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Consider the following:


The idea is to define a function MyFunction which will return res when applied on data. Please note that for every recursive step the function will delete the first element of a 4-tuple and append one element to this tuple.

I had the following idea which worked but maybe somebody has a much shorter version:

ListBuilder[data_] := Module[
  {data1 = data, data2, data3, data30},

  data2 = 
   Join[{data1}, {data1 /. {a_, b_, c_, d_, e_, f_, g_, h_} :> 
       Flatten@{b, c, d, e, f, g, h, 
         Round@RandomVariate[NormalDistribution[1000, 0.1*1000], 1]}}];

  data3 = 
   Join[data2, {data2[[-1]] /. {a_, b_, c_, d_, e_, f_, g_, h_} :> 
       Flatten@{b, c, d, e, f, g, h, 
         Round@RandomVariate[NormalDistribution[1000, 0.1*1000], 1]}}];

  While[Length@data3 < 8, 
   data3 = Join[
     data3, {data3[[-1]] /. {a_, b_, c_, d_, e_, f_, g_, h_} :> 
        Flatten@{b, c, d, e, f, g, h, 
          Round@RandomVariate[NormalDistribution[1000, 0.1*1000], 


NumData = Round@RandomVariate[NormalDistribution[1000, 0.1*1000], 8]


It would also be interesting to allow list building with more than one argument (e.g. answer by YvesKlett.)

Does anyone have an idea?

share|improve this question
Just added the part for two arguments, you might want to add that to your question as well... – Yves Klett Aug 21 '12 at 6:25
up vote 5 down vote accepted
data = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, r, s, t, u, w, z};
Partition[data, 4, 1]
{{a, b, c, d}, {b, c, d, e}, {c, d, e, f}, {d, e, f, g}, {e, f, g, h},
 {f, g, h, i}, {g, h, i, j}, {h, i, j, k}, {i, j, k, l}, {j, k, l, m},
 {k, l, m, n}, {l, m, n, o}, {m, n, o, p}, {n, o, p, r}, {o, p, r, s},
 {p, r, s, t}, {r, s, t, u}, {s, t, u, w}, {t, u, w, z}}
share|improve this answer
+1 Clean, straightforward interpretation of the problem (even though it's not recursive). – DavidC Aug 20 '12 at 17:21
@DavidCarraher Thanks ! I think the term recursive is not crucial here, since Partition is the best approach IMO. – Artes Aug 20 '12 at 18:38
I fully agree with you. – DavidC Aug 20 '12 at 18:40
You don't have q and v? – Per Alexandersson Aug 21 '12 at 13:39
@Artes Sorry, I did not pay attention! :-/ Best answer to my question! – John Aug 22 '12 at 2:36

Few more alternatives using various combinations of FoldList, NestList, Join and PadRight:

ClearAll[dropAddToList1a, dropAddToList2a, dropAddToList3a];
dropAddToList1a[list_, n_, μ_: 0, σ_: 1, r_: 0.1] :=
  FoldList[Join[Rest[#1], {#2}] &, list, 
     Round[RandomVariate[NormalDistribution[μ, σ]], r] & /@  Range[n]];
dropAddToList2a[list_, n_, μ_: 0, σ_: 1, r_: 0.1] :=
     {Round[RandomVariate[NormalDistribution[μ, σ]], r]}] &, list, n];
dropAddToList3a[list_, n_, μ_: 0, σ_: 1, r_: 0.1] :=
  NestList[PadRight[Rest[#1], Length@list, 
     {Round[RandomVariate[NormalDistribution[μ, σ]], r]}] &, list, n];
dropAddToList1a[data, 3, 0, 1]
 (* {{a,b,c,d},{b,c,d,-0.3},{c,d,-0.3,0.7},{d,-0.3,0.7,-0.2}}*)
dropAddToList2a[data, 3]
 (* {{a,b,c,d},{b,c,d,0.},{c,d,0.,-1.2},{d,0.,-1.2,-0.9}}*)
dropAddToList3a[data, 3, 3]
 (* {{a,b,c,d},{b,c,d,4.2},{c,d,4.2,2.2},{d,4.2,2.2,2.7}} *)
dropAddToList3a[data, 3, 100, 50, 10]
 (* {{a,b,c,d},{b,c,d,110},{c,d,110,170},{d,110,170,120}} *)

Drop and append k elements:

ClearAll[dropAddToList1b, dropAddToList2b, dropAddToList3b];
dropAddToList1b[list_, n_, k_: 1, μ_: 0, σ_: 1, r_: 0.1] :=
   FoldList[Join[Drop[#1, k], #2] &, list,
     Round[RandomVariate[NormalDistribution[μ, σ], k], r] & /@  Range[n]];
dropAddToList2b[list_, n_, k_: 1, μ_: 0, σ_: 1, r_: 0.1] :=
   NestList[Join[Drop[#1, k],
     Round[RandomVariate[NormalDistribution[μ, σ], k],  r]] &, list, n];
dropAddToList3b[list_, n_, k_: 1, μ_: 0, σ_: 1, r_: 0.1] :=  
   NestList[PadRight[Drop[#1, k], Length@list,
     Round[RandomVariate[NormalDistribution[μ, σ], k], r]] &, list, n];
dropAddToList1b[data, 3]
 (* {{a,b,c,d},{b,c,d,0.1},{c,d,0.1,1.5},{d,0.1,1.5,-1.5}} *)
dropAddToList2b[data, 3, 2]
 (* {{a,b,c,d},{c,d,-0.3,-0.1},{-0.3,-0.1,0.2,-1.6},{0.2,-1.6,-1.7,-0.1}}*)
dropAddToList3b[data, 3, 1, 100, 10, 10]
 (* {{a,b,c,d},{b,c,d,90},{c,d,90,100},{d,90,100,110}}*)
share|improve this answer
... and now you´ll have to benchmark them all ;-) – Yves Klett Aug 21 '12 at 11:18
@Yves, traditional threshold for number of answers before attempting benchmarks is 7:) – kglr Aug 21 '12 at 14:37
you´ll have to watch edits to this post then ;-) – Yves Klett Aug 21 '12 at 14:39

If you want recursive (not performance, mind):

dat = Round@RandomVariate[NormalDistribution[1000, 0.1*1000], 8];

rec[dat_] := 
 Append[dat[[2 ;; -1]], 
  Round@RandomVariate[NormalDistribution[1000, 0.1*1000]]]

NestList[rec, dat, 10]//Grid

Mathematica graphics

ach well, here cometh the color:

dat = MapIndexed[Style[#, Hue[#2[[1]]/8]] &, 
   Round@RandomVariate[NormalDistribution[1000, 0.1*1000], 8]];

NestList[rec, dat, 10] // Grid

Mathematica graphics

As for a two-argument function:

k = 1;
dat = Round@RandomVariate[NormalDistribution[1000, k*1000], 8];

rec[dat_, k_] := 
 Append[dat[[2 ;; -1]], 
  Round@RandomVariate[NormalDistribution[1000, k*1000]]]

NestList[rec[#, k] &, dat, 10] // Grid

Mathematica graphics

share|improve this answer
Works perfect and at least five times faster than my approach. However, I have one more question: What if rec needs two arguments. For instance: rec[dat_,k_]:= Append[dat[[2 ;; -1]], Round@RandomVariate[NormalDistribution[1000, k*1000]]]? – John Aug 20 '12 at 17:38
I like your answer since it follows my idea, but I have to admit that Artes is right: the problem is not recursive if you use Partition. – John Aug 22 '12 at 2:38
@John Not to worry... – Yves Klett Aug 22 '12 at 5:11

Something like

SeedRandom[30, Method -> "MKL"]; (* for reproducibility *)
NestList[Append[Rest[#], RandomChoice[CharacterRange["f", "z"]]] &,
         {"a", "b", "c", "d", "e"}, 6]
{{"a", "b", "c", "d", "e"}, {"b", "c", "d", "e", "u"},
 {"c", "d", "e", "u", "s"}, {"d", "e", "u", "s", "f"},
 {"e", "u", "s", "f", "k"}, {"u", "s", "f", "k", "g"},
 {"s", "f", "k", "g", "n"}}

I suppose.

share|improve this answer
Not sure if I should remove my answer because it so much resembles yours? – Yves Klett Aug 20 '12 at 17:11
Well, your call... – J. M. Aug 20 '12 at 17:21

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