How do I use Holt-Winters Seasonal Dampened Method to compute a two-month sales projection in PHP?

Question

Holt-Winters is introduced here:

http://en.wikipedia.org/wiki/Holt-Winters

The Seasonal Dampened version of it is discussed here (scroll down the page):

http://otexts.com/fpp/7/5/

In a nutshell, it basically looks at 3 things:

• long-term trend
• short-term trend
• seasonal trend

It also doesn't average those together, because really what you need is weighted averaging, where seasonal and short-term are more significant than long-term trend, naturally, with financial data trends.

Given $anYear1 and $anYear2, how do I apply the Holt-Winters Seasonal Dampened Method to forecast 2 more months past the end of $anYear2? Assume $anYear1 is an array of 12 numbers. Assume $anYear2 is an array of a range of 0 to 12 numbers.

So, I can fill it with random data like so:

<?php

$anYear1 = array();
$anYear2 = array();
$nStop = 10; // so we need 11 and 12 of the year
for ($i = 1; $i <= 12; $i++) {
  $anYear1[$i] = rand(200,500);
  if ($i <= $nStop) {
    // give it a natural lift like real financial data
    $anYear2[$i] = rand(400,700); 
  }
}
$nSeasonRange = 4; // 4 months in a business quarter

Therefore, I want to create a function like so:

function forecastHoltWinters($anYear1, $anYear2, $nSeasonRange = 4) {
  ///////////////////
  // DO MAGIC HERE //
  ///////////////////

  // an array with 2 numbers, indicating 2 months forward from end of $anYear2
  return $anForecast;
}

$anForecast = forecastHoltWinters($anYear1, $anYear2, $nSeasonRange);
echo "YEAR 1\n";
print_r($anYear1);
echo "\n\nYEAR 2\n"
print_r($anYear2);
echo "\n\nTWO MONTHS FORECAST\n";
print_r($anForecast);

Note: I have found a Github example here, but it doesn't show how to do a projection. It is also discussed here.

Answer 1

I found a way to adapt Ian Barber's function to do what I needed.

<?php

error_reporting(E_ALL);
ini_set('display_errors','On');

$anYear1 = array();
$anYear2 = array();
$nStop = 10;
for($i = 1; $i <= 12; $i++) {
    $anYear1[$i] = rand(100,400);
    if ($i <= $nStop) {
        $anYear2[$i+12] = rand(200,600);
    }
}

print_r($anYear1);
print_r($anYear2);
$anData = array_merge($anYear1,$anYear2);
print_r(forecastHoltWinters($anData));

function forecastHoltWinters($anData, $nForecast = 2, $nSeasonLength = 4, $nAlpha = 0.2, $nBeta = 0.01, $nGamma = 0.01, $nDevGamma = 0.1) {

    // Calculate an initial trend level
    $nTrend1 = 0;
    for($i = 0; $i < $nSeasonLength; $i++) {
      $nTrend1 += $anData[$i];
    }
    $nTrend1 /= $nSeasonLength;

    $nTrend2 = 0;
    for($i = $nSeasonLength; $i < 2*$nSeasonLength; $i++) {
      $nTrend2 += $anData[$i];
    }
    $nTrend2 /= $nSeasonLength;

    $nInitialTrend = ($nTrend2 - $nTrend1) / $nSeasonLength;

    // Take the first value as the initial level
    $nInitialLevel = $anData[0];

    // Build index
    $anIndex = array();
    foreach($anData as $nKey => $nVal) {
      $anIndex[$nKey] = $nVal / ($nInitialLevel + ($nKey + 1) * $nInitialTrend);
    }

    // Build season buffer
    $anSeason = array_fill(0, count($anData), 0);
    for($i = 0; $i < $nSeasonLength; $i++) {
      $anSeason[$i] = ($anIndex[$i] + $anIndex[$i+$nSeasonLength]) / 2;
    }

    // Normalise season
    $nSeasonFactor = $nSeasonLength / array_sum($anSeason);
    foreach($anSeason as $nKey => $nVal) {
      $anSeason[$nKey] *= $nSeasonFactor;
    }

    $anHoltWinters = array();
    $anDeviations = array();
    $nAlphaLevel = $nInitialLevel;
    $nBetaTrend = $nInitialTrend;
    foreach($anData as $nKey => $nVal) {
      $nTempLevel = $nAlphaLevel;
      $nTempTrend = $nBetaTrend;

      $nAlphaLevel = $nAlpha * $nVal / $anSeason[$nKey] + (1.0 - $nAlpha) * ($nTempLevel + $nTempTrend);
      $nBetaTrend = $nBeta * ($nAlphaLevel - $nTempLevel) + ( 1.0 - $nBeta ) * $nTempTrend;

      $anSeason[$nKey + $nSeasonLength] = $nGamma * $nVal / $nAlphaLevel + (1.0 - $nGamma) * $anSeason[$nKey];

      $anHoltWinters[$nKey] = ($nAlphaLevel + $nBetaTrend * ($nKey + 1)) * $anSeason[$nKey];
      $anDeviations[$nKey] = $nDevGamma * abs($nVal - $anHoltWinters[$nKey]) + (1-$nDevGamma) 
                  * (isset($anDeviations[$nKey - $nSeasonLength]) ? $anDeviations[$nKey - $nSeasonLength] : 0);
    }

    $anForecast = array();
    $nLast = end($anData);
    for($i = 1; $i <= $nForecast; $i++) {
       $nComputed = round($nAlphaLevel + $nBetaTrend * $anSeason[$nKey + $i]);
       if ($nComputed < 0) { // wildly off due to outliers
         $nComputed = $nLast;
       }
       $anForecast[] = $nComputed;
    }

    return $anForecast;
}