NBD-Model.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "NDB Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Sources**:\n",
    "- [Illustrating the Performance of the NBD as a Benchmark Model for Customer-Base Analysis](http://www.brucehardie.com/notes/005/)\n",
    "- [\"Counting Your Customers\" the Easy Way: An Alternative to the Pareto/NBD Model](https://brucehardie.com/papers/018/)\n",
    "- [A Note on Implementing the Pareto/NBD Model in MATLAB](https://www.brucehardie.com/notes/008/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.optimize import minimize\n",
    "from scipy.special import beta, gamma, gammaln, factorial, hyp2f1\n",
    "from scipy.stats import chisquare, chi2\n",
    "\n",
    "import polars as pl\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib_inline\n",
    "from IPython.display import display_markdown\n",
    "\n",
    "from utils import CDNOW, bic\n",
    "\n",
    "matplotlib_inline.backend_inline.set_matplotlib_formats('svg')\n",
    "plt.rcParams[\"axes.spines.right\"] = False\n",
    "plt.rcParams[\"axes.spines.top\"] = False"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = (\n",
    "    CDNOW(master=False, calib_p=273)\n",
    "    .rfm_summary()\n",
    "    .select('P1X', 't_x', 'T')\n",
    ")\n",
    "\n",
    "# x: 'repeat_trans' = “repeat frequency”, number of repeat transactions made by a customer in a specified time period.\n",
    "# t_x: 'last_purch' (in weeks) = Time of last calibration period repeat purchase \n",
    "# T: 'T' (in weeks) = length of time over which we have had an opportunity to observe any repeat purchasing behavior. \n",
    "# 'first_purch' = Time of first purchase (in weeks)\n",
    "repeat_trans, last_purch, T = np.hsplit(data.collect().to_numpy(), 3)\n",
    "num_customers = len(repeat_trans)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# NBD Model - timing-model equivalent of the basic NBD model\n",
    "def nbd_timing_params(x, T):\n",
    "    def log_likelihood(params):\n",
    "        r, alpha = params\n",
    "        return -np.sum(gammaln(r+x)-gammaln(r)+r*np.log(alpha)-(r+x)*np.log(alpha+T))\n",
    "    return minimize(log_likelihood, x0=[0.1,0.1], bounds=[(1e-6, np.inf), (1e-6, np.inf)])\n",
    "\n",
    "def nbd_pmf(x, t, r, alpha):\n",
    "    # P(X(T)=x)\n",
    "    # np.exp(gammaln(r+x)-gammaln(r))/factorial(x) * (alpha/(alpha+t))**r * (t/(alpha+t))**x\n",
    "    return gamma(r + x)/(gamma(r)*factorial(x)) * (alpha/(alpha+t))**r * (t/(alpha+t))**x\n",
    "\n",
    "# For a randomly-chosen individual, the expected number of transactions in a time period of length t is given by the mean of the NBD\n",
    "def nbd_E_X_t(t, r, alpha):\n",
    "    return r/alpha * t\n",
    "\n",
    "def nbd_E_Y_X(x, T, t, r, alpha):\n",
    "    return (r+x)/(alpha+T) * t\n",
    "\n",
    "# NBD Model - standard counting form\n",
    "def nbd_params(x, T):\n",
    "    def log_likelihood(params):\n",
    "        r, alpha = params\n",
    "        pmf = nbd_pmf(x, T, r, alpha)\n",
    "        return -np.sum(np.log(pmf))\n",
    "    return minimize(log_likelihood, x0=[0.1,0.1], bounds=[(1e-6, np.inf), (1e-6, np.inf)])\n",
    "\n",
    "# BG/NBD Model\n",
    "def bgnbd_params(x, t_x, T):\n",
    "    def log_likelihood(params):\n",
    "        r, alpha, a, b = params\n",
    "        likelihood_function = (\n",
    "            beta(a,b+x)/beta(a,b) * \n",
    "            (gamma(r+x)*alpha**r)/(gamma(r)*(alpha+T)**(r+x))\n",
    "        )\n",
    "        likelihood_function += (\n",
    "            np.where(x > 0,\n",
    "                beta(a+1, b+x-1)/beta(a,b) *\n",
    "                (gamma(r+x)*alpha**r)/(gamma(r)*(alpha+t_x)**(r+x)),\n",
    "                0)\n",
    "        )\n",
    "        return -np.sum(np.log(likelihood_function))\n",
    "    \n",
    "    bnds = [(1e-6, np.inf) for _ in range(4)]\n",
    "    guess = [0.01 for _ in range(4)]\n",
    "    return minimize(log_likelihood, x0=guess, bounds=bnds)\n",
    "\n",
    "# expected number of purchases in a time period of length t\n",
    "def bgnbd_E_X_t(t, r, alpha, a, b):\n",
    "    return (a+b-1)/(a-1) * (1 - (alpha/(alpha+t))**r * hyp2f1(r, b, a+b-1, t/(alpha+t)))\n",
    "\n",
    "# probability of observing x purchases in a time period of length t\n",
    "def bgnbd_pmf(f_x, t, r, alpha, a, b):\n",
    "    A = np.cumsum(\n",
    "        gamma(r+f_x[:-1])/(gamma(r)*factorial(f_x[:-1])) *\n",
    "        (t/(alpha+t))**f_x[:-1], axis=0\n",
    "    )\n",
    "    pmf = (\n",
    "        beta(a,b+f_x)/beta(a,b) * \n",
    "        gamma(r+f_x)/(gamma(r)*factorial(f_x)) * \n",
    "        (alpha/(alpha+t))**r * \n",
    "        (t/(alpha+t))**f_x\n",
    "    )\n",
    "    pmf[1:] += (\n",
    "        beta(a+1,b+f_x[1:]-1)/beta(a,b) *\n",
    "        (1 - (alpha/(alpha+t))**r * A)        \n",
    "    )\n",
    "    return pmf\n",
    "\n",
    "def bgnbd_E_Y_X(x, t_x, T, t, r, alpha, a, b):\n",
    "    h2f1_cust = hyp2f1(r+x, b+x, a+b+x-1, t/(alpha + T + t))\n",
    "    return (\n",
    "        (a + b + x - 1) / (a - 1) * (1 - ((alpha + T) / (alpha + T + t))**(r + x) * h2f1_cust) /\n",
    "        (1 + (x > 0) * a / (b + x - 1) * ((alpha + T) / (alpha + t_x))**(r + x))\n",
    "    )\n",
    "\n",
    "# Pareto/NBD Model\n",
    "def paretonbd_params(x, t_x, T):\n",
    "    \n",
    "    def log_likelihood(params):\n",
    "        r, alpha, s, beta = params\n",
    "        \n",
    "        maxab = np.max((alpha, beta))\n",
    "        absab = np.abs(alpha - beta)\n",
    "        param2 = s + 1\n",
    "        if alpha < beta:\n",
    "            param2 = r + x\n",
    "            \n",
    "        part1 = (alpha**r * beta**s / gamma(r)) * gamma(r+x)\n",
    "        part2 = 1/((alpha+T)**(r+x) * (beta+T)**s)\n",
    "\n",
    "        if absab == 0:\n",
    "            F1 = 1/((maxab+t_x)**(r+s+x))\n",
    "            F2 = 1/((maxab+T)**(r+s+x))\n",
    "        else:\n",
    "            F1 = hyp2f1(r+s+x, param2, r+s+x+1, absab/(maxab+t_x)) / ((maxab+t_x)**(r+s+x))\n",
    "            F2 = hyp2f1(r+s+x, param2, r+s+x+1, absab/(maxab+T)) / ((maxab+T)**(r+s+x))        \n",
    "        \n",
    "        return -np.sum(np.log(part1*(part2+(s/(r+s+x))*(F1-F2))))\n",
    "    \n",
    "    bnds = [(1e-6, 20) for _ in range(4)]\n",
    "    guess = [0.01 for _ in range(4)]\n",
    "    return minimize(log_likelihood, x0=guess, bounds=bnds)\n",
    "\n",
    "# distribution of the number of transactions in the interval (0, t] for a randomly-chosen individual\n",
    "def paretonbd_pmf(f_x, t, r, alpha, s, beta_param):\n",
    "    maxab = np.max((alpha, beta_param))\n",
    "    absab = np.abs(alpha - beta_param)\n",
    "    param2 = s + 1\n",
    "    if alpha < beta_param:\n",
    "        param2 = r + f_x    \n",
    "    B2i = np.cumsum(\n",
    "        gamma(r+s+f_x)/gamma(r+s) * t**f_x / factorial(f_x) *\n",
    "        hyp2f1(r+s+f_x, param2, r+s+f_x+1, absab/(maxab+t)) / \n",
    "        (maxab+t)**(r+s+f_x),\n",
    "        axis=0\n",
    "    )\n",
    "    pmf = (\n",
    "        gamma(r+f_x)/(gamma(r)*factorial(f_x)) *\n",
    "        (alpha/(alpha+t))**r * \n",
    "        (t/(alpha+t))**f_x *\n",
    "        (beta_param/(beta_param+t))**s +\n",
    "        alpha**r * beta_param**s *\n",
    "        beta(r+f_x,s+1)/beta(r,s) *\n",
    "        (hyp2f1(r+s, param2, r+s+f_x+1, absab/maxab) / maxab**(r+s) - B2i)\n",
    "    )\n",
    "    return pmf\n",
    "\n",
    "def paretonbd_E_X_t(t, r, alpha, s, beta_param):\n",
    "    return r*beta_param/(alpha*(s-1)) * (1 - (beta_param/(beta_param + t))**(s-1))\n",
    "\n",
    "def paretonbd_pactive(x, t_x, T, r, alpha, s, beta_param):\n",
    "    maxab = np.max((alpha, beta_param))\n",
    "    absab = np.abs(alpha - beta_param)\n",
    "    param2 = s + 1\n",
    "    if alpha < beta_param:\n",
    "        param2 = r + x\n",
    "\n",
    "    F0 = (alpha + T)**(r + x) * (beta_param + T)**s\n",
    "\n",
    "    if absab == 0:\n",
    "        F1 = 1/((maxab+t_x)**(r+s+x))\n",
    "        F2 = 1/((maxab+T)**(r+s+x))\n",
    "    else:\n",
    "        F1 = hyp2f1(r+s+x, param2, r+s+x+1, absab/(maxab+t_x)) / ((maxab+t_x)**(r+s+x))\n",
    "        F2 = hyp2f1(r+s+x, param2, r+s+x+1, absab/(maxab+T)) / ((maxab+T)**(r+s+x))       \n",
    "\n",
    "    return (1 + (s / (r+s+x)) * F0 * (F1-F2))**(-1)\n",
    "\n",
    "def paretonbd_E_Y_X(x, t_x, T, t, r, alpha, s, beta_param):\n",
    "    return (\n",
    "        (r + x) * (beta_param + T) / ((alpha + T) * (s - 1)) *\n",
    "        (1 - ((beta_param + T) / (beta_param + T + t))**(s - 1)) *\n",
    "        paretonbd_pactive(x, t_x, T, r, alpha, s, beta_param)\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "**NBD - Timing-Model Equivalent:**\n",
       "\n",
       "Parameter Estimates:\n",
       "- $r$ = 0.3848\n",
       "- $\\alpha$ = 12.0718\n",
       "\n",
       "Log-Likelihood = -9763.6576\n",
       "\n",
       "BIC = 19542.8"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "**NBD - Standard Counting Method:**\n",
       "\n",
       "Parameter Estimates:\n",
       "- $r$ = 0.3848\n",
       "- $\\alpha$ = 12.0718\n",
       "\n",
       "Log-Likelihood = -3193.0587\n",
       "\n",
       "BIC = 6401.6"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "**BG/NBD:**\n",
       "\n",
       "Parameter Estimates:\n",
       "- $r$ = 0.2426\n",
       "- $\\alpha$ = 4.4136\n",
       "- $a$ = 0.7929\n",
       "- $b$ = 2.4258\n",
       "\n",
       "Log-Likelihood = -9582.4292\n",
       "\n",
       "BIC = 19195.9"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "**Pareto/NBD:**\n",
       "\n",
       "Parameter Estimates:\n",
       "- $r$ = 0.5533\n",
       "- $\\alpha$ = 10.5781\n",
       "- $s$ = 0.6064\n",
       "- $\\beta$ = 11.6726\n",
       "\n",
       "Log-Likelihood = -9594.9762\n",
       "\n",
       "BIC = 19221.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "res = nbd_timing_params(repeat_trans, T)\n",
    "r_nbd, alpha_nbd = res.x\n",
    "ll = res.fun\n",
    "\n",
    "display_markdown(f'''**NBD - Timing-Model Equivalent:**\n",
    "\n",
    "Parameter Estimates:\n",
    "- $r$ = {r_nbd:0.4f}\n",
    "- $\\\\alpha$ = {alpha_nbd:0.4f}\n",
    "\n",
    "Log-Likelihood = {-ll:0.4f}\n",
    "\n",
    "BIC = {bic(2, num_customers, ll):.1f}''', raw=True)\n",
    "\n",
    "res = nbd_params(repeat_trans, T)\n",
    "r, alpha = res.x\n",
    "ll = res.fun\n",
    "\n",
    "display_markdown(f'''**NBD - Standard Counting Method:**\n",
    "\n",
    "Parameter Estimates:\n",
    "- $r$ = {r:0.4f}\n",
    "- $\\\\alpha$ = {alpha:0.4f}\n",
    "\n",
    "Log-Likelihood = {-ll:0.4f}\n",
    "\n",
    "BIC = {bic(2, num_customers, ll):.1f}''', raw=True)\n",
    "\n",
    "res = bgnbd_params(repeat_trans, last_purch, T)\n",
    "r_bgnbd, alpha_bgnbd, a_bgnbd, b_bgnbd = res.x\n",
    "ll = res.fun\n",
    "\n",
    "display_markdown(f'''**BG/NBD:**\n",
    "\n",
    "Parameter Estimates:\n",
    "- $r$ = {r_bgnbd:0.4f}\n",
    "- $\\\\alpha$ = {alpha_bgnbd:0.4f}\n",
    "- $a$ = {a_bgnbd:0.4f}\n",
    "- $b$ = {b_bgnbd:0.4f}\n",
    "\n",
    "Log-Likelihood = {-ll:0.4f}\n",
    "\n",
    "BIC = {bic(4, num_customers, ll):.1f}''', raw=True)\n",
    "\n",
    "res = paretonbd_params(repeat_trans, last_purch, T)\n",
    "r_pareto, alpha_pareto, s, beta_param = res.x\n",
    "ll = res.fun\n",
    "\n",
    "display_markdown(f'''**Pareto/NBD:**\n",
    "\n",
    "Parameter Estimates:\n",
    "- $r$ = {r_pareto:0.4f}\n",
    "- $\\\\alpha$ = {alpha_pareto:0.4f}\n",
    "- $s$ = {s:0.4f}\n",
    "- $\\\\beta$ = {beta_param:0.4f}\n",
    "\n",
    "Log-Likelihood = {-ll:0.4f}\n",
    "\n",
    "BIC = {bic(4, num_customers, ll):.1f}''', raw=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Predicted Distribution of Transactions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def nbd_predict_trans_dist(f_x, n_s, T, r, alpha):\n",
    "    '''\n",
    "    Parameters:\n",
    "    f_x:    frequency of repeat transactions (right-censored)\n",
    "    n_s:    the number of customers who made their first purchase on day s\n",
    "    T:      the number of days (in weeks) during which repeat transactions could have occurred in the calibration period, \n",
    "            it follows that the “time of first purchase” is simply: calibration week - T.\n",
    "            T = t - s/7 (in weeks), T = effective calibration period, s = time of trial purchase, t = duration of calibration period \n",
    "    r, alpha: NBD model parameters\n",
    "    '''\n",
    "    pmf = nbd_pmf(f_x, T, r, alpha)\n",
    "    pmf[-1] = 1 - np.sum(pmf[:-1], axis=0)\n",
    "    return np.dot(pmf, n_s)\n",
    "\n",
    "def bgnbd_predict_trans_dist(f_x, n_s, T, r, alpha, a, b):\n",
    "    pmf = bgnbd_pmf(f_x, T, r, alpha, a, b)\n",
    "    pmf[-1] = 1 - np.sum(pmf[:-1], axis=0)\n",
    "    return np.dot(pmf, n_s)\n",
    "\n",
    "def paretonbd_redict_trans_dist(f_x, n_s, t, r, alpha, s, beta_param):\n",
    "    pmf = paretonbd_pmf(f_x, t, r, alpha, s, beta_param)\n",
    "    pmf[-1] = 1 - np.sum(pmf[:-1], axis=0)\n",
    "    return np.dot(pmf, n_s)"
   ]
  },
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   "cell_type": "code",
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   "source": [
    "num_repeats, repeat_freq = np.unique(repeat_trans, return_counts=True)\n",
    "\n",
    "# right-censored distribution in which counts greater than 7 are collapsed into a 7+ bin\n",
    "num_repeats_censored = num_repeats[:8].copy()\n",
    "repeat_freq_censored = repeat_freq[:8].copy()\n",
    "repeat_freq_censored[-1] = np.sum(repeat_freq) - np.sum(repeat_freq[:7])\n",
    "\n",
    "# n_s is the number of customers who made their first purchase on day s\n",
    "# T_unique is the unique t - s/7 weeks within which to make repeat purchases\n",
    "T_unique, n_s = np.unique(T, return_counts=True)\n",
    "\n",
    "nbd_E_f_x = nbd_predict_trans_dist(num_repeats_censored.reshape(-1,1), n_s, T_unique, r_nbd, alpha_nbd)\n",
    "bgnbd_E_f_x = bgnbd_predict_trans_dist(num_repeats_censored.reshape(-1,1), n_s, T_unique, r_bgnbd, alpha_bgnbd, a_bgnbd, b_bgnbd)\n",
    "paretonbd_E_f_x = paretonbd_redict_trans_dist(num_repeats_censored.reshape(-1,1), n_s, T_unique, r_pareto, alpha_pareto, s, beta_param)\n",
    "\n",
    "bar_width = 0.2\n",
    "plt.figure(figsize=(9,5), dpi=100)\n",
    "plt.bar(num_repeats_censored - bar_width*1.5, repeat_freq_censored, width=bar_width, label='Actual - $f_x$', color='black')\n",
    "plt.bar(num_repeats_censored - bar_width/2, bgnbd_E_f_x, width=bar_width, label='BG/NBD - $E(f_x)$', color='grey', edgecolor='black', linewidth=0.5)\n",
    "plt.bar(num_repeats_censored + bar_width/2, paretonbd_E_f_x, width=bar_width, label='Pareto/NBD - $E(f_x)$', color='lightgrey', edgecolor='black', linewidth=0.5)\n",
    "plt.bar(num_repeats_censored + bar_width*1.5, nbd_E_f_x, width=bar_width, label='NBD - $E(f_x)$', color='white', edgecolor='black', linewidth=0.5)\n",
    "plt.xlabel(\"Number of Repeat Transactions\")\n",
    "plt.ylabel(\"Frequecy\")\n",
    "plt.title('Predicted versus Actual Frequency of Repeat Transaction', pad=30)\n",
    "plt.xticks(num_repeats_censored, [0,1,2,3,4,5,6,'7+'])\n",
    "plt.ylim(0, 1500)\n",
    "plt.xlim(0-bar_width*2.5, 8)\n",
    "plt.legend(loc=7, frameon=False);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "**NBD:**\n",
       "\n",
       "- Test Statistics = 10.27\n",
       "- df = 5\n",
       "- Critical Value = 11.070\n",
       "- p-Value = 0.068\n",
       "\n",
       "**BG/NBD:**\n",
       "\n",
       "- Test Statistics = 4.82\n",
       "- df = 3\n",
       "- Critical Value = 7.815\n",
       "- p-Value = 0.185\n",
       "\n",
       "**Pareto/NBD:**\n",
       "\n",
       "- Test Statistics = 11.56\n",
       "- df = 3\n",
       "- Critical Value = 7.815\n",
       "- p-Value = 0.009"
      ]
     },
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   "source": [
    "test_stat_nbd, p_value_nbd = chisquare(repeat_freq_censored, nbd_E_f_x, ddof=2)\n",
    "critical_val_nbd = chi2.isf(0.05, df=5)\n",
    "\n",
    "test_stat_bgnbd, p_value_bgnbd = chisquare(repeat_freq_censored, bgnbd_E_f_x, ddof=4)\n",
    "critical_val_bgnbd = chi2.isf(0.05, df=3)\n",
    "\n",
    "test_stat_paretonbd, p_value_paretonbd = chisquare(repeat_freq_censored, paretonbd_E_f_x, ddof=4)\n",
    "critical_val_paretonbd = chi2.isf(0.05, df=3)\n",
    "\n",
    "display_markdown(f'''**NBD:**\n",
    "\n",
    "- Test Statistics = {test_stat_nbd:.2f}\n",
    "- df = {5}\n",
    "- Critical Value = {critical_val_nbd:.3f}\n",
    "- p-Value = {p_value_nbd:.3f}\n",
    "\n",
    "**BG/NBD:**\n",
    "\n",
    "- Test Statistics = {test_stat_bgnbd:.2f}\n",
    "- df = {3}\n",
    "- Critical Value = {critical_val_bgnbd:.3f}\n",
    "- p-Value = {p_value_bgnbd:.3f}\n",
    "\n",
    "**Pareto/NBD:**\n",
    "\n",
    "- Test Statistics = {test_stat_paretonbd:.2f}\n",
    "- df = {3}\n",
    "- Critical Value = {critical_val_paretonbd:.3f}\n",
    "- p-Value = {p_value_paretonbd:.3f}''', raw=True)"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Aggregate Sales Forecast"
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  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def nbd_cum_repeat_trans(n_s, s, t, r, alpha, period=7):\n",
    "    '''\n",
    "    Parameters:\n",
    "    n_s:    the number of customers who made their first purchase on day s\n",
    "    s:      range of time of trial purchase (in weeks if 'period' = 7)\n",
    "    t:      range of the forecast horizon (in weeks if 'period' = 7)\n",
    "    r, alpha: NBD model parameters    \n",
    "    period: 7 = weekly\n",
    "    '''\n",
    "    nbd_mean = nbd_E_X_t(t.reshape(-1,1)-s, r, alpha)\n",
    "    nbd_mean = np.where(nbd_mean > 0, nbd_mean, 0)\n",
    "    cum_rpt_sales = np.dot(nbd_mean, n_s)\n",
    "    return cum_rpt_sales[period-1::period]\n",
    "\n",
    "def bgnbd_cum_repeat_trans(n_s, s, t, r, alpha, a, b, period=7):\n",
    "    E_X_t = bgnbd_E_X_t(t, r, alpha, a, b)\n",
    "    s = np.arange(np.max(s)*period - 1, -1, -1, dtype=np.int32)\n",
    "    t = np.arange(np.max(t)*period, dtype=np.int32).reshape(-1,1)\n",
    "    index = np.clip(t - s, 0, len(E_X_t))\n",
    "    E_X_t = np.where(t > s, E_X_t[index-1], 0)\n",
    "    return np.dot(E_X_t, n_s)[6::7]\n",
    "\n",
    "def paretonbd_cum_repeat_trans(n_s, s, t, r, alpha, s_param, beta_param, period=7):\n",
    "    E_X_t = paretonbd_E_X_t(t, r, alpha, s_param, beta_param)\n",
    "    s = np.arange(np.max(s)*period - 1, -1, -1, dtype=np.int32)\n",
    "    t = np.arange(np.max(t)*period, dtype=np.int32).reshape(-1,1)\n",
    "    index = np.clip(t - s, 0, len(E_X_t))\n",
    "    E_X_t = np.where(t > s, E_X_t[index-1], 0)\n",
    "    return np.dot(E_X_t, n_s)[6::7]"
   ]
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   "execution_count": 9,
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   "source": [
    "actual_cum_repeat, actual_wkly_sales, actual_cum_repeat = np.hsplit(CDNOW(master=False, calib_p=273).repeat_sales().to_numpy(), 3)\n",
    "\n",
    "forecast_horizon = np.arange(1/7, 78, 1/7) # day-by-day in weeks\n",
    "calib_p = 39 # in weeks\n",
    "\n",
    "nbd_cum_repeat = nbd_cum_repeat_trans(n_s, calib_p-T_unique, forecast_horizon, r_nbd, alpha_nbd)\n",
    "nbd_wkly_repeat = np.diff(nbd_cum_repeat, prepend=0)\n",
    "\n",
    "bgnbd_cum_repeat = bgnbd_cum_repeat_trans(n_s, calib_p-T_unique, forecast_horizon, r_bgnbd, alpha_bgnbd, a_bgnbd, b_bgnbd)\n",
    "bgnbd_wkly_repeat = np.diff(bgnbd_cum_repeat, prepend=0)\n",
    "\n",
    "paretonbd_cum_repeat = paretonbd_cum_repeat_trans(n_s, calib_p-T_unique, forecast_horizon, r_pareto, alpha_pareto, s, beta_param)\n",
    "paretonbd_wkly_repeat = np.diff(paretonbd_cum_repeat, prepend=0)\n",
    "\n",
    "forecast_horizon = np.arange(78)\n",
    "plt.figure(figsize=(8,5), dpi=100)\n",
    "plt.plot(forecast_horizon, actual_cum_repeat, color='black', linestyle='solid', linewidth=0.75, label='Actual')\n",
    "plt.plot(forecast_horizon, nbd_cum_repeat, color='black', linestyle='dotted', linewidth=0.75, label='NBD')\n",
    "plt.plot(forecast_horizon, bgnbd_cum_repeat, color='black', linestyle='dashed', linewidth=0.75, label='BG/NBD')\n",
    "plt.plot(forecast_horizon, paretonbd_cum_repeat, color='black', linestyle=(0, (5, 10)), linewidth=0.75, label='Pareto/NBD')\n",
    "plt.xlabel(\"Week\")\n",
    "plt.ylabel(\"# of Transactions\")\n",
    "plt.title('Predicted versus Actual Cumulative Repeat Transactions', pad=30)\n",
    "plt.ylim(0,6000)\n",
    "plt.xlim(0, 80)\n",
    "plt.legend(loc=4, frameon=False);\n",
    "\n",
    "plt.figure(figsize=(8,5), dpi=100)\n",
    "plt.plot(forecast_horizon, actual_wkly_sales, color='black', linestyle='solid', linewidth=0.75, label='Actual')\n",
    "plt.plot(forecast_horizon, nbd_wkly_repeat, color='black', linestyle='dotted', linewidth=0.75, label='NBD')\n",
    "plt.plot(forecast_horizon, bgnbd_wkly_repeat, color='black', linestyle='dashed', linewidth=0.75, label='BG/NBD')\n",
    "plt.plot(forecast_horizon, paretonbd_wkly_repeat, color='black', linestyle=(0, (5, 10)), linewidth=0.75, label='Pareto/NBD')\n",
    "plt.xlabel(\"Week\")\n",
    "plt.ylabel(\"# of Transactions\")\n",
    "plt.title('Predicted versus Actual Weekly Repeat Transactions', pad=30)\n",
    "plt.ylim(0,150)\n",
    "plt.xlim(0, 80)\n",
    "plt.legend(loc=1, frameon=False);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conditional Expectations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "actual_ce = (\n",
    "    CDNOW(master=False, calib_p=273)\n",
    "    .rfm_summary()\n",
    "    .group_by('P1X').agg(pl.col('P2X').mean().alias('Actual CE'))\n",
    "    .sort('P1X').select('Actual CE')\n",
    ")\n",
    "\n",
    "t = 39 # the length of the period over which we wish to make the conditional forecast\n",
    "nbd_ce = nbd_E_Y_X(repeat_trans, T, t, r_nbd, alpha_nbd)\n",
    "bgnbd_ce = bgnbd_E_Y_X(repeat_trans, last_purch, T, t, r_bgnbd, alpha_bgnbd, a_bgnbd, b_bgnbd)\n",
    "paretonbd_ce = paretonbd_E_Y_X(repeat_trans, last_purch, T, t, r_pareto, alpha_pareto, s, beta_param)\n",
    "\n",
    "nbd_ce, bgnbd_ce, paretonbd_ce, actual_ce  =  np.hsplit(\n",
    "    pl.DataFrame({'x': repeat_trans.astype(np.int32).flatten(), \n",
    "                  'NBD - E(Y|X)': nbd_ce.flatten(),\n",
    "                  'BG/NBD - E(Y|X)': bgnbd_ce.flatten(),\n",
    "                  'Pareto/NBD -  E(Y|X)': paretonbd_ce.flatten()})\n",
    "    .group_by('x')\n",
    "    .agg(pl.col('NBD - E(Y|X)', 'BG/NBD - E(Y|X)', 'Pareto/NBD -  E(Y|X)').mean())\n",
    "    .sort('x').hstack(actual_ce.collect()).drop('x').to_numpy(), 4\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
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     "metadata": {},
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   ],
   "source": [
    "censor = 7\n",
    "\n",
    "actual_ce_censored = actual_ce[:censor+1].copy()\n",
    "actual_ce_censored[-1] = np.dot(repeat_freq[censor:], actual_ce[censor:]) / np.sum(repeat_freq[censor:])\n",
    "\n",
    "nbd_ce_censored = nbd_ce[:censor+1].copy()\n",
    "nbd_ce_censored[-1] = np.dot(repeat_freq[censor:], nbd_ce[censor:]) / np.sum(repeat_freq[censor:])\n",
    "\n",
    "bgnbd_ce_censored = bgnbd_ce[:censor+1].copy()\n",
    "bgnbd_ce_censored[-1] = np.dot(repeat_freq[censor:], bgnbd_ce[censor:]) / np.sum(repeat_freq[censor:])\n",
    "\n",
    "paretonbd_ce_censored = paretonbd_ce[:censor+1].copy()\n",
    "paretonbd_ce_censored[-1] = np.dot(repeat_freq[censor:], paretonbd_ce[censor:]) / np.sum(repeat_freq[censor:])\n",
    "\n",
    "plt.figure(figsize=(8,5), dpi=100)\n",
    "plt.plot(num_repeats_censored, actual_ce_censored, color='black', linestyle='solid', linewidth=0.75, label='Actual')\n",
    "plt.plot(num_repeats_censored, nbd_ce_censored, color='black', linestyle='dotted', marker='x', markersize=5, markeredgewidth=0.4, linewidth=0.75, label='NBD')\n",
    "plt.plot(num_repeats_censored, bgnbd_ce_censored, color='black', linestyle='dashed', marker='d', fillstyle='none', markeredgewidth=0.5, linewidth=0.75, label='BG/NBD')\n",
    "plt.plot(num_repeats_censored, paretonbd_ce_censored, color='black', linestyle=(0, (5, 10)), marker='*', fillstyle='none', markeredgewidth=0.5, linewidth=0.75, label='Pareto/NBD')\n",
    "plt.xlabel(\"# Transactions in Weeks 1−39\")\n",
    "plt.ylabel(\"Expected # Transactions in Weeks 40−78\")\n",
    "plt.title('Conditional Expectations', pad=30)\n",
    "plt.xticks(num_repeats_censored, [0,1,2,3,4,5,6,'7+'])\n",
    "plt.ylim(0,10)\n",
    "plt.xlim(-0.25, 7.25)\n",
    "plt.legend(loc=2, frameon=False);   "
   ]
  }
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