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Added Kulczynski Measure and Imbalance Ratio as quality metrics #882

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Added Kulczynski Measure and Imbalance Ratio as quality metrics #882

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1 change: 1 addition & 0 deletions docs/sources/CHANGELOG.md
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Expand Up @@ -21,6 +21,7 @@ The CHANGELOG for the current development version is available at

- The `mlxtend.evaluate.bootstrap_point632_score` now supports `fit_params`. ([#861](https://github.com/rasbt/mlxtend/pull/861))
- The `mlxtend/plotting/decision_regions.py` function now has a `contourf_kwargs` for matplotlib to change the look of the decision boundaries if desired. ([#881](https://github.com/rasbt/mlxtend/pull/881) via [[pbloem](https://github.com/pbloem)])
- The `mlxtend.frequent_patterns.metrics` provides **Kulczynski metric** and **Imbalance Ratio** metrics as `kulczynski_measure` and `imbalance_ratio` ([#840](https://github.com/rasbt/mlxtend/issues/840))

##### Changes

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389 changes: 389 additions & 0 deletions docs/sources/user_guide/frequent_patterns/metrics.ipynb
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Evaluating quality of Association Rules"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Overview"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A strong association rule may or may not be interesting for a specific application. Some measures have been developed to help evaluate association rules. `mlxtend` implements two such measures, Kulczynski Measure and Imbalance Ratio."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Kulczynski Measure:\n",
"\n",
"The Kulczynski measure $K_{A,B}$ can be interpreted as the average between the confidence that $A ⇒ B$ and the confidence that $B ⇒ A$\n",
"\n",
"The Kulczynski measure $K_{A,B} ∈ [0, 1]$ of the itemsets $A ⊆ I$ and\n",
"$B ⊆ I$ such that $A ∩ B = \\varnothing$ is given by\n",
"\n",
"$$K_{A,B} = \\frac{V_{A⇒B} + V_{B⇒A}}{2}$$\n",
"\n",
"$$K_{A,B} = \\frac{1}{2} \\Bigg[\\frac{sup(A \\cup B)}{sup(A)} + \\frac{sup(A \\cup B)}{sup(B)} \\Bigg]$$\n",
"\n",
"- If $K_{A,B} = 0$, then $A ⊆ T$ implies that $B \\nsubseteq T$ for any transaction $T$\n",
"- If $K_{A,B} = 1$, then $A ⊆ T$ implies that $B ⊆ T$ for any transaction $T$\n",
"- Note that the Kulczynski measure is symmetric: $K_{A,B} = K_{B,A}$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Imbalance Ratio:\n",
"The imbalance ratio $I_{A,B}$ can be interpreted as the ratio between the absolute difference between the support count of $A$ and the support count of $B$ and the number of transactions that contain $A$, $B$, or both $A$ and $B$\n",
"- The imbalance ratio $I_{A,B} ∈ [0, 1]$ of the itemsets $A ⊆ I$ and $B ⊆ I$ is given by\n",
"\n",
"$$I_{A,B} =\\frac{|N_A − N_B|}{N_A + N_B − N_{A∪B}}$$\n",
"- If $I_{A,B} = 0$, then $A$ and $B$ have the same support\n",
"- If $I_{A,B} = 1$, then either $A$ or $B$ has zero support\n",
"- Note that the imbalance ratio is symmetric: $I_{A,B} = I_{B,A}$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## References\n",
"\n",
"[1] Chapter 6 of J. Han, M. Kamber, J. Pei, “Data Mining: Concepts and Techniques”, 3rd edition, Elsevier/Morgan Kaufmann, 2012"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 1 -- Evaluate Kulczynski Measure of an Association rule:\n"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
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"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
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"\n",
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"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>antecedents</th>\n",
" <th>consequents</th>\n",
" <th>antecedent support</th>\n",
" <th>consequent support</th>\n",
" <th>support</th>\n",
" <th>confidence</th>\n",
" <th>lift</th>\n",
" <th>leverage</th>\n",
" <th>conviction</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>(Eggs)</td>\n",
" <td>(Kidney Beans)</td>\n",
" <td>0.8</td>\n",
" <td>1.0</td>\n",
" <td>0.8</td>\n",
" <td>1.00</td>\n",
" <td>1.00</td>\n",
" <td>0.00</td>\n",
" <td>inf</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>(Kidney Beans)</td>\n",
" <td>(Eggs)</td>\n",
" <td>1.0</td>\n",
" <td>0.8</td>\n",
" <td>0.8</td>\n",
" <td>0.80</td>\n",
" <td>1.00</td>\n",
" <td>0.00</td>\n",
" <td>1.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>(Eggs)</td>\n",
" <td>(Onion)</td>\n",
" <td>0.8</td>\n",
" <td>0.6</td>\n",
" <td>0.6</td>\n",
" <td>0.75</td>\n",
" <td>1.25</td>\n",
" <td>0.12</td>\n",
" <td>1.6</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>(Onion)</td>\n",
" <td>(Eggs)</td>\n",
" <td>0.6</td>\n",
" <td>0.8</td>\n",
" <td>0.6</td>\n",
" <td>1.00</td>\n",
" <td>1.25</td>\n",
" <td>0.12</td>\n",
" <td>inf</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>(Milk)</td>\n",
" <td>(Kidney Beans)</td>\n",
" <td>0.6</td>\n",
" <td>1.0</td>\n",
" <td>0.6</td>\n",
" <td>1.00</td>\n",
" <td>1.00</td>\n",
" <td>0.00</td>\n",
" <td>inf</td>\n",
" </tr>\n",
" <tr>\n",
" <th>5</th>\n",
" <td>(Onion)</td>\n",
" <td>(Kidney Beans)</td>\n",
" <td>0.6</td>\n",
" <td>1.0</td>\n",
" <td>0.6</td>\n",
" <td>1.00</td>\n",
" <td>1.00</td>\n",
" <td>0.00</td>\n",
" <td>inf</td>\n",
" </tr>\n",
" <tr>\n",
" <th>6</th>\n",
" <td>(Yogurt)</td>\n",
" <td>(Kidney Beans)</td>\n",
" <td>0.6</td>\n",
" <td>1.0</td>\n",
" <td>0.6</td>\n",
" <td>1.00</td>\n",
" <td>1.00</td>\n",
" <td>0.00</td>\n",
" <td>inf</td>\n",
" </tr>\n",
" <tr>\n",
" <th>7</th>\n",
" <td>(Eggs, Onion)</td>\n",
" <td>(Kidney Beans)</td>\n",
" <td>0.6</td>\n",
" <td>1.0</td>\n",
" <td>0.6</td>\n",
" <td>1.00</td>\n",
" <td>1.00</td>\n",
" <td>0.00</td>\n",
" <td>inf</td>\n",
" </tr>\n",
" <tr>\n",
" <th>8</th>\n",
" <td>(Eggs, Kidney Beans)</td>\n",
" <td>(Onion)</td>\n",
" <td>0.8</td>\n",
" <td>0.6</td>\n",
" <td>0.6</td>\n",
" <td>0.75</td>\n",
" <td>1.25</td>\n",
" <td>0.12</td>\n",
" <td>1.6</td>\n",
" </tr>\n",
" <tr>\n",
" <th>9</th>\n",
" <td>(Onion, Kidney Beans)</td>\n",
" <td>(Eggs)</td>\n",
" <td>0.6</td>\n",
" <td>0.8</td>\n",
" <td>0.6</td>\n",
" <td>1.00</td>\n",
" <td>1.25</td>\n",
" <td>0.12</td>\n",
" <td>inf</td>\n",
" </tr>\n",
" <tr>\n",
" <th>10</th>\n",
" <td>(Eggs)</td>\n",
" <td>(Onion, Kidney Beans)</td>\n",
" <td>0.8</td>\n",
" <td>0.6</td>\n",
" <td>0.6</td>\n",
" <td>0.75</td>\n",
" <td>1.25</td>\n",
" <td>0.12</td>\n",
" <td>1.6</td>\n",
" </tr>\n",
" <tr>\n",
" <th>11</th>\n",
" <td>(Onion)</td>\n",
" <td>(Eggs, Kidney Beans)</td>\n",
" <td>0.6</td>\n",
" <td>0.8</td>\n",
" <td>0.6</td>\n",
" <td>1.00</td>\n",
" <td>1.25</td>\n",
" <td>0.12</td>\n",
" <td>inf</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" antecedents consequents antecedent support \\\n",
"0 (Eggs) (Kidney Beans) 0.8 \n",
"1 (Kidney Beans) (Eggs) 1.0 \n",
"2 (Eggs) (Onion) 0.8 \n",
"3 (Onion) (Eggs) 0.6 \n",
"4 (Milk) (Kidney Beans) 0.6 \n",
"5 (Onion) (Kidney Beans) 0.6 \n",
"6 (Yogurt) (Kidney Beans) 0.6 \n",
"7 (Eggs, Onion) (Kidney Beans) 0.6 \n",
"8 (Eggs, Kidney Beans) (Onion) 0.8 \n",
"9 (Onion, Kidney Beans) (Eggs) 0.6 \n",
"10 (Eggs) (Onion, Kidney Beans) 0.8 \n",
"11 (Onion) (Eggs, Kidney Beans) 0.6 \n",
"\n",
" consequent support support confidence lift leverage conviction \n",
"0 1.0 0.8 1.00 1.00 0.00 inf \n",
"1 0.8 0.8 0.80 1.00 0.00 1.0 \n",
"2 0.6 0.6 0.75 1.25 0.12 1.6 \n",
"3 0.8 0.6 1.00 1.25 0.12 inf \n",
"4 1.0 0.6 1.00 1.00 0.00 inf \n",
"5 1.0 0.6 1.00 1.00 0.00 inf \n",
"6 1.0 0.6 1.00 1.00 0.00 inf \n",
"7 1.0 0.6 1.00 1.00 0.00 inf \n",
"8 0.6 0.6 0.75 1.25 0.12 1.6 \n",
"9 0.8 0.6 1.00 1.25 0.12 inf \n",
"10 0.6 0.6 0.75 1.25 0.12 1.6 \n",
"11 0.8 0.6 1.00 1.25 0.12 inf "
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import pandas as pd\n",
"from mlxtend.preprocessing import TransactionEncoder\n",
"from mlxtend.frequent_patterns import apriori, association_rules\n",
"from mlxtend.frequent_patterns import metrics\n",
"\n",
"dataset = [['Milk', 'Onion', 'Nutmeg', 'Kidney Beans', 'Eggs', 'Yogurt'],\n",
" ['Dill', 'Onion', 'Nutmeg', 'Kidney Beans', 'Eggs', 'Yogurt'],\n",
" ['Milk', 'Apple', 'Kidney Beans', 'Eggs'],\n",
" ['Milk', 'Unicorn', 'Corn', 'Kidney Beans', 'Yogurt'],\n",
" ['Corn', 'Onion', 'Onion', 'Kidney Beans', 'Ice cream', 'Eggs']]\n",
"\n",
"te = TransactionEncoder()\n",
"te_ary = te.fit_transform(dataset)\n",
"df = pd.DataFrame(te_ary, columns=te.columns_)\n",
"freq_items = apriori(df, min_support=0.6, use_colnames=True)\n",
"rules = association_rules(freq_items, metric=\"confidence\", min_threshold=0.7)\n",
"rules"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.875"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = frozenset(['Onion'])\n",
"b = frozenset(['Kidney Beans', 'Eggs'])\n",
"metrics.kulczynski_measure(rules, a, b)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 2 -- Evaluate Imabalance Ratio of an Association rule:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.2500000000000001"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = frozenset(['Onion'])\n",
"b = frozenset(['Kidney Beans', 'Eggs'])\n",
"metrics.imbalance_ratio(freq_items, a, b)"
]
}
],
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