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<!DOCTYPE html>
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<title>Multilevel Elasticities for a Single SKU - Part III. - Dr. Juan Camilo Orduz</title>
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<article class="article">
<span class="article-duration">7 min read</span>
<h1 class="article-title">Multilevel Elasticities for a Single SKU - Part III.</h1>
<span class="article-date">2024-07-19</span>
<div class="article-content">
<p>In this notebook we continue our simulation study for elasticities, see:</p>
<ul>
<li><p><a href="https://juanitorduz.github.io/multilevel_elasticities_single_sku/">Multilevel Elasticities for a Single SKU - Part I</a></p></li>
<li><p><a href="https://juanitorduz.github.io/multilevel_elasticities_single_sku_2/">Multilevel Elasticities for a Single SKU - Part II</a></p></li>
</ul>
<p>for an introduction to the problem and some models. We extend the covariance model to allow two covariance components on both the intercepts and slopes (coefficient of the <code>median_ income</code> variable). Also, to abstract from a specific framework, we do the implementation in <a href="https://num.pyro.ai/">NumPyro</a>. The corresponding <a href="https://docs.pymc.io/">PyMC</a> is very similar.</p>
<div id="prepare-notebook" class="section level2">
<h2>Prepare Notebook</h2>
<pre class="python"><code>import arviz as az
import jax.numpy as jnp
import matplotlib.pyplot as plt
import numpy as np
import numpyro
import numpyro.distributions as dist
import polars as pl
import seaborn as sns
from jax import random
from numpyro.infer import MCMC, NUTS, Predictive
from sklearn.preprocessing import LabelEncoder
az.style.use("arviz-darkgrid")
plt.rcParams["figure.figsize"] = [12, 7]
plt.rcParams["figure.dpi"] = 100
plt.rcParams["figure.facecolor"] = "white"
numpyro.set_host_device_count(n=4)
rng_key = random.PRNGKey(seed=42)
%load_ext autoreload
%autoreload 2
%config InlineBackend.figure_format = "retina"</code></pre>
</div>
<div id="read-data" class="section level2">
<h2>Read Data</h2>
<p>We read the data from cour case study. YOu can find an explicit data generating process in the first notebook <a href="https://juanitorduz.github.io/multilevel_elasticities_single_sku/">Multilevel Elasticities for a Single SKU - Part I</a>.</p>
<pre class="python"><code>market_df = pl.read_csv(
"https://raw.githubusercontent.com/juanitorduz/website_projects/master/data/multilevel_elasticities_single_sku_data.csv",
schema_overrides={"region_id": pl.Categorical},
)
market_df.head()</code></pre>
<center>
<div>
<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe thead th {
text-align: left;
font-size: 15px;
}
.dataframe tbody tr th {
vertical-align: top;
font-size: 15px;
}
.dataframe tbody tr td {
vertical-align: top;
font-size: 15px;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<small>shape: (5, 10)</small>
<table border="1" class="dataframe">
<thead>
<tr>
<th>
item_id
</th>
<th>
price
</th>
<th>
quantities
</th>
<th>
time_step
</th>
<th>
store_id
</th>
<th>
region_store_id
</th>
<th>
region_id
</th>
<th>
median_income
</th>
<th>
log_price
</th>
<th>
log_quantities
</th>
</tr>
<tr>
<td>
i64
</td>
<td>
f64
</td>
<td>
f64
</td>
<td>
i64
</td>
<td>
i64
</td>
<td>
str
</td>
<td>
cat
</td>
<td>
f64
</td>
<td>
f64
</td>
<td>
f64
</td>
</tr>
</thead>
<tbody>
<tr>
<td>
0
</td>
<td>
1.335446
</td>
<td>
6.170862
</td>
<td>
16
</td>
<td>
0
</td>
<td>
"r-0_s-0"
</td>
<td>
"0"
</td>
<td>
5.873343
</td>
<td>
0.289265
</td>
<td>
1.819839
</td>
</tr>
<tr>
<td>
0
</td>
<td>
1.702792
</td>
<td>
3.715124
</td>
<td>
15
</td>
<td>
0
</td>
<td>
"r-0_s-0"
</td>
<td>
"0"
</td>
<td>
5.873343
</td>
<td>
0.532269
</td>
<td>
1.312412
</td>
</tr>
<tr>
<td>
0
</td>
<td>
1.699778
</td>
<td>
3.290962
</td>
<td>
14
</td>
<td>
0
</td>
<td>
"r-0_s-0"
</td>
<td>
"0"
</td>
<td>
5.873343
</td>
<td>
0.530498
</td>
<td>
1.19118
</td>
</tr>
<tr>
<td>
0
</td>
<td>
1.335844
</td>
<td>
5.702928
</td>
<td>
13
</td>
<td>
0
</td>
<td>
"r-0_s-0"
</td>
<td>
"0"
</td>
<td>
5.873343
</td>
<td>
0.289563
</td>
<td>
1.74098
</td>
</tr>
<tr>
<td>
0
</td>
<td>
1.517213
</td>
<td>
4.264949
</td>
<td>
12
</td>
<td>
0
</td>
<td>
"r-0_s-0"
</td>
<td>
"0"
</td>
<td>
5.873343
</td>
<td>
0.416875
</td>
<td>
1.45043
</td>
</tr>
</tbody>
</table>
</div>
</center>
<p>Recall that our objective is to estimate the elasticity of a single SKU across different regions.</p>
<pre class="python"><code>g = sns.lmplot(
data=market_df,
x="log_price",
y="log_quantities",
hue="region_id",
height=8,
aspect=1.2,
scatter=False,
)
g.set_axis_labels(x_var="Log Prices", y_var="Log Quantities")
legend = g.legend
legend.set_title(title="Region ID", prop={"size": 10})
g.fig.suptitle(
"Log Prices vs Log Quantities by Region", y=1.05, fontsize=18, fontweight="bold"
);</code></pre>
<center>
<img src="../images/multilevel_elasticities_single_sku_3_files/multilevel_elasticities_single_sku_3_6_1.png" style="width: 900px;"/>
</center>
<p>We expect this elasticity to be influenced by the region’s median income.</p>
<pre class="python"><code>fig, ax = plt.subplots(figsize=(9, 6))
(
market_df.group_by("region_id")
.agg(pl.col("median_income").mean())
.to_pandas()
.pipe(
(sns.barplot, "data"), x="region_id", y="median_income", hue="region_id", ax=ax
)
)
ax.set(xlabel="Region ID", ylabel="Median Income")
ax.set_title(label="Median Income by Region", fontsize=18, fontweight="bold");</code></pre>
<center>
<img src="../images/multilevel_elasticities_single_sku_3_files/multilevel_elasticities_single_sku_3_8_0.png" style="width: 900px;"/>
</center>
</div>
<div id="model-specification" class="section level2">
<h2>Model Specification</h2>
<p>As described in the previous notebooks, we can use a multilevel covariance model to estimate the elasticity by constraining how the intercept and slops vary. In the first notebook we did this just for the <code>median_income</code> variable. Here we extend it so that we have two covariance components for the intercept and two for the slope. We do thin in the non-centered parametrization (see <a href="https://fehiepsi.github.io/rethinking-numpyro/14-adventures-in-covariance.html">Statistical Rethinking in NumPyro - Chapter 14. Adventures in Covariance</a>). Here is a mathematical description of the model:</p>
<p><span class="math display">\[\begin{align*}
\log(q) & \sim \text{Normal}(\mu, \sigma ^2) \\
\mu & = \alpha + \beta \log(p) \\
\alpha & = \alpha_{\text{intercept}, \text{region}} + \alpha_{\text{slope}, \text{region}} m \\
\beta & = \beta_{\text{intercept}, \text{region}} + \beta_{\text{slope}, \text{region}} m \\
\alpha_{\text{intercept}, \text{region}} & = z'_{\text{intercept}}[0] \\
\alpha_{\text{slope}, \text{region}} & = z'_{\text{slope}}[0] \\
\beta_{\text{intercept}, \text{region}} & = z'_{\text{intercept}}[1] \\
\beta_{\text{slope}, \text{region}} & = z'_{\text{slope}}[1] \\
z'_{\text{intercept}} &= \text{diag}(\sigma_{\alpha_{\text{intercept}}}^2, \sigma_{\beta_{\text{intercept}}}^2) L_{\text{intercept}} \: z_{\text{intercept}} \\
z'_{\text{slope}} &= \text{diag}(\sigma_{\alpha_{\text{slope}}}^2, \sigma_{\beta_{\text{slope}}}^2) L_{\text{slope}} \: z_{\text{slope}} \\
L_{\text{slope}} & \sim \text{LKJCholesky}(1) \\
z_{\text{intercept}} & \sim \text{Normal}(0, 1) \\
z_{\text{slope}} & \sim \text{Normal}(0, 1)
\end{align*}\]</span></p>
<p>We prepare the data (very similar to the previous notebooks) and then implement the model in <code>NumPyro</code>.</p>
<pre class="python"><code>obs_idx = np.arange(market_df.shape[0])
# For the region_id we use a label encoder to transform the categorical
# variable into integers (in more applications these tags won't be always integers so
# it is good to get used to these encoders.
regions_encoder = LabelEncoder()
regions_encoder.fit(market_df["region_id"].to_numpy())
regions_idx = regions_encoder.transform(market_df["region_id"].to_numpy())
regions = regions_encoder.classes_
log_price = market_df["log_price"].to_jax()
log_quantities = market_df["log_quantities"].to_jax()
# This is just an auxiliary variable to indicate we are modeling
# two covariance components.
effect = ["intercept", "median_income"]</code></pre>
<p>Besides the usual data preparation, we need a map from the encoded category regions to the corresponding median income values:</p>
<pre class="python"><code>region_median_income_mapping_df = market_df.select(
["region_id", "median_income"]
).unique()
region_median_income_mapping_df = region_median_income_mapping_df.with_columns(
region_idx=regions_encoder.transform(
region_median_income_mapping_df["region_id"].to_numpy()
)
).sort("region_idx")
median_income = region_median_income_mapping_df["median_income"].to_jax()</code></pre>
<p>Now we implement the model:</p>
<pre class="python"><code>def model(log_price, regions_idx, median_income, log_quantities=None):
dim = 2
effects = 2
n_obs = log_price.size
n_regions = np.unique(regions_idx).size
# Prior for the degrees of freedom of the Student-T distribution.
nu = numpyro.sample("nu", dist.Gamma(concentration=20.0, rate=3))
# Prior for the scale parameter of the Student-T distribution.
sigma = numpyro.sample("sigma", dist.Exponential(rate=1.0 / 0.5))
# concentration prior for the LKJ prior (this value assumes uniform correlation),
concentration = jnp.ones(1)
with numpyro.plate("effect", effects, dim=-1):
# standard deviation of the effect.
sigma_effect = numpyro.sample(
"sigma_effect", dist.HalfNormal(scale=0.2 * jnp.ones(dim))
)
# correlation matrix
chol_corr = numpyro.sample(
"chol_corr",
dist.LKJCholesky(dimension=dim, concentration=concentration),
)
with numpyro.plate("regions", n_regions, dim=-2):
# z is the latent variable that we will use to compute the covariance matrix
# in the non-centered parameterization.
z = numpyro.sample("z", dist.Normal(loc=0, scale=1))
# We compute the covariance matrix using the cholesky decomposition
chol_cov = z @ (sigma_effect[None, ...] * chol_corr)
# We extract the region effects
alpha_region_effect = chol_cov[:, :, 0]
beta_region_effect = chol_cov[:, :, 1]
# Global intercept component (intercept and median income term)
alpha_region_intercept = alpha_region_effect[0, :]
alpha_region_median_income = alpha_region_effect[1, :]
# Global slope factor of `log_price` (intercept and median income term)
beta_region_intercept = beta_region_effect[0, :]
beta_region_median_income = beta_region_effect[1, :]
# Region specific intercept and slope
alpha_region = numpyro.deterministic(
"alpha_region",
alpha_region_intercept + alpha_region_median_income * median_income,
)
beta_region = numpyro.deterministic(
"beta_region",
beta_region_intercept + beta_region_median_income * median_income,
)
# Likelihood mean
mu = alpha_region[regions_idx] + beta_region[regions_idx] * log_price
with numpyro.plate("data", n_obs):
# We use a Student-T likelihood to model the residuals.
numpyro.sample(
"log_quantities",
dist.StudentT(df=nu, loc=mu, scale=sigma),
obs=log_quantities,
)</code></pre>
<p>Let’s visualize the model:</p>
<pre class="python"><code>numpyro.render_model(
model=model,
model_kwargs={
"log_price": log_price,
"regions_idx": regions_idx,
"median_income": median_income,
"log_quantities": log_quantities,
},
render_distributions=True,
render_params=True,
)</code></pre>
<center>
<img src="../images/multilevel_elasticities_single_sku_3_files/multilevel_elasticities_single_sku_3_18_0.svg" style="width: 900px;"/>
</center>
</div>
<div id="inference" class="section level2">
<h2>Inference</h2>
<p>We now use NUTS to sample from the posterior distribution of the model.</p>
<pre class="python"><code>sampler = NUTS(model)
mcmc = MCMC(
sampler=sampler,
num_warmup=1_000,
num_samples=4_000,
num_chains=4,
)
rng_key, rng_subkey = random.split(rng_key)
mcmc.run(rng_subkey, log_price, regions_idx, median_income, log_quantities)</code></pre>
<p>We collect the results in an <code>az.InferenceData</code> object:</p>
<pre class="python"><code>idata = az.from_numpyro(
mcmc,
coords={"effect": effect, "regions": regions},
dims={
"sigma_effect": ["effect"],
"z": ["regions", "effect"],
"alpha_region": ["regions"],
"beta_region": ["regions"],
},
)</code></pre>
</div>
<div id="model-diagnostics" class="section level2">
<h2>Model Diagnostics</h2>
<pre class="python"><code># Count divergences
idata.sample_stats.diverging.sum().item()</code></pre>
<pre><code>0</code></pre>
<pre class="python"><code>az.summary(idata, var_names=["~z", "~chol_corr"])</code></pre>
<center>
<div>
<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th>
</th>
<th>
mean
</th>
<th>
sd
</th>
<th>
hdi_3%
</th>
<th>
hdi_97%
</th>
<th>
mcse_mean
</th>
<th>
mcse_sd
</th>
<th>
ess_bulk
</th>
<th>
ess_tail
</th>
<th>
r_hat
</th>
</tr>
</thead>
<tbody>
<tr>
<th>
alpha_region[0]
</th>
<td>
2.752
</td>
<td>
0.052
</td>
<td>
2.654
</td>
<td>
2.848
</td>
<td>
0.000
</td>
<td>
0.000
</td>
<td>
18213.0
</td>
<td>
11582.0
</td>
<td>
1.0
</td>
</tr>
<tr>
<th>
alpha_region[1]
</th>
<td>
2.994
</td>
<td>
0.056
</td>
<td>
2.890
</td>
<td>
3.100
</td>
<td>
0.000
</td>
<td>
0.000
</td>
<td>
16905.0
</td>
<td>
11383.0
</td>
<td>
1.0
</td>
</tr>
<tr>
<th>
alpha_region[2]
</th>
<td>
2.440
</td>
<td>
0.035
</td>
<td>
2.376
</td>
<td>
2.507
</td>
<td>
0.000
</td>
<td>
0.000
</td>
<td>
17436.0
</td>
<td>
12368.0
</td>
<td>
1.0
</td>
</tr>
<tr>
<th>
alpha_region[3]
</th>
<td>
3.606
</td>
<td>
0.139
</td>
<td>
3.344
</td>
<td>
3.867
</td>
<td>
0.001
</td>
<td>
0.001
</td>
<td>
16645.0
</td>
<td>
11384.0
</td>
<td>
1.0
</td>
</tr>
<tr>
<th>
alpha_region[4]
</th>
<td>
2.989
</td>
<td>
0.052
</td>
<td>
2.894
</td>
<td>
3.088
</td>
<td>
0.000
</td>
<td>
0.000
</td>
<td>
18849.0
</td>
<td>
11999.0
</td>
<td>
1.0
</td>
</tr>
<tr>
<th>
alpha_region[5]
</th>
<td>
2.827
</td>
<td>
0.057
</td>
<td>
2.720
</td>
<td>
2.935
</td>
<td>
0.000
</td>
<td>
0.000
</td>
<td>
16608.0
</td>
<td>
11715.0
</td>
<td>
1.0
</td>
</tr>
<tr>
<th>
alpha_region[6]
</th>
<td>
3.082
</td>
<td>
0.049
</td>
<td>
2.994
</td>
<td>
3.178
</td>
<td>
0.000
</td>
<td>
0.000
</td>
<td>
18674.0
</td>
<td>
13358.0
</td>
<td>
1.0
</td>
</tr>
<tr>
<th>
alpha_region[7]
</th>
<td>
2.343
</td>
<td>
0.029
</td>
<td>
2.291
</td>
<td>
2.398
</td>
<td>
0.000
</td>
<td>
0.000
</td>
<td>
18050.0
</td>
<td>
13305.0
</td>
<td>
1.0
</td>
</tr>
<tr>
<th>
alpha_region[8]
</th>
<td>
2.541
</td>
<td>
0.039
</td>
<td>
2.470
</td>
<td>
2.615
</td>
<td>
0.000
</td>
<td>
0.000
</td>
<td>
16718.0
</td>
<td>
12278.0
</td>
<td>
1.0
</td>
</tr>
<tr>
<th>
beta_region[0]
</th>
<td>
-3.558
</td>
<td>
0.124
</td>
<td>
-3.789
</td>
<td>
-3.324
</td>
<td>
0.001
</td>
<td>
0.001
</td>
<td>
18427.0
</td>
<td>
11093.0
</td>
<td>
1.0
</td>
</tr>
<tr>
<th>
beta_region[1]
</th>
<td>
-4.074
</td>
<td>
0.127
</td>
<td>
-4.321
</td>
<td>
-3.844
</td>
<td>
0.001
</td>
<td>
0.001
</td>
<td>
17216.0
</td>
<td>
11973.0
</td>
<td>
1.0
</td>