tomazos has asked for the wisdom of the Perl Monks concerning the following question:

Please consider the following function called inaccuracy. It looks at a long array called @sales which contains a series of integers:

sub inaccuracy {
  my ($base, $multiple, $log) = @_;
  
  my $inaccuracy = 0;
  my $interval = 0;
  
  foreach my $sale (@sales) {

    $interval++;

    $inaccuracy +=
     ( $sale -
       $base -
       $multiple * log ($interval) / log ($log)
     ) ** 2;

  }

  return $inaccuracy;
}

[download]

The challenge is to write a function called estimate that analyses the data in @sales and comes up with values for $base, $multiple and $log such that when they are passed to inaccuracy they minimize the value returned by inaccuracy.

Test data is provided below if anyone is interested in giving this a shot... 

@sales = (12920, 12640, 12540, 12680, 12775, 12525, 12570, 12670,
12680, 12775, 12885, 12770, 12585, 12605, 12560, 12635, 12590,
12560, 12535, 12465, 12465, 12455, 12570, 12365, 12505, 12555,
12530, 12280, 12320, 11930, 12070, 12155, 11735, 11625, 11765,
11695, 11740, 11820, 11965, 11925, 12000, 11950, 11995, 11695,
11515, 11705, 11795, 11915, 11750, 11725, 11585, 11905, 12070,
11810, 11735, 11555, 11415, 11260, 11050, 11025, 10820, 10790,
10900, 10980, 11000, 10900, 10625, 10655, 10670, 10690, 10670,
10675, 10535, 10395, 10475, 10380, 10330, 10375, 10325, 10190,
10210, 10375, 10310, 10260, 10375, 10255, 10520, 10260, 10185,
10060, 9870, 10030, 10030, 9975, 9770, 9715, 9400, 9215, 9075,
9125, 9150, 9005, 8445, 8160, 8020, 8140, 8140, 8125, 7940, 7950,
7920, 7895, 7850, 7715, 7635, 7545, 7550, 7620, 7560, 7510, 7485,
7460, 7465, 7425, 7310, 7250, 7160, 7135, 7235, 7260, 7640, 7505,
7410, 7425, 7545, 7405, 7505, 7420, 7765, 7695, 7740, 7975, 7675,
7655, 7835, 8120, 7735, 7680, 7775, 7960, 8025, 8125, 8330, 8700,
8765, 8855, 8985, 9120, 9095, 9110, 9250, 9505, 9595, 9640, 9640,
9710, 9855, 9950, 10020, 10105, 10160, 10115, 10205, 10275, 10130,
10220, 10220, 10310, 10335, 10430, 10270, 9930, 9910, 9795, 10040,
10060, 10240, 10215, 9965, 9480, 9400, 9555, 9455, 9595, 9615,
9090, 9140, 8910, 9000, 8950, 8740, 8670, 8525, 8575, 8225, 8225,
8090, 7920, 7755, 7775, 8040, 7740, 7650, 7605, 7745, 7675, 7650,
7690, 7430, 7410, 7320, 7310, 7380, 7310, 7235, 7305, 7395, 7390,
7255, 7350, 7205, 7260, 7210, 7460, 7240, 7125, 6825, 6895, 6825,
7025, 7120, 7170, 6965, 7020, 7140, 7215, 7125, 6990, 6965, 6970,
7020, 7470, 7470, 7405, 7345, 7370, 7440, 7415, 7575, 7570, 7635,
7475, 7560, 7725, 7610, 7545, 7640, 7620, 7505, 7485, 7505, 7435,
7635, 7590, 7545, 7660, 7545, 7435, 7505, 7475, 7400, 7380, 7420,
7450, 7125, 7210, 7520, 7565, 7565, 7680, 7930, 7940, 7985, 7960,
7885, 7950, 8150, 8270, 8540, 8590, 8540, 8490, 8095, 8450, 8265,
8425, 8575, 8480, 8325, 8300, 8235, 8190, 8045, 8260, 8050, 8140,
8090, 8020, 8085, 7970, 8015, 7900, 7855, 7695, 7770, 7795, 7725,
7825, 7825, 7710, 7715, 7760, 7690, 7665, 7590, 7595, 7375, 7250,
7155, 7065, 6830, 6300, 6115, 5865, 5815, 5605, 5465, 5235, 5155,
4790, 4670, 4765, 4810, 4730, 4380, 4305, 4060, 3895, 3945, 4105,
4330, 4305, 4290, 4485, 4100, 4055, 4035, 4035, 4100, 4035, 4350,
4415, 4530, 4555, 4525, 4500, 4520, 4450, 4305, 4280, 4330, 4445,
4400, 4445, 4450, 4445, 4495, 4450, 4515, 4540, 4660, 4805, 4805,
4895, 4920, 4990, 5060, 5015, 5080, 5265);

[download]

Replies are listed 'Best First'.
Re: Numerical Analysis Challenge
by Masem (Monsignor) on Jan 03, 2002 at 00:41 UTC
    Basically, you're trying to find the regresssion for the function: 
    sales = base_sale + K * log( i )
    where K = multiple/log( logbase ). There's no way by regression alone to determine multiple or logbase separately.

    The above regression is not strictly linear, but if you tranform i to x via 

     x = log i
    (or 
    i = e^x
    ), then you get: 
    sales = base_sale + K * x
    which is a straight linear regression once you apply the transform correctly to your interval variable. Thus, you simply have to do some stat summation over your data set, and you're all set; the exact equations for that should be in any numerical math text (I don't see any Perl modules that do regression easily, but the routine isn't hard for this).

    -----------------------------------------------------
    Dr. Michael K. Neylon - mneylon-pm@masemware.com || "You've left the lens cap of your mind on again, Pinky" - The Brain
    "I can see my house from here!"
    It's not what you know, but knowing how to find it if you don't know that's important

[reply]
      Oh course, Masem is largely right; however it looks to me like there is a module Statistics::OLS that will do linear least squares, which is exactly what you would want. I have never used it though, so YMMV for sure.

      Scott

[reply]
[reply]
      Dooh. You are right: $multiple and $log combine to give a single constant.

      Transforming on x = log i turns into into a straight linear regression and from some java code I found laying around the net I've got the alrgorithym for getting the line of best fit.

      Theoretically I should be all set. Thanks for your help.

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