Center-gap GPQR (correlated)

In this example, the data generating process is $$Y = \cos(2 \pi (X+0.1)) + \epsilon, \quad \epsilon \sim \mathcal{N}(0, X+0.1).$$

Central quantile $Q_{\tau_0}(x)$ and gaps $\Delta Q_{\tau_i}(x)$ of $Y$ are modeled by linear combination of latent GP $g_j(x)$: $$Q_{\tau_0}(x) = g_0(x; \theta), \quad \Delta Q_{\tau_i}(x) = \log \left(1 + \exp \sum_j a_{ij}g_j(x)\right),$$ where $$g_0(x;\theta) \sim \mathcal{N}(\cos(2 \pi x + \theta) + c_0, k(x, x’)), \quad g_j(x) \sim \mathcal{N}(c_j, k(x, x’)).$$

Parameters $\theta$, $a_{ij}$ and $c_i$ are learned by maximizing the marginal likelihood.

import os

import torch
from torch.distributions import Normal
from gpytorch.variational import CholeskyVariationalDistribution
from gpytorch.variational import VariationalStrategy
from gpytorch.means import ConstantMean, Mean
from gpytorch.kernels import RBFKernel, ScaleKernel
from gpytorch.mlls import VariationalELBO
import matplotlib.pyplot as plt

from gpytorch_qr.means import CenterGapMean
from gpytorch_qr.models import CenterGapQuantileGP
from gpytorch_qr.variational import CenterGapLMCVariationalStrategy
from gpytorch_qr.likelihoods import CenterGapQuantileLikelihood

try:
    import sys

    sys.path.insert(0, os.path.abspath("../.."))

    import config_notebook
except ImportError:
    print("Output will not be deterministic SVG.")

torch.manual_seed(42)
device = torch.device("cuda" if torch.cuda.is_available() else "cpu")

n_epochs = int(os.getenv("GPYTORCHQR_N_EPOCHS", 10000))

Data preparation

def f(x):
    return torch.cos((x + 0.1) * 2 * 3.14)


def std(x):
    return x + 0.1


x_range = torch.linspace(0, 1, 100).reshape(-1, 1).to(device)
x = x_range.repeat(5, 1)
y = (f(x) + torch.randn(x.shape, device=device).mul(std(x))).squeeze()

q = torch.tensor([0.1, 0.25, 0.5, 0.75, 0.9]).to(device)
true_quantiles = f(x_range) + std(x_range) * Normal(0, 1).icdf(q)

x_pred = torch.linspace(0, 1.5, 100).reshape(-1, 1).to(device)

plt.scatter(x.cpu(), y.cpu(), c="k", marker=".")
plt.plot(x_range.cpu(), true_quantiles.cpu(), "--", c="gray")
plt.show()

Prior mean

class PriorMean(Mean):
    def __init__(self, batch_shape=torch.Size([])):
        super().__init__()
        self.batch_shape = batch_shape
        self.theta = torch.nn.Parameter(torch.tensor(0.0))
        if len(batch_shape) == 0:
            self.register_parameter("offset", torch.nn.Parameter(torch.tensor(0.0)))
        else:
            self.register_parameter(
                "offset", torch.nn.Parameter(torch.zeros(*batch_shape))
            )

    def forward(self, x):
        # x: (N, D)
        m = torch.cos(2 * 3.14 * x + self.theta)  # (N, D)
        ret = m + self.offset.reshape(*self.offset.shape, 1, 1)  # (B, N, D)
        return ret.squeeze(-1)  # (B, N)


prior_mean = PriorMean().to(device)

plt.scatter(x.cpu(), y.cpu(), c="k", marker=".")
plt.plot(x_pred.cpu(), prior_mean(x_pred).detach().cpu())
plt.show()

Define models and likelihoods

class MyGP(CenterGapQuantileGP):
    def __init__(
        self,
        inducing_points,
        num_quantiles,
        num_lower_quantiles,
        num_latents,
    ):
        N, D = inducing_points.size()
        variational_distribution = CholeskyVariationalDistribution(
            N,
            batch_shape=torch.Size([num_latents]),
        )
        variational_strategy = CenterGapLMCVariationalStrategy(
            VariationalStrategy(
                self,
                inducing_points,
                variational_distribution,
                learn_inducing_locations=True,
            ),
            num_quantiles,
            num_latents,
            num_quantiles=[num_quantiles],
            num_lower_quantiles=[num_lower_quantiles],
        )

        mean = CenterGapMean(
            PriorMean(batch_shape=torch.Size([1])),
            ConstantMean(batch_shape=torch.Size([num_latents - 1])),
        )
        covar = ScaleKernel(
            RBFKernel(ard_num_dims=D, batch_shape=torch.Size([num_latents])),
            batch_shape=torch.Size([num_latents]),
        )
        super().__init__(
            variational_strategy, mean, covar, [num_quantiles], [num_lower_quantiles]
        )


inducing_points = torch.linspace(0, 1, 10).reshape(-1, 1).to(device)
central_q_index = (q - 0.5).abs().argmin().item()
num_latents = len(q) - 2
gp = MyGP(inducing_points, len(q), central_q_index, num_latents).to(device)
likelihood = CenterGapQuantileLikelihood(q, central_q_index).to(device)

Train

gp.train()
likelihood.train()
mll = VariationalELBO(likelihood, gp, num_data=y.numel())
optimizer = torch.optim.Adam(
    list(gp.parameters()) + list(likelihood.parameters()),
    lr=0.001,
)

for _ in range(n_epochs):
    output = gp(x)
    loss = -mll(output, y)
    loss.backward()
    optimizer.step()
    optimizer.zero_grad()

Evaluate

gp.eval()
with torch.no_grad():
    mean_q = gp.mean_quantiles_mc(x_pred)
    lower_q, upper_q = gp.quantile_quantiles_mc(
        x_pred, torch.tensor([0.025, 0.975]).to(device)
    )

Plot result

colors = plt.cm.tab10.colors

plt.scatter(x.cpu(), y.cpu(), c="gray", marker=".", alpha=0.1)
plt.plot(x_range.cpu(), true_quantiles.cpu(), "--", c="k")

for i in range(len(q)):
    plt.plot(x_pred.cpu(), mean_q[:, i].cpu(), color=colors[i])
    plt.fill_between(
        x_pred.cpu().squeeze(),
        lower_q[:, i].cpu(),
        upper_q[:, i].cpu(),
        color=colors[i],
        alpha=0.3,
    )
plt.show()