离散数据结构上的高斯过程#

此示例说明了如何使用高斯过程对非固定长度特征向量形式的数据进行回归和分类任务。这是通过使用直接作用于离散结构（例如变长序列、树和图）的核函数来实现的。

具体来说，这里的输入变量是一些基因序列，存储为由字母“A”、“T”、“C”和“G”组成的变长字符串，而输出变量分别是回归和分类任务中的浮点数和真/假标签。

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所有字母对进行二元逐字母核函数的积分。

此示例将生成三个图形。

在第一个图形中，我们使用颜色图可视化核函数的值，即序列的相似性。此处颜色越亮表示相似性越高。

在第二个图形中，我们展示了对包含 6 个序列的数据集进行的一些回归结果。我们使用第 1、2、4 和 5 个序列作为训练集来预测第 3 和第 6 个序列。

在第三个图形中，我们演示了一个分类模型，该模型通过训练 6 个序列并在另外 5 个序列上进行预测。这里的真实值只是序列中是否存在至少一个“A”。模型在这里进行了四个正确的分类，并失败了一个。

# 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性矩阵#

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:442: 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 0.239 秒）