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MaxHalford/funnel.ipynb

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Decomposition without and with gaps
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funnel.ipynb
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ratio.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Ratio decomposition"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Without gaps"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Some dummy data."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>period</th>\n",
" <th>group</th>\n",
" <th>volume</th>\n",
" <th>rate</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>1999</td>\n",
" <td>A</td>\n",
" <td>1000</td>\n",
" <td>3.7</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>1999</td>\n",
" <td>B</td>\n",
" <td>900</td>\n",
" <td>3.5</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>1999</td>\n",
" <td>C</td>\n",
" <td>1200</td>\n",
" <td>4.5</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>2000</td>\n",
" <td>A</td>\n",
" <td>1000</td>\n",
" <td>6.4</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>2000</td>\n",
" <td>B</td>\n",
" <td>900</td>\n",
" <td>3.9</td>\n",
" </tr>\n",
" <tr>\n",
" <th>5</th>\n",
" <td>2000</td>\n",
" <td>C</td>\n",
" <td>800</td>\n",
" <td>4.6</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" period group volume rate\n",
"0 1999 A 1000 3.7\n",
"1 1999 B 900 3.5\n",
"2 1999 C 1200 4.5\n",
"3 2000 A 1000 6.4\n",
"4 2000 B 900 3.9\n",
"5 2000 C 800 4.6"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(data := pd.DataFrame(\n",
" [\n",
" (1999, 'A', 1000, 3.7),\n",
" (1999, 'B', 900, 3.5),\n",
" (1999, 'C', 1200, 4.5),\n",
" (2000, 'A', 1000, 6.4),\n",
" (2000, 'B', 900, 3.9),\n",
" (2000, 'C', 800, 4.6),\n",
" ],\n",
" columns=['period', 'group', 'volume', 'rate']\n",
"))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Aggregated rates at each period:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"period\n",
"1999 3.951613\n",
"2000 5.033333\n",
"dtype: float64"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(period_rates := (\n",
" data\n",
" .groupby('period')\n",
" .apply(lambda x: x['rate'] * x['volume'] / x['volume'].sum()).groupby('period')\n",
" .sum()\n",
"))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Rate difference between periods to explain:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"period\n",
"1999 NaN\n",
"2000 1.08172\n",
"dtype: float64"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"period_rates.diff()\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We want to explain the difference through the inner and mix contributions of each group.\n",
"\n",
"Implementation available here: https://maxhalford.github.io/blog/kpi-evolution-decomposition/#implementation-1"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>period</th>\n",
" <th>group</th>\n",
" <th>volume</th>\n",
" <th>rate</th>\n",
" <th>total</th>\n",
" <th>period_rate</th>\n",
" <th>share</th>\n",
" <th>rate_lag</th>\n",
" <th>share_lag</th>\n",
" <th>period_rate_lag</th>\n",
" <th>inner</th>\n",
" <th>mix</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>1999</td>\n",
" <td>A</td>\n",
" <td>1000</td>\n",
" <td>3.7</td>\n",
" <td>3700.0</td>\n",
" <td>3.951613</td>\n",
" <td>0.322581</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>1999</td>\n",
" <td>B</td>\n",
" <td>900</td>\n",
" <td>3.5</td>\n",
" <td>3150.0</td>\n",
" <td>3.951613</td>\n",
" <td>0.290323</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>1999</td>\n",
" <td>C</td>\n",
" <td>1200</td>\n",
" <td>4.5</td>\n",
" <td>5400.0</td>\n",
" <td>3.951613</td>\n",
" <td>0.387097</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>2000</td>\n",
" <td>A</td>\n",
" <td>1000</td>\n",
" <td>6.4</td>\n",
" <td>6400.0</td>\n",
" <td>5.033333</td>\n",
" <td>0.370370</td>\n",
" <td>3.7</td>\n",
" <td>0.322581</td>\n",
" <td>3.951613</td>\n",
" <td>1.000000</td>\n",
" <td>0.176822</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>2000</td>\n",
" <td>B</td>\n",
" <td>900</td>\n",
" <td>3.9</td>\n",
" <td>3510.0</td>\n",
" <td>5.033333</td>\n",
" <td>0.333333</td>\n",
" <td>3.5</td>\n",
" <td>0.290323</td>\n",
" <td>3.951613</td>\n",
" <td>0.133333</td>\n",
" <td>0.150538</td>\n",
" </tr>\n",
" <tr>\n",
" <th>5</th>\n",
" <td>2000</td>\n",
" <td>C</td>\n",
" <td>800</td>\n",
" <td>4.6</td>\n",
" <td>3680.0</td>\n",
" <td>5.033333</td>\n",
" <td>0.296296</td>\n",
" <td>4.5</td>\n",
" <td>0.387097</td>\n",
" <td>3.951613</td>\n",
" <td>0.029630</td>\n",
" <td>-0.408602</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" period group volume rate total period_rate share rate_lag \\\n",
"0 1999 A 1000 3.7 3700.0 3.951613 0.322581 NaN \n",
"1 1999 B 900 3.5 3150.0 3.951613 0.290323 NaN \n",
"2 1999 C 1200 4.5 5400.0 3.951613 0.387097 NaN \n",
"3 2000 A 1000 6.4 6400.0 5.033333 0.370370 3.7 \n",
"4 2000 B 900 3.9 3510.0 5.033333 0.333333 3.5 \n",
"5 2000 C 800 4.6 3680.0 5.033333 0.296296 4.5 \n",
"\n",
" share_lag period_rate_lag inner mix \n",
"0 NaN NaN NaN NaN \n",
"1 NaN NaN NaN NaN \n",
"2 NaN NaN NaN NaN \n",
"3 0.322581 3.951613 1.000000 0.176822 \n",
"4 0.290323 3.951613 0.133333 0.150538 \n",
"5 0.387097 3.951613 0.029630 -0.408602 "
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"decomp = data.copy()\n",
"decomp['total'] = decomp.eval('volume * rate')\n",
"decomp['period_rate'] = (\n",
" decomp.groupby('period')['total'].transform('sum') /\n",
" decomp.groupby('period')['volume'].transform('sum')\n",
")\n",
"decomp['share'] = (\n",
" decomp['volume'] /\n",
" decomp.groupby('period')['volume'].transform('sum')\n",
")\n",
"decomp['rate_lag'] = decomp.groupby('group')['rate'].shift(1)\n",
"decomp['share_lag'] = decomp.groupby('group')['share'].shift(1)\n",
"decomp['period_rate_lag'] = decomp.groupby('group')['period_rate'].shift(1)\n",
"decomp['inner'] = decomp.eval('share * (rate - rate_lag)')\n",
"decomp['mix'] = decomp.eval('(share - share_lag) * rate_lag')\n",
"decomp"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can verify the contributions sum up to the rate difference:"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Inner: 1.163\n",
"Mix: -0.081\n"
]
}
],
"source": [
"import math\n",
"\n",
"assert math.isclose(decomp['inner'].sum() + decomp['mix'].sum(), period_rates.diff().sum())\n",
"print(\n",
" f'Inner: {decomp[\"inner\"].sum():.3f}\\n'\n",
" f'Mix: {decomp[\"mix\"].sum():.3f}'\n",
")\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## With gaps"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>period</th>\n",
" <th>group</th>\n",
" <th>volume</th>\n",
" <th>rate</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>1999</td>\n",
" <td>A</td>\n",
" <td>1000</td>\n",
" <td>3.7</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>1999</td>\n",
" <td>C</td>\n",
" <td>1200</td>\n",
" <td>4.5</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>2000</td>\n",
" <td>B</td>\n",
" <td>900</td>\n",
" <td>3.9</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>2000</td>\n",
" <td>C</td>\n",
" <td>800</td>\n",
" <td>4.6</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" period group volume rate\n",
"0 1999 A 1000 3.7\n",
"1 1999 C 1200 4.5\n",
"2 2000 B 900 3.9\n",
"3 2000 C 800 4.6"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(data := pd.DataFrame(\n",
" [\n",
" (1999, 'A', 1000, 3.7),\n",
" (1999, 'C', 1200, 4.5),\n",
" (2000, 'B', 900, 3.9),\n",
" (2000, 'C', 800, 4.6),\n",
" ],\n",
" columns=['period', 'group', 'volume', 'rate']\n",
"))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The trick is to add a row for each period where the group is not present."
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>period</th>\n",
" <th>group</th>\n",
" <th>volume</th>\n",
" <th>rate</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>1999</td>\n",
" <td>A</td>\n",
" <td>1000.0</td>\n",
" <td>3.7</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>1999</td>\n",
" <td>C</td>\n",
" <td>1200.0</td>\n",
" <td>4.5</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>1999</td>\n",
" <td>B</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>2000</td>\n",
" <td>A</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>2000</td>\n",
" <td>C</td>\n",
" <td>800.0</td>\n",
" <td>4.6</td>\n",
" </tr>\n",
" <tr>\n",
" <th>5</th>\n",
" <td>2000</td>\n",
" <td>B</td>\n",
" <td>900.0</td>\n",
" <td>3.9</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" period group volume rate\n",
"0 1999 A 1000.0 3.7\n",
"1 1999 C 1200.0 4.5\n",
"2 1999 B 0.0 0.0\n",
"3 2000 A 0.0 0.0\n",
"4 2000 C 800.0 4.6\n",
"5 2000 B 900.0 3.9"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"periods = pd.Series(data.period.unique(), name='period').to_frame()\n",
"groups = pd.Series(data.group.unique(), name='group').to_frame()\n",
"product = periods.merge(groups, how='cross')\n",
"data = pd.merge(data, product, left_on=['period', 'group'], right_on=['period', 'group'], how='right')\n",
"data['volume'] = data['volume'].fillna(0)\n",
"data['rate'] = data['rate'].fillna(0)\n",
"data\n"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"period\n",
"1999 4.136364\n",
"2000 4.229412\n",
"dtype: float64"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(period_rates := (\n",
" data\n",
" .groupby('period')\n",
" .apply(lambda x: x['rate'] * x['volume'] / x['volume'].sum()).groupby('period')\n",
" .sum()\n",
"))\n"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"period\n",
"1999 NaN\n",
"2000 0.093048\n",
"dtype: float64"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"period_rates.diff()\n"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>period</th>\n",
" <th>group</th>\n",
" <th>volume</th>\n",
" <th>rate</th>\n",
" <th>total</th>\n",
" <th>period_rate</th>\n",
" <th>share</th>\n",
" <th>rate_lag</th>\n",
" <th>share_lag</th>\n",
" <th>period_rate_lag</th>\n",
" <th>inner</th>\n",
" <th>mix</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>1999</td>\n",
" <td>A</td>\n",
" <td>1000.0</td>\n",
" <td>3.7</td>\n",
" <td>3700.0</td>\n",
" <td>4.136364</td>\n",
" <td>0.454545</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>1999</td>\n",
" <td>C</td>\n",
" <td>1200.0</td>\n",
" <td>4.5</td>\n",
" <td>5400.0</td>\n",
" <td>4.136364</td>\n",
" <td>0.545455</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>1999</td>\n",
" <td>B</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" <td>4.136364</td>\n",
" <td>0.000000</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" <td>NaN</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>2000</td>\n",
" <td>A</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" <td>0.0</td>\n",
" <td>4.229412</td>\n",
" <td>0.000000</td>\n",
" <td>3.7</td>\n",
" <td>0.454545</td>\n",
" <td>4.136364</td>\n",
" <td>-0.000000</td>\n",
" <td>-1.681818</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>2000</td>\n",
" <td>C</td>\n",
" <td>800.0</td>\n",
" <td>4.6</td>\n",
" <td>3680.0</td>\n",
" <td>4.229412</td>\n",
" <td>0.470588</td>\n",
" <td>4.5</td>\n",
" <td>0.545455</td>\n",
" <td>4.136364</td>\n",
" <td>0.047059</td>\n",
" <td>-0.336898</td>\n",
" </tr>\n",
" <tr>\n",
" <th>5</th>\n",
" <td>2000</td>\n",
" <td>B</td>\n",
" <td>900.0</td>\n",
" <td>3.9</td>\n",
" <td>3510.0</td>\n",
" <td>4.229412</td>\n",
" <td>0.529412</td>\n",
" <td>0.0</td>\n",
" <td>0.000000</td>\n",
" <td>4.136364</td>\n",
" <td>2.064706</td>\n",
" <td>0.000000</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" period group volume rate total period_rate share rate_lag \\\n",
"0 1999 A 1000.0 3.7 3700.0 4.136364 0.454545 NaN \n",
"1 1999 C 1200.0 4.5 5400.0 4.136364 0.545455 NaN \n",
"2 1999 B 0.0 0.0 0.0 4.136364 0.000000 NaN \n",
"3 2000 A 0.0 0.0 0.0 4.229412 0.000000 3.7 \n",
"4 2000 C 800.0 4.6 3680.0 4.229412 0.470588 4.5 \n",
"5 2000 B 900.0 3.9 3510.0 4.229412 0.529412 0.0 \n",
"\n",
" share_lag period_rate_lag inner mix \n",
"0 NaN NaN NaN NaN \n",
"1 NaN NaN NaN NaN \n",
"2 NaN NaN NaN NaN \n",
"3 0.454545 4.136364 -0.000000 -1.681818 \n",
"4 0.545455 4.136364 0.047059 -0.336898 \n",
"5 0.000000 4.136364 2.064706 0.000000 "
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"decomp = data.copy()\n",
"decomp['total'] = decomp.eval('volume * rate')\n",
"decomp['period_rate'] = (\n",
" decomp.groupby('period')['total'].transform('sum') /\n",
" decomp.groupby('period')['volume'].transform('sum')\n",
")\n",
"decomp['share'] = (\n",
" decomp['volume'] /\n",
" decomp.groupby('period')['volume'].transform('sum')\n",
")\n",
"decomp['rate_lag'] = decomp.groupby('group')['rate'].shift(1)\n",
"decomp['share_lag'] = decomp.groupby('group')['share'].shift(1)\n",
"decomp['period_rate_lag'] = decomp.groupby('group')['period_rate'].shift(1)\n",
"decomp['inner'] = decomp.eval('share * (rate - rate_lag)')\n",
"decomp['mix'] = decomp.eval('(share - share_lag) * rate_lag')\n",
"decomp\n"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Inner: 2.112\n",
"Mix: -2.019\n"
]
}
],
"source": [
"import math\n",
"\n",
"assert math.isclose(decomp['inner'].sum() + decomp['mix'].sum(), period_rates.diff().sum())\n",
"print(\n",
" f'Inner: {decomp.query(\"period == 2000\")[\"inner\"].sum():.3f}\\n'\n",
" f'Mix: {decomp.query(\"period == 2000\")[\"mix\"].sum():.3f}'\n",
")\n"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.0"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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