离散数据结构上的高斯过程#

此示例说明了如何在高斯过程上使用回归和分类任务，这些数据并非以固定长度特征向量形式存在。这通过使用直接在离散结构（如可变长度序列、树和图）上操作的核函数实现。

具体而言，这里输入变量是作为可变长度字符串存储的一些基因序列，由字母“A”、“T”、“C”和“G”组成，而输出变量在回归和分类任务中分别为浮点数和 True/False 标签。

基因序列之间的核函数是使用 R 卷积 [1] 定义的，通过对一对字符串中的所有字母对整合二进制字母级核函数。

此示例将生成三张图。

在第一张图中，我们使用颜色图可视化核函数的值，即序列的相似度。这里，颜色越亮表示相似度越高。

在第二张图中，我们展示了对 6 个序列数据集的一些回归结果。这里我们使用第 1、2、4 和 5 个序列作为训练集，对第 3 和 6 个序列进行预测。

在第三张图中，我们通过在 6 个序列上训练并对另外 5 个序列进行预测来演示一个分类模型。这里的真实情况是序列中是否至少有一个“A”。模型在此做出了四次正确分类，一次失败。

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

import numpy as np

from sklearn.base import clone
from sklearn.gaussian_process import GaussianProcessClassifier, GaussianProcessRegressor
from sklearn.gaussian_process.kernels import GenericKernelMixin, Hyperparameter, Kernel


class SequenceKernel(GenericKernelMixin, Kernel):
    """
    A minimal (but valid) convolutional kernel for sequences of variable
    lengths."""

    def __init__(self, baseline_similarity=0.5, baseline_similarity_bounds=(1e-5, 1)):
        self.baseline_similarity = baseline_similarity
        self.baseline_similarity_bounds = baseline_similarity_bounds

    @property
    def hyperparameter_baseline_similarity(self):
        return Hyperparameter(
            "baseline_similarity", "numeric", self.baseline_similarity_bounds
        )

    def _f(self, s1, s2):
        """
        kernel value between a pair of sequences
        """
        return sum(
            [1.0 if c1 == c2 else self.baseline_similarity for c1 in s1 for c2 in s2]
        )

    def _g(self, s1, s2):
        """
        kernel derivative between a pair of sequences
        """
        return sum([0.0 if c1 == c2 else 1.0 for c1 in s1 for c2 in s2])

    def __call__(self, X, Y=None, eval_gradient=False):
        if Y is None:
            Y = X

        if eval_gradient:
            return (
                np.array([[self._f(x, y) for y in Y] for x in X]),
                np.array([[[self._g(x, y)] for y in Y] for x in X]),
            )
        else:
            return np.array([[self._f(x, y) for y in Y] for x in X])

    def diag(self, X):
        return np.array([self._f(x, x) for x in X])

    def is_stationary(self):
        return False

    def clone_with_theta(self, theta):
        cloned = clone(self)
        cloned.theta = theta
        return cloned


kernel = SequenceKernel()

核函数下的序列相似度矩阵#

import matplotlib.pyplot as plt

X = np.array(["AGCT", "AGC", "AACT", "TAA", "AAA", "GAACA"])

K = kernel(X)
D = kernel.diag(X)

plt.figure(figsize=(8, 5))
plt.imshow(np.diag(D**-0.5).dot(K).dot(np.diag(D**-0.5)))
plt.xticks(np.arange(len(X)), X)
plt.yticks(np.arange(len(X)), X)
plt.title("Sequence similarity under the kernel")
plt.show()

回归#

X = np.array(["AGCT", "AGC", "AACT", "TAA", "AAA", "GAACA"])
Y = np.array([1.0, 1.0, 2.0, 2.0, 3.0, 3.0])

training_idx = [0, 1, 3, 4]
gp = GaussianProcessRegressor(kernel=kernel)
gp.fit(X[training_idx], Y[training_idx])

plt.figure(figsize=(8, 5))
plt.bar(np.arange(len(X)), gp.predict(X), color="b", label="prediction")
plt.bar(training_idx, Y[training_idx], width=0.2, color="r", alpha=1, label="training")
plt.xticks(np.arange(len(X)), X)
plt.title("Regression on sequences")
plt.legend()
plt.show()

分类#

X_train = np.array(["AGCT", "CGA", "TAAC", "TCG", "CTTT", "TGCT"])
# whether there are 'A's in the sequence
Y_train = np.array([True, True, True, False, False, False])

gp = GaussianProcessClassifier(kernel)
gp.fit(X_train, Y_train)

X_test = ["AAA", "ATAG", "CTC", "CT", "C"]
Y_test = [True, True, False, False, False]

plt.figure(figsize=(8, 5))
plt.scatter(
    np.arange(len(X_train)),
    [1.0 if c else -1.0 for c in Y_train],
    s=100,
    marker="o",
    edgecolor="none",
    facecolor=(1, 0.75, 0),
    label="training",
)
plt.scatter(
    len(X_train) + np.arange(len(X_test)),
    [1.0 if c else -1.0 for c in Y_test],
    s=100,
    marker="o",
    edgecolor="none",
    facecolor="r",
    label="truth",
)
plt.scatter(
    len(X_train) + np.arange(len(X_test)),
    [1.0 if c else -1.0 for c in gp.predict(X_test)],
    s=100,
    marker="x",
    facecolor="b",
    linewidth=2,
    label="prediction",
)
plt.xticks(np.arange(len(X_train) + len(X_test)), np.concatenate((X_train, X_test)))
plt.yticks([-1, 1], [False, True])
plt.title("Classification on sequences")
plt.legend()
plt.show()

/home/circleci/project/sklearn/gaussian_process/kernels.py:440: ConvergenceWarning:

The optimal value found for dimension 0 of parameter baseline_similarity is close to the specified lower bound 1e-05. Decreasing the bound and calling fit again may find a better value.

脚本总运行时间： (0 分钟 0.218 秒)

离散数据结构上的高斯过程#

核函数下的序列相似度矩阵#

回归#

分类#

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