Modifying a list of matrices with conditional statement

Question

I have a list of around 100 matrices, that looks like this

A={{{425060., 2.14235*10^6, 0.49, 0.01, 0.38, 0.620161, 
   20.}, {1.24808*10^6, 1.53025*10^6, 0.04, 0.07, 0.31, 0.320312, 
   20.}, {7.39304*10^6, 1.40204*10^6, 0.83, 0.45, 0.09, 0.94842, 20.} .... {4.27537*10^6, 
 1.62124*10^6, 0.24, 0.28, 0.62, 0.721388, 20.1}, {3.27776*10^6, 
 2.25816*10^6, 0.01, 0.21, 0.72, 0.750067, 20.1}, {3.0814*10^6, 
 1.95624*10^6, 0.02, 0.19, 0.72, 0.744916, 20.1}, {2.42706*10^6, 
 1.25729*10^6, 0.03, 0.17, 0.7, 0.720972, 20.1}, {4.47196*10^6, 
 1.23247*10^6, 0.24, 0.27, 0.57, 0.674833, 20.1}, {1.57132*10^7, 
 1.65019*10^6, 0.76, 0.92, 0.5, 1.29383, 20.1} ...... 

Each matrix has an unknown number of rows. Each matrix holds information about a group of particles at a given time. Columns 3,4,5 are the x,y and z coordinate of each particle. For each matrix, I want to compare the Euclidean distance between every two particles, and keep the particle whose distance from all other particles in that matrix is greater than 0.01. I want to use the following algorithm, I pick a matrix, then pick its first row, then I check whether EuclideanDistance[{#3, #4, #5}, {from all other particles}]> 0.01 then that row (that particle) gets copied into another matrix, if not, then ignore both the elements for further computation. This way a new list of matrix will be created with particles which are at least 0.01 units away from each other.

EDIT : I am getting error messages EuclideanDistance::argr: EuclideanDistance called with 1 argument; 2 arguments are expected. >> Pick::incomp: Expressions {{1.13266*10^7,1.40618*10^6,0.82,0.67,0.35,1.11526,22.7}} and {EuclideanDistance[{0.82,0.67,0.35}]>0.01} have incompatible shapes. >> EuclideanDistance::argr: EuclideanDistance called with 1 argument; 2 arguments are expected. >> Pick::incomp: Expressions {{1.13266*10^7,1.38136*10^6,0.82,0.67,0.35,1.11526,22.8}} and {EuclideanDistance[{0.82,0.67,0.35}]>0.01} have incompatible shapes

this is happening when there is only one element in the matrix, as EuclideanDistance requires two arguments. Is there any way I can have the Euclidean distance skip over these single elements, yet the fina list of matrices will have those single elements?

Answer 1

Taking one the matrices out of your 100, say

aA = {{4.27537*10^6, 1.62124*10^6, 0.24, 0.28, 0.62, 0.721388, 20.1}, 
      {3.27776*10^6, 2.25816*10^6, 0.01, 0.21, 0.72, 0.750067, 20.1},
      {3.0814*10^6, 1.95624*10^6, 0.02, 0.19, 0.72, 0.744916,  20.1}, 
      {2.42706*10^6, 1.25729*10^6, 0.03, 0.17, 0.7, 0.720972,  20.1}, 
      {4.47196*10^6, 1.23247*10^6, 0.24, 0.27, 0.57, 0.674833, 20.1}, 
      {1.57132*10^7, 1.65019*10^6, 0.76, 0.92, 0.5, 1.29383,   20.1}};

Create a Nearest function for the matrix based on columns {3,4,5}:

nf = Nearest[aA[[All, {3, 4, 5}]]];

so that

nf[u, 2]

gives the two elements that are closest to u; one of the those two is, of course, u.

Then, for a threshold distance t

Pick[aA, EuclideanDistance @@ nf[#, 2] > t & /@ aA[[All, {3, 4, 5}]]]

picks the elements of the matrix whose {x,y,z} coordinates are at least t distance far from the {x,y,z} coordinates of the other elements.

t = .03;
Pick[aA, EuclideanDistance @@ nf[#, 2] > t & /@ aA[[All, {3, 4, 5}]]]

{{4.27537*10^6, 1.62124*10^6, 0.24, 0.28, 0.62, 0.721388, 20.1}, {4.47196*10^6, 1.23247*10^6, 0.24, 0.27, 0.57, 0.674833, 20.1}, {1.57132*10^7, 1.65019*10^6, 0.76, 0.92, 0.5, 1.29383, 20.1}}

Note: You could also use

Select[aA, (EuclideanDistance @@ nf[#[[{3, 4, 5}]], 2] > t) &]
(* same output *)

Update: a function that can be Mapped to the entire list of matrices:

ClearAll[pF];
pF[t_] := With[{nfunc = Nearest[#[[All, {3, 4, 5}]]], mat = #},
    Pick[mat, EuclideanDistance @@ nfunc[#, 2] > t & /@ mat[[All, {3, 4, 5}]]]] &;

Example:

pF[.03] @ aA
(* same result as above *)

aB = {{425060., 2.14235*10^6, 0.49, 0.01, 0.38, 0.620161, 20.}, 
      {1.24808*10^6, 1.53025*10^6, 0.04, 0.07, 0.31, 0.320312, 20.}, 
      {7.39304*10^6, 1.40204*10^6, 0.83, 0.45, 0.09, 0.94842,  20.}}; 

A={aB,aA};

result = pF[.03] /@ A;
TableForm[TableForm /@ result, TableSpacing -> 3 ]