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Recreation and Estimation.html
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Recreation and Estimation.html
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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<title>Guadagni Little 1983</title>
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<h1 id="I.-Importation-of-the-modules">I. Importation of the modules<a class="anchor-link" href="#I.-Importation-of-the-modules">¶</a></h1>
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<div class=" highlight hl-ipython3"><pre><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">scipy.stats</span> <span class="k">as</span> <span class="nn">stats</span>
<span class="kn">import</span> <span class="nn">itertools</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">exp</span>
<span class="kn">from</span> <span class="nn">IPython.core.pylabtools</span> <span class="k">import</span> <span class="n">figsize</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="k">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="n">figsize</span><span class="p">(</span><span class="mf">12.5</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
<span class="n">pd</span><span class="o">.</span><span class="n">set_option</span><span class="p">(</span><span class="s">"display.precision"</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
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<h1 id="II.-Data-generating-process">II. Data-generating process<a class="anchor-link" href="#II.-Data-generating-process">¶</a></h1><p>Here we define variables and code that will allows us to generate the data.</p>
<h2 id="A.-Observational-Space">A. Observational Space<a class="anchor-link" href="#A.-Observational-Space">¶</a></h2><p>We only consider two observational spaces: the training group and the holdout group.</p>
<ul>
<li>$N_{Group}: \text{Number of households in the training/holdout groups}$</li>
<li>$T_{Group}: \text{Number of weeks in the training/holdout groups}$</li>
</ul>
<p>We do not model the 25 weeks used to calibrate the loyalty of the household (we instead initialize it from the first purchase observed, see below): instead, we model the behavior of individuals on 120 weeks and discard the first 10. We also assume that all households make one purchase a week.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">N_Training</span> <span class="o">=</span> <span class="n">N_Holdout</span> <span class="o">=</span> <span class="mi">300</span>
<span class="n">N_Total</span> <span class="o">=</span> <span class="n">N_Training</span> <span class="o">+</span> <span class="n">N_Holdout</span>
<span class="n">T_Training</span> <span class="o">=</span> <span class="n">T_Holdout</span> <span class="o">=</span> <span class="mi">60</span>
<span class="n">T_Total</span> <span class="o">=</span> <span class="n">T_Training</span> <span class="o">+</span> <span class="n">T_Holdout</span>
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<h2 id="B.-Product-space">B. Product space<a class="anchor-link" href="#B.-Product-space">¶</a></h2><p>Here we define:</p>
<ul>
<li>$V_{Sizes}: \text{Vector of sizes}$</li>
<li>$V_{Brands}: \text{Vector of brands}$</li>
</ul>
<p>For simplicity, we assume that all brands are available in all sizes.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">V_Sizes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">V_Brands</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
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<h2 id="C.-Attribute-Space">C. Attribute Space<a class="anchor-link" href="#C.-Attribute-Space">¶</a></h2><p>Here we define several important constants used to generate the data.</p>
<h3 id="1°)-Loyalty">1°) Loyalty<a class="anchor-link" href="#1°)-Loyalty">¶</a></h3><ul>
<li>$\alpha_B: \text{Smoothing constant of the loyalty for brand}$</li>
<li>$\alpha_S: \text{Smoothing constant of the loyalty for size}$</li>
</ul>
<h3 id="2°)-Prices">2°) Prices<a class="anchor-link" href="#2°)-Prices">¶</a></h3><ul>
<li>$M_{PriceM}: \text{Price mean matrix for the Brand/Size pairs}$</li>
<li>$M_{PriceSD}: \text{Price SD matrix for the Brand/Size pairs}$</li>
</ul>
<h3 id="3°)-Promotion">3°) Promotion<a class="anchor-link" href="#3°)-Promotion">¶</a></h3><ul>
<li>$M_{PromoP}: \text{Probability that a Brand/Size pair is promoted on a given week}$</li>
<li>$M_{PromoM}: \text{Promotion discount mean matrix for the Brand/Size pairs}$</li>
<li>$M_{PromoSD}: \text{Promotion discount SD matrix for the Brand/Size pairs}$</li>
</ul>
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<div class=" highlight hl-ipython3"><pre><span class="n">alpha_B</span> <span class="o">=</span> <span class="mf">0.875</span>
<span class="n">alpha_S</span> <span class="o">=</span> <span class="mf">0.812</span>
<span class="n">M_priceM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">3.25</span><span class="p">,</span> <span class="mf">2.75</span><span class="p">,</span> <span class="mf">2.65</span><span class="p">,</span> <span class="mf">4.20</span><span class="p">,</span> <span class="mf">1.95</span><span class="p">],[</span><span class="mf">3.00</span><span class="p">,</span> <span class="mf">2.50</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">]])</span> <span class="c">#Price is expressed in $/oz in the paper</span>
<span class="n">M_priceSD</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.12</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.06</span><span class="p">,</span> <span class="mf">0.07</span><span class="p">,</span> <span class="mf">0.09</span><span class="p">],[</span><span class="mf">0.03</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.07</span><span class="p">,</span> <span class="mf">0.09</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">]])</span> <span class="c">#We assume very small price variations</span>
<span class="n">M_promoP</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.35</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">,</span> <span class="mf">0.20</span><span class="p">,</span> <span class="mf">0.12</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.14</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">,</span> <span class="mf">0.18</span><span class="p">,</span> <span class="mf">0.21</span><span class="p">,</span> <span class="mf">0.35</span><span class="p">]])</span> <span class="c">#Probability of observing a promotion</span>
<span class="n">M_promoM</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.15</span><span class="p">,</span> <span class="mf">0.19</span><span class="p">,</span> <span class="mf">0.12</span><span class="p">,</span> <span class="mf">0.26</span><span class="p">,</span> <span class="mf">0.32</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.17</span><span class="p">,</span> <span class="mf">0.28</span><span class="p">,</span> <span class="mf">0.07</span><span class="p">,</span> <span class="mf">0.12</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">]])</span> <span class="c">#Promotion is expressed in percentage</span>
<span class="n">M_promoSD</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.03</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.02</span><span class="p">,</span> <span class="mf">0.06</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.03</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.04</span><span class="p">]])</span> <span class="c">#We assume very small variations in magnitude</span>
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<h2 id="D.-Utility-Function">D. Utility Function<a class="anchor-link" href="#D.-Utility-Function">¶</a></h2><p>Here we define the utility function and utility parameters of the household i for the brand-size option k at times t:</p>
<h3 id="1°)-Formulation:">1°) Formulation:<a class="anchor-link" href="#1°)-Formulation:">¶</a></h3><p>$U_{i, k, t}= \beta^{0}x^{0}_{i, k} + \beta^{1}x^{1}_{k, t} + \beta^{2}x^{2}_{k, t} + \beta^{3}x^{3}_{k, t} + \beta^{4}x^{4}_{i, k, t} + \beta^{5}x^{5}_{i, k, t} + \beta^{6}x^{6}_{i, k, t} + \beta^{7}x^{7}_{i, k, t} +\epsilon_{i, t}$</p>
<h3 id="2°)-Coefficients-description:">2°) Coefficients description:<a class="anchor-link" href="#2°)-Coefficients-description:">¶</a></h3><ul>
<li>$\beta^{0}$: Fixed utility constants of the brand-size pairs</li>
<li>$\beta^{1}$: Price factor</li>
<li>$\beta^{2}$: Promotion factor</li>
<li>$\beta^{3}$: Promotion discount factor</li>
<li>$\beta^{4}$: Previous promotion purchase factor</li>
<li>$\beta^{5}$: Second previous promotion purchase factor</li>
<li>$\beta^{6}$: Brand Loyalty factor</li>
<li>$\beta^{7}$: Size Loyalty factor</li>
</ul>
<h3 id="3°)-Coefficients-determination:">3°) Coefficients determination:<a class="anchor-link" href="#3°)-Coefficients-determination:">¶</a></h3>
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<h4 id="a.-Paper-coefficients:">a. Paper coefficients:<a class="anchor-link" href="#a.-Paper-coefficients:">¶</a></h4>
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<p>Using those results in the choice of the 9th option more than 99% of the time, because its price is lower than the other alternatives. For this reason, these coefficients are not used.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">b0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.28</span><span class="p">,</span> <span class="mf">0.48</span><span class="p">,</span> <span class="mf">0.99</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.25</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.83</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.06</span><span class="p">,</span> <span class="mf">0.38</span><span class="p">,</span> <span class="mf">0.56</span><span class="p">,</span> <span class="mf">0.33</span><span class="p">]])</span>
<span class="n">b1</span> <span class="o">=</span> <span class="o">-</span><span class="mf">29.94</span>
<span class="n">b2</span> <span class="o">=</span> <span class="mf">2.11</span>
<span class="n">b3</span> <span class="o">=</span> <span class="mf">29.21</span>
<span class="n">b4</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.22</span>
<span class="n">b5</span><span class="o">=</span> <span class="o">-</span><span class="mf">0.46</span>
<span class="n">b6</span> <span class="o">=</span> <span class="mf">3.91</span>
<span class="n">b7</span> <span class="o">=</span> <span class="mf">2.97</span>
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<h4 id="b.-Made-up-coefficients:">b. Made-up coefficients:<a class="anchor-link" href="#b.-Made-up-coefficients:">¶</a></h4>
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<p>They ensure that the multinomial logit estimation can recover estimates (enough variability in choices)</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">b0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">],[</span><span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">]])</span>
<span class="n">b1</span> <span class="o">=</span> <span class="o">-</span><span class="mf">1.7</span>
<span class="n">b2</span> <span class="o">=</span> <span class="mf">0.3</span>
<span class="n">b3</span> <span class="o">=</span> <span class="mf">1.2</span>
<span class="n">b4</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.5</span>
<span class="n">b5</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.2</span>
<span class="n">b6</span> <span class="o">=</span> <span class="mf">2.5</span>
<span class="n">b7</span> <span class="o">=</span> <span class="mf">1.6</span>
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<h1 id="III.-Dataframes-generation">III. Dataframes generation<a class="anchor-link" href="#III.-Dataframes-generation">¶</a></h1><p>Here we initialize four dataframes:</p>
<ul>
<li>$DF_{Supply}: \text{Panel data of Brand/Size pairs characteristics across weeks}$</li>
<li>$DF_{Demand}: \text{Panel data of household demand across weeks}$</li>
<li>$DF_{BLoyalty}: \text{Panel data of the Brand loyalties across households and weeks}$</li>
<li>$DF_{SLoyalty}: \text{Panel data of the Brand loyalties across households and weeks}$</li>
</ul>
<h2 id="A.-Generation-of-the-Supply-dataframe">A. Generation of the Supply dataframe<a class="anchor-link" href="#A.-Generation-of-the-Supply-dataframe">¶</a></h2>
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<h3 id="1°)-Initialization">1°) Initialization<a class="anchor-link" href="#1°)-Initialization">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="n">weeks</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">T_Total</span><span class="o">+</span><span class="mi">1</span><span class="p">)]</span>
<span class="n">week_pairs_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="n">weeks</span><span class="p">,</span> <span class="n">V_Sizes</span><span class="p">,</span> <span class="n">V_Brands</span><span class="p">)]</span>
<span class="n">weeks_list</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([</span><span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">week_pairs_list</span><span class="p">])</span>
<span class="n">sizes_list</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([</span><span class="n">i</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">week_pairs_list</span><span class="p">])</span>
<span class="n">brands_list</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([</span><span class="n">i</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">week_pairs_list</span><span class="p">])</span>
<span class="n">price_list</span> <span class="o">=</span> <span class="n">promo_list</span> <span class="o">=</span> <span class="n">promosize_list</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([</span><span class="n">np</span><span class="o">.</span><span class="n">NaN</span><span class="p">])</span>
<span class="n">DF_Supply</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">concat</span><span class="p">([</span><span class="n">weeks_list</span><span class="p">,</span> <span class="n">sizes_list</span><span class="p">,</span> <span class="n">brands_list</span><span class="p">,</span> <span class="n">price_list</span><span class="p">,</span> <span class="n">promo_list</span><span class="p">,</span> <span class="n">promosize_list</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">colsTime</span> <span class="o">=</span> <span class="p">[</span><span class="s">"Week"</span><span class="p">,</span> <span class="s">"Size"</span><span class="p">,</span> <span class="s">"Brand"</span><span class="p">,</span> <span class="s">"BasePrice"</span><span class="p">,</span> <span class="s">"Promo"</span><span class="p">,</span> <span class="s">"PromoSize"</span><span class="p">]</span>
<span class="n">DF_Supply</span><span class="o">.</span><span class="n">columns</span> <span class="o">=</span> <span class="n">colsTime</span>
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<h3 id="2°)-Definition-of-the-data-generating-functions">2°) Definition of the data-generating functions<a class="anchor-link" href="#2°)-Definition-of-the-data-generating-functions">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">gen_random_price</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Given the Size and the Brand, return a</span>
<span class="sd"> random price for the time period</span>
<span class="sd"> """</span>
<span class="n">sizeid</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="s">"Size"</span><span class="p">]</span>
<span class="n">brandid</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="s">"Brand"</span><span class="p">]</span>
<span class="n">price_mean</span> <span class="o">=</span> <span class="n">M_priceM</span><span class="p">[</span><span class="n">sizeid</span><span class="p">][</span><span class="n">brandid</span><span class="p">]</span>
<span class="n">price_SD</span> <span class="o">=</span> <span class="n">M_priceSD</span><span class="p">[</span><span class="n">sizeid</span><span class="p">][</span><span class="n">brandid</span><span class="p">]</span>
<span class="k">return</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">price_mean</span><span class="p">,</span> <span class="n">price_SD</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">gen_random_promo</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Given the Size and the Brand, return a</span>
<span class="sd"> discount or not for the time period</span>
<span class="sd"> """</span>
<span class="n">sizeid</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="s">"Size"</span><span class="p">]</span>
<span class="n">brandid</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="s">"Brand"</span><span class="p">]</span>
<span class="n">promo_prob</span> <span class="o">=</span> <span class="n">M_promoP</span><span class="p">[</span><span class="n">sizeid</span><span class="p">][</span><span class="n">brandid</span><span class="p">]</span>
<span class="k">return</span> <span class="n">stats</span><span class="o">.</span><span class="n">bernoulli</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">promo_prob</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">gen_random_discount</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> If the product is discounted on that time period,</span>
<span class="sd"> return a discount amount given the Size and the Brand</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">x</span><span class="p">[</span><span class="s">"Promo"</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">sizeid</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="s">"Size"</span><span class="p">]</span>
<span class="n">brandid</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="s">"Brand"</span><span class="p">]</span>
<span class="n">promo_mean</span> <span class="o">=</span> <span class="n">M_promoM</span><span class="p">[</span><span class="n">sizeid</span><span class="p">][</span><span class="n">brandid</span><span class="p">]</span>
<span class="n">promo_SD</span> <span class="o">=</span> <span class="n">M_promoSD</span><span class="p">[</span><span class="n">sizeid</span><span class="p">][</span><span class="n">brandid</span><span class="p">]</span>
<span class="k">return</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">promo_mean</span><span class="p">,</span> <span class="n">promo_SD</span><span class="p">)</span>
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<h3 id="3°)-Generation-of-the-BasePrice,-Promotion-Status-and-Discount-Amount-for-all-the-weeks-and-Brand/Size-pairs">3°) Generation of the BasePrice, Promotion Status and Discount Amount for all the weeks and Brand/Size pairs<a class="anchor-link" href="#3°)-Generation-of-the-BasePrice,-Promotion-Status-and-Discount-Amount-for-all-the-weeks-and-Brand/Size-pairs">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="n">DF_Supply</span><span class="p">[</span><span class="s">"Option"</span><span class="p">]</span> <span class="o">=</span> <span class="n">DF_Supply</span><span class="p">[</span><span class="s">"Size"</span><span class="p">]</span><span class="o">*</span><span class="mi">5</span> <span class="o">+</span> <span class="n">DF_Supply</span><span class="p">[</span><span class="s">"Brand"</span><span class="p">]</span>
<span class="n">DF_Supply</span><span class="p">[</span><span class="s">"BasePrice"</span><span class="p">]</span> <span class="o">=</span> <span class="n">DF_Supply</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">gen_random_price</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">DF_Supply</span><span class="p">[</span><span class="s">"Promo"</span><span class="p">]</span> <span class="o">=</span> <span class="n">DF_Supply</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">gen_random_promo</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">DF_Supply</span><span class="p">[</span><span class="s">"PromoSize"</span><span class="p">]</span> <span class="o">=</span> <span class="n">DF_Supply</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">gen_random_discount</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
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<h3 id="4°)-Fully-generated-"Supply"-dataset:">4°) Fully generated "Supply" dataset:<a class="anchor-link" href="#4°)-Fully-generated-"Supply"-dataset:">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="n">DF_Supply</span><span class="o">.</span><span class="n">head</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
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<div style="max-height:1000px;max-width:1500px;overflow:auto;">
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>Week</th>
<th>Size</th>
<th>Brand</th>
<th>BasePrice</th>
<th>Promo</th>
<th>PromoSize</th>
<th>Option</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td> 1</td>
<td> 0</td>
<td> 0</td>
<td> 3.22</td>
<td> 0</td>
<td> 0</td>
<td> 0</td>
</tr>
<tr>
<th>1</th>
<td> 1</td>
<td> 0</td>
<td> 1</td>
<td> 2.69</td>
<td> 0</td>
<td> 0</td>
<td> 1</td>
</tr>
<tr>
<th>2</th>
<td> 1</td>
<td> 0</td>
<td> 2</td>
<td> 2.59</td>
<td> 0</td>
<td> 0</td>
<td> 2</td>
</tr>
<tr>
<th>3</th>
<td> 1</td>
<td> 0</td>
<td> 3</td>
<td> 4.23</td>
<td> 0</td>
<td> 0</td>
<td> 3</td>
</tr>
<tr>
<th>4</th>
<td> 1</td>
<td> 0</td>
<td> 4</td>
<td> 1.78</td>
<td> 0</td>
<td> 0</td>
<td> 4</td>
</tr>
<tr>
<th>5</th>
<td> 1</td>
<td> 1</td>
<td> 0</td>
<td> 3.02</td>
<td> 0</td>
<td> 0</td>
<td> 5</td>
</tr>
<tr>
<th>6</th>
<td> 1</td>
<td> 1</td>
<td> 1</td>
<td> 2.51</td>
<td> 0</td>
<td> 0</td>
<td> 6</td>
</tr>
<tr>
<th>7</th>
<td> 1</td>
<td> 1</td>
<td> 2</td>
<td> 2.06</td>
<td> 0</td>
<td> 0</td>
<td> 7</td>
</tr>
<tr>
<th>8</th>
<td> 1</td>
<td> 1</td>
<td> 3</td>
<td> 1.43</td>
<td> 0</td>
<td> 0</td>
<td> 8</td>
</tr>
<tr>
<th>9</th>
<td> 1</td>
<td> 1</td>
<td> 4</td>
<td> 2.11</td>
<td> 0</td>
<td> 0</td>
<td> 9</td>
</tr>
</tbody>
</table>
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<h2 id="B.-Initialization-of-the-Demand-dataframe">B. Initialization of the Demand dataframe<a class="anchor-link" href="#B.-Initialization-of-the-Demand-dataframe">¶</a></h2>
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<h3 id="1°)-Simulated-choices-for-the-first-two-weeks,-and-missing-values-for-the-following-ones">1°) Simulated choices for the first two weeks, and missing values for the following ones<a class="anchor-link" href="#1°)-Simulated-choices-for-the-first-two-weeks,-and-missing-values-for-the-following-ones">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="n">households</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N_Total</span><span class="o">+</span><span class="mi">1</span><span class="p">)]</span>
<span class="n">households_weeks_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="n">weeks</span><span class="p">,</span> <span class="n">households</span><span class="p">)]</span>
<span class="n">weeks_list</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([</span><span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">households_weeks_list</span><span class="p">])</span>
<span class="n">households_list</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([</span><span class="n">i</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">households_weeks_list</span><span class="p">])</span>
<span class="n">sizechoice_list</span> <span class="o">=</span> <span class="n">brandchoice_list</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([</span><span class="mi">0</span><span class="p">])</span>
<span class="n">DF_Demand</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">concat</span><span class="p">([</span><span class="n">weeks_list</span><span class="p">,</span> <span class="n">households_list</span><span class="p">,</span> <span class="n">sizechoice_list</span><span class="p">,</span> <span class="n">brandchoice_list</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">colsPurchase</span> <span class="o">=</span> <span class="p">[</span><span class="s">"Week"</span><span class="p">,</span> <span class="s">"Household"</span><span class="p">,</span><span class="s">"Size"</span><span class="p">,</span> <span class="s">"Brand"</span><span class="p">]</span>
<span class="n">DF_Demand</span><span class="o">.</span><span class="n">columns</span> <span class="o">=</span> <span class="n">colsPurchase</span>
<span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="mi">1</span><span class="p">,</span> <span class="s">"Size"</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="n">N_Total</span><span class="p">)</span>
<span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="mi">1</span><span class="p">,</span> <span class="s">"Brand"</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="n">N_Total</span><span class="p">)</span>
<span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="mi">2</span><span class="p">,</span> <span class="s">"Size"</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="n">N_Total</span><span class="p">)</span>
<span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="mi">2</span><span class="p">,</span> <span class="s">"Brand"</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="n">N_Total</span><span class="p">)</span>
<span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[:,</span> <span class="s">"Option"</span><span class="p">]</span> <span class="o">=</span> <span class="n">DF_Demand</span><span class="p">[</span><span class="s">"Size"</span><span class="p">]</span><span class="o">*</span><span class="mi">5</span> <span class="o">+</span> <span class="n">DF_Demand</span><span class="p">[</span><span class="s">"Brand"</span><span class="p">]</span>
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<h3 id="2°)-Definition-of-the-data-generating-functions-for-the-following-weeks">2°) Definition of the data-generating functions for the following weeks<a class="anchor-link" href="#2°)-Definition-of-the-data-generating-functions-for-the-following-weeks">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">update_prev_promo</span><span class="p">(</span><span class="n">week</span><span class="p">,</span> <span class="n">past</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Return the brand code if the past purchase was a promotional purchase, otherwise returns a number greater than 10</span>
<span class="sd"> """</span>
<span class="n">colname</span> <span class="o">=</span> <span class="s">"PrevPromo{0}"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">past</span><span class="p">)</span>
<span class="n">promo_history</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">DF_Supply</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Supply</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="p">(</span><span class="n">week</span><span class="o">-</span><span class="n">past</span><span class="p">),</span>
<span class="p">[</span><span class="s">"Brand"</span><span class="p">,</span> <span class="s">"Promo"</span><span class="p">]])</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s">"Brand"</span><span class="p">])</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span><span class="o">.</span><span class="n">reset_index</span><span class="p">()</span>
<span class="k">if</span> <span class="n">week</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
<span class="n">DF_Demand</span><span class="p">[</span><span class="n">colname</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">NaN</span> <span class="c">#Initialize the past brand column</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">merge</span><span class="p">(</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="p">(</span><span class="n">week</span><span class="o">-</span><span class="n">past</span><span class="p">)],</span> <span class="n">promo_history</span><span class="p">,</span> <span class="n">on</span><span class="o">=</span><span class="p">[</span><span class="s">"Brand"</span><span class="p">],</span> <span class="n">how</span><span class="o">=</span><span class="s">"left"</span><span class="p">)</span><span class="o">.</span><span class="n">sort</span><span class="p">([</span><span class="s">"Week"</span><span class="p">,</span> <span class="s">"Household"</span><span class="p">])</span>
<span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="n">week</span><span class="p">,</span> <span class="n">colname</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">out</span><span class="p">[</span><span class="s">"Promo"</span><span class="p">]</span><span class="o">==</span><span class="mi">0</span><span class="p">)</span><span class="o">*</span><span class="mi">10</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">out</span><span class="p">[</span><span class="s">"Brand"</span><span class="p">])</span>
<span class="k">def</span> <span class="nf">update_all</span><span class="p">(</span><span class="n">week</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Update all the previous period variables, as well as the loyalty</span>
<span class="sd"> """</span>
<span class="n">update_prev_promo</span><span class="p">(</span><span class="n">week</span><span class="p">,</span> <span class="n">past</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">update_prev_promo</span><span class="p">(</span><span class="n">week</span><span class="p">,</span> <span class="n">past</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">update_BLoyalty</span><span class="p">(</span><span class="n">week</span><span class="p">)</span>
<span class="n">update_SLoyalty</span><span class="p">(</span><span class="n">week</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">gen_choices</span><span class="p">(</span><span class="n">week</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Generate the utility for brand-size options for a given week, and generate a choice of size and brand for all households.</span>
<span class="sd"> """</span>
<span class="c">#Updating the Demand DataFrame to reflect past choices</span>
<span class="n">update_all</span><span class="p">(</span><span class="n">week</span><span class="p">)</span>
<span class="n">mask_week</span> <span class="o">=</span> <span class="p">(</span><span class="n">DF_Supply</span><span class="p">[</span><span class="s">"Week"</span><span class="p">]</span> <span class="o">==</span> <span class="n">week</span><span class="p">)</span>
<span class="c">#Dataframe of utilities, one row per household</span>
<span class="n">utilities</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="mi">10</span><span class="p">]</span><span class="o">*</span><span class="n">N_Total</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s">"Option_{0}"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)])</span>
<span class="n">current_week</span> <span class="o">=</span> <span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="n">week</span><span class="p">]</span>
<span class="c">#Looping over sizes and brands to create the utility for each option</span>
<span class="k">for</span> <span class="n">size</span> <span class="ow">in</span> <span class="n">V_Sizes</span><span class="p">:</span>
<span class="n">mask_size</span> <span class="o">=</span> <span class="p">(</span><span class="n">DF_Supply</span><span class="p">[</span><span class="s">"Size"</span><span class="p">]</span> <span class="o">==</span> <span class="n">size</span><span class="p">)</span>
<span class="k">for</span> <span class="n">brand</span> <span class="ow">in</span> <span class="n">V_Brands</span><span class="p">:</span>
<span class="n">option_number</span> <span class="o">=</span> <span class="n">size</span><span class="o">*</span><span class="mi">5</span><span class="o">+</span><span class="n">brand</span>
<span class="n">mask_brand</span> <span class="o">=</span> <span class="p">(</span><span class="n">DF_Supply</span><span class="p">[</span><span class="s">"Brand"</span><span class="p">]</span> <span class="o">==</span> <span class="n">brand</span><span class="p">)</span>
<span class="c">#Product infos from the Supply dataframe</span>
<span class="n">product_infos</span> <span class="o">=</span> <span class="n">DF_Supply</span><span class="p">[</span><span class="n">mask_week</span> <span class="o">&</span> <span class="n">mask_brand</span> <span class="o">&</span> <span class="n">mask_size</span><span class="p">]</span>
<span class="c">#Option-specific utility components (Scalar)</span>
<span class="n">utility1</span> <span class="o">=</span> <span class="n">b0</span><span class="p">[</span><span class="n">size</span><span class="p">][</span><span class="n">brand</span><span class="p">]</span> <span class="o">+</span> <span class="n">b1</span><span class="o">*</span><span class="n">product_infos</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="s">"BasePrice"</span><span class="p">]</span> <span class="o">+</span> \
<span class="n">b2</span><span class="o">*</span><span class="n">product_infos</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="s">"Promo"</span><span class="p">]</span> <span class="o">+</span> <span class="n">b3</span><span class="o">*</span><span class="n">product_infos</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="s">"PromoSize"</span><span class="p">]</span>
<span class="c">#Household-option-specific utility components (Vector of households)</span>
<span class="n">utility2</span> <span class="o">=</span> <span class="n">b4</span><span class="o">*</span><span class="p">(</span><span class="n">current_week</span><span class="p">[</span><span class="s">"PrevPromo1"</span><span class="p">]</span><span class="o">==</span><span class="n">brand</span><span class="p">)</span> <span class="o">+</span> <span class="n">b5</span><span class="o">*</span><span class="p">(</span><span class="n">current_week</span><span class="p">[</span><span class="s">"PrevPromo2"</span><span class="p">]</span><span class="o">==</span><span class="n">brand</span><span class="p">)</span> <span class="o">+</span> \
<span class="n">b6</span><span class="o">*</span><span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="n">week</span><span class="p">,</span> <span class="n">brand</span><span class="p">]</span> <span class="o">+</span> \
<span class="n">b7</span><span class="o">*</span><span class="n">DF_SLoyalty</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_SLoyalty</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="n">week</span><span class="p">,</span> <span class="n">size</span><span class="p">]</span>
<span class="c">#Assigning the utility in the matrix</span>
<span class="n">utilities</span><span class="p">[</span><span class="s">"Option_{0}"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">option_number</span><span class="p">)]</span> <span class="o">=</span> <span class="n">utility1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">utility2</span><span class="p">)</span>
<span class="c">#Converting the utilities to exp-utilities</span>
<span class="n">utilities</span> <span class="o">=</span> <span class="n">utilities</span><span class="o">.</span><span class="n">applymap</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
<span class="n">utilities</span><span class="p">[</span><span class="s">"TOTAL"</span><span class="p">]</span> <span class="o">=</span> <span class="n">utilities</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="nb">sum</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="c">#Converting the utilities to choice probabilities </span>
<span class="n">proba</span> <span class="o">=</span> <span class="n">utilities</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">/</span><span class="n">x</span><span class="p">[</span><span class="s">"TOTAL"</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="c">#Formulating the choice: random sampling of options weighted by choice probabilities</span>
<span class="n">choice</span> <span class="o">=</span> <span class="n">proba</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span> <span class="n">p</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="c">#Updating the dataframe with the choices</span>
<span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="n">week</span><span class="p">,</span> <span class="s">"Option"</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
<span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="n">week</span><span class="p">,</span> <span class="s">"Size"</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span><span class="o">//</span><span class="mi">5</span>
<span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="n">week</span><span class="p">,</span> <span class="s">"Brand"</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span><span class="o">%</span><span class="k">5</span>
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<h2 id="C.-Initialization-of-the-Brand-Loyalty-and-Size-Loyalty-dataframes:">C. Initialization of the Brand Loyalty and Size Loyalty dataframes:<a class="anchor-link" href="#C.-Initialization-of-the-Brand-Loyalty-and-Size-Loyalty-dataframes:">¶</a></h2>
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<div class=" highlight hl-ipython3"><pre><span class="n">households</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N_Total</span><span class="o">+</span><span class="mi">1</span><span class="p">)]</span>
<span class="n">households_weeks_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="n">weeks</span><span class="p">,</span> <span class="n">households</span><span class="p">)]</span>
<span class="n">weeks_list</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([</span><span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">households_weeks_list</span><span class="p">])</span>
<span class="n">households_list</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">([</span><span class="n">i</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">households_weeks_list</span><span class="p">])</span>
<span class="n">DF_BLoyalty</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">concat</span><span class="p">([</span><span class="n">weeks_list</span><span class="p">,</span> <span class="n">households_list</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s">"Week"</span><span class="p">,</span> <span class="s">"Household"</span><span class="p">]</span>
<span class="k">for</span> <span class="n">brand</span> <span class="ow">in</span> <span class="n">V_Brands</span><span class="p">:</span>
<span class="n">DF_BLoyalty</span><span class="p">[</span><span class="n">brand</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">NaN</span>
<span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s">"Week"</span><span class="p">,</span> <span class="s">"Household"</span><span class="p">]</span><span class="o">+</span><span class="nb">list</span><span class="p">(</span><span class="n">V_Brands</span><span class="p">)</span>
<span class="n">DF_SLoyalty</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">concat</span><span class="p">([</span><span class="n">weeks_list</span><span class="p">,</span> <span class="n">households_list</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">DF_SLoyalty</span><span class="o">.</span><span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s">"Week"</span><span class="p">,</span> <span class="s">"Household"</span><span class="p">]</span>
<span class="k">for</span> <span class="n">size</span> <span class="ow">in</span> <span class="n">V_Sizes</span><span class="p">:</span>
<span class="n">DF_SLoyalty</span><span class="p">[</span><span class="n">size</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">NaN</span>
<span class="n">DF_SLoyalty</span><span class="o">.</span><span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s">"Week"</span><span class="p">,</span> <span class="s">"Household"</span><span class="p">]</span><span class="o">+</span><span class="nb">list</span><span class="p">(</span><span class="n">V_Sizes</span><span class="p">)</span>
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<h3 id="1°)-Generating-Brand-and-Size-Loyalties-in-accordance-to-the-first-choice-of-each-household-(Week-1)">1°) Generating Brand and Size Loyalties in accordance to the first choice of each household (Week 1)<a class="anchor-link" href="#1°)-Generating-Brand-and-Size-Loyalties-in-accordance-to-the-first-choice-of-each-household-(Week-1)">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="k">for</span> <span class="n">brand</span> <span class="ow">in</span> <span class="n">V_Brands</span><span class="p">:</span>
<span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="mi">1</span><span class="p">,</span> <span class="n">brand</span><span class="p">]</span> <span class="o">=</span> \
<span class="n">alpha_B</span><span class="o">*</span><span class="p">(</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="mi">1</span><span class="p">,</span> <span class="s">"Brand"</span><span class="p">]</span><span class="o">==</span><span class="n">brand</span><span class="p">)</span> <span class="o">+</span> \
<span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">alpha_B</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">V_Brands</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="mi">1</span><span class="p">,</span> <span class="s">"Brand"</span><span class="p">]</span><span class="o">!=</span><span class="n">brand</span><span class="p">)</span>
<span class="k">for</span> <span class="n">size</span> <span class="ow">in</span> <span class="n">V_Sizes</span><span class="p">:</span>
<span class="n">DF_SLoyalty</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_SLoyalty</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="mi">1</span><span class="p">,</span> <span class="n">size</span><span class="p">]</span> <span class="o">=</span> \
<span class="n">alpha_S</span><span class="o">*</span><span class="p">(</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="mi">1</span><span class="p">,</span> <span class="s">"Size"</span><span class="p">]</span><span class="o">==</span><span class="n">size</span><span class="p">)</span> <span class="o">+</span> \
<span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">alpha_S</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="mi">1</span><span class="p">,</span> <span class="s">"Size"</span><span class="p">]</span><span class="o">!=</span><span class="n">size</span><span class="p">)</span>
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<h3 id="2°)-Loyalty-generating-functions">2°) Loyalty generating functions<a class="anchor-link" href="#2°)-Loyalty-generating-functions">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">update_BLoyalty</span><span class="p">(</span><span class="n">week</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
<span class="k">for</span> <span class="n">brand</span> <span class="ow">in</span> <span class="n">V_Brands</span><span class="p">:</span>
<span class="n">new_loyalty</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">alpha_B</span><span class="o">*</span><span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="p">(</span><span class="n">week</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">brand</span><span class="p">]</span> <span class="o">+</span> \
<span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">alpha_B</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="p">(</span><span class="n">week</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="s">"Brand"</span><span class="p">]</span><span class="o">==</span><span class="n">brand</span><span class="p">))</span>
<span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="n">week</span><span class="p">,</span> <span class="n">brand</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_loyalty</span>
<span class="k">def</span> <span class="nf">update_SLoyalty</span><span class="p">(</span><span class="n">week</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
<span class="k">for</span> <span class="n">size</span> <span class="ow">in</span> <span class="n">V_Sizes</span><span class="p">:</span>
<span class="n">new_loyalty</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">alpha_S</span><span class="o">*</span><span class="n">DF_SLoyalty</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_SLoyalty</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="p">(</span><span class="n">week</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">size</span><span class="p">]</span> <span class="o">+</span> \
<span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">alpha_S</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_Demand</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="p">(</span><span class="n">week</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="s">"Size"</span><span class="p">]</span><span class="o">==</span><span class="n">size</span><span class="p">))</span>
<span class="n">DF_SLoyalty</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">DF_SLoyalty</span><span class="o">.</span><span class="n">Week</span><span class="o">==</span><span class="n">week</span><span class="p">,</span> <span class="n">size</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_loyalty</span>
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<h3 id="3°)-Generation-of-the-loyalty-data-for-week-2">3°) Generation of the loyalty data for week 2<a class="anchor-link" href="#3°)-Generation-of-the-loyalty-data-for-week-2">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="n">update_BLoyalty</span><span class="p">(</span><span class="n">week</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">update_SLoyalty</span><span class="p">(</span><span class="n">week</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
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<h3 id="4°)-Fully-generated-loyalty-data-for-the-first-two-weeks">4°) Fully generated loyalty data for the first two weeks<a class="anchor-link" href="#4°)-Fully-generated-loyalty-data-for-the-first-two-weeks">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="n">DF_BLoyalty</span><span class="o">.</span><span class="n">head</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
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<th></th>
<th>Week</th>
<th>Household</th>
<th>0</th>
<th>1</th>
<th>2</th>
<th>3</th>
<th>4</th>
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