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 @@ +<!-- + Licensed to the Apache Software Foundation (ASF) under one + or more contributor license agreements. See the NOTICE file + distributed with this work for additional information + regarding copyright ownership. The ASF licenses this file + to you under the Apache License, Version 2.0 (the + "License"); you may not use this file except in compliance + with the License. You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, + software distributed under the License is distributed on an + "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY + KIND, either express or implied. See the License for the + specific language governing permissions and limitations + 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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