Support Vector Regression (SVR)

Last updated 2020-06-03 Read time 4 min

Summary

Support Vector Regression extends SVMs to regression, treating errors within an ε-insensitive tube as zero to reduce outlier impact.
Kernel methods enable flexible non-linear relationships while keeping the model compact via support vectors.
Hyperparameters C, epsilon, and gamma govern the balance between generalization and smoothness.
Feature scaling is essential; wrapping preprocessing and learning in a pipeline ensures consistent transformations.

Intuition #

This method should be interpreted through its assumptions, data conditions, and how parameter choices affect generalization.

Detailed Explanation #

Mathematical formulation #

The optimization problem is

$$ \min_{\mathbf{w}, b, \boldsymbol{\xi}, \boldsymbol{\xi}^*} \frac{1}{2} \|\mathbf{w}\|^2 + C \sum_{i=1}^{n} (\xi_i + \xi_i^*) $$

subject to

$$ \begin{aligned} y_i - (\mathbf{w}^\top \phi(\mathbf{x}_i) + b) &\le \epsilon + \xi_i, \\ (\mathbf{w}^\top \phi(\mathbf{x}_i) + b) - y_i &\le \epsilon + \xi_i^*, \\ \xi_i, \xi_i^* &\ge 0, \end{aligned} $$

where $\phi$ maps inputs into a feature space via the chosen kernel. Solving the dual yields the support vectors and coefficients.

Experiments with Python #

This example demonstrates SVR combined with StandardScaler in a pipeline.

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from __future__ import annotations

import japanize_matplotlib
import matplotlib.pyplot as plt
import numpy as np
from sklearn.metrics import mean_squared_error
from sklearn.model_selection import train_test_split
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import StandardScaler
from sklearn.svm import SVR

def run_svr_demo(
    *,
    n_samples: int = 300,
    train_size: float = 0.75,
    xlabel: str = "input x",
    ylabel: str = "output y",
    label_train: str = "train samples",
    label_test: str = "test samples",
    label_pred: str = "SVR prediction",
    label_truth: str = "ground truth",
    title: str | None = None,
) -> dict[str, float]:
    """Train SVR on synthetic nonlinear data, plot fit, and report metrics.

    Args:
        n_samples: Number of data points sampled from the underlying function.
        train_size: Fraction of data used for training.
        xlabel: X-axis label for the plot.
        ylabel: Y-axis label for the plot.
        label_train: Legend label for training samples.
        label_test: Legend label for test samples.
        label_pred: Legend label for the SVR prediction line.
        label_truth: Legend label for the ground-truth curve.
        title: Optional title for the plot.

    Returns:
        Dictionary containing training and test RMSE values.
    """
    japanize_matplotlib.japanize()
    rng = np.random.default_rng(seed=42)

    X = np.linspace(0.0, 6.0, n_samples, dtype=float)
    y_true = np.sin(X) * 1.5 + 0.3 * np.cos(2 * X)
    y_noisy = y_true + rng.normal(scale=0.2, size=X.shape)

    X_train, X_test, y_train, y_test, y_true_train, y_true_test = train_test_split(
        X[:, np.newaxis],
        y_noisy,
        y_true,
        train_size=train_size,
        random_state=42,
        shuffle=True,
    )

    svr = make_pipeline(
        StandardScaler(),
        SVR(kernel="rbf", C=10.0, epsilon=0.1, gamma="scale"),
    )
    svr.fit(X_train, y_train)

    train_pred = svr.predict(X_train)
    test_pred = svr.predict(X_test)

    grid = np.linspace(0.0, 6.0, 400, dtype=float)[:, np.newaxis]
    grid_truth = np.sin(grid.ravel()) * 1.5 + 0.3 * np.cos(2 * grid.ravel())
    grid_pred = svr.predict(grid)

    fig, ax = plt.subplots(figsize=(10, 5))
    ax.scatter(X_train, y_train, color="#1f77b4", alpha=0.6, label=label_train)
    ax.scatter(X_test, y_test, color="#ff7f0e", alpha=0.6, label=label_test)
    ax.plot(grid, grid_truth, color="#2ca02c", linewidth=2, label=label_truth)
    ax.plot(grid, grid_pred, color="#d62728", linewidth=2, linestyle="--", label=label_pred)
    ax.set_xlabel(xlabel)
    ax.set_ylabel(ylabel)
    if title:
        ax.set_title(title)
    ax.legend()
    fig.tight_layout()
    plt.show()

    return {
        "train_rmse": float(mean_squared_error(y_train, train_pred, squared=False)),
        "test_rmse": float(mean_squared_error(y_test, test_pred, squared=False)),
    }

metrics = run_svr_demo()
print(f"Train RMSE: {metrics['train_rmse']:.3f}")
print(f"Test RMSE: {metrics['test_rmse']:.3f}")

Reading the results #

The pipeline scales training data using its mean and variance, then applies the same transform to the test set.
pred contains predictions for the test features; tuning epsilon and C adjusts the trade-off between overfitting and underfitting.
Increasing the RBF kernel’s gamma focuses on local patterns, whereas smaller values produce smoother functions.

References #

Smola, A. J., & Schölkopf, B. (2004). A Tutorial on Support Vector Regression. Statistics and Computing, 14(3), 199–222.
Vapnik, V. (1995). The Nature of Statistical Learning Theory. Springer.