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Revision as of 16:28, 23 July 2021
You are encouraged to solve this task according to the task description, using any language you may know.
W Edwards Deming was an American statistician and management guru who used physical demonstrations to illuminate his teachings. In one demonstration Deming repeatedly dropped marbles through a funnel at a target, marking where they landed, and observing the resulting pattern. He applied a sequence of "rules" to try to improve performance. In each case the experiment begins with the funnel positioned directly over the target.
- Rule 1: The funnel remains directly above the target.
- Rule 2: Adjust the funnel position by shifting the target to compensate after each drop. E.g. If the last drop missed 1 cm east, move the funnel 1 cm to the west of its current position.
- Rule 3: As rule 2, but first move the funnel back over the target, before making the adjustment. E.g. If the funnel is 2 cm north, and the marble lands 3 cm north, move the funnel 3 cm south of the target.
- Rule 4: The funnel is moved directly over the last place a marble landed.
Apply the four rules to the set of 50 pseudorandom displacements provided (e.g in the Racket solution) for the dxs and dys. Output: calculate the mean and standard-deviations of the resulting x and y values for each rule.
Note that rules 2, 3, and 4 give successively worse results. Trying to deterministically compensate for a random process is counter-productive, but -- according to Deming -- quite a popular pastime: see the Further Information, below for examples.
Stretch goal 1: Generate fresh pseudorandom data. The radial displacement of the drop from the funnel position is given by a Gaussian distribution (standard deviation is 1.0) and the angle of displacement is uniformly distributed.
Stretch goal 2: Show scatter plots of all four results.
- Further information
- Further explanation and interpretation
- Video demonstration of the funnel experiment at the Mayo Clinic.
Ada
<lang Ada>with Ada.Numerics.Elementary_Functions; with Ada.Text_IO;
procedure Demings_Funnel is
type Float_List is array (Positive range <>) of Float;
Dxs : constant Float_List :=
(-0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087);
Dys : constant Float_List :=
( 0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032);
type Rule_Access is access function (Z, Dz : Float) return Float;
function Funnel (List : in Float_List;
Rule : in Rule_Access)
return Float_List
is
Correc : Float := 0.0;
Result : Float_List (List'Range);
begin
for I in List'Range loop
Result (I) := Correc + List (I);
Correc := Rule (Correc, List (I));
end loop;
return Result;
end Funnel;
function Mean (List : in Float_List)
return Float
is
Sum : Float := 0.0;
begin
for Value of List loop
Sum := Sum + Value;
end loop;
return Sum / Float (List'Length);
end Mean;
function Stddev (List : in Float_List)
return Float
is
use Ada.Numerics.Elementary_Functions;
M : constant Float := Mean (List);
Sum : Float := 0.0;
begin
for F of List loop
Sum := Sum + (F - M) * (F - M);
end loop;
return Sqrt (Sum / Float (List'Length));
end Stddev;
procedure Experiment (Label : in String;
Rule : in Rule_Access)
is
package Float_IO is new Ada.Text_IO.Float_IO (Float);
use Ada.Text_IO;
use Float_IO;
Rxs : constant Float_List := Funnel (Dxs, Rule);
Rys : constant Float_List := Funnel (Dys, Rule);
begin
Default_Exp := 0;
Default_Fore := 4;
Default_Aft := 4;
Put_Line (Label & " : x y");
Put ("Mean: "); Put (Mean (Rxs)); Put (Mean (Rys)); New_Line;
Put ("StdDev: "); Put (Stddev (Rxs)); Put (Stddev (Rys)); New_Line;
New_Line;
end Experiment;
function Rule_1 (Z, Dz : Float) return Float is (0.0); function Rule_2 (Z, Dz : Float) return Float is (-Dz); function Rule_3 (Z, Dz : Float) return Float is (-Z - Dz); function Rule_4 (Z, Dz : Float) return Float is (Z + Dz);
begin
Experiment ("Rule 1", Rule_1'Access);
Experiment ("Rule 2", Rule_2'Access);
Experiment ("Rule 3", Rule_3'Access);
Experiment ("Rule 4", Rule_4'Access);
end Demings_Funnel;</lang>
D
<lang d>import std.stdio, std.math, std.algorithm, std.range, std.typecons;
auto mean(T)(in T[] xs) pure nothrow @nogc {
return xs.sum / xs.length;
}
auto stdDev(T)(in T[] xs) pure nothrow {
immutable m = xs.mean; return sqrt(xs.map!(x => (x - m) ^^ 2).sum / xs.length);
}
alias TF = double function(in double, in double) pure nothrow @nogc;
auto funnel(T)(in T[] dxs, in T[] dys, in TF rule) {
T x = 0, y = 0; immutable(T)[] rxs, rys;
foreach (const dx, const dy; zip(dxs, dys)) {
immutable rx = x + dx;
immutable ry = y + dy;
x = rule(x, dx);
y = rule(y, dy);
rxs ~= rx;
rys ~= ry;
}
return tuple!("x", "y")(rxs, rys);
}
void experiment(T)(in string label,
in T[] dxs, in T[] dys, in TF rule) {
//immutable (rxs, rys) = funnel(dxs, dys, rule);
immutable rs = funnel(dxs, dys, rule);
label.writeln;
writefln("Mean x, y: %.4f, %.4f", rs.x.mean, rs.y.mean);
writefln("Std dev x, y: %.4f, %.4f", rs.x.stdDev, rs.y.stdDev);
writeln;
}
void main() {
immutable dxs = [
-0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087];
immutable dys = [
0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032];
static assert(dxs.length == dys.length);
experiment("Rule 1:", dxs, dys, (z, dz) => 0.0);
experiment("Rule 2:", dxs, dys, (z, dz) => -dz);
experiment("Rule 3:", dxs, dys, (z, dz) => -(z + dz));
experiment("Rule 4:", dxs, dys, (z, dz) => z + dz);
}</lang>
- Output:
Rule 1: Mean x, y: 0.0004, 0.0702 Std dev x, y: 0.7153, 0.6462 Rule 2: Mean x, y: 0.0008, -0.0103 Std dev x, y: 1.0371, 0.8999 Rule 3: Mean x, y: 0.0438, -0.0063 Std dev x, y: 7.9871, 4.7784 Rule 4: Mean x, y: 3.1341, 5.4210 Std dev x, y: 1.5874, 3.9304
Elixir
<lang elixir>defmodule Deming do
def funnel(dxs, rule) do
{_, rxs} = Enum.reduce(dxs, {0, []}, fn dx,{x,rxs} ->
{rule.(x, dx), [x + dx | rxs]}
end)
rxs
end
def mean(xs), do: Enum.sum(xs) / length(xs)
def stddev(xs) do
m = mean(xs)
Enum.reduce(xs, 0.0, fn x,sum -> sum + (x-m)*(x-m) / length(xs) end)
|> :math.sqrt
end
def experiment(label, dxs, dys, rule) do
{rxs, rys} = {funnel(dxs, rule), funnel(dys, rule)}
IO.puts label
:io.format "Mean x, y : ~7.4f, ~7.4f~n", [mean(rxs), mean(rys)]
:io.format "Std dev x, y : ~7.4f, ~7.4f~n~n", [stddev(rxs), stddev(rys)]
end
end
dxs = [ -0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087]
dys = [ 0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032]
Deming.experiment("Rule 1:", dxs, dys, fn _z, _dz -> 0 end) Deming.experiment("Rule 2:", dxs, dys, fn _z, dz -> -dz end) Deming.experiment("Rule 3:", dxs, dys, fn z, dz -> -(z+dz) end) Deming.experiment("Rule 4:", dxs, dys, fn z, dz -> z+dz end)</lang>
- Output:
Rule 1: Mean x, y : 0.0004, 0.0702 Std dev x, y : 0.7153, 0.6462 Rule 2: Mean x, y : 0.0009, -0.0103 Std dev x, y : 1.0371, 0.8999 Rule 3: Mean x, y : 0.0439, -0.0063 Std dev x, y : 7.9871, 4.7784 Rule 4: Mean x, y : 3.1341, 5.4210 Std dev x, y : 1.5874, 3.9304
Factor
<lang factor>USING: combinators formatting generalizations grouping.extras io kernel math math.statistics sequences ;
- show ( seq1 seq2 -- )
[ [ mean ] bi@ ] [ [ population-std ] bi@ ] 2bi "Mean x, y : %.4f, %.4f\nStd dev x, y : %.4f, %.4f\n" printf ;
{
-0.533 0.270 0.859 -0.043 -0.205 -0.127 -0.071 0.275
1.251 -0.231 -0.401 0.269 0.491 0.951 1.150 0.001
-0.382 0.161 0.915 2.080 -2.337 0.034 -0.126 0.014
0.709 0.129 -1.093 -0.483 -1.193 0.020 -0.051 0.047
-0.095 0.695 0.340 -0.182 0.287 0.213 -0.423 -0.021
-0.134 1.798 0.021 -1.099 -0.361 1.636 -1.134 1.315
0.201 0.034 0.097 -0.170 0.054 -0.553 -0.024 -0.181
-0.700 -0.361 -0.789 0.279 -0.174 -0.009 -0.323 -0.658
0.348 -0.528 0.881 0.021 -0.853 0.157 0.648 1.774
-1.043 0.051 0.021 0.247 -0.310 0.171 0.000 0.106
0.024 -0.386 0.962 0.765 -0.125 -0.289 0.521 0.017
0.281 -0.749 -0.149 -2.436 -0.909 0.394 -0.113 -0.598
0.443 -0.521 -0.799 0.087
} {
0.136 0.717 0.459 -0.225 1.392 0.385 0.121 -0.395
0.490 -0.682 -0.065 0.242 -0.288 0.658 0.459 0.000
0.426 0.205 -0.765 -2.188 -0.742 -0.010 0.089 0.208
0.585 0.633 -0.444 -0.351 -1.087 0.199 0.701 0.096
-0.025 -0.868 1.051 0.157 0.216 0.162 0.249 -0.007
0.009 0.508 -0.790 0.723 0.881 -0.508 0.393 -0.226
0.710 0.038 -0.217 0.831 0.480 0.407 0.447 -0.295
1.126 0.380 0.549 -0.445 -0.046 0.428 -0.074 0.217
-0.822 0.491 1.347 -0.141 1.230 -0.044 0.079 0.219
0.698 0.275 0.056 0.031 0.421 0.064 0.721 0.104
-0.729 0.650 -1.103 0.154 -1.720 0.051 -0.385 0.477
1.537 -0.901 0.939 -0.411 0.341 -0.411 0.106 0.224
-0.947 -1.424 -0.542 -1.032
} {
[ "Rule 1:" print ] [ "Rule 2:" print [ [ [ swap neg + ] 2clump-map ] [ first suffix ] bi ] bi@ ] [ "Rule 3:" print [ 0 [ - neg ] accumulate* ] bi@ ] [ "Rule 4:" print [ cum-sum ] bi@ ]
} [ show ] map-compose 2cleave</lang>
- Output:
Rule 1: Mean x, y : 0.0004, 0.0702 Std dev x, y : 0.7153, 0.6462 Rule 2: Mean x, y : 0.0009, -0.0103 Std dev x, y : 1.0371, 0.8999 Rule 3: Mean x, y : 0.0439, -0.0063 Std dev x, y : 7.9871, 4.7784 Rule 4: Mean x, y : 3.1341, 5.4210 Std dev x, y : 1.5874, 3.9304
Go
<lang go>package main
import (
"fmt" "math"
)
type rule func(float64, float64) float64
var dxs = []float64{
-0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087,
}
var dys = []float64{
0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032,
}
func funnel(fa []float64, r rule) []float64 {
x := 0.0
result := make([]float64, len(fa))
for i, f := range fa {
result[i] = x + f
x = r(x, f)
}
return result
}
func mean(fa []float64) float64 {
sum := 0.0
for _, f := range fa {
sum += f
}
return sum / float64(len(fa))
}
func stdDev(fa []float64) float64 {
m := mean(fa)
sum := 0.0
for _, f := range fa {
sum += (f - m) * (f - m)
}
return math.Sqrt(sum / float64(len(fa)))
}
func experiment(label string, r rule) {
rxs := funnel(dxs, r)
rys := funnel(dys, r)
fmt.Println(label, " : x y")
fmt.Printf("Mean : %7.4f, %7.4f\n", mean(rxs), mean(rys))
fmt.Printf("Std Dev : %7.4f, %7.4f\n", stdDev(rxs), stdDev(rys))
fmt.Println()
}
func main() {
experiment("Rule 1", func(_, _ float64) float64 {
return 0.0
})
experiment("Rule 2", func(_, dz float64) float64 {
return -dz
})
experiment("Rule 3", func(z, dz float64) float64 {
return -(z + dz)
})
experiment("Rule 4", func(z, dz float64) float64 {
return z + dz
})
}</lang>
- Output:
Rule 1 : x y Mean : 0.0004, 0.0702 Std Dev : 0.7153, 0.6462 Rule 2 : x y Mean : 0.0009, -0.0103 Std Dev : 1.0371, 0.8999 Rule 3 : x y Mean : 0.0439, -0.0063 Std Dev : 7.9871, 4.7784 Rule 4 : x y Mean : 3.1341, 5.4210 Std Dev : 1.5874, 3.9304
Haskell
<lang haskell>import Data.List (mapAccumL, genericLength) import Text.Printf
funnel :: (Num a) => (a -> a -> a) -> [a] -> [a] funnel rule = snd . mapAccumL (\x dx -> (rule x dx, x + dx)) 0
mean :: (Fractional a) => [a] -> a mean xs = sum xs / genericLength xs
stddev :: (Floating a) => [a] -> a stddev xs = sqrt $ sum [(x-m)**2 | x <- xs] / genericLength xs where
m = mean xs
experiment :: String -> [Double] -> [Double] -> (Double -> Double -> Double) -> IO () experiment label dxs dys rule = do
let rxs = funnel rule dxs
rys = funnel rule dys
putStrLn label
printf "Mean x, y : %7.4f, %7.4f\n" (mean rxs) (mean rys)
printf "Std dev x, y : %7.4f, %7.4f\n" (stddev rxs) (stddev rys)
putStrLn ""
dxs = [ -0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087]
dys = [ 0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032]
main :: IO () main = do
experiment "Rule 1:" dxs dys (\_ _ -> 0) experiment "Rule 2:" dxs dys (\_ dz -> -dz) experiment "Rule 3:" dxs dys (\z dz -> -(z+dz)) experiment "Rule 4:" dxs dys (\z dz -> z+dz)</lang>
- Output:
Rule 1: Mean x, y : 0.0004, 0.0702 Std dev x, y : 0.7153, 0.6462 Rule 2: Mean x, y : 0.0009, -0.0103 Std dev x, y : 1.0371, 0.8999 Rule 3: Mean x, y : 0.0439, -0.0063 Std dev x, y : 7.9871, 4.7784 Rule 4: Mean x, y : 3.1341, 5.4210 Std dev x, y : 1.5874, 3.9304
J
<lang J> dx=:".0 :0-.LF
_0.533 0.270 0.859 _0.043 _0.205 _0.127 _0.071 0.275 1.251 _0.231 _0.401 0.269 0.491 0.951 1.150 0.001 _0.382 0.161 0.915 2.080 _2.337 0.034 _0.126 0.014 0.709 0.129 _1.093 _0.483 _1.193 0.020 _0.051 0.047 _0.095 0.695 0.340 _0.182 0.287 0.213 _0.423 _0.021 _0.134 1.798 0.021 _1.099 _0.361 1.636 _1.134 1.315 0.201 0.034 0.097 _0.170 0.054 _0.553 _0.024 _0.181 _0.700 _0.361 _0.789 0.279 _0.174 _0.009 _0.323 _0.658 0.348 _0.528 0.881 0.021 _0.853 0.157 0.648 1.774 _1.043 0.051 0.021 0.247 _0.310 0.171 0.000 0.106 0.024 _0.386 0.962 0.765 _0.125 _0.289 0.521 0.017 0.281 _0.749 _0.149 _2.436 _0.909 0.394 _0.113 _0.598 0.443 _0.521 _0.799 0.087
)
dy=:".0 :0-.LF
0.136 0.717 0.459 _0.225 1.392 0.385 0.121 _0.395 0.490 _0.682 _0.065 0.242 _0.288 0.658 0.459 0.000 0.426 0.205 _0.765 _2.188 _0.742 _0.010 0.089 0.208 0.585 0.633 _0.444 _0.351 _1.087 0.199 0.701 0.096 _0.025 _0.868 1.051 0.157 0.216 0.162 0.249 _0.007 0.009 0.508 _0.790 0.723 0.881 _0.508 0.393 _0.226 0.710 0.038 _0.217 0.831 0.480 0.407 0.447 _0.295 1.126 0.380 0.549 _0.445 _0.046 0.428 _0.074 0.217 _0.822 0.491 1.347 _0.141 1.230 _0.044 0.079 0.219 0.698 0.275 0.056 0.031 0.421 0.064 0.721 0.104 _0.729 0.650 _1.103 0.154 _1.720 0.051 _0.385 0.477 1.537 _0.901 0.939 _0.411 0.341 _0.411 0.106 0.224 _0.947 _1.424 _0.542 _1.032
)
Rule1=: ] Rule2=: -/\.&.|. Rule3=: ]-0,}: Rule4=: ]+0,}:
smoutput ' Rule 1 (x,y):' smoutput ' Mean: ',":dx ,&mean&Rule1 dy smoutput ' Std dev: ',":dx ,&stddev&Rule1 dy smoutput ' ' smoutput ' Rule 2 (x,y):' smoutput ' Mean: ',":dx ,&mean&Rule2 dy smoutput ' Std dev: ',":dx ,&stddev&Rule2 dy smoutput ' ' smoutput ' Rule 3 (x,y):' smoutput ' Mean: ',":dx ,&mean&Rule3 dy smoutput ' Std dev: ',":dx ,&stddev&Rule3 dy smoutput ' ' smoutput ' Rule 4 (x,y):' smoutput ' Mean: ',":dx ,&mean&Rule4 dy smoutput ' Std dev: ',":dx ,&stddev&Rule4 dy</lang>
Displayed result:
Rule 1 (x,y): Mean: 0.0004 0.07023 Std dev: 0.718875 0.649462 Rule 2 (x,y): Mean: 0.04386 _0.0063 Std dev: 8.02735 4.80249 Rule 3 (x,y): Mean: 0.00087 _0.01032 Std dev: 1.04236 0.904482 Rule 4 (x,y): Mean: _7e_5 0.15078 Std dev: 0.990174 0.918942
Author's note: these numbers are different from those of other implementations. I claim that this represents errors in the other implementations and invite proof that I am wrong.
Java
<lang java>import static java.lang.Math.*; import java.util.Arrays; import java.util.function.BiFunction;
public class DemingsFunnel {
public static void main(String[] args) {
double[] dxs = {
-0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087};
double[] dys = {
0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032};
experiment("Rule 1:", dxs, dys, (z, dz) -> 0.0);
experiment("Rule 2:", dxs, dys, (z, dz) -> -dz);
experiment("Rule 3:", dxs, dys, (z, dz) -> -(z + dz));
experiment("Rule 4:", dxs, dys, (z, dz) -> z + dz);
}
static void experiment(String label, double[] dxs, double[] dys,
BiFunction<Double, Double, Double> rule) {
double[] resx = funnel(dxs, rule);
double[] resy = funnel(dys, rule);
System.out.println(label);
System.out.printf("Mean x, y: %.4f, %.4f%n", mean(resx), mean(resy));
System.out.printf("Std dev x, y: %.4f, %.4f%n", stdDev(resx), stdDev(resy));
System.out.println();
}
static double[] funnel(double[] input, BiFunction<Double, Double, Double> rule) {
double x = 0;
double[] result = new double[input.length];
for (int i = 0; i < input.length; i++) {
double rx = x + input[i];
x = rule.apply(x, input[i]);
result[i] = rx;
}
return result;
}
static double mean(double[] xs) {
return Arrays.stream(xs).sum() / xs.length;
}
static double stdDev(double[] xs) {
double m = mean(xs);
return sqrt(Arrays.stream(xs).map(x -> pow((x - m), 2)).sum() / xs.length);
}
}</lang>
Rule 1: Mean x, y: 0,0004, 0,0702 Std dev x, y: 0,7153, 0,6462 Rule 2: Mean x, y: 0,0009, -0,0103 Std dev x, y: 1,0371, 0,8999 Rule 3: Mean x, y: 0,0439, -0,0063 Std dev x, y: 7,9871, 4,7784 Rule 4: Mean x, y: 3,1341, 5,4210 Std dev x, y: 1,5874, 3,9304
Julia
<lang julia># Run from Julia REPL to see the plots. using Statistics, Distributions, Plots
const racket_xdata = [-0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275, 1.251, -0.231,
-0.401, 0.269, 0.491, 0.951, 1.150, 0.001, -0.382, 0.161, 0.915, 2.080, -2.337,
0.034, -0.126, 0.014, 0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051,
0.047, -0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021, -0.134, 1.798,
0.021, -1.099, -0.361, 1.636, -1.134, 1.315, 0.201, 0.034, 0.097, -0.170, 0.054,
-0.553, -0.024, -0.181, -0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323,
-0.658, 0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774, -1.043, 0.051,
0.021, 0.247, -0.310, 0.171, 0.000, 0.106, 0.024, -0.386, 0.962, 0.765, -0.125,
-0.289, 0.521, 0.017, 0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087]
const racket_ydata = [0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395, 0.490, -0.682, -0.065,
0.242, -0.288, 0.658, 0.459, 0.000, 0.426, 0.205, -0.765, -2.188, -0.742, -0.010,
0.089, 0.208, 0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096, -0.025,
-0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007, 0.009, 0.508, -0.790, 0.723,
0.881, -0.508, 0.393, -0.226, 0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447,
-0.295, 1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217, -0.822, 0.491,
1.347, -0.141, 1.230, -0.044, 0.079, 0.219, 0.698, 0.275, 0.056, 0.031, 0.421, 0.064,
0.721, 0.104, -0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477, 1.537,
-0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224, -0.947, -1.424, -0.542, -1.032]
const rules = [(x, y, dx, dy) -> [0, 0], (x, y, dx, dy) -> [-dx, -dy],
(x, y, dx, dy) -> [-x - dx, -y - dy], (x, y, dx, dy) -> [x + dx, y + dy]]
const plots, colors = plot(layout=(1,2)), [:red, :green, :blue, :yellow]
function makedata()
radius_angles = zip(rand(Normal(), 100), rand(Uniform(-π, π), 100)) zip([z[1] * cos(z[2]) for z in radius_angles], [z[1] * sin(z[2]) for z in radius_angles])
end
function testfunnel(useracket=true)
for (i, rule) in enumerate(rules)
origin = [0.0, 0.0]
xvec, yvec = Float64[], Float64[]
for point in (useracket ? zip(racket_xdata, racket_ydata) : makedata())
push!(xvec, origin[1] + point[1])
push!(yvec, origin[2] + point[2])
origin .= rule(origin[1], origin[2], point[1], point[2])
end
println("Rule $i results:")
println("mean x: ", round(mean(xvec), digits=4), " std x: ", round(std(xvec, corrected=false), digits=4),
" mean y: ", round(mean(yvec), digits=4), " std y: ", round(std(yvec, corrected=false), digits=4))
scatter!(xvec, yvec, color=colors[i], subplot=(useracket ? 1 : 2),
title= useracket ? "Racket Data" : "Random Data", label="Rule $i")
end
end
println("\nUsing Racket data.") testfunnel() println("\nUsing new data.") testfunnel(false) display(plots)
</lang>
- Output:
Using Racket data. Rule 1 results: mean x: 0.0004 std x: 0.7153 mean y: 0.0702 std y: 0.6462 Rule 2 results: mean x: 0.0009 std x: 1.0371 mean y: -0.0103 std y: 0.8999 Rule 3 results: mean x: 0.0439 std x: 7.9871 mean y: -0.0063 std y: 4.7784 Rule 4 results: mean x: 3.1341 std x: 1.5874 mean y: 5.421 std y: 3.9304 Using new data. Rule 1 results: mean x: -0.0814 std x: 0.7761 mean y: -0.0187 std y: 0.799 Rule 2 results: mean x: 0.0009 std x: 0.9237 mean y: 0.0028 std y: 0.9626 Rule 3 results: mean x: 0.0123 std x: 4.7695 mean y: 0.0658 std y: 3.7198 Rule 4 results: mean x: -6.7132 std x: 4.5367 mean y: 1.632 std y: 2.0975
Kotlin
<lang scala>// version 1.1.3
typealias Rule = (Double, Double) -> Double
val dxs = doubleArrayOf(
-0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087
)
val dys = doubleArrayOf(
0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032
)
fun funnel(da: DoubleArray, rule: Rule): DoubleArray {
var x = 0.0
val result = DoubleArray(da.size)
for ((i, d) in da.withIndex()) {
result[i] = x + d
x = rule(x, d)
}
return result
}
fun mean(da: DoubleArray) = da.average()
fun stdDev(da: DoubleArray): Double {
val m = mean(da)
return Math.sqrt(da.map { (it - m) * (it - m) }.average())
}
fun experiment(label: String, rule: Rule) {
val rxs = funnel(dxs, rule)
val rys = funnel(dys, rule)
println("$label : x y")
println("Mean : ${"%7.4f, %7.4f".format(mean(rxs), mean(rys))}")
println("Std Dev : ${"%7.4f, %7.4f".format(stdDev(rxs), stdDev(rys))}")
println()
}
fun main(args: Array<String>) {
experiment("Rule 1") { _, _ -> 0.0 }
experiment("Rule 2") { _, dz -> -dz }
experiment("Rule 3") { z, dz -> -(z + dz) }
experiment("Rule 4") { z, dz -> z + dz }
}</lang>
- Output:
Rule 1 : x y Mean : 0.0004, 0.0702 Std Dev : 0.7153, 0.6462 Rule 2 : x y Mean : 0.0009, -0.0103 Std Dev : 1.0371, 0.8999 Rule 3 : x y Mean : 0.0439, -0.0063 Std Dev : 7.9871, 4.7784 Rule 4 : x y Mean : 3.1341, 5.4210 Std Dev : 1.5874, 3.9304
Mathematica
<lang Mathematica> </lang>
- Output:
Nim
<lang Nim>import stats, strformat
type Rule = proc(x, y: float): float
const Dxs = [-0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087]
const Dys = [ 0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032]
func funnel(a: openArray[float]; rule: Rule): seq[float] =
var x = 0.0 result.setlen(a.len) for i, val in a: result[i] = x + val x = rule(x, val)
proc experiment(label: string; r: Rule) =
let rxs = funnel(Dxs, r)
let rys = funnel(Dys, r)
echo label
echo fmt"Mean x, y : {rxs.mean:7.4f} {rys.mean:7.4f}"
echo fmt"Std dev x, y : {rxs.standardDeviation:7.4f} {rys.standardDeviation:7.4f}"
echo ""
experiment("Rule 1", proc(z, dz: float): float = 0.0)
experiment("Rule 2", proc(z, dz: float): float = -dz)
experiment("Rule 3", proc(z, dz: float): float = -(z + dz))
experiment("Rule 4", proc(z, dz: float): float = z + dz)</lang>
- Output:
Rule 1 Mean x, y : 0.0004 0.0702 Std dev x, y : 0.7153 0.6462 Rule 2 Mean x, y : 0.0009 -0.0103 Std dev x, y : 1.0371 0.8999 Rule 3 Mean x, y : 0.0439 -0.0063 Std dev x, y : 7.9871 4.7784 Rule 4 Mean x, y : 3.1341 5.4210 Std dev x, y : 1.5874 3.9304
PARI/GP
- This is a work-in-progress.
<lang parigp>drop(drops, rule, rnd)={
my(v=vector(drops),target=0); v[1]=rule(target, 0); for(i=2,drops, target=rule(target, v[i-1]); v[i]=rnd(n)+target ); v
}; R=[-.533-.136*I,.27-.717*I,.859-.459*I,-.043+.225*I,-.205-1.39*I,-.127-.385*I,-.071-.121*I,.275+.395*I,1.25-.490*I,-.231+.682*I,-.401+.0650*I,.269-.242*I,.491+.288*I,.951-.658*I,1.15-.459*I,.001,-.382-.426*I,.161-.205*I,.915+.765*I,2.08+2.19*I,-2.34+.742*I,.034+.0100*I,-.126-.0890*I,.014-.208*I,.709-.585*I,.129-.633*I,-1.09+.444*I,-.483+.351*I,-1.19+1.09*I,.02-.199*I,-.051-.701*I,.047-.0960*I,-.095+.0250*I,.695+.868*I,.34-1.05*I,-.182-.157*I,.287-.216*I,.213-.162*I,-.423-.249*I,-.021+.00700*I,-0.134-.00900*I,1.8-.508*I,.021+.790*I,-1.1-.723*I,-.361-.881*I,1.64+.508*I,-1.13-.393*I,1.32+.226*I,.201-.710*I,.034-.0380*I,.097+.217*I,-.17-.831*I,.054-.480*I,-.553-.407*I,-.024-.447*I,-.181+.295*I,-.7-1.13*I,-.361-.380*I,-.789-.549*I,.279+.445*I,-.174+.0460*I,-.009-.428*I,-.323+.0740*I,-.658-.217*I,.348+.822*I,-.528-.491*I,.881-1.35*I,.021+.141*I,-.853-1.23*I,.157+.0440*I,.648-.0790*I,1.77-.219*I,-1.04-.698*I,.051-.275*I,.021-.0560*I,.247-.0310*I,-.31-.421*I,.171-.0640*I,-.721*I,.106-.104*I,.024+.729*I,-.386-.650*I,.962+1.10*I,.765-.154*I,-.125+1.72*I,-.289-.0510*I,.521+.385*I,.017-.477*I,.281-1.54*I,-.749+.901*I,-.149-.939*I,-2.44+.411*I,-.909-.341*I,.394+.411*I,-.113-.106*I,-.598-.224*I,.443+.947*I,-.521+1.42*I,-.799+.542*I,.087+1.03*I]; rule1(target, result)=0; rule2(target, result)=target-result; rule3(target, result)=-result; rule4(target, result)=result; mean(v)=sum(i=1,#v,v[i])/#v; stdev(v,mu=mean(v))=sqrt(sum(i=1,#v,(v[i]-mu)^2)/#v); main()={
my(V);
V=apply(f->drop(100,f,n->R[n]), [rule1, rule2, rule3, rule4]);
for(i=1,4,
print("Method #"i);
print("Means: ", mean(real(V[i])), "\t", mean(imag(V[i])));
print("StDev: ", stdev(real(V[i])), "\t", stdev(imag(V[i])));
print()
)
}</lang>
Perl
<lang perl>@dx = qw<
-0.533 0.270 0.859 -0.043 -0.205 -0.127 -0.071 0.275
1.251 -0.231 -0.401 0.269 0.491 0.951 1.150 0.001
-0.382 0.161 0.915 2.080 -2.337 0.034 -0.126 0.014
0.709 0.129 -1.093 -0.483 -1.193 0.020 -0.051 0.047
-0.095 0.695 0.340 -0.182 0.287 0.213 -0.423 -0.021
-0.134 1.798 0.021 -1.099 -0.361 1.636 -1.134 1.315
0.201 0.034 0.097 -0.170 0.054 -0.553 -0.024 -0.181
-0.700 -0.361 -0.789 0.279 -0.174 -0.009 -0.323 -0.658
0.348 -0.528 0.881 0.021 -0.853 0.157 0.648 1.774
-1.043 0.051 0.021 0.247 -0.310 0.171 0.000 0.106
0.024 -0.386 0.962 0.765 -0.125 -0.289 0.521 0.017
0.281 -0.749 -0.149 -2.436 -0.909 0.394 -0.113 -0.598
0.443 -0.521 -0.799 0.087>;
@dy = qw<
0.136 0.717 0.459 -0.225 1.392 0.385 0.121 -0.395
0.490 -0.682 -0.065 0.242 -0.288 0.658 0.459 0.000
0.426 0.205 -0.765 -2.188 -0.742 -0.010 0.089 0.208
0.585 0.633 -0.444 -0.351 -1.087 0.199 0.701 0.096
-0.025 -0.868 1.051 0.157 0.216 0.162 0.249 -0.007
0.009 0.508 -0.790 0.723 0.881 -0.508 0.393 -0.226
0.710 0.038 -0.217 0.831 0.480 0.407 0.447 -0.295
1.126 0.380 0.549 -0.445 -0.046 0.428 -0.074 0.217
-0.822 0.491 1.347 -0.141 1.230 -0.044 0.079 0.219
0.698 0.275 0.056 0.031 0.421 0.064 0.721 0.104
-0.729 0.650 -1.103 0.154 -1.720 0.051 -0.385 0.477
1.537 -0.901 0.939 -0.411 0.341 -0.411 0.106 0.224
-0.947 -1.424 -0.542 -1.032>;
sub mean { my $s; $s += $_ for @_; $s / @_ } sub stddev { sqrt( mean(map { $_**2 } @_) - mean(@_)**2) }
@rules = ( sub { 0 }, sub { -$_[1] }, sub { -$_[0] - $_[1] }, sub { $_[0] + $_[1] } );
for (@rules) {
print "Rule " . ++$cnt . "\n";
my @ddx; my $tx = 0;
for my $x (@dx) { push @ddx, $tx + $x; $tx = &$_($tx, $x) }
my @ddy; my $ty = 0;
for my $y (@dy) { push @ddy, $ty + $y; $ty = &$_($ty, $y) }
printf "Mean x, y : %7.4f %7.4f\n", mean(@ddx), mean(@ddy); printf "Std dev x, y : %7.4f %7.4f\n", stddev(@ddx), stddev(@ddy);
}</lang>
- Output:
Rule 1 Mean x, y : 0.0004 0.0702 Std dev x, y : 0.7153 0.6462 Rule 2 Mean x, y : 0.0009 -0.0103 Std dev x, y : 1.0371 0.8999 Rule 3 Mean x, y : 0.0439 -0.0063 Std dev x, y : 7.9871 4.7784 Rule 4 Mean x, y : 3.1341 5.4210
Std dev x, y : 1.5874 3.9304
Phix
<lang Phix>function funnel(sequence dxs, integer rule)
atom x:=0.0
sequence rxs = {}
for i=1 to length(dxs) do
atom dx = dxs[i]
rxs = append(rxs,x + dx)
switch rule
case 2: x = -dx
case 3: x = -(x+dx)
case 4: x = x+dx
end switch
end for
return rxs
end function
function mean(sequence xs)
return sum(xs)/length(xs)
end function
function stddev(sequence xs)
atom m = mean(xs) return sqrt(sum(sq_power(sq_sub(xs,m),2))/length(xs))
end function
procedure experiment(integer n, sequence dxs, dys)
sequence rxs = funnel(dxs,n),
rys = funnel(dys,n)
printf(1,"Mean x, y : %7.4f, %7.4f\n",{mean(rxs), mean(rys)})
printf(1,"Std dev x, y : %7.4f, %7.4f\n",{stddev(rxs), stddev(rys)})
end procedure
constant dxs = {-0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087}
constant dys = { 0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032}
for i=1 to 4 do
experiment(i, dxs, dys)
end for</lang>
- Output:
Mean x, y : 0.0004, 0.0702 Std dev x, y : 0.7153, 0.6462 Mean x, y : 0.0009, -0.0103 Std dev x, y : 1.0371, 0.8999 Mean x, y : 0.0439, -0.0063 Std dev x, y : 7.9871, 4.7784 Mean x, y : 3.1341, 5.4210 Std dev x, y : 1.5874, 3.9304
Python
<lang python>import math
dxs = [-0.533, 0.27, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275, 1.251,
-0.231, -0.401, 0.269, 0.491, 0.951, 1.15, 0.001, -0.382, 0.161, 0.915,
2.08, -2.337, 0.034, -0.126, 0.014, 0.709, 0.129, -1.093, -0.483, -1.193,
0.02, -0.051, 0.047, -0.095, 0.695, 0.34, -0.182, 0.287, 0.213, -0.423,
-0.021, -0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315, 0.201,
0.034, 0.097, -0.17, 0.054, -0.553, -0.024, -0.181, -0.7, -0.361, -0.789,
0.279, -0.174, -0.009, -0.323, -0.658, 0.348, -0.528, 0.881, 0.021, -0.853,
0.157, 0.648, 1.774, -1.043, 0.051, 0.021, 0.247, -0.31, 0.171, 0.0, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017, 0.281, -0.749,
-0.149, -2.436, -0.909, 0.394, -0.113, -0.598, 0.443, -0.521, -0.799,
0.087]
dys = [0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395, 0.49, -0.682,
-0.065, 0.242, -0.288, 0.658, 0.459, 0.0, 0.426, 0.205, -0.765, -2.188,
-0.742, -0.01, 0.089, 0.208, 0.585, 0.633, -0.444, -0.351, -1.087, 0.199,
0.701, 0.096, -0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.79, 0.723, 0.881, -0.508, 0.393, -0.226, 0.71, 0.038,
-0.217, 0.831, 0.48, 0.407, 0.447, -0.295, 1.126, 0.38, 0.549, -0.445,
-0.046, 0.428, -0.074, 0.217, -0.822, 0.491, 1.347, -0.141, 1.23, -0.044,
0.079, 0.219, 0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.65, -1.103, 0.154, -1.72, 0.051, -0.385, 0.477, 1.537, -0.901,
0.939, -0.411, 0.341, -0.411, 0.106, 0.224, -0.947, -1.424, -0.542, -1.032]
def funnel(dxs, rule):
x, rxs = 0, []
for dx in dxs:
rxs.append(x + dx)
x = rule(x, dx)
return rxs
def mean(xs): return sum(xs) / len(xs)
def stddev(xs):
m = mean(xs) return math.sqrt(sum((x-m)**2 for x in xs) / len(xs))
def experiment(label, rule):
rxs, rys = funnel(dxs, rule), funnel(dys, rule) print label print 'Mean x, y : %.4f, %.4f' % (mean(rxs), mean(rys)) print 'Std dev x, y : %.4f, %.4f' % (stddev(rxs), stddev(rys)) print
experiment('Rule 1:', lambda z, dz: 0)
experiment('Rule 2:', lambda z, dz: -dz)
experiment('Rule 3:', lambda z, dz: -(z+dz))
experiment('Rule 4:', lambda z, dz: z+dz)</lang>
- Output:
Rule 1: Mean x, y : 0.0004, 0.0702 Std dev x, y : 0.7153, 0.6462 Rule 2: Mean x, y : 0.0009, -0.0103 Std dev x, y : 1.0371, 0.8999 Rule 3: Mean x, y : 0.0439, -0.0063 Std dev x, y : 7.9871, 4.7784 Rule 4: Mean x, y : 3.1341, 5.4210 Std dev x, y : 1.5874, 3.9304
Alternative: [Generates pseudo-random data and gives some interpretation.] The funnel experiment is performed in one dimension. The other dimension would act similarly. <lang python>from random import gauss from math import sqrt from pprint import pprint as pp
NMAX=50
def statscreator():
sum_ = sum2 = n = 0
def stats(x):
nonlocal sum_, sum2, n
sum_ += x
sum2 += x*x
n += 1.0
return sum_/n, sqrt(sum2/n - sum_*sum_/n/n)
return stats
def drop(target, sigma=1.0):
'Drop ball at target' return gauss(target, sigma)
def deming(rule, nmax=NMAX):
Simulate Demings funnel in 1D.
stats = statscreator()
target = 0
for i in range(nmax):
value = drop(target)
mean, sdev = stats(value)
target = rule(target, value)
if i == nmax - 1:
return mean, sdev
def d1(target, value):
Keep Funnel over target.
return target
def d2(target, value):
The new target starts at the center, 0,0 then is adjusted to be the previous target _minus_ the offset of the new drop from the previous target. return -value # - (target - (target - value)) = - value
def d3(target, value):
The new target starts at the center, 0,0 then is adjusted to be the previous target _minus_ the offset of the new drop from the center, 0.0. return target - value
def d4(target, value):
(Dumb). The new target is where it last dropped. return value
def printit(rule, trials=5):
print('\nDeming simulation. %i trials using rule %s:\n %s'
% (trials, rule.__name__.upper(), rule.__doc__))
for i in range(trials):
print(' Mean: %7.3f, Sdev: %7.3f' % deming(rule))
if __name__ == '__main__':
rcomments = [ (d1, 'Should have smallest deviations ~1.0, and be centered on 0.0'),
(d2, 'Should be centred on 0.0 with larger deviations than D1'),
(d3, 'Should be centred on 0.0 with larger deviations than D1'),
(d4, 'Center wanders all over the place, with deviations to match!'),
]
for rule, comment in rcomments:
printit(rule)
print(' %s\n' % comment)</lang>
- Output:
Deming simulation. 5 trials using rule D1:
Keep Funnel over target.
Mean: -0.161, Sdev: 0.942
Mean: -0.092, Sdev: 0.924
Mean: -0.199, Sdev: 1.079
Mean: -0.256, Sdev: 0.820
Mean: -0.211, Sdev: 0.971
Should have smallest deviations ~1.0, and be centered on 0.0
Deming simulation. 5 trials using rule D2:
The new target starts at the center, 0,0 then is adjusted to
be the previous target _minus_ the offset of the new drop from the
previous target.
Mean: -0.067, Sdev: 4.930
Mean: 0.035, Sdev: 4.859
Mean: -0.080, Sdev: 2.575
Mean: 0.147, Sdev: 4.948
Mean: 0.050, Sdev: 4.149
Should be centred on 0.0 with larger deviations than D1
Deming simulation. 5 trials using rule D3:
The new target starts at the center, 0,0 then is adjusted to
be the previous target _minus_ the offset of the new drop from the
center, 0.0.
Mean: 0.006, Sdev: 1.425
Mean: -0.039, Sdev: 1.436
Mean: 0.030, Sdev: 1.305
Mean: 0.009, Sdev: 1.419
Mean: 0.001, Sdev: 1.479
Should be centred on 0.0 with larger deviations than D1
Deming simulation. 5 trials using rule D4:
(Dumb). The new target is where it last dropped.
Mean: 5.252, Sdev: 2.839
Mean: 1.403, Sdev: 3.073
Mean: -1.525, Sdev: 3.650
Mean: 3.844, Sdev: 2.715
Mean: -7.697, Sdev: 3.715
Center wanders all over the place, with deviations to match!
Racket
The stretch solutions can be obtained by uncommenting radii etc. (delete the 4 semi-colons) to generate fresh data, and scatter-plots can be obtained by deleting the #; . <lang racket>#lang racket (require math/distributions math/statistics plot)
(define dxs '(-0.533 0.270 0.859 -0.043 -0.205 -0.127 -0.071 0.275 1.251 -0.231
-0.401 0.269 0.491 0.951 1.150 0.001 -0.382 0.161 0.915 2.080 -2.337
0.034 -0.126 0.014 0.709 0.129 -1.093 -0.483 -1.193 0.020 -0.051
0.047 -0.095 0.695 0.340 -0.182 0.287 0.213 -0.423 -0.021 -0.134 1.798
0.021 -1.099 -0.361 1.636 -1.134 1.315 0.201 0.034 0.097 -0.170 0.054
-0.553 -0.024 -0.181 -0.700 -0.361 -0.789 0.279 -0.174 -0.009 -0.323
-0.658 0.348 -0.528 0.881 0.021 -0.853 0.157 0.648 1.774 -1.043 0.051
0.021 0.247 -0.310 0.171 0.000 0.106 0.024 -0.386 0.962 0.765 -0.125
-0.289 0.521 0.017 0.281 -0.749 -0.149 -2.436 -0.909 0.394 -0.113 -0.598
0.443 -0.521 -0.799 0.087))
(define dys '(0.136 0.717 0.459 -0.225 1.392 0.385 0.121 -0.395 0.490 -0.682 -0.065
0.242 -0.288 0.658 0.459 0.000 0.426 0.205 -0.765 -2.188 -0.742 -0.010
0.089 0.208 0.585 0.633 -0.444 -0.351 -1.087 0.199 0.701 0.096 -0.025
-0.868 1.051 0.157 0.216 0.162 0.249 -0.007 0.009 0.508 -0.790 0.723
0.881 -0.508 0.393 -0.226 0.710 0.038 -0.217 0.831 0.480 0.407 0.447
-0.295 1.126 0.380 0.549 -0.445 -0.046 0.428 -0.074 0.217 -0.822 0.491
1.347 -0.141 1.230 -0.044 0.079 0.219 0.698 0.275 0.056 0.031 0.421 0.064
0.721 0.104 -0.729 0.650 -1.103 0.154 -1.720 0.051 -0.385 0.477 1.537
-0.901 0.939 -0.411 0.341 -0.411 0.106 0.224 -0.947 -1.424 -0.542 -1.032))
- (define radii (map abs (sample (normal-dist 0 1) 100)))
- (define angles (sample (uniform-dist (- pi) pi) 100))
- (define dxs (map (λ (r theta) (* r (cos theta))) radii angles))
- (define dys (map (λ (r theta) (* r (sin theta))) radii angles))
(define (funnel dxs rule)
(let ([x 0])
(for/fold ([rxs null])
([dx dxs])
(let ([rx (+ x dx)])
(set! x (rule x dx))
(cons rx rxs)))))
(define (experiment label rule)
(define (p s) (real->decimal-string s 4))
(let ([rxs (funnel dxs rule)]
[rys (funnel dys rule)])
(displayln label)
(printf "Mean x, y : ~a, ~a\n" (p (mean rxs)) (p (mean rys)))
(printf "Std dev x, y: ~a, ~a\n\n" (p (stddev rxs)) (p (stddev rys)))
#;(plot (points (map vector rxs rys)
#:x-min -15 #:x-max 15 #:y-min -15 #:y-max 15))))
(experiment "Rule 1:" (λ (z dz) 0)) (experiment "Rule 2:" (λ (z dz) (- dz))) (experiment "Rule 3:" (λ (z dz) (- (+ z dz)))) (experiment "Rule 4:" (λ (z dz) (+ z dz))) </lang>
- Output:
Rule 1: Mean x, y : 0.0004, 0.0702 Std dev x, y: 0.7153, 0.6462 Rule 2: Mean x, y : 0.0009, -0.0103 Std dev x, y: 1.0371, 0.8999 Rule 3: Mean x, y : 0.0439, -0.0063 Std dev x, y: 7.9871, 4.7784 Rule 4: Mean x, y : 3.1341, 5.4210 Std dev x, y: 1.5874, 3.9304
Raku
(formerly Perl 6)
<lang perl6>sub mean { @_ R/ [+] @_ } sub stddev {
# <(x - <x>)²> = <x²> - <x>² sqrt( mean(@_ »**» 2) - mean(@_)**2 )
}
constant @dz = <
-0.533 0.270 0.859 -0.043 -0.205 -0.127 -0.071 0.275
1.251 -0.231 -0.401 0.269 0.491 0.951 1.150 0.001
-0.382 0.161 0.915 2.080 -2.337 0.034 -0.126 0.014
0.709 0.129 -1.093 -0.483 -1.193 0.020 -0.051 0.047
-0.095 0.695 0.340 -0.182 0.287 0.213 -0.423 -0.021
-0.134 1.798 0.021 -1.099 -0.361 1.636 -1.134 1.315
0.201 0.034 0.097 -0.170 0.054 -0.553 -0.024 -0.181
-0.700 -0.361 -0.789 0.279 -0.174 -0.009 -0.323 -0.658
0.348 -0.528 0.881 0.021 -0.853 0.157 0.648 1.774
-1.043 0.051 0.021 0.247 -0.310 0.171 0.000 0.106
0.024 -0.386 0.962 0.765 -0.125 -0.289 0.521 0.017
0.281 -0.749 -0.149 -2.436 -0.909 0.394 -0.113 -0.598
0.443 -0.521 -0.799 0.087
> Z+ (1i X* <
0.136 0.717 0.459 -0.225 1.392 0.385 0.121 -0.395
0.490 -0.682 -0.065 0.242 -0.288 0.658 0.459 0.000
0.426 0.205 -0.765 -2.188 -0.742 -0.010 0.089 0.208
0.585 0.633 -0.444 -0.351 -1.087 0.199 0.701 0.096
-0.025 -0.868 1.051 0.157 0.216 0.162 0.249 -0.007
0.009 0.508 -0.790 0.723 0.881 -0.508 0.393 -0.226
0.710 0.038 -0.217 0.831 0.480 0.407 0.447 -0.295
1.126 0.380 0.549 -0.445 -0.046 0.428 -0.074 0.217
-0.822 0.491 1.347 -0.141 1.230 -0.044 0.079 0.219
0.698 0.275 0.056 0.031 0.421 0.064 0.721 0.104
-0.729 0.650 -1.103 0.154 -1.720 0.051 -0.385 0.477
1.537 -0.901 0.939 -0.411 0.341 -0.411 0.106 0.224
-0.947 -1.424 -0.542 -1.032
>);
constant @rule = -> \z, \dz { 0 }, -> \z, \dz { -dz }, -> \z, \dz { -z - dz }, -> \z, \dz { z + dz },
for @rule {
say "Rule $(++$):";
my $target = 0i;
my @z = gather for @dz -> $dz {
take $target + $dz; $target = .($target, $dz)
} printf "Mean x, y : %7.4f %7.4f\n", mean(@z».re), mean(@z».im); printf "Std dev x, y : %7.4f %7.4f\n", stddev(@z».re), stddev(@z».im);
}</lang>
- Output:
Rule 1: Mean x, y : 0.0004 0.0702 Std dev x, y : 0.7153 0.6462 Rule 2: Mean x, y : 0.0009 -0.0103 Std dev x, y : 1.0371 0.8999 Rule 3: Mean x, y : 0.0439 -0.0063 Std dev x, y : 7.9871 4.7784 Rule 4: Mean x, y : 3.1341 5.4210 Std dev x, y : 1.5874 3.9304
Ruby
<lang ruby>def funnel(dxs, &rule)
x, rxs = 0, [] for dx in dxs rxs << (x + dx) x = rule[x, dx] end rxs
end
def mean(xs) xs.inject(:+) / xs.size end
def stddev(xs)
m = mean(xs)
Math.sqrt(xs.inject(0.0){|sum,x| sum + (x-m)**2} / xs.size)
end
def experiment(label, dxs, dys, &rule)
rxs, rys = funnel(dxs, &rule), funnel(dys, &rule) puts label puts 'Mean x, y : %7.4f, %7.4f' % [mean(rxs), mean(rys)] puts 'Std dev x, y : %7.4f, %7.4f' % [stddev(rxs), stddev(rys)] puts
end
dxs = [ -0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087]
dys = [ 0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032]
experiment('Rule 1:', dxs, dys) {|z, dz| 0} experiment('Rule 2:', dxs, dys) {|z, dz| -dz} experiment('Rule 3:', dxs, dys) {|z, dz| -(z+dz)} experiment('Rule 4:', dxs, dys) {|z, dz| z+dz}</lang>
- Output:
Rule 1: Mean x, y : 0.0004, 0.0702 Std dev x, y : 0.7153, 0.6462 Rule 2: Mean x, y : 0.0009, -0.0103 Std dev x, y : 1.0371, 0.8999 Rule 3: Mean x, y : 0.0439, -0.0063 Std dev x, y : 7.9871, 4.7784 Rule 4: Mean x, y : 3.1341, 5.4210 Std dev x, y : 1.5874, 3.9304
Sidef
<lang ruby>func x̄(a) {
a.sum / a.len
}
func σ(a) {
sqrt(x̄(a.map{.**2}) - x̄(a)**2)
}
const Δ = (%n<
-0.533 0.270 0.859 -0.043 -0.205 -0.127 -0.071 0.275
1.251 -0.231 -0.401 0.269 0.491 0.951 1.150 0.001
-0.382 0.161 0.915 2.080 -2.337 0.034 -0.126 0.014
0.709 0.129 -1.093 -0.483 -1.193 0.020 -0.051 0.047
-0.095 0.695 0.340 -0.182 0.287 0.213 -0.423 -0.021
-0.134 1.798 0.021 -1.099 -0.361 1.636 -1.134 1.315
0.201 0.034 0.097 -0.170 0.054 -0.553 -0.024 -0.181
-0.700 -0.361 -0.789 0.279 -0.174 -0.009 -0.323 -0.658
0.348 -0.528 0.881 0.021 -0.853 0.157 0.648 1.774
-1.043 0.051 0.021 0.247 -0.310 0.171 0.000 0.106
0.024 -0.386 0.962 0.765 -0.125 -0.289 0.521 0.017
0.281 -0.749 -0.149 -2.436 -0.909 0.394 -0.113 -0.598
0.443 -0.521 -0.799 0.087
> ~Z+ %n<
0.136 0.717 0.459 -0.225 1.392 0.385 0.121 -0.395
0.490 -0.682 -0.065 0.242 -0.288 0.658 0.459 0.000
0.426 0.205 -0.765 -2.188 -0.742 -0.010 0.089 0.208
0.585 0.633 -0.444 -0.351 -1.087 0.199 0.701 0.096
-0.025 -0.868 1.051 0.157 0.216 0.162 0.249 -0.007
0.009 0.508 -0.790 0.723 0.881 -0.508 0.393 -0.226
0.710 0.038 -0.217 0.831 0.480 0.407 0.447 -0.295
1.126 0.380 0.549 -0.445 -0.046 0.428 -0.074 0.217
-0.822 0.491 1.347 -0.141 1.230 -0.044 0.079 0.219
0.698 0.275 0.056 0.031 0.421 0.064 0.721 0.104
-0.729 0.650 -1.103 0.154 -1.720 0.051 -0.385 0.477
1.537 -0.901 0.939 -0.411 0.341 -0.411 0.106 0.224
-0.947 -1.424 -0.542 -1.032
>.map{ .i })
const rules = [
{ 0 },
- Output:
Rule 1: Mean x, y : 0.0004 0.0702 Std dev x, y : 0.7153 0.6462 Rule 2: Mean x, y : 0.0009 -0.0103 Std dev x, y : 1.0371 0.8999 Rule 3: Mean x, y : 0.0439 -0.0063 Std dev x, y : 7.9871 4.7784 Rule 4: Mean x, y : 3.1341 5.4210 Std dev x, y : 1.5874 3.9304
Swift
<lang swift>import Foundation
let dxs = [
-0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275, 1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001, -0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014, 0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047, -0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021, -0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315, 0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181, -0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658, 0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774, -1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106, 0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017, 0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598, 0.443, -0.521, -0.799, 0.087
]
let dys = [
0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395, 0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000, 0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208, 0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096, -0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007, 0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226, 0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295, 1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217, -0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219, 0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104, -0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477, 1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224, -0.947, -1.424, -0.542, -1.032
]
extension Collection where Element: FloatingPoint {
@inlinable
public func mean() -> Element {
return reduce(0, +) / Element(count)
}
@inlinable
public func stdDev() -> Element {
let m = mean()
return map({ ($0 - m) * ($0 - m) }).mean().squareRoot()
}
}
typealias Rule = (Double, Double) -> Double
func funnel(_ arr: [Double], rule: Rule) -> [Double] {
var x = 0.0 var res = [Double](repeating: 0, count: arr.count)
for (i, d) in arr.enumerated() {
res[i] = x + d
x = rule(x, d)
}
return res
}
func experiment(label: String, rule: Rule) {
let rxs = funnel(dxs, rule: rule) let rys = funnel(dys, rule: rule)
print("\(label)\t: x y")
print("Mean\t:\(String(format: "%7.4f, %7.4f", rxs.mean(), rys.mean()))")
print("Std Dev\t:\(String(format: "%7.4f, %7.4f", rxs.stdDev(), rys.stdDev()))")
print()
}
experiment(label: "Rule 1", rule: {_, _ in 0 }) experiment(label: "Rule 2", rule: {_, dz in -dz }) experiment(label: "Rule 3", rule: {z, dz in -(z + dz) }) experiment(label: "Rule 4", rule: {z, dz in z + dz })</lang>
- Output:
Rule 1 : x y Mean : 0.0004, 0.0702 Std Dev : 0.7153, 0.6462 Rule 2 : x y Mean : 0.0009, -0.0103 Std Dev : 1.0371, 0.8999 Rule 3 : x y Mean : 0.0439, -0.0063 Std Dev : 7.9871, 4.7784 Rule 4 : x y Mean : 3.1341, 5.4210 Std Dev : 1.5874, 3.9304
Tcl
<lang tcl>package require Tcl 8.6 namespace path {tcl::mathop tcl::mathfunc}
proc funnel {items rule} {
set x 0.0
set result {}
foreach item $items {
lappend result [+ $x $item] set x [apply $rule $x $item]
} return $result
}
proc mean {items} {
/ [+ {*}$items] [double [llength $items]]
} proc stddev {items} {
set m [mean $items]
sqrt [mean [lmap x $items {** [- $x $m] 2}]]
}
proc experiment {label dxs dys rule} {
set rxs [funnel $dxs $rule] set rys [funnel $dys $rule] puts $label puts [format "Mean x, y : %7.4f, %7.4f" [mean $rxs] [mean $rys]] puts [format "Std dev x, y : %7.4f, %7.4f" [stddev $rxs] [stddev $rys]] puts ""
}
set dxs {
-0.533 0.270 0.859 -0.043 -0.205 -0.127 -0.071 0.275 1.251 -0.231 -0.401 0.269 0.491 0.951 1.150 0.001 -0.382 0.161 0.915 2.080 -2.337 0.034 -0.126 0.014 0.709 0.129 -1.093 -0.483 -1.193 0.020 -0.051 0.047 -0.095 0.695 0.340 -0.182 0.287 0.213 -0.423 -0.021 -0.134 1.798 0.021 -1.099 -0.361 1.636 -1.134 1.315 0.201 0.034 0.097 -0.170 0.054 -0.553 -0.024 -0.181 -0.700 -0.361 -0.789 0.279 -0.174 -0.009 -0.323 -0.658 0.348 -0.528 0.881 0.021 -0.853 0.157 0.648 1.774 -1.043 0.051 0.021 0.247 -0.310 0.171 0.000 0.106 0.024 -0.386 0.962 0.765 -0.125 -0.289 0.521 0.017 0.281 -0.749 -0.149 -2.436 -0.909 0.394 -0.113 -0.598 0.443 -0.521 -0.799 0.087
} set dys {
0.136 0.717 0.459 -0.225 1.392 0.385 0.121 -0.395 0.490 -0.682 -0.065 0.242 -0.288 0.658 0.459 0.000 0.426 0.205 -0.765 -2.188 -0.742 -0.010 0.089 0.208 0.585 0.633 -0.444 -0.351 -1.087 0.199 0.701 0.096 -0.025 -0.868 1.051 0.157 0.216 0.162 0.249 -0.007 0.009 0.508 -0.790 0.723 0.881 -0.508 0.393 -0.226 0.710 0.038 -0.217 0.831 0.480 0.407 0.447 -0.295 1.126 0.380 0.549 -0.445 -0.046 0.428 -0.074 0.217 -0.822 0.491 1.347 -0.141 1.230 -0.044 0.079 0.219 0.698 0.275 0.056 0.031 0.421 0.064 0.721 0.104 -0.729 0.650 -1.103 0.154 -1.720 0.051 -0.385 0.477 1.537 -0.901 0.939 -0.411 0.341 -0.411 0.106 0.224 -0.947 -1.424 -0.542 -1.032
}
puts "USING STANDARD DATA" experiment "Rule 1:" $dxs $dys {{z dz} {expr {0}}} experiment "Rule 2:" $dxs $dys {{z dz} {expr {-$dz}}} experiment "Rule 3:" $dxs $dys {{z dz} {expr {-($z+$dz)}}} experiment "Rule 4:" $dxs $dys {{z dz} {expr {$z+$dz}}}</lang> The first stretch goal:
<lang tcl>package require math::constants package require simulation::random
math::constants::constants degtorad
set rng(radius) [simulation::random::prng_Normal 0.0 1.0] set rng(angle) [simulation::random::prng_Uniform 0.0 360.0] set dxs [set dys {}] for {set i 0} {$i < 500} {incr i} {
set r [$rng(radius)]
set theta [expr {[$rng(angle)] * $degtorad}]
lappend dxs [expr {$r * cos($theta)}]
lappend dys [expr {$r * sin($theta)}]
}
puts "USING RANDOM DATA" experiment "Rule 1:" $dxs $dys {{z dz} {expr {0}}} experiment "Rule 2:" $dxs $dys {{z dz} {expr {-$dz}}} experiment "Rule 3:" $dxs $dys {{z dz} {expr {-($z+$dz)}}} experiment "Rule 4:" $dxs $dys {{z dz} {expr {$z+$dz}}}</lang>
- Output:
USING STANDARD DATA Rule 1: Mean x, y : 0.0004, 0.0702 Std dev x, y : 0.7153, 0.6462 Rule 2: Mean x, y : 0.0009, -0.0103 Std dev x, y : 1.0371, 0.8999 Rule 3: Mean x, y : 0.0439, -0.0063 Std dev x, y : 7.9871, 4.7784 Rule 4: Mean x, y : 3.1341, 5.4210 Std dev x, y : 1.5874, 3.9304 USING RANDOM DATA Rule 1: Mean x, y : 0.0053, 0.0112 Std dev x, y : 0.4954, 0.5082 Rule 2: Mean x, y : -0.0012, -0.0002 Std dev x, y : 0.6914, 0.7331 Rule 3: Mean x, y : -0.0132, 0.0098 Std dev x, y : 9.3480, 5.0290 Rule 4: Mean x, y : -6.3314, -4.0168 Std dev x, y : 3.2387, 4.4825
Wren
<lang ecmascript>import "/math" for Nums import "/fmt" for Fmt
var dxs = [
-0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087
]
var dys = [
0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032
]
var funnel = Fn.new { |fa, r|
var x = 0
var res = List.filled(fa.count, 0)
for (i in 0...fa.count) {
var f = fa[i]
res[i] = x + f
x = r.call(x, f)
}
return res
}
var experiment = Fn.new { |label, r|
var rxs = funnel.call(dxs, r)
var rys = funnel.call(dys, r)
System.print("%(label) : x y")
System.print("Mean : %(Fmt.f(7, Nums.mean(rxs), 4)), %(Fmt.f(7, Nums.mean(rys), 4))")
System.print("Std Dev : %(Fmt.f(7, Nums.popStdDev(rxs), 4)), %(Fmt.f(7, Nums.popStdDev(rys), 4))")
System.print()
}
experiment.call("Rule 1") { |z, dz| 0 } experiment.call("Rule 2") { |z, dz| -dz } experiment.call("Rule 3") { |z, dz| -(z + dz) } experiment.call("Rule 4") { |z, dz| z + dz }</lang>
- Output:
Rule 1 : x y Mean : 0.0004, 0.0702 Std Dev : 0.7153, 0.6462 Rule 2 : x y Mean : 0.0009, -0.0103 Std Dev : 1.0371, 0.8999 Rule 3 : x y Mean : 0.0439, -0.0063 Std Dev : 7.9871, 4.7784 Rule 4 : x y Mean : 3.1341, 5.4210 Std Dev : 1.5874, 3.9304
zkl
<lang zkl>fcn funnel(dxs, rule){
x:=0.0; rxs:=L();
foreach dx in (dxs){
rxs.append(x + dx);
x = rule(x,dx);
}
rxs
}
fcn mean(xs){ xs.sum(0.0)/xs.len() }
fcn stddev(xs){
m:=mean(xs);
(xs.reduce('wrap(sum,x){ sum + (x-m)*(x-m) },0.0)/xs.len()).sqrt();
}
fcn experiment(label,dxs,dys,rule){
rxs:=funnel(dxs,rule); rys:=funnel(dys,rule); label.println(); "Mean x, y : %7.4f, %7.4f".fmt(mean(rxs), mean(rys)) .println(); "Std dev x, y : %7.4f, %7.4f".fmt(stddev(rxs),stddev(rys)).println(); println();
}</lang> <lang zkl>dxs:=T( -0.533, 0.270, 0.859, -0.043, -0.205, -0.127, -0.071, 0.275,
1.251, -0.231, -0.401, 0.269, 0.491, 0.951, 1.150, 0.001,
-0.382, 0.161, 0.915, 2.080, -2.337, 0.034, -0.126, 0.014,
0.709, 0.129, -1.093, -0.483, -1.193, 0.020, -0.051, 0.047,
-0.095, 0.695, 0.340, -0.182, 0.287, 0.213, -0.423, -0.021,
-0.134, 1.798, 0.021, -1.099, -0.361, 1.636, -1.134, 1.315,
0.201, 0.034, 0.097, -0.170, 0.054, -0.553, -0.024, -0.181,
-0.700, -0.361, -0.789, 0.279, -0.174, -0.009, -0.323, -0.658,
0.348, -0.528, 0.881, 0.021, -0.853, 0.157, 0.648, 1.774,
-1.043, 0.051, 0.021, 0.247, -0.310, 0.171, 0.000, 0.106,
0.024, -0.386, 0.962, 0.765, -0.125, -0.289, 0.521, 0.017,
0.281, -0.749, -0.149, -2.436, -0.909, 0.394, -0.113, -0.598,
0.443, -0.521, -0.799, 0.087);
dys:=T( 0.136, 0.717, 0.459, -0.225, 1.392, 0.385, 0.121, -0.395,
0.490, -0.682, -0.065, 0.242, -0.288, 0.658, 0.459, 0.000,
0.426, 0.205, -0.765, -2.188, -0.742, -0.010, 0.089, 0.208,
0.585, 0.633, -0.444, -0.351, -1.087, 0.199, 0.701, 0.096,
-0.025, -0.868, 1.051, 0.157, 0.216, 0.162, 0.249, -0.007,
0.009, 0.508, -0.790, 0.723, 0.881, -0.508, 0.393, -0.226,
0.710, 0.038, -0.217, 0.831, 0.480, 0.407, 0.447, -0.295,
1.126, 0.380, 0.549, -0.445, -0.046, 0.428, -0.074, 0.217,
-0.822, 0.491, 1.347, -0.141, 1.230, -0.044, 0.079, 0.219,
0.698, 0.275, 0.056, 0.031, 0.421, 0.064, 0.721, 0.104,
-0.729, 0.650, -1.103, 0.154, -1.720, 0.051, -0.385, 0.477,
1.537, -0.901, 0.939, -0.411, 0.341, -0.411, 0.106, 0.224,
-0.947, -1.424, -0.542, -1.032);
experiment("Rule 1:", dxs, dys, fcn(z,dz){ 0.0 }); experiment("Rule 2:", dxs, dys, fcn(z,dz){ -dz }); experiment("Rule 3:", dxs, dys, fcn(z,dz){ -(z+dz) }); experiment("Rule 4:", dxs, dys, fcn(z,dz){ z+dz });</lang>
- Output:
Rule 1: Mean x, y : 0.0004, 0.0702 Std dev x, y : 0.7153, 0.6462 Rule 2: Mean x, y : 0.0009, -0.0103 Std dev x, y : 1.0371, 0.8999 Rule 3: Mean x, y : 0.0439, -0.0063 Std dev x, y : 7.9871, 4.7784 Rule 4: Mean x, y : 3.1341, 5.4210 Std dev x, y : 1.5874, 3.9304