'm going to give some of these a try! :D Thanks! (Not sure which answer to accept but I want to accept one eventually)! $\endgroup$ Nov 20, 2014 at 2:17

You have the import.

dates = Import["http://www.sudomemo.net/statistics/firstSeenDump.php", "JSON"]

Gather them up by day and count how many in each day.

dailyGather = GatherBy[dates, Take[DateList[#], 3] &];
dailyVisits = {Take[DateList[dailyGather[[#, 1]]], 3],Length[dailyGather[[#]]]} & 
 /@ Range[First@Dimensions[dailyGather]];

Plot the visits.

DateListPlot[dailyVisits, PlotRange -> All]

Calc and plot the pace of visits.

paceOfVisits = MovingMap[#[[2]] - #[[1]] &, dailyVisits, 2];
DateListPlot[paceOfVisits, PlotRange -> All]

Calc and plot the change in pace of visits.

changeOfPaceOfVisits = MovingMap[#[[2]] - #[[1]] &, paceOfVisits, 2];
DateListPlot[changeOfPaceOfVisits, PlotRange -> All]

Look at smoothing the change in pace with average.

 DateListPlot[MovingAverage[changeOfPaceOfVisits, averageN], 
  PlotRange -> {Automatic, {-100, 100}}], {averageN, 3, 25, 1}]


You can try more smoothers with MovingMap instead of MovingAverage in the Manipulate.

 DateListPlot[MovingMap[smoother, changeOfPaceOfVisits, averageN], 
  PlotRange -> {Automatic, {-100, 100}}], 
 {smoother, {Mean, Median, Quantile[#, .8] & -> "80% Quantile", 
   TrimmedMean[#, .1] & -> "10% Trimmed Mean"}, SetterBar}, 
 {averageN, 3, 30, 1}]
  • $\begingroup$ MovingMap! That exists! $\endgroup$
    – Aisamu
    Nov 20, 2014 at 1:44
  • 1
    $\begingroup$ @Aisamu Yes, In version 10. $\endgroup$
    – Edmund
    Nov 20, 2014 at 1:50
  • $\begingroup$ It even does some nice DateList conversion automagically! (#[[2]]-#[[1]] evaluates to 3600633600, instead of {0, 1, -23} on the first term of paceOfVisits) $\endgroup$
    – Aisamu
    Nov 20, 2014 at 2:00
  • $\begingroup$ Oh wow, thanks! :D These are both neat, I'm going to give some of these a try! :D Thanks! (Not sure which answer to accept but I want to accept one eventually)! $\endgroup$ Nov 20, 2014 at 2:17
  • 1
    $\begingroup$ @Aisamu Try more smoothers with `MovingMap'. See the update. $\endgroup$
    – Edmund
    Nov 20, 2014 at 2:47

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