Remove outliers from a 3D list

Questions
1

Hi,

For removing outliers for a 1D list of data e.g. {5,3,8,10,8,2}, I did the following procedure:

{min, max} = {Mean[data] - 2*StandardDeviation[data], 
  Mean[data] + 2*StandardDeviation[data]}

The outliers are : Select[data, Or[# > max, # < min] &]

Now if we have a 3D list of data e.g., {{-1.197, -1.169, 0.424}, {-3.597, 1.220, 2.234},…}

How is it possible to fit a 3D ellipse to this data?

Can we remove the outlier with the help of the fitted ellipse? I enclosed the real data.

I appreciate your help.

Question.nb (802 KB)

2

Check out this post .

3

It would be best to define what an “inlier” is first or at least call the odd values “potential outliers”. The point is that if one is doing real science, then all “outliers” need to be explained as opposed to just finding them and tossing them out.

4

Thank you so much, Jim.

If we study a three-dimensional list data, (u,v,w) separately, then we get the below results:

u = {-10, -2, 0, -3, 1, 2, 6, 14, 5, 4, 8, 11, 9, 3, 7};

{min, max} = {Mean[u] - 2*StandardDeviation[u], 
    Mean[u] + 2*StandardDeviation[u]} // N;

Select[u, Or[# > max, # < min] &];

ListPlot[u]

v.png
v.png649×361 3.06 KB

v = {-19, 5, 1, 1.5, 3.5, -3, 0, 7, 6, -11, 25, 17, 2, 7.5, 4};

{min, max} = {Mean[v] - 2*StandardDeviation[v], 
    Mean[v] + 2*StandardDeviation[v]} // N;

Select[v, Or[# > max, # < min] &];

ListPlot[v]

vv.png
vv.png679×361 3.82 KB

w = {-5, 8, -16, 9, 13, 6, -26, 15, 14, 6, 15, -2, 6, 3, 10};

{min, max} = {Mean[w] - 2*StandardDeviation[w], 
    Mean[w] + 2*StandardDeviation[w]} // N;

Select[w, Or[# > max, # < min] &];

w.png
w.png658×361 3.14 KB

Now for all of the lists in a 3D list we have :

transpose.jpg
transpose.jpg1292×191 53.2 KB

How do I remove these three elements sublists from the data?

I appreciate your kindness and help.
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=v.png&userId=1822915
[2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=vv.png&userId=1822915
[3]: https://community.wolfram.com//c/portal/getImageAttachment?filename=w.png&userId=1822915
[4]: https://community.wolfram.com//c/portal/getImageAttachment?filename=transpose.jpg&userId=1822915

5

Hi Alex,

Here is one way

{min, max} = {Mean[data] - 2*StandardDeviation[data], Mean[data] + 2*StandardDeviation[data]};
limits = Transpose[{min, max}];

data // Select[
 Between[#[[1]], limits[[1]]] && Between[#[[2]], limits[[2]]] && Between[#[[2]], limits[[2]]] &]
6

Dear Rohit, you always are great. Thank you so much.

I got a small error in the output.

Rohit.png
Rohit.png1831×449 15.1 KB

7

Hello Alex,

basically and in principle I do share Jim’s point of view. But just to get things done, here is a “quick and dirty” approach (you are asking for outliers in 3D):

u = {-10, -2, 0, -3, 1, 2, 6, 14, 5, 4, 8, 11, 9, 3, 7};
v = {-19, 5, 1, 1.5, 3.5, -3, 0, 7, 6, -11, 25, 17, 2, 7.5, 4};
w = {-5, 8, -16, 9, 13, 6, -26, 15, 14, 6, 15, -2, 6, 3, 10};
pts = Transpose[{u, v, w}];
anomPts = FindAnomalies[pts, PerformanceGoal -> "Quality", Method -> "Multinormal"];
Graphics3D[{Point[pts], Red, Opacity[.2], Sphere[anomPts, 1]}, Boxed -> True, Axes -> True]

outliers.png
outliers.png359×378 4.07 KB

8

Not sure what you mean. In the image you annotated with outliers, {2, -3, 6} is not an outlier, so it is present in the result.

9

I am sorry, Rohit.

I mean was (6,0,-26) .

10

I do believe in outliers. But I also believe that when doing science (or even making a decision for a business) one must not just look for and toss “inconvenient” observations.

Context matters. Are the dimensions in the same units? Are the dimensions of equal importance? How is the data collected? Are the “potential outliers” just in a region of space not explored as intensively as other regions?

There is no algorithm void of subject matter knowledge that will appropriately find outliers. Such algorithms ignorant of how the data was collected might help “round up the usual suspects” but each suspected outlier needs to be vetted (with vetting also for some observations that “seem OK”).

And if one has less than 50 observations, one probably has no business looking for an automated outlier detection algorithm.

11

Ah. My careless copy/paste error. Should be

data // Select[
  Between[#[[1]], limits[[1]]] && Between[#[[2]], limits[[2]]] && Between[#[[3]], limits[[3]]] &]
12

Hi Henrik,

Thank you so much for the interesting method.

I get different results every time I run the program.

6860Untitled000.png
6860Untitled000.png1307×662 24.2 KB

13

I appreciate your help, Rohit.

14

Hi Jim,

Thank you so much for your useful explains.
I have more than 5000 data for finding outliers. I just wanted to learn with the lists including small numbers of elements.

15

Dear Henrik,

Alex is right. I got different results every time.

Using the " Seed" command can help in this case?

16

Hi Mohammad,

Take a look at Andreas Lauschke’s livecoding session on anomaly detection on YouTube.