Module: HoltWinters

Defined in:
lib/holt_winters.rb,
lib/holt_winters/version.rb

Overview

Given a time series, say a complete monthly data for 12 months, the Holt-Winters smoothing and forecasting

technique is built on the following formulae (multiplicative version):

St[i] = alpha * y[i] / It[i - period] + (1.0 - alpha) * (St[i - 1] + Bt[i - 1])
Bt[i] = gamma * (St[i] - St[i - 1]) + (1 - gamma) * Bt[i - 1]
It[i] = beta * y[i] / St[i] + (1.0 - beta) * It[i - period]
Ft[i + m] = (St[i] + (m * Bt[i])) * It[i - period + m]

Note: Many authors suggest calculating initial values of St, Bt and It in a variety of ways, but
some of them are incorrect e.g. determination of It parameter using regression. I have used
the NIST recommended methods.

For more details, see:
http://adorio-research.org/wordpress/?p=1230
http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc435.htm

Constant Summary collapse

VERSION = 
"0.0.0"

Class Method Summary collapse

Class Method Details

.forecast(y, alpha, beta, gamma, period, m) ⇒ Object

Calculate initial values and return the forecast for m periods.

y           Time series array
alpha       Level smoothing coefficient
beta        Trend smoothing coefficient (increasing beta tightens fit)
gamma       Seasonal smoothing coefficient
period      A complete season's data consists of L periods. And we need
            to estimate the trend factor from one period to the next. To
            accomplish this, it is advisable to use two complete seasons;
            that is, 2L periods.
m           Extrapolated future data points
            - 4 quarterly
            - 7 weekly
            - 12 monthly



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# File 'lib/holt_winters.rb', line 34

def forecast(y, alpha, beta, gamma, period, m)
  return nil if y.empty?

  seasons = y.size / period
  a0 = initial_level(y, period)
  b0 = initial_trend(y, period)

  seasonal = seasonal_indicies(y, period, seasons)

  holt_winters(y, a0, b0, alpha, beta, gamma, seasonal, period, m);
end

.holt_winters(y, a0, b0, alpha, beta, gamma, seasonal, period, m) ⇒ Object 



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# File 'lib/holt_winters.rb', line 46

def holt_winters(y, a0, b0, alpha, beta, gamma, seasonal, period, m)
  st = Array.new(y.length, 0.0)
  bt = Array.new(y.length, 0.0)
  it = Array.new(y.length, 0.0)
  ft = Array.new(y.length + m, 0.0)

  st[1] = a0
  bt[1] = b0

  (0..period - 1).each do |i|
    it[i] = seasonal[i]
  end

  ft[m] = (st[0] + (m * bt[0])) * it[0] # This is actually 0 since bt[0] = 0
  ft[m + 1] = (st[1] + (m * bt[1])) * it[1] # Forecast starts from period + 2

  (2..(y.size - 1)).each do |i|
    # Calculate overall smoothing
    if (i - period) >= 0
     st[i] = alpha * y[i] / it[i - period] + (1.0 - alpha) * (st[i - 1] + bt[i - 1])
    else
      st[i] = alpha * y[i] + (1.0 - alpha) * (st[i - 1] + bt[i - 1])
    end

    # Calculate trend smoothing
    bt[i] = gamma * (st[i] - st[i - 1]) + (1 - gamma) * bt[i - 1]

    # Calculate seasonal smoothing
    if (i - period) >= 0
      it[i] = beta * y[i] / st[i] + (1.0 - beta) * it[i - period]
    end

    # Calculate forecast
    if (i + m) >= period
      ft[i + m] = (st[i] + (m * bt[i])) * it[i - period + m]
    end
  end

  ft
end

.initial_level(y, period) ⇒ Object

See: robjhyndman.com/researchtips/hw-initialization/ 1st period’s average can be taken. But y works better.



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# File 'lib/holt_winters.rb', line 89

def initial_level(y, period)
  y.first
end

.initial_trend(y, period) ⇒ Object

See: www.itl.nist.gov/div898/handbook/pmc/section4/pmc435.htm



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# File 'lib/holt_winters.rb', line 95

def initial_trend(y, period)
  sum = 0

  (0..period - 1).each do |i|
    sum += (y[period + i] - y[i])
  end

  sum / (period * period)
end

.seasonal_indicies(y, period, seasons) ⇒ Object

See: www.itl.nist.gov/div898/handbook/pmc/section4/pmc435.htm



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# File 'lib/holt_winters.rb', line 107

def seasonal_indicies(y, period, seasons)
  seasonal_average = Array.new(seasons, 0.0)
  seasonal_indices = Array.new(period, 0.0)
  averaged_observations = Array.new(y.size, 0.0)

  (0..seasons - 1).each do |i|
    (0..period - 1).each do |j|
      seasonal_average[i] += y[(i * period) + j]
    end
    seasonal_average[i] /= period
  end

  (0..seasons - 1).each do |i|
    (0..period - 1).each do |j|
      averaged_observations[(i * period) + j] = y[(i * period) + j] / seasonal_average[i]
    end
  end

  (0..period - 1).each do |i|
    (0..seasons - 1).each do |j|
      seasonal_indices[i] += averaged_observations[(j * period) + i]
    end
    seasonal_indices[i] /= seasons
  end

  seasonal_indices
end