How to detect the singular value in 3-dimension with wavelet analysis

Question

Actually this question is related with this post,but this is about point of 3-dimension

data = ReplacePart[
   Catenate[
    Table[{i, j, Sin[j^2 + i]}, {i, 0, Pi, 0.2}, {j, 0, Pi, 0.2}]], 
   100 -> {1.2, 0.6, 1.5}];
ListPlot3D[data, Mesh -> None, InterpolationOrder -> 2, 
 ColorFunction -> "SouthwestColors"]

I add a singular value in position $100$ manully,but I don't know how to detect it.

Answer 1

In general, detecting anomalies is pretty straightforward assuming you know the distribution of your data.

In this case, I will estimate the distribution for each dimension in your data, then calculate the probabilities retroactively.

Find Distributions

dataPDFs = Table[
  Block[{dist},
    dist = FindDistribution[data[[All,i]], MaxItems->1];
    PDF[dist,#] &/@ data[[All,i]]
  ]
  ,{i,1,3}] 

We have a distribution for each dimension, then plug each value back into the dimensions to find the probability that the value belongs to the distribution.

To visualize what we just did:

Column[ListPlot[#, PlotStyle -> {Red, PointSize -> Large}] & /@ dataPDFs]

Outlier is immediately evident.

We can search our results for probabilities less than a given threshold:

Position[dataPDFs, x_ /; x < .01]
(* {{3, 100}} *)

Outlier found in dimension three, position 100