Class: IrtRuby::ThreeParameterModel

Inherits:
Object
  • Object
show all
Defined in:
lib/irt_ruby/three_parameter_model.rb

Overview

A class representing the Three-Parameter model (3PL) for Item Response Theory. Incorporates:

  • Adaptive learning rate

  • Missing data handling

  • Parameter clamping for discrimination, guessing

  • Multiple convergence checks

  • Separate gradient calculation & updates

Constant Summary collapse

MISSING_STRATEGIES = 
%i[ignore treat_as_incorrect treat_as_correct].freeze

Instance Method Summary collapse

Constructor Details

#initialize(data, max_iter: 1000, tolerance: 1e-6, param_tolerance: 1e-6, learning_rate: 0.01, decay_factor: 0.5, missing_strategy: :ignore) ⇒ ThreeParameterModel

Returns a new instance of ThreeParameterModel.

Raises:

  • (ArgumentError) 


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# File 'lib/irt_ruby/three_parameter_model.rb', line 14

def initialize(data,
               max_iter: 1000,
               tolerance: 1e-6,
               param_tolerance: 1e-6,
               learning_rate: 0.01,
               decay_factor: 0.5,
               missing_strategy: :ignore)
  @data = data
  @data_array = data.to_a
  num_rows = @data_array.size
  num_cols = @data_array.first.size

  raise ArgumentError, "missing_strategy must be one of #{MISSING_STRATEGIES}" unless MISSING_STRATEGIES.include?(missing_strategy)

  @missing_strategy = missing_strategy

  # Initialize parameters
  @abilities       = Array.new(num_rows)  { rand(-0.25..0.25) }
  @difficulties    = Array.new(num_cols)  { rand(-0.25..0.25) }
  @discriminations = Array.new(num_cols)  { rand(0.5..1.5) }
  @guessings       = Array.new(num_cols)  { rand(0.0..0.3) }

  @max_iter        = max_iter
  @tolerance       = tolerance
  @param_tolerance = param_tolerance
  @learning_rate   = learning_rate
  @decay_factor    = decay_factor
end

Instance Method Details

#apply_gradient_update(ga, gd, gdisc, gc) ⇒ Object 



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# File 'lib/irt_ruby/three_parameter_model.rb', line 117

def apply_gradient_update(ga, gd, gdisc, gc)
  old_a    = @abilities.dup
  old_d    = @difficulties.dup
  old_disc = @discriminations.dup
  old_c    = @guessings.dup

  @abilities.each_index do |i|
    @abilities[i] += @learning_rate * ga[i]
  end

  @difficulties.each_index do |j|
    @difficulties[j] += @learning_rate * gd[j]
  end

  @discriminations.each_index do |j|
    @discriminations[j] += @learning_rate * gdisc[j]
    @discriminations[j] = 0.01 if @discriminations[j] < 0.01
    @discriminations[j] = 5.0  if @discriminations[j] > 5.0
  end

  @guessings.each_index do |j|
    @guessings[j] += @learning_rate * gc[j]
    @guessings[j] = 0.0  if @guessings[j] < 0.0
    @guessings[j] = 0.35 if @guessings[j] > 0.35
  end

  [old_a, old_d, old_disc, old_c]
end

#average_param_update(old_a, old_d, old_disc, old_c) ⇒ Object 



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# File 'lib/irt_ruby/three_parameter_model.rb', line 146

def average_param_update(old_a, old_d, old_disc, old_c)
  deltas = []
  @abilities.each_with_index       { |x, i| deltas << (x - old_a[i]).abs }
  @difficulties.each_with_index    { |x, j| deltas << (x - old_d[j]).abs }
  @discriminations.each_with_index { |x, j| deltas << (x - old_disc[j]).abs }
  @guessings.each_with_index       { |x, j| deltas << (x - old_c[j]).abs }
  deltas.sum / deltas.size
end

#compute_gradient ⇒ Object 



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# File 'lib/irt_ruby/three_parameter_model.rb', line 87

def compute_gradient
  grad_abilities       = Array.new(@abilities.size, 0.0)
  grad_difficulties    = Array.new(@difficulties.size, 0.0)
  grad_discriminations = Array.new(@discriminations.size, 0.0)
  grad_guessings       = Array.new(@guessings.size, 0.0)

  @data_array.each_with_index do |row, i|
    row.each_with_index do |resp, j|
      value, skip = resolve_missing(resp)
      next if skip

      theta = @abilities[i]
      a     = @discriminations[j]
      b     = @difficulties[j]
      c     = @guessings[j]

      prob  = probability(theta, a, b, c)
      error = value - prob

      grad_abilities[i]       += error * a * (1 - c)
      grad_difficulties[j]    -= error * a * (1 - c)
      grad_discriminations[j] += error * (theta - b) * (1 - c)

      grad_guessings[j]       += error * 1.0
    end
  end

  [grad_abilities, grad_difficulties, grad_discriminations, grad_guessings]
end

#fit ⇒ Object 



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# File 'lib/irt_ruby/three_parameter_model.rb', line 155

def fit
  prev_ll = log_likelihood

  @max_iter.times do
    ga, gd, gdisc, gc = compute_gradient
    old_a, old_d, old_disc, old_c = apply_gradient_update(ga, gd, gdisc, gc)

    curr_ll     = log_likelihood
    param_delta = average_param_update(old_a, old_d, old_disc, old_c)

    if curr_ll < prev_ll
      @abilities       = old_a
      @difficulties    = old_d
      @discriminations = old_disc
      @guessings       = old_c
      @learning_rate  *= @decay_factor
    else
      ll_diff = (curr_ll - prev_ll).abs
      break if ll_diff < @tolerance && param_delta < @param_tolerance

      prev_ll = curr_ll
    end
  end

  {
    abilities: @abilities,
    difficulties: @difficulties,
    discriminations: @discriminations,
    guessings: @guessings
  }
end

#log_likelihood ⇒ Object 



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# File 'lib/irt_ruby/three_parameter_model.rb', line 65

def log_likelihood
  ll = 0.0
  @data_array.each_with_index do |row, i|
    row.each_with_index do |resp, j|
      value, skip = resolve_missing(resp)
      next if skip

      prob = probability(@abilities[i],
                         @discriminations[j],
                         @difficulties[j],
                         @guessings[j])

      ll += if value == 1
              Math.log(prob + 1e-15)
            else
              Math.log((1 - prob) + 1e-15)
            end
    end
  end
  ll
end

#probability(theta, a, b, c) ⇒ Object

Probability for the 3PL model: c + (1-c)*sigmoid(a*(θ - b))



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# File 'lib/irt_ruby/three_parameter_model.rb', line 48

def probability(theta, a, b, c)
  c + ((1.0 - c) * sigmoid(a * (theta - b)))
end

#resolve_missing(resp) ⇒ Object 



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# File 'lib/irt_ruby/three_parameter_model.rb', line 52

def resolve_missing(resp)
  return [resp, false] unless resp.nil?

  case @missing_strategy
  when :ignore
    [nil, true]
  when :treat_as_incorrect
    [0, false]
  when :treat_as_correct
    [1, false]
  end
end

#sigmoid(x) ⇒ Object 



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# File 'lib/irt_ruby/three_parameter_model.rb', line 43

def sigmoid(x)
  1.0 / (1.0 + Math.exp(-x))
end