BG-BB-Model.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import polars as pl\n",
    "import numpy as np\n",
    "from utils import Donation\n",
    "from scipy.optimize import minimize\n",
    "from scipy.special import gammaln, comb, hyp2f1\n",
    "from scipy.special import beta as beta_fn\n",
    "from scipy.stats import beta as beta_dist\n",
    "from scipy.integrate import quad\n",
    "\n",
    "import altair as alt\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib_inline\n",
    "from IPython.display import display_markdown\n",
    "\n",
    "matplotlib_inline.backend_inline.set_matplotlib_formats('svg')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Source**: \n",
    "- [Customer-Base Analysis in a Discrete-Time Noncontractual Setting](http://www.brucehardie.com/papers/020/)\n",
    "- [Implementing the BG/BB Model for Customer-Base Analysis in Excel](http://www.brucehardie.com/notes/010/)\n",
    "- [Implementing the $S_{BB}-G/B$ Model in MATLAB](http://www.brucehardie.com/notes/023/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = Donation()\n",
    "rfm_summary_calib = data.p1x_data()\n",
    "rfm_array_calib = rfm_summary_calib.collect().to_numpy()\n",
    "rfm_summary_valid = data.p2x_data()\n",
    "p1x, t_x, _, num_donors = [*rfm_array_calib.T]\n",
    "\n",
    "n = 6\n",
    "n * (n + 1)/2 + 1 # Possible recency/frequency patterns in calibration period\n",
    "years = data.years"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rfcalib_cross_tab(df, values, title, subtitle=None, color_range=[0,1]):\n",
    "    tx_year_map = {tx: int(year) for tx, year in enumerate(years)}\n",
    "    \n",
    "    return (\n",
    "        df\n",
    "        .with_columns(pl.col(\"t_x\").replace(tx_year_map).alias(\"Year\"))\n",
    "        .sort('Year')\n",
    "        .pivot(on='Year', index='P1X', values=values)\n",
    "        .sort('P1X')\n",
    "        .style.tab_header(title=title, subtitle=subtitle)\n",
    "        .tab_stub(rowname_col=\"P1X\")\n",
    "        .tab_stubhead(label='P1X')\n",
    "        .fmt_number(decimals=2)   \n",
    "        .tab_spanner(\n",
    "            label=\"Year of last transaction\",\n",
    "            columns=years[:7]\n",
    "        ).data_color(\n",
    "            domain=color_range,\n",
    "            palette=[\"white\", \"rebeccapurple\"],\n",
    "            na_color=\"white\",\n",
    "        ).sub_missing(\n",
    "            columns=pl.col('*'),\n",
    "            missing_text=\"\"\n",
    "        )            \n",
    "    )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Parameter Estimation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "$\\alpha$ = 1.2035\n",
       "\n",
       "$\\beta$ = 0.7497\n",
       "\n",
       "$\\gamma$ = 0.6568\n",
       "\n",
       "$\\delta$ = 2.7840\n",
       "\n",
       "Log-Likelihood = -33225.5813"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def bgbb_est(rfm_data, guess={'alpha': 1, 'beta': 0.5, 'gamma': 0.5, 'delta': 2.5}):\n",
    "    p1x, t_x, n, num_donors = [*rfm_data.T]\n",
    "\n",
    "    def log_likelihood(param):\n",
    "        alpha, beta, gamma, delta = param\n",
    "        B_alpha_beta = beta_fn(alpha, beta)\n",
    "        B_gamma_delta = beta_fn(gamma, delta)\n",
    "        \n",
    "        A1 = beta_fn(alpha+p1x,beta+n-p1x)/B_alpha_beta * \\\n",
    "                 beta_fn(gamma,delta+n)/B_gamma_delta\n",
    "        i = np.arange(6).reshape(-1,1)\n",
    "        A2 = beta_fn(alpha+p1x, beta+t_x-p1x+i)/B_alpha_beta* \\\n",
    "                  beta_fn(gamma+1, delta+t_x+i)/B_gamma_delta        \n",
    "        A2 = np.where(i <=  (n - t_x - 1), A2, 0)\n",
    "\n",
    "        return -np.sum(num_donors * np.log(A1 + np.sum(A2, axis=0)))\n",
    "    \n",
    "    bnds = [[0, np.inf] for _ in range(4)]\n",
    "    return minimize(log_likelihood, x0=list(guess.values()), bounds=bnds)\n",
    "\n",
    "# Sample parameters\n",
    "# alpha = 1.20352083040498\n",
    "# beta = 0.749714243061896\n",
    "# gamma = 0.656712169147878\n",
    "# delta = 2.78340801635898\n",
    "\n",
    "res = bgbb_est(rfm_array_calib)\n",
    "alpha, beta, gamma, delta = res.x\n",
    "ll = res.fun\n",
    "\n",
    "display_markdown(f'''$\\\\alpha$ = {alpha:0.4f}\n",
    "\n",
    "$\\\\beta$ = {beta:0.4f}\n",
    "\n",
    "$\\\\gamma$ = {gamma:0.4f}\n",
    "\n",
    "$\\\\delta$ = {delta:0.4f}\n",
    "\n",
    "Log-Likelihood = {-ll:0.4f}''', raw=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Likelihood Function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Likelihood function for a randomly chosen customer with purchase history ($x, t_{x}, n$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "B_alpha_beta = beta_fn(alpha, beta)\n",
    "B_gamma_delta = beta_fn(gamma, delta)\n",
    "\n",
    "A1 = beta_fn(alpha+p1x, beta+n-p1x)/B_alpha_beta * beta_fn(gamma, delta+n)/B_gamma_delta\n",
    "i = np.arange(6).reshape(-1,1)\n",
    "A1a = beta_fn(alpha+p1x, beta+t_x-p1x+i)/B_alpha_beta * beta_fn(gamma+1, delta+t_x+i)/B_gamma_delta  \n",
    "A1a = np.where(i <= (n - t_x - 1), A1a, 0)\n",
    "L = A1 + np.sum(A1a, axis=0)\n",
    "\n",
    "L_df = rfm_summary_calib.collect().hstack([pl.Series('Likelihood', L)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### In-Sample Model Fit Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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         {
          "Actual Vs Estimated": "Actual",
          "No of people": 3464,
          "P1X": 0
         },
         {
          "Actual Vs Estimated": "Actual",
          "No of people": 1823,
          "P1X": 1
         },
         {
          "Actual Vs Estimated": "Actual",
          "No of people": 1430,
          "P1X": 2
         },
         {
          "Actual Vs Estimated": "Actual",
          "No of people": 1085,
          "P1X": 3
         },
         {
          "Actual Vs Estimated": "Actual",
          "No of people": 1036,
          "P1X": 4
         },
         {
          "Actual Vs Estimated": "Actual",
          "No of people": 1063,
          "P1X": 5
         },
         {
          "Actual Vs Estimated": "Actual",
          "No of people": 1203,
          "P1X": 6
         },
         {
          "Actual Vs Estimated": "Model",
          "No of people": 3454.821413754415,
          "P1X": 0
         },
         {
          "Actual Vs Estimated": "Model",
          "No of people": 1888.6822804489507,
          "P1X": 1
         },
         {
          "Actual Vs Estimated": "Model",
          "No of people": 1348.9084408980011,
          "P1X": 2
         },
         {
          "Actual Vs Estimated": "Model",
          "No of people": 1113.3917799917174,
          "P1X": 3
         },
         {
          "Actual Vs Estimated": "Model",
          "No of people": 1017.9234186689152,
          "P1X": 4
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         {
          "Actual Vs Estimated": "Model",
          "No of people": 1027.158561881708,
          "P1X": 5
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          "Actual Vs Estimated": "Model",
          "No of people": 1253.1141043562939,
          "P1X": 6
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        "x": {
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         "field": "P1X",
         "title": "No. of repeat transactions",
         "type": "ordinal"
        },
        "xOffset": {
         "field": "Actual Vs Estimated",
         "type": "nominal"
        },
        "y": {
         "field": "No of people",
         "title": "No. of people",
         "type": "quantitative"
        }
       },
       "height": 250,
       "mark": {
        "type": "bar"
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       "title": "Predicted vs. Actual Frequency of Repeat Transactions (Calibration Period) in 1996 to 2001",
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     "execution_count": 6,
     "metadata": {},
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   ],
   "source": [
    "x = np.arange(n+1)\n",
    "A1 = comb(n,x)*beta_fn(alpha+x, beta+n-x)/B_alpha_beta*beta_fn(gamma, delta+n)/B_gamma_delta \n",
    "i = np.arange(n).reshape(-1,1)\n",
    "A2 = comb(i,x)*beta_fn(alpha+x, beta+i-x)/B_alpha_beta * beta_fn(gamma+1, delta+i)/B_gamma_delta\n",
    "P_X_n = A1 + np.sum(A2,axis=0)\n",
    "\n",
    "model_repeat_calib = pl.DataFrame({'Model': P_X_n * np.sum(rfm_array_calib[:,3])})\n",
    "\n",
    "actual_model_repeat_calib = (\n",
    "    rfm_summary_calib\n",
    "    .group_by('P1X')\n",
    "    .agg((pl.col('Count').sum()).alias('Actual'))\n",
    "    .sort('P1X')\n",
    "    .collect()\n",
    "    .hstack(model_repeat_calib)\n",
    "    .unpivot(on=['Actual', 'Model'], index='P1X', value_name='No of people', variable_name='Actual Vs Estimated')\n",
    ")\n",
    "\n",
    "(\n",
    "    alt.Chart(actual_model_repeat_calib).mark_bar()\n",
    "    .encode(x=alt.X('P1X:O', title='No. of repeat transactions', axis=alt.Axis(labelAngle=0)), \n",
    "         y=alt.Y('No of people:Q', title='No. of people'), \n",
    "         color='Actual Vs Estimated:N', \n",
    "         xOffset='Actual Vs Estimated')\n",
    "    .properties(\n",
    "        width=650,\n",
    "        height=250,\n",
    "        title='Predicted vs. Actual Frequency of Repeat Transactions (Calibration Period) in 1996 to 2001'\n",
    "    ).configure_view(stroke=None).configure_axisY(grid=False).configure_axisX(grid=False)   \n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calibration Period Model Fit Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       },
       "data": {
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       "datasets": {
        "data-f44c44e05e8d6e8abaa2c3ccb4aac799": [
         {
          "Actual Vs Estimated": "Actual",
          "No of people": 6746,
          "P2X": 0
         },
         {
          "Actual Vs Estimated": "Actual",
          "No of people": 926,
          "P2X": 1
         },
         {
          "Actual Vs Estimated": "Actual",
          "No of people": 772,
          "P2X": 2
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         {
          "Actual Vs Estimated": "Actual",
          "No of people": 812,
          "P2X": 3
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         {
          "Actual Vs Estimated": "Actual",
          "No of people": 886,
          "P2X": 4
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         {
          "Actual Vs Estimated": "Actual",
          "No of people": 962,
          "P2X": 5
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         {
          "Actual Vs Estimated": "Model",
          "No of people": 6964.680084067298,
          "P2X": 0
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         {
          "Actual Vs Estimated": "Model",
          "No of people": 793.9169831014856,
          "P2X": 1
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         {
          "Actual Vs Estimated": "Model",
          "No of people": 745.7096030251939,
          "P2X": 2
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         {
          "Actual Vs Estimated": "Model",
          "No of people": 749.4486922983122,
          "P2X": 3
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         {
          "Actual Vs Estimated": "Model",
          "No of people": 810.2922483979518,
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         {
          "Actual Vs Estimated": "Model",
          "No of people": 1039.9523891097565,
          "P2X": 5
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        ]
       },
       "encoding": {
        "color": {
         "field": "Actual Vs Estimated",
         "type": "nominal"
        },
        "x": {
         "axis": {
          "labelAngle": 0
         },
         "field": "P2X",
         "title": "No. of repeat transactions",
         "type": "ordinal"
        },
        "xOffset": {
         "field": "Actual Vs Estimated",
         "type": "nominal"
        },
        "y": {
         "field": "No of people",
         "title": "No. of people",
         "type": "quantitative"
        }
       },
       "height": 250,
       "mark": {
        "type": "bar"
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       "title": "Predicted vs. Actual Frequency of Repeat Transactions (Validation Period) in 2002-2006",
       "width": 650
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       "\n",
       "If you see this message, it means the renderer has not been properly enabled\n",
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     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_star = 5\n",
    "x_star = np.arange(n_star+1)\n",
    "\n",
    "A1 = comb(n_star,x_star)*beta_fn(alpha+x_star, beta+n_star-x_star)/B_alpha_beta * beta_fn(gamma, delta+n+n_star)/B_gamma_delta\n",
    "A1 += np.where(x_star == 0, 1 - beta_fn(gamma, delta+n)/B_gamma_delta, 0)\n",
    "i = np.arange(n_star).reshape(-1,1)\n",
    "A2 = comb(i,x_star)*beta_fn(alpha+x_star, beta+i-x_star)/B_alpha_beta * beta_fn(gamma+1, delta+n+i)/B_gamma_delta\n",
    "P_X_n_star = A1 + np.sum(A2, axis=0)\n",
    "\n",
    "valid_repeat_count = rfm_summary_valid.collect().to_numpy()[:,2]\n",
    "model_repeat_valid = pl.DataFrame({'Model': P_X_n_star * np.sum(valid_repeat_count)})\n",
    "\n",
    "actual_model_repeat_valid = (\n",
    "    rfm_summary_valid\n",
    "    .group_by('P2X')\n",
    "    .agg((pl.col('Count').sum()).alias('Actual'))\n",
    "    .sort('P2X')\n",
    "    .collect()\n",
    "    .hstack(model_repeat_valid)\n",
    "    .unpivot(on=['Actual', 'Model'], index='P2X', value_name='No of people', variable_name='Actual Vs Estimated')\n",
    ")\n",
    "\n",
    "(\n",
    "    alt.Chart(actual_model_repeat_valid).mark_bar()\n",
    "    .encode(x=alt.X('P2X:O', title='No. of repeat transactions', axis=alt.Axis(labelAngle=0)), \n",
    "         y=alt.Y('No of people:Q', title='No. of people'), \n",
    "         color='Actual Vs Estimated:N', \n",
    "         xOffset='Actual Vs Estimated')\n",
    "    .properties(\n",
    "        width=650,\n",
    "        height=250,\n",
    "        title='Predicted vs. Actual Frequency of Repeat Transactions (Validation Period) in 2002-2006'\n",
    "    ).configure_view(stroke=None).configure_axisY(grid=False).configure_axisX(grid=False)   \n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tracking Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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          "Actual Vs Model": "Actual",
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          "Year": "1997"
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          "Actual Vs Model": "Actual",
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          "Actual Vs Model": "Actual",
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          "Year": "1996"
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          "Actual Vs Model": "Model",
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          "Year": "1997"
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         {
          "Actual Vs Model": "Model",
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          "Actual Vs Model": "Model",
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         {
          "Actual Vs Model": "Model",
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          "Actual Vs Model": "Model",
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          "Year": "2003"
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         {
          "Actual Vs Model": "Model",
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          "Year": "2004"
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         {
          "Actual Vs Model": "Model",
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          "Year": "2005"
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        "x": {
         "axis": {
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         "field": "Year",
         "type": "nominal"
        },
        "y": {
         "field": "Repeat Trans",
         "title": "No. of repeat transactions",
         "type": "quantitative"
        }
       },
       "height": 250,
       "mark": {
        "type": "line"
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       "title": "Predicted vs. Actual Annual Repeat Transactions",
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       "<VegaLite 5 object>\n",
       "\n",
       "If you see this message, it means the renderer has not been properly enabled\n",
       "for the frontend that you are using. For more information, see\n",
       "https://altair-viz.github.io/user_guide/display_frontends.html#troubleshooting\n"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "act_yearly_repeat = data.data.select(pl.col('*').exclude('ID', '1995')).sum().collect().to_numpy()\n",
    "act_cum_repeat = act_yearly_repeat.cumsum()\n",
    "\n",
    "A1 = alpha / (alpha + beta)\n",
    "A2 = 1/(gamma-1)\n",
    "n_trans = np.arange(1, len(years))\n",
    "A3 = np.exp(gammaln(delta+n_trans+1)+gammaln(gamma+delta)-gammaln(delta)-gammaln(gamma+delta+n_trans))\n",
    "E_X_n = A1 * (delta*A2-A2*A3)\n",
    "\n",
    "est_cum_repeat = np.sum(rfm_array_calib[:,3]) * E_X_n\n",
    "est_yearly_repeat = np.diff(est_cum_repeat, prepend=0)\n",
    "\n",
    "yearly_repeat = pl.DataFrame({'Year': years[1:], 'Actual': act_yearly_repeat.flatten(), 'Model': est_yearly_repeat.flatten()})\n",
    "yearly_repeat = yearly_repeat.unpivot(on=['Actual', 'Model'], index='Year', variable_name='Actual Vs Model', value_name='Repeat Trans')\n",
    "cum_repeat = pl.DataFrame({'Year': years[1:], 'Actual': act_cum_repeat.flatten(), 'Model': est_cum_repeat.flatten()})\n",
    "cum_repeat = cum_repeat.unpivot(on=['Actual', 'Model'], index='Year', variable_name='Actual Vs Model', value_name='Repeat Trans')\n",
    "\n",
    "(\n",
    "    alt.Chart(yearly_repeat).mark_line().encode(\n",
    "        x=alt.X('Year', axis=alt.Axis(labelAngle=0)),\n",
    "        y=alt.Y('Repeat Trans', title='No. of repeat transactions'),\n",
    "        strokeDash='Actual Vs Model'\n",
    "    ).properties(\n",
    "            width=650,\n",
    "            height=250,\n",
    "            title='Predicted vs. Actual Annual Repeat Transactions'\n",
    "    ).configure_view(stroke=None).configure_axisY(grid=False).configure_axisX(grid=False)  \n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.vegalite.v5+json": {
       "$schema": "https://vega.github.io/schema/vega-lite/v5.20.1.json",
       "config": {
        "axisX": {
         "grid": false
        },
        "axisY": {
         "grid": false
        },
        "view": {
         "continuousHeight": 300,
         "continuousWidth": 300,
         "stroke": null
        }
       },
       "data": {
        "name": "data-8a008725596d24418e8f47d884237772"
       },
       "datasets": {
        "data-8a008725596d24418e8f47d884237772": [
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 5652,
          "Year": "1996"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 4674,
          "Year": "1997"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 4019,
          "Year": "1998"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 3552,
          "Year": "1999"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 3555,
          "Year": "2000"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 3163,
          "Year": "2001"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 3110,
          "Year": "2002"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 2938,
          "Year": "2003"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 2703,
          "Year": "2004"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 2573,
          "Year": "2005"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 1936,
          "Year": "2006"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 5535.843383078184,
          "Year": "1996"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 4717.085277607378,
          "Year": "1997"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 4147.649177161176,
          "Year": "1998"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 3724.6914916715887,
          "Year": "1999"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 3395.911107604541,
          "Year": "2000"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 3131.665175319089,
          "Year": "2001"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 2913.792665903082,
          "Year": "2002"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 2730.4932740715703,
          "Year": "2003"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 2573.738392477888,
          "Year": "2004"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 2437.85930580736,
          "Year": "2005"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 2318.729566927206,
          "Year": "2006"
         }
        ]
       },
       "encoding": {
        "color": {
         "field": "Actual Vs Model",
         "type": "nominal"
        },
        "x": {
         "axis": {
          "labelAngle": 0
         },
         "field": "Year",
         "type": "nominal"
        },
        "xOffset": {
         "field": "Actual Vs Model",
         "type": "nominal"
        },
        "y": {
         "field": "Repeat Trans",
         "title": "No. of repeat transactions",
         "type": "quantitative"
        }
       },
       "height": 250,
       "mark": {
        "type": "bar"
       },
       "title": "Predicted vs. Actual Annual Repeat Transactions",
       "width": 650
      },
      "text/plain": [
       "<VegaLite 5 object>\n",
       "\n",
       "If you see this message, it means the renderer has not been properly enabled\n",
       "for the frontend that you are using. For more information, see\n",
       "https://altair-viz.github.io/user_guide/display_frontends.html#troubleshooting\n"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(\n",
    "    alt.Chart(yearly_repeat).mark_bar().encode(\n",
    "        x=alt.X('Year', axis=alt.Axis(labelAngle=0)),\n",
    "        y=alt.Y('Repeat Trans', title='No. of repeat transactions'),\n",
    "        color='Actual Vs Model',\n",
    "        xOffset='Actual Vs Model'\n",
    "    ).properties(\n",
    "            width=650,\n",
    "            height=250,\n",
    "            title='Predicted vs. Actual Annual Repeat Transactions'\n",
    "    ).configure_view(stroke=None).configure_axisY(grid=False).configure_axisX(grid=False)  \n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.vegalite.v5+json": {
       "$schema": "https://vega.github.io/schema/vega-lite/v5.20.1.json",
       "config": {
        "axisX": {
         "grid": false
        },
        "axisY": {
         "grid": false
        },
        "view": {
         "continuousHeight": 300,
         "continuousWidth": 300,
         "stroke": null
        }
       },
       "data": {
        "name": "data-e72d163db56a537ebb7e68e622e62c12"
       },
       "datasets": {
        "data-e72d163db56a537ebb7e68e622e62c12": [
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 5652,
          "Year": "1996"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 10326,
          "Year": "1997"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 14345,
          "Year": "1998"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 17897,
          "Year": "1999"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 21452,
          "Year": "2000"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 24615,
          "Year": "2001"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 27725,
          "Year": "2002"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 30663,
          "Year": "2003"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 33366,
          "Year": "2004"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 35939,
          "Year": "2005"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 37875,
          "Year": "2006"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 5535.843383078184,
          "Year": "1996"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 10252.928660685562,
          "Year": "1997"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 14400.577837846738,
          "Year": "1998"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 18125.269329518327,
          "Year": "1999"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 21521.180437122868,
          "Year": "2000"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 24652.845612441957,
          "Year": "2001"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 27566.63827834504,
          "Year": "2002"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 30297.13155241661,
          "Year": "2003"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 32870.8699448945,
          "Year": "2004"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 35308.72925070186,
          "Year": "2005"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 37627.45881762906,
          "Year": "2006"
         }
        ]
       },
       "encoding": {
        "strokeDash": {
         "field": "Actual Vs Model",
         "type": "nominal"
        },
        "x": {
         "axis": {
          "labelAngle": 0
         },
         "field": "Year",
         "type": "nominal"
        },
        "y": {
         "field": "Repeat Trans",
         "title": "Cumulative no. of repeat transactions",
         "type": "quantitative"
        }
       },
       "height": 250,
       "mark": {
        "type": "line"
       },
       "title": "Predicted vs. Actual Cumulative Repeat Transactions",
       "width": 650
      },
      "text/plain": [
       "<VegaLite 5 object>\n",
       "\n",
       "If you see this message, it means the renderer has not been properly enabled\n",
       "for the frontend that you are using. For more information, see\n",
       "https://altair-viz.github.io/user_guide/display_frontends.html#troubleshooting\n"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(\n",
    "    alt.Chart(cum_repeat).mark_line().encode(\n",
    "        x=alt.X('Year', axis=alt.Axis(labelAngle=0)),\n",
    "        y=alt.Y('Repeat Trans', title='Cumulative no. of repeat transactions'),\n",
    "        strokeDash='Actual Vs Model'\n",
    "    ).properties(\n",
    "            width=650,\n",
    "            height=250,\n",
    "            title='Predicted vs. Actual Cumulative Repeat Transactions'\n",
    "    ).configure_view(stroke=None).configure_axisY(grid=False).configure_axisX(grid=False)  \n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.vegalite.v5+json": {
       "$schema": "https://vega.github.io/schema/vega-lite/v5.20.1.json",
       "config": {
        "axisX": {
         "grid": false
        },
        "axisY": {
         "grid": false
        },
        "view": {
         "continuousHeight": 300,
         "continuousWidth": 300,
         "stroke": null
        }
       },
       "data": {
        "name": "data-e72d163db56a537ebb7e68e622e62c12"
       },
       "datasets": {
        "data-e72d163db56a537ebb7e68e622e62c12": [
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 5652,
          "Year": "1996"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 10326,
          "Year": "1997"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 14345,
          "Year": "1998"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 17897,
          "Year": "1999"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 21452,
          "Year": "2000"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 24615,
          "Year": "2001"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 27725,
          "Year": "2002"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 30663,
          "Year": "2003"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 33366,
          "Year": "2004"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 35939,
          "Year": "2005"
         },
         {
          "Actual Vs Model": "Actual",
          "Repeat Trans": 37875,
          "Year": "2006"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 5535.843383078184,
          "Year": "1996"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 10252.928660685562,
          "Year": "1997"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 14400.577837846738,
          "Year": "1998"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 18125.269329518327,
          "Year": "1999"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 21521.180437122868,
          "Year": "2000"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 24652.845612441957,
          "Year": "2001"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 27566.63827834504,
          "Year": "2002"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 30297.13155241661,
          "Year": "2003"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 32870.8699448945,
          "Year": "2004"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 35308.72925070186,
          "Year": "2005"
         },
         {
          "Actual Vs Model": "Model",
          "Repeat Trans": 37627.45881762906,
          "Year": "2006"
         }
        ]
       },
       "encoding": {
        "color": {
         "field": "Actual Vs Model",
         "type": "nominal"
        },
        "x": {
         "axis": {
          "labelAngle": 0
         },
         "field": "Year",
         "type": "nominal"
        },
        "xOffset": {
         "field": "Actual Vs Model",
         "type": "nominal"
        },
        "y": {
         "field": "Repeat Trans",
         "title": "Cumulative no. of repeat transactions",
         "type": "quantitative"
        }
       },
       "height": 250,
       "mark": {
        "type": "bar"
       },
       "title": "Predicted vs. Actual Cumulative Repeat Transactions",
       "width": 650
      },
      "text/plain": [
       "<VegaLite 5 object>\n",
       "\n",
       "If you see this message, it means the renderer has not been properly enabled\n",
       "for the frontend that you are using. For more information, see\n",
       "https://altair-viz.github.io/user_guide/display_frontends.html#troubleshooting\n"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(\n",
    "    alt.Chart(cum_repeat).mark_bar().encode(\n",
    "        x=alt.X('Year', axis=alt.Axis(labelAngle=0)),\n",
    "        y=alt.Y('Repeat Trans', title='Cumulative no. of repeat transactions'),\n",
    "        color='Actual Vs Model',\n",
    "        xOffset='Actual Vs Model'\n",
    "    ).properties(\n",
    "            width=650,\n",
    "            height=250,\n",
    "            title='Predicted vs. Actual Cumulative Repeat Transactions'\n",
    "    ).configure_view(stroke=None).configure_axisY(grid=False).configure_axisX(grid=False)  \n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Conditional Expectations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_star = 5\n",
    "\n",
    "A2 = beta_fn(alpha+p1x+1, beta+n-p1x)/B_alpha_beta\n",
    "A3 = delta/(gamma-1)*np.exp(gammaln(gamma+delta)-gammaln(delta+1))*(np.exp(gammaln(1+delta+n)-\\\n",
    "     gammaln(gamma+delta+n))-np.exp(gammaln(1+delta+n+n_star)-gammaln(gamma+delta+n+n_star)))\n",
    "\n",
    "ce = A2 * A3 / L\n",
    "\n",
    "exp_total = ce * num_donors\n",
    "\n",
    "ce_df = (\n",
    "    data.rfm_data()\n",
    "    .group_by('P1X', 't_x', 'np1x')\n",
    "    .agg(pl.col('P2X').sum().alias('Actual Total - Valid'))\n",
    "    .sort(['t_x', 'P1X'], descending=True)\n",
    "    .collect()\n",
    "    .hstack([pl.Series('Exp Total', exp_total)])\n",
    "    .hstack([pl.Series('Conditional Expectation', ce)])\n",
    ")\n",
    "\n",
    "# Actual total 2002-2006 donations by p1x / tx\n",
    "actual_ce_mat = ce_df.sort('t_x').pivot(index='P1X', on='t_x', values='Actual Total - Valid').sort('P1X').fill_null(0).to_numpy()\n",
    "p1x_frequency = actual_ce_mat[:,0]\n",
    "actual_ce_mat = actual_ce_mat[:,1:]\n",
    "\n",
    "# Expected total 2002-2006 donations by p1x / tx\n",
    "est_ce_mat = ce_df.sort('t_x').pivot(index='P1X', on='t_x', values='Exp Total').sort('P1X').fill_null(0).to_numpy()[:,1:]\n",
    "\n",
    "# Number of Donors\n",
    "num_donors_mat = rfm_summary_calib.collect().sort('t_x').pivot(index='P1X', on='t_x', values='Count').sort('P1X').fill_null(0).to_numpy()[:,1:]\n",
    "\n",
    "# CE by Frequency\n",
    "actual_ce_freq = np.sum(actual_ce_mat, axis=1) / np.sum(num_donors_mat, axis=1)\n",
    "est_ce_freq = np.sum(est_ce_mat, axis=1) / np.sum(num_donors_mat, axis=1)\n",
    "ce_freq = pl.DataFrame({'x': p1x_frequency, 'Actual': actual_ce_freq, 'Model': est_ce_freq})\n",
    "ce_freq = ce_freq.unpivot(index='x', on=['Actual', 'Model'], variable_name='Actual Vs Model', value_name='CE by Freq')\n",
    "\n",
    "# CE by Recency\n",
    "actual_ce_rec = np.sum(actual_ce_mat, axis=0) / np.sum(num_donors_mat, axis=0)\n",
    "est_ce_rec = np.sum(est_ce_mat, axis=0) / np.sum(num_donors_mat, axis=0)\n",
    "ce_rec = pl.DataFrame({'t_x': years[:len(p1x_frequency)], 'Actual': actual_ce_rec, 'Model': est_ce_rec}).with_columns(pl.col('t_x').cast(pl.Int16))\n",
    "ce_rec = ce_rec.unpivot(index='t_x', on=['Actual', 'Model'], variable_name='Actual Vs Model', value_name='CE by Rec')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.vegalite.v5+json": {
       "$schema": "https://vega.github.io/schema/vega-lite/v5.20.1.json",
       "config": {
        "axisX": {
         "grid": false
        },
        "axisY": {
         "grid": false
        },
        "view": {
         "continuousHeight": 300,
         "continuousWidth": 300,
         "stroke": null
        }
       },
       "data": {
        "name": "data-e4a788215c7f13c15c587a146b87b567"
       },
       "datasets": {
        "data-e4a788215c7f13c15c587a146b87b567": [
         {
          "Actual Vs Model": "Actual",
          "CE by Freq": 0.17436489607390301,
          "x": 0
         },
         {
          "Actual Vs Model": "Actual",
          "CE by Freq": 0.4328030718595721,
          "x": 1
         },
         {
          "Actual Vs Model": "Actual",
          "CE by Freq": 0.8125874125874126,
          "x": 2
         },
         {
          "Actual Vs Model": "Actual",
          "CE by Freq": 1.3898617511520737,
          "x": 3
         },
         {
          "Actual Vs Model": "Actual",
          "CE by Freq": 2.0617760617760617,
          "x": 4
         },
         {
          "Actual Vs Model": "Actual",
          "CE by Freq": 2.642521166509878,
          "x": 5
         },
         {
          "Actual Vs Model": "Actual",
          "CE by Freq": 3.5344970906068163,
          "x": 6
         },
         {
          "Actual Vs Model": "Model",
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   "metadata": {},
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       "    <td colspan=\"8\" class=\"gt_heading gt_title gt_font_normal\">Expected Number of Repeat Transactions in 2002–2006</td>\n",
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       "    <td style=\"color: #000000; background-color: #d7cae5;\" class=\"gt_row gt_right\">1.03</td>\n",
       "    <td style=\"color: #000000; background-color: #baa3d1;\" class=\"gt_row gt_right\">1.80</td>\n",
       "    <td style=\"color: #000000; background-color: #ab8fc7;\" class=\"gt_row gt_right\">2.19</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #FFFFFF; background-color: #663399;\" class=\"gt_row gt_left gt_stub\">4</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #e9e1f0;\" class=\"gt_row gt_right\">0.58</td>\n",
       "    <td style=\"color: #000000; background-color: #b197cb;\" class=\"gt_row gt_right\">2.03</td>\n",
       "    <td style=\"color: #000000; background-color: #9775ba;\" class=\"gt_row gt_right\">2.71</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">5</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #baa3d1;\" class=\"gt_row gt_right\">1.81</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #835aad;\" class=\"gt_row gt_right\">3.23</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">6</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #6f409f;\" class=\"gt_row gt_right\">3.75</td>\n",
       "  </tr>\n",
       "</tbody>\n",
       "\n",
       "\n",
       "</table>\n",
       "\n",
       "</div>\n",
       "        "
      ],
      "text/plain": [
       "GT(_tbl_data=shape: (7, 8)\n",
       "┌─────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┐\n",
       "│ P1X ┆ 1995     ┆ 1996     ┆ 1997     ┆ 1998     ┆ 1999     ┆ 2000     ┆ 2001     │\n",
       "│ --- ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      │\n",
       "│ i8  ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      │\n",
       "╞═════╪══════════╪══════════╪══════════╪══════════╪══════════╪══════════╪══════════╡\n",
       "│ 0   ┆ 0.072867 ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     │\n",
       "│ 1   ┆ null     ┆ 0.085696 ┆ 0.314228 ┆ 0.593827 ┆ 0.83936  ┆ 1.02165  ┆ 1.147848 │\n",
       "│ 2   ┆ null     ┆ null     ┆ 0.119117 ┆ 0.536089 ┆ 1.057563 ┆ 1.442999 ┆ 1.668778 │\n",
       "│ 3   ┆ null     ┆ null     ┆ null     ┆ 0.223582 ┆ 1.034528 ┆ 1.804653 ┆ 2.189709 │\n",
       "│ 4   ┆ null     ┆ null     ┆ null     ┆ null     ┆ 0.583154 ┆ 2.029963 ┆ 2.710639 │\n",
       "│ 5   ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ 1.81286  ┆ 3.231569 │\n",
       "│ 6   ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ 3.7525   │\n",
       "└─────┴──────────┴──────────┴──────────┴──────────┴──────────┴──────────┴──────────┘, _body=<great_tables._gt_data.Body object at 0x1191b8050>, _boxhead=Boxhead([ColInfo(var='P1X', type=<ColInfoTypeEnum.stub: 2>, column_label='P1X', column_align='right', column_width=None), ColInfo(var='1995', type=<ColInfoTypeEnum.default: 1>, column_label='1995', column_align='right', column_width=None), ColInfo(var='1996', type=<ColInfoTypeEnum.default: 1>, column_label='1996', column_align='right', column_width=None), ColInfo(var='1997', type=<ColInfoTypeEnum.default: 1>, column_label='1997', column_align='right', column_width=None), ColInfo(var='1998', type=<ColInfoTypeEnum.default: 1>, column_label='1998', column_align='right', column_width=None), ColInfo(var='1999', type=<ColInfoTypeEnum.default: 1>, column_label='1999', column_align='right', column_width=None), ColInfo(var='2000', type=<ColInfoTypeEnum.default: 1>, column_label='2000', column_align='right', column_width=None), ColInfo(var='2001', type=<ColInfoTypeEnum.default: 1>, column_label='2001', column_align='right', column_width=None)]), _stub=<great_tables._gt_data.Stub object at 0x1191b0550>, _spanners=Spanners([SpannerInfo(spanner_id='Year of last transaction', spanner_level=0, spanner_label='Year of last transaction', spanner_units=None, spanner_pattern=None, vars=['1995', '1996', '1997', '1998', '1999', '2000', '2001'], built=None)]), _heading=Heading(title='Expected Number of Repeat Transactions in 2002–2006', subtitle='as a Function of Recency and Frequency', preheader=None), _stubhead='P1X', _source_notes=[], _footnotes=[], _styles=[StyleInfo(locname=LocBody(columns='P1X', rows=[0]), grpname=None, colname='P1X', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#ffffff')]), StyleInfo(locname=LocBody(columns='P1X', rows=[1]), 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styles=[CellStyleText(color='#FFFFFF', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#663399')]), StyleInfo(locname=LocBody(columns='P1X', rows=[5]), grpname=None, colname='P1X', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='P1X', rows=[6]), grpname=None, colname='P1X', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1995', rows=[0]), grpname=None, colname='1995', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#fcfbfd')]), StyleInfo(locname=LocBody(columns='1995', rows=[1]), grpname=None, colname='1995', rownum=1, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1995', rows=[2]), grpname=None, colname='1995', rownum=2, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1995', rows=[3]), grpname=None, colname='1995', rownum=3, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1995', rows=[4]), grpname=None, colname='1995', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1995', rows=[5]), grpname=None, colname='1995', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1995', rows=[6]), grpname=None, colname='1995', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1996', rows=[0]), grpname=None, colname='1996', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1996', rows=[1]), grpname=None, colname='1996', rownum=1, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#fcfbfd')]), StyleInfo(locname=LocBody(columns='1996', rows=[2]), grpname=None, colname='1996', rownum=2, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1996', rows=[3]), grpname=None, colname='1996', rownum=3, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1996', rows=[4]), grpname=None, colname='1996', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1996', rows=[5]), grpname=None, colname='1996', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1996', rows=[6]), grpname=None, colname='1996', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1997', rows=[0]), grpname=None, colname='1997', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1997', rows=[1]), grpname=None, colname='1997', rownum=1, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#f3eff7')]), StyleInfo(locname=LocBody(columns='1997', rows=[2]), grpname=None, colname='1997', rownum=2, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#faf9fc')]), StyleInfo(locname=LocBody(columns='1997', rows=[3]), grpname=None, colname='1997', rownum=3, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1997', rows=[4]), grpname=None, colname='1997', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1997', rows=[5]), grpname=None, colname='1997', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1997', rows=[6]), grpname=None, colname='1997', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1998', rows=[0]), grpname=None, colname='1998', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1998', rows=[1]), grpname=None, colname='1998', rownum=1, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#e8e1f0')]), StyleInfo(locname=LocBody(columns='1998', rows=[2]), grpname=None, colname='1998', rownum=2, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#eae4f1')]), StyleInfo(locname=LocBody(columns='1998', rows=[3]), grpname=None, colname='1998', rownum=3, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#f6f4f9')]), StyleInfo(locname=LocBody(columns='1998', rows=[4]), grpname=None, colname='1998', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1998', rows=[5]), grpname=None, colname='1998', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1998', rows=[6]), grpname=None, colname='1998', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1999', rows=[0]), grpname=None, colname='1999', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1999', rows=[1]), grpname=None, colname='1999', rownum=1, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#dfd4ea')]), StyleInfo(locname=LocBody(columns='1999', rows=[2]), grpname=None, colname='1999', rownum=2, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#d7c9e4')]), StyleInfo(locname=LocBody(columns='1999', rows=[3]), grpname=None, colname='1999', rownum=3, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#d7cae5')]), StyleInfo(locname=LocBody(columns='1999', rows=[4]), grpname=None, colname='1999', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#e9e1f0')]), StyleInfo(locname=LocBody(columns='1999', rows=[5]), grpname=None, colname='1999', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1999', rows=[6]), grpname=None, colname='1999', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='2000', rows=[0]), grpname=None, colname='2000', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='2000', rows=[1]), grpname=None, colname='2000', rownum=1, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#d8cbe5')]), StyleInfo(locname=LocBody(columns='2000', rows=[2]), grpname=None, colname='2000', rownum=2, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#c8b5da')]), StyleInfo(locname=LocBody(columns='2000', rows=[3]), 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      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rfcalib_cross_tab(ce_df, \n",
    "                  values='Conditional Expectation',\n",
    "                  title='Expected Number of Repeat Transactions in 2002–2006',\n",
    "                  subtitle='as a Function of Recency and Frequency',\n",
    "                  color_range=[0,4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### P(Alive) as a Function of Recency and Frequency"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       " #sxblvtuzzo .gt_row_group_first td { border-top-width: 2px !important; }\n",
       " #sxblvtuzzo .gt_row_group_first th { border-top-width: 2px !important; }\n",
       " #sxblvtuzzo .gt_striped { background-color: rgba(128,128,128,0.05) !important; }\n",
       " #sxblvtuzzo .gt_table_body { border-top-style: solid !important; border-top-width: 2px !important; border-top-color: #D3D3D3 !important; border-bottom-style: solid !important; border-bottom-width: 2px !important; border-bottom-color: #D3D3D3 !important; }\n",
       " #sxblvtuzzo .gt_sourcenotes { color: #333333 !important; background-color: #FFFFFF !important; border-bottom-style: none !important; border-bottom-width: 2px !important; border-bottom-color: #D3D3D3 !important; border-left-style: none !important; border-left-width: 2px !important; border-left-color: #D3D3D3 !important; border-right-style: none !important; border-right-width: 2px !important; border-right-color: #D3D3D3 !important; }\n",
       " #sxblvtuzzo .gt_sourcenote { font-size: 90% !important; padding-top: 4px !important; padding-bottom: 4px !important; padding-left: 5px !important; padding-right: 5px !important; text-align: left !important; }\n",
       " #sxblvtuzzo .gt_left { text-align: left !important; }\n",
       " #sxblvtuzzo .gt_center { text-align: center !important; }\n",
       " #sxblvtuzzo .gt_right { text-align: right !important; font-variant-numeric: tabular-nums !important; }\n",
       " #sxblvtuzzo .gt_font_normal { font-weight: normal !important; }\n",
       " #sxblvtuzzo .gt_font_bold { font-weight: bold !important; }\n",
       " #sxblvtuzzo .gt_font_italic { font-style: italic !important; }\n",
       " #sxblvtuzzo .gt_super { font-size: 65% !important; }\n",
       " #sxblvtuzzo .gt_footnote_marks { font-size: 75% !important; vertical-align: 0.4em !important; position: initial !important; }\n",
       " #sxblvtuzzo .gt_asterisk { font-size: 100% !important; vertical-align: 0 !important; }\n",
       " \n",
       "</style>\n",
       "<table class=\"gt_table\" data-quarto-disable-processing=\"false\" data-quarto-bootstrap=\"false\">\n",
       "<thead>\n",
       "\n",
       "  <tr class=\"gt_heading\">\n",
       "    <td colspan=\"8\" class=\"gt_heading gt_title gt_font_normal\">P(Alive in 2002) as a Function of Recency and Frequency</td>\n",
       "  </tr>\n",
       "<tr class=\"gt_col_headings gt_spanner_row\">\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_left\" rowspan=\"2\" colspan=\"1\" scope=\"col\" id=\"P1X\">P1X</th>\n",
       "  <th class=\"gt_center gt_columns_top_border gt_column_spanner_outer\" rowspan=\"1\" colspan=\"7\" scope=\"colgroup\" id=\"Year of last transaction\">\n",
       "    <span class=\"gt_column_spanner\">Year of last transaction</span>\n",
       "  </th>\n",
       "</tr>\n",
       "<tr class=\"gt_col_headings\">\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1995\">1995</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1996\">1996</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1997\">1997</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1998\">1998</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1999\">1999</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"2000\">2000</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"2001\">2001</th>\n",
       "</tr>\n",
       "</thead>\n",
       "<tbody class=\"gt_table_body\">\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #ffffff;\" class=\"gt_row gt_left gt_stub\">0</th>\n",
       "    <td style=\"color: #000000; background-color: #eee9f4;\" class=\"gt_row gt_right\">0.11</td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #FFFFFF; background-color: #663399;\" class=\"gt_row gt_left gt_stub\">1</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #f4f1f8;\" class=\"gt_row gt_right\">0.07</td>\n",
       "    <td style=\"color: #000000; background-color: #d8cbe5;\" class=\"gt_row gt_right\">0.25</td>\n",
       "    <td style=\"color: #000000; background-color: #b59dce;\" class=\"gt_row gt_right\">0.48</td>\n",
       "    <td style=\"color: #000000; background-color: #9774ba;\" class=\"gt_row gt_right\">0.68</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #8056ab;\" class=\"gt_row gt_right\">0.83</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #7141a0;\" class=\"gt_row gt_right\">0.93</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">2</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #f5f1f8;\" class=\"gt_row gt_right\">0.07</td>\n",
       "    <td style=\"color: #000000; background-color: #d1c2e1;\" class=\"gt_row gt_right\">0.30</td>\n",
       "    <td style=\"color: #000000; background-color: #a587c3;\" class=\"gt_row gt_right\">0.59</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #845bad;\" class=\"gt_row gt_right\">0.80</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #7141a0;\" class=\"gt_row gt_right\">0.93</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">3</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #f0ecf5;\" class=\"gt_row gt_right\">0.10</td>\n",
       "    <td style=\"color: #000000; background-color: #bca5d2;\" class=\"gt_row gt_right\">0.44</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #8a63b1;\" class=\"gt_row gt_right\">0.77</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #7141a0;\" class=\"gt_row gt_right\">0.93</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">4</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #e0d6eb;\" class=\"gt_row gt_right\">0.20</td>\n",
       "    <td style=\"color: #000000; background-color: #9471b8;\" class=\"gt_row gt_right\">0.70</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #7141a0;\" class=\"gt_row gt_right\">0.93</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">5</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #af95ca;\" class=\"gt_row gt_right\">0.52</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #7141a0;\" class=\"gt_row gt_right\">0.93</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">6</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #7141a0;\" class=\"gt_row gt_right\">0.93</td>\n",
       "  </tr>\n",
       "</tbody>\n",
       "\n",
       "\n",
       "</table>\n",
       "\n",
       "</div>\n",
       "        "
      ],
      "text/plain": [
       "GT(_tbl_data=shape: (7, 8)\n",
       "┌─────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┐\n",
       "│ P1X ┆ 1995     ┆ 1996     ┆ 1997     ┆ 1998     ┆ 1999     ┆ 2000     ┆ 2001     │\n",
       "│ --- ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      │\n",
       "│ i8  ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      │\n",
       "╞═════╪══════════╪══════════╪══════════╪══════════╪══════════╪══════════╪══════════╡\n",
       "│ 0   ┆ 0.108144 ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     │\n",
       "│ 1   ┆ null     ┆ 0.069464 ┆ 0.254709 ┆ 0.481348 ┆ 0.680373 ┆ 0.828135 ┆ 0.930429 │\n",
       "│ 2   ┆ null     ┆ null     ┆ 0.066414 ┆ 0.298897 ┆ 0.589645 ┆ 0.804546 ┆ 0.930429 │\n",
       "│ 3   ┆ null     ┆ null     ┆ null     ┆ 0.095002 ┆ 0.439581 ┆ 0.766815 ┆ 0.930429 │\n",
       "│ 4   ┆ null     ┆ null     ┆ null     ┆ null     ┆ 0.200168 ┆ 0.696787 ┆ 0.930429 │\n",
       "│ 5   ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ 0.521956 ┆ 0.930429 │\n",
       "│ 6   ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ 0.930429 │\n",
       "└─────┴──────────┴──────────┴──────────┴──────────┴──────────┴──────────┴──────────┘, _body=<great_tables._gt_data.Body object at 0x1191acfc0>, _boxhead=Boxhead([ColInfo(var='P1X', type=<ColInfoTypeEnum.stub: 2>, column_label='P1X', column_align='right', column_width=None), ColInfo(var='1995', type=<ColInfoTypeEnum.default: 1>, column_label='1995', column_align='right', column_width=None), ColInfo(var='1996', type=<ColInfoTypeEnum.default: 1>, column_label='1996', column_align='right', column_width=None), ColInfo(var='1997', type=<ColInfoTypeEnum.default: 1>, column_label='1997', column_align='right', column_width=None), ColInfo(var='1998', type=<ColInfoTypeEnum.default: 1>, column_label='1998', column_align='right', column_width=None), ColInfo(var='1999', type=<ColInfoTypeEnum.default: 1>, column_label='1999', column_align='right', column_width=None), ColInfo(var='2000', type=<ColInfoTypeEnum.default: 1>, column_label='2000', column_align='right', column_width=None), ColInfo(var='2001', type=<ColInfoTypeEnum.default: 1>, column_label='2001', column_align='right', column_width=None)]), _stub=<great_tables._gt_data.Stub object at 0x1191ad220>, _spanners=Spanners([SpannerInfo(spanner_id='Year of last transaction', spanner_level=0, spanner_label='Year of last transaction', spanner_units=None, spanner_pattern=None, vars=['1995', '1996', '1997', '1998', '1999', '2000', '2001'], built=None)]), _heading=Heading(title='P(Alive in 2002) as a Function of Recency and Frequency', subtitle=None, preheader=None), _stubhead='P1X', _source_notes=[], _footnotes=[], _styles=[StyleInfo(locname=LocBody(columns='P1X', rows=[0]), grpname=None, colname='P1X', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#ffffff')]), StyleInfo(locname=LocBody(columns='P1X', rows=[1]), grpname=None, colname='P1X', 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weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#eee9f4')]), StyleInfo(locname=LocBody(columns='1995', rows=[1]), grpname=None, colname='1995', rownum=1, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1995', rows=[2]), grpname=None, colname='1995', rownum=2, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1995', rows=[3]), grpname=None, colname='1995', rownum=3, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, 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StyleInfo(locname=LocBody(columns='1996', rows=[3]), grpname=None, colname='1996', rownum=3, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1996', rows=[4]), grpname=None, colname='1996', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1996', rows=[5]), grpname=None, colname='1996', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1996', rows=[6]), grpname=None, colname='1996', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1997', rows=[0]), grpname=None, colname='1997', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1997', rows=[1]), grpname=None, colname='1997', rownum=1, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#d8cbe5')]), StyleInfo(locname=LocBody(columns='1997', rows=[2]), grpname=None, colname='1997', rownum=2, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#f5f1f8')]), StyleInfo(locname=LocBody(columns='1997', rows=[3]), grpname=None, colname='1997', rownum=3, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1997', rows=[4]), grpname=None, colname='1997', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1997', rows=[5]), grpname=None, colname='1997', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1997', rows=[6]), grpname=None, colname='1997', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1998', rows=[0]), grpname=None, colname='1998', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1998', rows=[1]), grpname=None, colname='1998', rownum=1, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#b59dce')]), StyleInfo(locname=LocBody(columns='1998', rows=[2]), grpname=None, colname='1998', rownum=2, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#d1c2e1')]), StyleInfo(locname=LocBody(columns='1998', rows=[3]), grpname=None, colname='1998', rownum=3, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#f0ecf5')]), StyleInfo(locname=LocBody(columns='1998', rows=[4]), grpname=None, colname='1998', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1998', rows=[5]), grpname=None, colname='1998', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1998', rows=[6]), grpname=None, colname='1998', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1999', rows=[0]), grpname=None, colname='1999', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1999', rows=[1]), grpname=None, colname='1999', rownum=1, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#9774ba')]), StyleInfo(locname=LocBody(columns='1999', rows=[2]), grpname=None, colname='1999', rownum=2, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#a587c3')]), StyleInfo(locname=LocBody(columns='1999', rows=[3]), grpname=None, colname='1999', rownum=3, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#bca5d2')]), StyleInfo(locname=LocBody(columns='1999', rows=[4]), grpname=None, colname='1999', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#e0d6eb')]), StyleInfo(locname=LocBody(columns='1999', rows=[5]), grpname=None, colname='1999', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1999', rows=[6]), grpname=None, colname='1999', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='2000', rows=[0]), grpname=None, colname='2000', rownum=0, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='2000', rows=[1]), grpname=None, colname='2000', rownum=1, colnum=None, styles=[CellStyleText(color='#FFFFFF', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#8056ab')]), StyleInfo(locname=LocBody(columns='2000', rows=[2]), grpname=None, colname='2000', rownum=2, colnum=None, styles=[CellStyleText(color='#FFFFFF', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#845bad')]), StyleInfo(locname=LocBody(columns='2000', rows=[3]), grpname=None, colname='2000', rownum=3, colnum=None, styles=[CellStyleText(color='#FFFFFF', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#8a63b1')]), StyleInfo(locname=LocBody(columns='2000', rows=[4]), grpname=None, colname='2000', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#9471b8')]), StyleInfo(locname=LocBody(columns='2000', rows=[5]), grpname=None, colname='2000', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#af95ca')]), StyleInfo(locname=LocBody(columns='2000', rows=[6]), grpname=None, colname='2000', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, 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      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A1 = np.exp(gammaln(alpha+p1x)+gammaln(beta+n-p1x)-gammaln(alpha+beta+n))/B_alpha_beta * \\\n",
    "                        np.exp(gammaln(gamma)+gammaln(delta+n+1)-gammaln(gamma+delta+n+1))/B_gamma_delta\n",
    "P_alive = A1 * L**-1\n",
    "\n",
    "rfcalib_cross_tab(rfm_summary_calib.collect().hstack(pl.DataFrame({'P(Alive)': P_alive})), \n",
    "                  values='P(Alive)',\n",
    "                  title='P(Alive in 2002) as a Function of Recency and Frequency',\n",
    "                  color_range=[0,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Posterior Mean of P as a Function of Recency and Frequency"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
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       "    <td colspan=\"8\" class=\"gt_heading gt_title gt_font_normal\">Posterior Mean of P as a Function of Recency and Frequency</td>\n",
       "  </tr>\n",
       "<tr class=\"gt_col_headings gt_spanner_row\">\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_left\" rowspan=\"2\" colspan=\"1\" scope=\"col\" id=\"P1X\">P1X</th>\n",
       "  <th class=\"gt_center gt_columns_top_border gt_column_spanner_outer\" rowspan=\"1\" colspan=\"7\" scope=\"colgroup\" id=\"Year of last transaction\">\n",
       "    <span class=\"gt_column_spanner\">Year of last transaction</span>\n",
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       "<tr class=\"gt_col_headings\">\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1995\">1995</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1996\">1996</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1997\">1997</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1998\">1998</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1999\">1999</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"2000\">2000</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"2001\">2001</th>\n",
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       "</thead>\n",
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       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">0</th>\n",
       "    <td style=\"color: #000000; background-color: #c8b6da;\" class=\"gt_row gt_right\">0.49</td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #FFFFFF; background-color: #663399;\" class=\"gt_row gt_left gt_stub\">1</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #a689c4;\" class=\"gt_row gt_right\">0.66</td>\n",
       "    <td style=\"color: #000000; background-color: #d1c1e0;\" class=\"gt_row gt_right\">0.44</td>\n",
       "    <td style=\"color: #000000; background-color: #e4dbed;\" class=\"gt_row gt_right\">0.34</td>\n",
       "    <td style=\"color: #000000; background-color: #ece6f3;\" class=\"gt_row gt_right\">0.30</td>\n",
       "    <td style=\"color: #000000; background-color: #efeaf5;\" class=\"gt_row gt_right\">0.28</td>\n",
       "    <td style=\"color: #000000; background-color: #f0ebf5;\" class=\"gt_row gt_right\">0.28</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">2</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #9672b9;\" class=\"gt_row gt_right\">0.75</td>\n",
       "    <td style=\"color: #000000; background-color: #bda7d3;\" class=\"gt_row gt_right\">0.54</td>\n",
       "    <td style=\"color: #000000; background-color: #d0c1e0;\" class=\"gt_row gt_right\">0.44</td>\n",
       "    <td style=\"color: #000000; background-color: #d7c9e4;\" class=\"gt_row gt_right\">0.41</td>\n",
       "    <td style=\"color: #000000; background-color: #d8cbe5;\" class=\"gt_row gt_right\">0.40</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">3</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #8d67b3;\" class=\"gt_row gt_right\">0.80</td>\n",
       "    <td style=\"color: #000000; background-color: #b096ca;\" class=\"gt_row gt_right\">0.61</td>\n",
       "    <td style=\"color: #000000; background-color: #bea8d3;\" class=\"gt_row gt_right\">0.54</td>\n",
       "    <td style=\"color: #000000; background-color: #c0abd5;\" class=\"gt_row gt_right\">0.53</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">4</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #8961b0;\" class=\"gt_row gt_right\">0.82</td>\n",
       "    <td style=\"color: #000000; background-color: #a485c2;\" class=\"gt_row gt_right\">0.68</td>\n",
       "    <td style=\"color: #000000; background-color: #a88bc5;\" class=\"gt_row gt_right\">0.65</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">5</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #875faf;\" class=\"gt_row gt_right\">0.83</td>\n",
       "    <td style=\"color: #000000; background-color: #906bb5;\" class=\"gt_row gt_right\">0.78</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">6</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #784ba5;\" class=\"gt_row gt_right\">0.91</td>\n",
       "  </tr>\n",
       "</tbody>\n",
       "\n",
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       "        "
      ],
      "text/plain": [
       "GT(_tbl_data=shape: (7, 8)\n",
       "┌─────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┐\n",
       "│ P1X ┆ 1995     ┆ 1996     ┆ 1997     ┆ 1998     ┆ 1999     ┆ 2000     ┆ 2001     │\n",
       "│ --- ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      │\n",
       "│ i8  ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      │\n",
       "╞═════╪══════════╪══════════╪══════════╪══════════╪══════════╪══════════╪══════════╡\n",
       "│ 0   ┆ 0.487723 ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     │\n",
       "│ 1   ┆ null     ┆ 0.663354 ┆ 0.442589 ┆ 0.340437 ┆ 0.297261 ┆ 0.281435 ┆ 0.277055 │\n",
       "│ 2   ┆ null     ┆ null     ┆ 0.751243 ┆ 0.544319 ┆ 0.44468  ┆ 0.410628 ┆ 0.402791 │\n",
       "│ 3   ┆ null     ┆ null     ┆ null     ┆ 0.797491 ┆ 0.611963 ┆ 0.541894 ┆ 0.528527 │\n",
       "│ 4   ┆ null     ┆ null     ┆ null     ┆ null     ┆ 0.817711 ┆ 0.677892 ┆ 0.654263 │\n",
       "│ 5   ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ 0.829248 ┆ 0.78     │\n",
       "│ 6   ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ 0.905736 │\n",
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    "l = 1\n",
    "m = 0\n",
    "alphal = alpha + l\n",
    "gammam = gamma + m\n",
    "\n",
    "B_alphal_beta = beta_fn(alphal, beta)\n",
    "B_gammam_delta = beta_fn(gammam, delta)\n",
    "\n",
    "A1 = beta_fn(alphal+p1x, beta+n-p1x)/B_alphal_beta * beta_fn(gammam, delta+n)/B_gammam_delta\n",
    "i = np.arange(6).reshape(-1,1)\n",
    "A2 = beta_fn(alphal+p1x, beta+t_x-p1x+i)/B_alphal_beta * beta_fn(gammam+1, delta+t_x+i)/B_gammam_delta  \n",
    "A2 = np.where(i <= (n - t_x - 1), A2, 0)\n",
    "L_lm = A1 + np.sum(A2, axis=0) \n",
    "\n",
    "E_P_Theta = (B_alphal_beta/B_alpha_beta) * (B_gammam_delta/B_gamma_delta) * (L_lm/L)\n",
    "\n",
    "rfcalib_cross_tab(rfm_summary_calib.collect().hstack(pl.DataFrame({'E_P_Theta': E_P_Theta})), \n",
    "                  values='E_P_Theta',\n",
    "                  title='Posterior Mean of P as a Function of Recency and Frequency',\n",
    "                  color_range=[0.2,1])"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Prior and Selected Posterior Distributions of (a) $P$ and (b) $\\Theta$"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A customer’s latent transaction and dropout probabilities."
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  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def marginal_posterior(x, tx, p_theta, theta=False):\n",
    "    i = np.arange(n).reshape(-1,1)\n",
    "    if not theta:\n",
    "        B1 = p_theta**(alpha+x-1)*(1-p_theta)**(beta+n-x-1)/B_alpha_beta * beta_fn(gamma, delta + n)/B_gamma_delta\n",
    "        B2 = p_theta**(alpha+x-1)*(1-p_theta)**(beta+tx-x+i-1)/B_alpha_beta * beta_fn(gamma+1, delta + tx + i)/B_gamma_delta\n",
    "    else:\n",
    "        B1 = beta_fn(alpha+x,beta+n-x)/B_alpha_beta * (p_theta**(gamma-1)*(1-p_theta)**(delta+n-1))/B_gamma_delta\n",
    "        B2 = beta_fn(alpha+x, beta+tx-x+i)/B_alpha_beta * (p_theta**gamma*(1-p_theta)**(delta+tx+i-1))/B_gamma_delta\n",
    "    B2 = np.where(i <= (n - tx - 1), B2, 0)\n",
    "    L = L_df.filter((pl.col('P1X') == x) & (pl.col('t_x') == tx)).select('Likelihood').item()\n",
    "    return (B1 + np.sum(B2, axis=0)) / L \n",
    "\n",
    "def compute_mean(x, tx, theta=False):\n",
    "    # Function for the marginal posterior distribution\n",
    "    def posterior_func(p_theta):\n",
    "        return p_theta * marginal_posterior(x, tx, p_theta, theta=theta)\n",
    "    \n",
    "    # Integrate over the range [0, 1] for p or theta\n",
    "    mean, _ = quad(posterior_func, 0, 1)\n",
    "    return mean"
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   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
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   "source": [
    "def annotate(ax, xy, xytext, str):\n",
    "    return ax.annotate(str,\n",
    "            xy=xy, xycoords='data',\n",
    "            xytext=xytext, textcoords='data',\n",
    "            arrowprops=dict(arrowstyle=\"->\", connectionstyle=\"arc3\"))\n",
    "        \n",
    "fig, ax = plt.subplots(figsize=(8, 5))\n",
    "p = np.linspace(beta_dist.ppf(0, alpha, beta),\n",
    "                beta_dist.ppf(0.99, alpha, beta), 100)\n",
    "ax.plot(p[1:-1], beta_dist.pdf(p[1:-1], alpha, beta), 'k', lw=1, alpha=0.6, label='Prior')\n",
    "ax.plot(p[1:], marginal_posterior(3,3, p[1:]), 'b--', lw=1, alpha=0.6, label='Posterior for $x = 3$, $t_{x} = 3$ (1998)')\n",
    "ax.plot(p, marginal_posterior(3,6, p), 'g--', lw=1, alpha=0.6, label='Posterior for $x = 3$, $t_{x} = 6$ (2001)')\n",
    "ax.set_xlabel('$p$')\n",
    "ax.set_ylabel('$f(p)$')\n",
    "ax.set_title('Prior and Selected Posterior Distributions of $P$')\n",
    "ax.legend()\n",
    "ax.set_xlim(0,1)\n",
    "ax.set_ylim(0,6)\n",
    "annotate(ax, (0.45, 2), (0.3, 2.5), f\"$E(P) = {compute_mean(3,6):.2f}$\")\n",
    "annotate(ax, (0.97, 4), (0.7, 5), f\"$E(P) = {compute_mean(3,3):.2f}$\")\n",
    "annotate(ax, (0.97, 2), (0.7, 3), f\"$E(P) = {beta_dist.stats(alpha, beta, moments='m'):.2f}$\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
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     "output_type": "execute_result"
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      "text/plain": [
       "<Figure size 800x500 with 1 Axes>"
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     "metadata": {},
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   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8, 5))\n",
    "p = np.linspace(beta_dist.ppf(0, alpha, beta),\n",
    "                beta_dist.ppf(0.99, alpha, beta), 100)\n",
    "ax.plot(p, beta_dist.pdf(p, gamma, delta), 'k', lw=1, alpha=0.6, label='Prior')\n",
    "ax.plot(p[1:], marginal_posterior(3,3, p[1:],theta=True), 'b--', lw=1, alpha=0.6, label='Posterior for $x = 3$, $t_{x} = 3$ (1998)')\n",
    "ax.plot(p[1:], marginal_posterior(3,6, p[1:],theta=True), 'g--', lw=1, alpha=0.6, label='Posterior for $x = 3$, $t_{x} = 6$ (2001)')\n",
    "ax.set_xlabel('$\\\\theta$')\n",
    "ax.set_ylabel('$f(\\\\theta)$')\n",
    "ax.set_title('Prior and Selected Posterior Distributions of $\\\\Theta$')\n",
    "ax.legend()\n",
    "ax.set_xlim(0,1)\n",
    "ax.set_ylim(0,14)\n",
    "annotate(ax, (0.02, 10), (0.1, 12), f\"$E(\\\\Theta) = {compute_mean(3,6,theta=True):.2f}$\")\n",
    "annotate(ax, (0.25, 2), (0.35, 3.5), f\"$E(\\\\Theta) = {compute_mean(3,3,theta=True):.2f}$\")\n",
    "annotate(ax, (0.015, 5), (0.1, 6.7), f\"$E(\\\\Theta) = {beta_dist.stats(gamma, delta, moments='m'):.2f}$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Conditional Penetration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Probability that a customer with purchase history ($x, t_{x}, n$) makes $x^{*}$ transactions in the interval $(n,n + n^{*}]$.\n",
    "\n",
    "The probability that a customer is active in the 2002–2006 period ($n^{*} = 5$) is computed as $1-P(X(n,n+n^{*})=0 \\mid x,t_{x}, n)$, where $x^{*}=0$, conditional on each of the 22 ($x,t_{x}$) patterns associated with $n = 6$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
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       " \n",
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       "<thead>\n",
       "\n",
       "  <tr class=\"gt_heading\">\n",
       "    <td colspan=\"8\" class=\"gt_heading gt_title gt_font_normal\">Probability of Being Active in 2002–2006</td>\n",
       "  </tr>\n",
       "  <tr class=\"gt_heading\">\n",
       "    <td colspan=\"8\" class=\"gt_heading gt_subtitle gt_font_normal gt_bottom_border\">as a Function of Recency and Frequency</td>\n",
       "  </tr>\n",
       "<tr class=\"gt_col_headings gt_spanner_row\">\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_left\" rowspan=\"2\" colspan=\"1\" scope=\"col\" id=\"P1X\">P1X</th>\n",
       "  <th class=\"gt_center gt_columns_top_border gt_column_spanner_outer\" rowspan=\"1\" colspan=\"7\" scope=\"colgroup\" id=\"Year of last transaction\">\n",
       "    <span class=\"gt_column_spanner\">Year of last transaction</span>\n",
       "  </th>\n",
       "</tr>\n",
       "<tr class=\"gt_col_headings\">\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1995\">1995</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1996\">1996</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1997\">1997</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1998\">1998</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"1999\">1999</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"2000\">2000</th>\n",
       "  <th class=\"gt_col_heading gt_columns_bottom_border gt_right\" rowspan=\"1\" colspan=\"1\" scope=\"col\" id=\"2001\">2001</th>\n",
       "</tr>\n",
       "</thead>\n",
       "<tbody class=\"gt_table_body\">\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #ffffff;\" class=\"gt_row gt_left gt_stub\">0</th>\n",
       "    <td style=\"color: #000000; background-color: #f8f5fa;\" class=\"gt_row gt_right\">0.05</td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #FFFFFF; background-color: #663399;\" class=\"gt_row gt_left gt_stub\">1</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #f8f5fa;\" class=\"gt_row gt_right\">0.05</td>\n",
       "    <td style=\"color: #000000; background-color: #e5dcee;\" class=\"gt_row gt_right\">0.17</td>\n",
       "    <td style=\"color: #000000; background-color: #cebdde;\" class=\"gt_row gt_right\">0.32</td>\n",
       "    <td style=\"color: #000000; background-color: #b9a2d0;\" class=\"gt_row gt_right\">0.46</td>\n",
       "    <td style=\"color: #000000; background-color: #aa8ec6;\" class=\"gt_row gt_right\">0.56</td>\n",
       "    <td style=\"color: #000000; background-color: #9f80bf;\" class=\"gt_row gt_right\">0.62</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">2</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #f7f4f9;\" class=\"gt_row gt_right\">0.05</td>\n",
       "    <td style=\"color: #000000; background-color: #dacde6;\" class=\"gt_row gt_right\">0.24</td>\n",
       "    <td style=\"color: #000000; background-color: #b59dce;\" class=\"gt_row gt_right\">0.48</td>\n",
       "    <td style=\"color: #000000; background-color: #9b79bc;\" class=\"gt_row gt_right\">0.66</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #8b64b2;\" class=\"gt_row gt_right\">0.76</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">3</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #f2eef6;\" class=\"gt_row gt_right\">0.09</td>\n",
       "    <td style=\"color: #000000; background-color: #c2aed7;\" class=\"gt_row gt_right\">0.40</td>\n",
       "    <td style=\"color: #000000; background-color: #9572b9;\" class=\"gt_row gt_right\">0.69</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #7f54aa;\" class=\"gt_row gt_right\">0.84</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">4</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #e2d8ec;\" class=\"gt_row gt_right\">0.19</td>\n",
       "    <td style=\"color: #000000; background-color: #9a78bc;\" class=\"gt_row gt_right\">0.66</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #784ba5;\" class=\"gt_row gt_right\">0.88</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">5</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #b197cb;\" class=\"gt_row gt_right\">0.51</td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #7446a2;\" class=\"gt_row gt_right\">0.91</td>\n",
       "  </tr>\n",
       "  <tr>\n",
       "    <th style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_left gt_stub\">6</th>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #000000; background-color: #FFFFFF;\" class=\"gt_row gt_right\"></td>\n",
       "    <td style=\"color: #FFFFFF; background-color: #7243a1;\" class=\"gt_row gt_right\">0.92</td>\n",
       "  </tr>\n",
       "</tbody>\n",
       "\n",
       "\n",
       "</table>\n",
       "\n",
       "</div>\n",
       "        "
      ],
      "text/plain": [
       "GT(_tbl_data=shape: (7, 8)\n",
       "┌─────┬─────────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┐\n",
       "│ P1X ┆ 1995    ┆ 1996     ┆ 1997     ┆ 1998     ┆ 1999     ┆ 2000     ┆ 2001     │\n",
       "│ --- ┆ ---     ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      ┆ ---      │\n",
       "│ i8  ┆ f64     ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      ┆ f64      │\n",
       "╞═════╪═════════╪══════════╪══════════╪══════════╪══════════╪══════════╪══════════╡\n",
       "│ 0   ┆ 0.04781 ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     │\n",
       "│ 1   ┆ null    ┆ 0.046646 ┆ 0.171041 ┆ 0.323233 ┆ 0.456882 ┆ 0.556106 ┆ 0.624799 │\n",
       "│ 2   ┆ null    ┆ null     ┆ 0.054124 ┆ 0.243587 ┆ 0.480533 ┆ 0.655667 ┆ 0.758256 │\n",
       "│ 3   ┆ null    ┆ null     ┆ null     ┆ 0.085556 ┆ 0.395873 ┆ 0.69057  ┆ 0.837916 │\n",
       "│ 4   ┆ null    ┆ null     ┆ null     ┆ null     ┆ 0.190051 ┆ 0.661568 ┆ 0.883402 │\n",
       "│ 5   ┆ null    ┆ null     ┆ null     ┆ null     ┆ null     ┆ 0.509775 ┆ 0.908715 │\n",
       "│ 6   ┆ null    ┆ null     ┆ null     ┆ null     ┆ null     ┆ null     ┆ 0.92319  │\n",
       "└─────┴─────────┴──────────┴──────────┴──────────┴──────────┴──────────┴──────────┘, _body=<great_tables._gt_data.Body object at 0x1197d5b50>, _boxhead=Boxhead([ColInfo(var='P1X', type=<ColInfoTypeEnum.stub: 2>, column_label='P1X', column_align='right', column_width=None), ColInfo(var='1995', type=<ColInfoTypeEnum.default: 1>, column_label='1995', column_align='right', column_width=None), ColInfo(var='1996', type=<ColInfoTypeEnum.default: 1>, column_label='1996', column_align='right', column_width=None), ColInfo(var='1997', type=<ColInfoTypeEnum.default: 1>, column_label='1997', column_align='right', column_width=None), ColInfo(var='1998', type=<ColInfoTypeEnum.default: 1>, column_label='1998', column_align='right', column_width=None), ColInfo(var='1999', type=<ColInfoTypeEnum.default: 1>, column_label='1999', column_align='right', column_width=None), ColInfo(var='2000', type=<ColInfoTypeEnum.default: 1>, column_label='2000', column_align='right', column_width=None), 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CellStyleFill(color='#c2aed7')]), StyleInfo(locname=LocBody(columns='1999', rows=[4]), grpname=None, colname='1999', rownum=4, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#e2d8ec')]), StyleInfo(locname=LocBody(columns='1999', rows=[5]), grpname=None, colname='1999', rownum=5, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), StyleInfo(locname=LocBody(columns='1999', rows=[6]), grpname=None, colname='1999', rownum=6, colnum=None, styles=[CellStyleText(color='#000000', font=None, size=None, align=None, v_align=None, style=None, weight=None, stretch=None, decorate=None, transform=None, whitespace=None), CellStyleFill(color='#FFFFFF')]), 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      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_star = 5\n",
    "x_star = 0\n",
    "\n",
    "A1 = beta_fn(alpha+p1x, beta+n-p1x)/B_alpha_beta * beta_fn(gamma, delta+n)/B_gamma_delta\n",
    "B1 = np.where(x_star == 0, 1-(A1/L), 0)\n",
    "A2 = comb(n_star, x_star) * beta_fn(alpha+p1x+x_star,beta+n-p1x+n_star-x_star)/B_alpha_beta * \\\n",
    "    beta_fn(gamma, delta+n+n_star)/beta_fn(gamma, delta)\n",
    "i = np.arange(n_star).reshape(-1,1)\n",
    "A2 += np.sum(comb(i,x_star)*beta_fn(alpha+p1x+x_star, beta+n-p1x+i-x_star)/B_alpha_beta * \\\n",
    "      beta_fn(gamma+1,delta+n+i)/B_gamma_delta, axis=0)\n",
    "\n",
    "prob_alive_valid = 1 - (B1 + A2/L)\n",
    "\n",
    "rfcalib_cross_tab(rfm_summary_calib.collect().hstack([pl.Series('Prob Alive in Valid Period', prob_alive_valid)]), \n",
    "                  values='Prob Alive in Valid Period',\n",
    "                  title='Probability of Being Active in 2002–2006',\n",
    "                  subtitle='as a Function of Recency and Frequency',\n",
    "                  color_range=[0,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Discounted Expected Residual Transactions (DERT)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assuming that there are $k$ transaction opportunities per year, an annual discount rate of $r$ maps to a discount rate of $d = (1+r)^{1/k} −1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5.90970778, 5.08930896, 4.26891013, 3.4485113 , 2.62811247,\n",
       "       1.80771365, 2.85502299, 3.19693293, 2.8420976 , 2.27253853,\n",
       "       1.60896852, 0.91839379, 1.62924881, 1.66552637, 1.32188456,\n",
       "       0.35211312, 0.84427117, 0.93520184, 0.18759408, 0.49486932,\n",
       "       0.13496036, 0.11475575])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = 0.1 # discount rate\n",
    "\n",
    "A1 = beta_fn(alpha+p1x+1, beta+n-p1x)/B_alpha_beta * beta_fn(gamma, delta+n+1)/(B_gamma_delta * (1 + d))\n",
    "A2 = hyp2f1(1, delta+n+1, gamma+delta+n+1, 1/(1+d)) / L\n",
    "DERT = A1 * A2\n",
    "DERT"
   ]
  }
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