离散数据结构上的高斯过程#

本示例说明了如何对非固定长度特征向量形式的数据使用高斯过程进行回归和分类任务。这是通过使用直接对离散结构（例如可变长度序列、树和图）进行操作的核函数来实现的。

具体来说，这里的输入变量是一些存储为由字母“A”、“T”、“C”和“G”组成的可变长度字符串的基因序列，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。

使用 R 卷积[1]通过在一对字符串中所有字母对上集成二进制字母核来定义基因序列之间的核。

本示例将生成三张图。

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

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

在第三张图中，我们通过对 6 个序列进行训练并对另外 5 个序列进行预测来演示分类模型。这里的真实情况是序列中是否至少有一个“A”。这里模型进行了四次正确分类，一次失败。

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 0.278 秒）