[GitHub] incubator-hivemall pull request #107: [WIP][HIVEMALL-132] Generalize f1score...

[myui <gi...@git.apache.org>]

Github user myui commented on a diff in the pull request:

    https://github.com/apache/incubator-hivemall/pull/107#discussion_r131559068
  
    --- Diff: docs/gitbook/eval/multilabel_classification_measures.md ---
    @@ -0,0 +1,148 @@
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    +  to you under the Apache License, Version 2.0 (the
    +  "License"); you may not use this file except in compliance
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    +  under the License.
    +-->
    +
    +<!-- toc -->
    +
    +# Multi-label classification
    +
    +
    +Multi-label classification problem is predicting the labels given categorized dataset.
    +Each sample $$i$$ has $$l_i$$ labels ($$0 \leq  l_i \leq |L| $$  )
    +, where $$L$$ is the number of unique labels in the geven dataset.
    +
    +This page focuses on evaluation of the results from such Multi-label classification problems.
    +
    +# Examples
    +
    +For the metrics explanation, this page introduces toy example data and two metrics.
    +
    +## Data
    +
    +The following table shows the sample of Multi-label classification's prediction.
    +Animal names represent the tags of blog post.
    +Left column includes supervised labels,
    +Right column includes are predicted labels by a Multi-label classifier.
    +
    +| truth labels| predicted labels |
    +|:---:|:---:|
    +|cat, dog | cat, bird |
    +| cat, bird | cat, dog |
    +| | cat |
    +| bird | bird |
    +| bird, cat | bird, cat |
    +| cat, dog, bird | cat, dog |
    +| dog | dog, bird|
    +
    +
    +# Evaluation metrics for multi-label classification
    +
    +Hivemall provises micro F1-score and micro F-measure.
    +
    +Given $$N$$ blog posts, we uses 
    +
    +Define $$L$$ is the set of the tag of blog posts, and 
    +$$l_i$$ is a tag set of $$i$$th document.
    +In the same manner,
    +$$p_i$$ is a predicted tag set of $$i$$th document.
    +
    +
    +
    +## Micro F1-score
    +
    +
    +F1-score is the harmonic mean of recall and precision.
    +
    +The value is computed by the following equation:
    +
    +$$
    +\mathrm{F}_1 = 2 \frac
    +{\sum_i |l_i \cap p_i |}
    +{ 2* \sum_i |l_i \cap p_i | + \sum_i |l_i - p_i | + \sum_i |p_i - l_i | }
    +$$
    +
    +The Following query shows the example to obtain F1-score.
    +
    +```sql
    +WITH data as (
    +  select array("cat", "dog")         as actual, array("cat", "bird") as predicted
    +union all
    +  select array("cat", "bird")        as actual, array("cat", "dog")  as predicted
    +union all
    +  select array()                     as actual, array("cat")         as predicted
    +union all
    +  select array("bird")               as actual, array("bird")        as predicted
    +union all
    +  select array("bird", "cat")        as actual, array("bird", "cat") as predicted
    +union all
    +  select array("cat", "dog", "bird") as actual, array("cat", "dog")  as predicted
    +union all
    +  select array("dog")                as actual, array("dog", "bird") as predicted
    +)
    +select
    +  f1score(actual, predicted)
    +from data
    +;
    +
    +--- 0.6956521739130435;
    +```
    +
    +
    +## Micro F-measure
    +
    +F-measure is generalized F1-score and the weighted harmonic mean of recall and precision.
    +
    +$$\beta$$ is the parameter to determine the weight of precision.
    +So, F1-score is the special case of F-measure given $$\beta=1$$.
    +
    +If $$\beta$$ is larger positive value than `1.0`, F-measure reaches to micro recall.
    +On the other hand,
    +if $$\beta$$ is smaller positive value than `1.0`, F-measure reaches to micro precision.
    +
    +The following query shows the example to obtain F-measure with $$\beta=2$$.
    +
    +$$
    +\mathrm{F}_{\beta} = (1+\beta^2) \frac
    +{\sum_i |l_i \cap p_i |}
    +{ \beta^2 (\sum_i |l_i \cap p_i | + \sum_i |p_i - l_i |) + \sum_i |l_i \cap p_i | + \sum_i |l_i - p_i |}
    +$$
    +
    +
    +```sql
    +WITH data as (
    +  select array("cat", "dog")         as actual, array("cat", "bird") as predicted
    +union all
    +  select array("cat", "bird")        as actual, array("cat", "dog")  as predicted
    +union all
    +  select array()                     as actual, array("cat")         as predicted
    +union all
    +  select array("bird")               as actual, array("bird")        as predicted
    +union all
    +  select array("bird", "cat")        as actual, array("bird", "cat") as predicted
    +union all
    +  select array("cat", "dog", "bird") as actual, array("cat", "dog")  as predicted
    +union all
    +  select array("dog")                as actual, array("dog", "bird") as predicted
    +)
    +select
    +  fmeasure(actual, predicted, 2)
    --- End diff --
    
    `fmeasure(actual, predicted, '-beta 2.0 -average macro')`


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