Normalized Function

A Normalized Function is a numeric-output function whose range has been transformed by a normalizing factor to produce values within a standardized range.

• AKA: Range-Normalized Function, Scaled Function, Normalization Function.
• Context:
  • It can (typically) map a raw function value onto a bounded interval, such as the unit interval or the [-1, 1] interval.
  • It can (typically) apply a normalizing factor derived from a maximum value, a minimum value, a reference value, or a theoretical bound.
  • It can (typically) preserve the relative ordering of its input values while rescaling their magnitude.
  • It can (typically) be defined as a ratio between an observed value and an ideal value or normalizing constant.
  • It can (often) improve the comparability of measures computed across datasets of different sizes.
  • It can (often) support cross-domain comparison by removing scale dependence.
  • It can (often) map values to a dimensionless quantity independent of the original measurement unit.
  • It can (often) be derived from a corresponding unnormalized function via normalization.
  • It can (often) bound its output value between a lower bound and an upper bound.
  • It can guarantee that its maximum attainable value equals a fixed reference, such as 1.
  • It can be constructed for a ranking metric, a similarity metric, or an information-theoretic measure.
  • It can be sensitive to the choice of normalization scheme, such as min-max normalization or z-score normalization.
  • It can support evaluation tasks that require a normalized score.
  • It can range from being a Linearly Normalized Function to being a Nonlinearly Normalized Function, depending on its transformation type.
  • It can range from being a Globally Normalized Function to being a Locally Normalized Function, depending on its normalization scope.
  • It can range from being a Bounded Normalized Function to being an Unbounded Normalized Function, depending on its range constraints.
  • It can range from being a Deterministic Normalized Function to being a Stochastic Normalized Function, depending on its computation model.
  • It can be undefined when its normalizing factor equals zero.
  • …
• Example(s):
  • Normalized Ranking Metrics, such as:
    • a Normalized Discounted Cumulative Gain, a normalized version of the discounted cumulative gain.
  • Normalized Information-Theoretic Measures, such as:
    • a Normalized Mutual Information Metric, a normalized version of the mutual information metric.
    • a Normalized Pointwise Mutual Information Measure, a normalized version of the pointwise mutual information measure.
  • Normalized Similarity Metrics, such as:
    • a Normalized Compression Distance, a normalized version of the compression distance.
    • a Normalized Levenshtein Distance, a normalized version of the Levenshtein distance.
  • Normalized Statistical Functions, such as:
    • a Standard Score Function (z-score), which normalizes by standard deviation.
    • a Min-Max Scaling Function, which normalizes onto the unit interval.
  • Normalized Probability Functions, such as:
    • a Softmax Function, which normalizes a real-valued vector into a probability distribution.
  • …
• Counter-Example(s):
  • a Normalized Vector, which is a vector rather than a function.
  • a Normalized Database Schema, which applies database normalization rather than a range transformation.
  • an Unnormalized Function, whose range is not transformed by a normalizing factor.
  • a Raw Score Function, which returns unscaled values.
  • …
• See: Function, Range, Normalization, Normalizing Factor, Discounted Cumulative Gain, Mutual Information Metric, Softmax Function, Min-Max Normalization, Z-Score Normalization.

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References

• • (Manning et al., 2008) ⇒ Christopher D. Manning, Prabhakar Raghavan, and Hinrich Schütze. (2008). “Introduction to Information Retrieval.” Cambridge University Press. — on normalized evaluation measures such as Normalized Discounted Cumulative Gain.