From 93f3278084664db46cdb230a95631e945da5e2a3 Mon Sep 17 00:00:00 2001 From: Will Dean Date: Wed, 15 Nov 2023 21:08:05 +0100 Subject: [PATCH] add distribution and doc --- docs/examples/linear-regression.md | 66 +- docs/images/linear-regression.png | Bin 932980 -> 1349591 bytes poetry.lock | 2372 +++++++++++++--------------- 3 files changed, 1198 insertions(+), 1240 deletions(-) diff --git a/docs/examples/linear-regression.md b/docs/examples/linear-regression.md index 36cd4a0..614f49d 100644 --- a/docs/examples/linear-regression.md +++ b/docs/examples/linear-regression.md @@ -2,12 +2,20 @@ comments: true --- +We can fit linear regression that includes a predictive distribution for new data using a conjugate prior. This example only has one covariate, but the same approach can be used for multiple covariates. + +## Simulate Data + +We are going to simulate data from a linear regression model. The true intercept is 3.5, the true slope is -2.0, and the true variance is 2.5. + ```python import numpy as np +import pandas as pd + import matplotlib.pyplot as plt -from conjugate.distributions import NormalInverseGamma -from conjugate.models import linear_regression +from conjugate.distributions import NormalInverseGamma, MultivariateStudentT +from conjugate.models import linear_regression, linear_regression_posterior_predictive intercept = 3.5 slope = -2.0 @@ -20,7 +28,13 @@ n_points = 100 x = np.linspace(-x_lim, x_lim, n_points) y = intercept + slope * x + rng.normal(scale=sigma, size=n_points) +``` + +## Define Prior and Find Posterior + +There needs to be a prior for the intercept, slope, and the variance. +```python prior = NormalInverseGamma( mu=np.array([0, 0]), delta_inverse=np.array([[1, 0], [0, 1]]), @@ -28,13 +42,44 @@ prior = NormalInverseGamma( beta=1, ) -X = np.stack([np.ones_like(x), x]).T -posterior = linear_regression( +def create_X(x: np.ndarray) -> np.ndarray: + return np.stack([np.ones_like(x), x]).T + +X = create_X(x) +posterior: NormalInverseGamma = linear_regression( X=X, y=y, normal_inverse_gamma_prior=prior, ) +``` + +## Posterior Predictive for New Data + +The multivariate student-t distribution is used for the posterior predictive distribution. We have to draw samples from it since the scipy implementation does not have a `ppf` method. + +```python + +# New Data +x_lim_new = 1.5 * x_lim +x_new = np.linspace(-x_lim_new, x_lim_new, 20) +X_new = create_X(x_new) +pp: MultivariateStudentT = linear_regression_posterior_predictive(normal_inverse_gamma=posterior, X=X_new) + +samples = pp.dist.rvs(5_000).T +df_samples = pd.DataFrame(samples, index=x_new) + + +``` + +## Plot Results + +We can see that the posterior predictive distribution begins to widen as we move away from the data. + +Overall, the posterior predictive distribution is a good fit for the data. The true line is within the 95% posterior predictive interval. + +```python + def plot_abline(intercept: float, slope: float, ax: plt.Axes = None, **kwargs): """Plot a line from slope and intercept""" @@ -76,7 +121,20 @@ plot_lines( plot_abline(intercept, slope, ax=ax, label="true", color="red") ax.set(xlabel="x", ylabel="y", title="Linear regression with conjugate prior") + +# New Data +ax.plot(x_new, pp.mu, color="green", label="posterior predictive mean") +df_quantile = df_samples.T.quantile([0.025, 0.975]).T +ax.fill_between( + x_new, + df_quantile[0.025], + df_quantile[0.975], + alpha=0.2, + color="green", + label="95% posterior predictive interval", +) ax.legend() +ax.set(xlim=(-x_lim_new, x_lim_new)) plt.show() ``` diff --git a/docs/images/linear-regression.png b/docs/images/linear-regression.png index 6d79a85d4d0196d61b28f6a34baf00b90c5f1bd4..b8e1548311b87038e00a7abcc405ac17050b7d8b 100644 GIT binary patch literal 1349591 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