「t-SNEの教師ありハイパーパラメーターチューニング」の続編です。同じ方法論が、 Isomap にも適用可能ということで、やってみました。

Isomap でパラメータを変化させる

Isomap は scikit-learn に実装されているので、それを使ってみましょう。お試し用のデータとして、scikit-learn で取得できる糖尿病データを用いてみます。

from sklearn.manifold import Isomap

import sklearn.datasets
import matplotlib.pyplot as plt

dataset = sklearn.datasets.load_diabetes()

mapper = Isomap()
embedding = mapper.fit_transform(dataset.data)
plt.scatter(embedding[:, 0], embedding[:, 1], c=dataset.target, alpha=0.5)
plt.colorbar()
plt.show()

ここで、点の色は dataset.target にある変数で、この値は Isomap の学習に用いられていないことに注意してください。 Isomap は「教師なし学習」の一種であり、学習に用いた高次元データの構造を可視化する働きをするのですが、その結果として、学習に用いていない変数との関係が見えてくることもあります。

さて、パラメータを変えると、このマッピングがどのくらい変わってしまうのかを見てみましょう。

n_neighbors

from sklearn.manifold import Isomap
import sklearn.datasets
import matplotlib.pyplot as plt

dataset = sklearn.datasets.load_diabetes()

for n_neighbors in (5, 10, 30, 50):
    mapper = Isomap(n_neighbors=n_neighbors)
    embedding = mapper.fit_transform(dataset.data)
    title='n_neighbors = {0}'.format(n_neighbors)
    plt.title(title)
    plt.scatter(embedding[:, 0], embedding[:, 1], c=dataset.target, alpha=0.5)
    plt.colorbar()
    plt.show()

eigen_solver

from sklearn.manifold import Isomap
import sklearn.datasets
import matplotlib.pyplot as plt

dataset = sklearn.datasets.load_diabetes()

for eigen_solver in ("arpack", "dense"):
    mapper = Isomap(eigen_solver=eigen_solver)
    embedding = mapper.fit_transform(dataset.data)
    title='eigen_solver = {0}'.format(eigen_solver)
    plt.title(title)
    plt.scatter(embedding[:, 0], embedding[:, 1], c=dataset.target, alpha=0.5)
    plt.colorbar()
    plt.show()

path_method

from sklearn.manifold import Isomap
import sklearn.datasets
import matplotlib.pyplot as plt

dataset = sklearn.datasets.load_diabetes()

for path_method in ["FW", "D"]:
    mapper = Isomap(path_method=path_method)
    embedding = mapper.fit_transform(dataset.data)
    title='path_method = {0}'.format(path_method)
    plt.title(title)
    plt.scatter(embedding[:, 0], embedding[:, 1], c=dataset.target, alpha=0.5)
    plt.colorbar()
    plt.show()

neighbors_algorithm

from sklearn.manifold import Isomap
import sklearn.datasets
import matplotlib.pyplot as plt

dataset = sklearn.datasets.load_diabetes()

for neighbors_algorithm in ["brute", "kd_tree", "ball_tree"]:
    mapper = Isomap(neighbors_algorithm=neighbors_algorithm)
    embedding = mapper.fit_transform(dataset.data)
    title='neighbors_algorithm = {0}'.format(neighbors_algorithm)
    plt.title(title)
    plt.scatter(embedding[:, 0], embedding[:, 1], c=dataset.target, alpha=0.5)
    plt.colorbar()
    plt.show()

t-SNE よりは、パラメータ依存が小さいように見えますね。なので、ぶっちゃけ、これ以降の解析の意味はあまりないのかもしれません。

回帰問題用スコア関数

教師なし学習である Isomap を「教師あり Isomap」にするためのスコア関数を設計しましょう。回帰問題の場合は、たとえば次のように作れると思います。マッピングされた２点に対して、目的変数の差を計算し、それを２点間のユークリッド距離で割った数値の総和です。

def regression_scorer(X, Y):
    sum = 0
    n1 = 0
    for x1, y1 in zip(X, Y):
        n1 += 1
        n2 = 0
        for x2, y2 in zip(X, Y):
            n2 += 1
            if n1 > n2:
                dist = ((x1[0] - x2[0])**2 + (x1[1] - x2[1])**2) + 1e-53
                sum += (y1 - y2)**2 / dist

    return sum / (len(Y) * (len(Y) - 1) / 2)

Optuna で最適化

パラメータの最適化といえば Optuna。Optuna と言えばパラメータの最適化。ということでインストールします。

!pip install optuna

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教師あり Isomap

Optuna を使って、Isomap を最適化するクラスを設計します。

import scipy

class SupervisedIsomap:
    def __init__(self, X, Y, scorer):
        self.X = X
        self.Y = Y
        self.scorer = scorer
        self.best_score = 1e53
        self.best_model = None

    def __call__(self, trial):
        n_neighbors = trial.suggest_int("n_neighbors", 5, 50)
        eigen_solver = trial.suggest_categorical("eigen_solver", ["arpack", "dense"])
        path_method = trial.suggest_categorical("path_method", ["FW", "D"])
        neighbors_algorithm = trial.suggest_categorical("neighbors_algorithm", ["brute", "kd_tree", "ball_tree"])

        mapper = Isomap(
            n_neighbors=n_neighbors, 
            eigen_solver=eigen_solver,
            path_method=path_method,
            neighbors_algorithm=neighbors_algorithm,
            )
        embedding = mapper.fit_transform(self.X)
        score = self.scorer(scipy.stats.zscore(embedding), self.Y)

        if self.best_score > score:
            self.best_score = score
            self.best_model = mapper
            
            print(self.best_model)
            title='trial={0}, score={1:.3e}'.format(trial.number, score)
            plt.title(title)
            plt.scatter(embedding[:, 0], embedding[:, 1], c=self.Y, alpha=0.5)
            plt.colorbar()
            plt.show()      

        return score

最適化計算の実行例です。いくらか省略しながら結果を示します。

import optuna

objective = SupervisedIsomap(dataset.data, dataset.target, regression_scorer)
study = optuna.create_study(direction="minimize")
study.optimize(objective, n_trials=100)

Isomap(eigen_solver='dense', n_neighbors=50, neighbors_algorithm='brute',
       path_method='FW')

Isomap(eigen_solver='dense', n_neighbors=41, neighbors_algorithm='brute',
       path_method='FW')

Isomap(eigen_solver='arpack', n_neighbors=41, neighbors_algorithm='brute',
       path_method='D')

Isomap(eigen_solver='arpack', n_neighbors=44, neighbors_algorithm='brute',
       path_method='D')

Isomap(eigen_solver='arpack', n_neighbors=44, neighbors_algorithm='ball_tree',
       path_method='D')

Isomap(eigen_solver='dense', n_neighbors=47, neighbors_algorithm='brute',
       path_method='FW')

目的変数の分離が最もしやすいパラメータが選択できたのではないかなと思います。

分類問題用スコア関数

回帰問題用スコア関数を少しだけ改変して、分類問題用スコア関数にします。分類問題と異なり、回帰問題では異なるクラス間の「差」は常に１であるとします。

def classification_scorer(X, Y, alpha=1e-3):
    sum = 0
    n1 = 0
    for x1, y1 in zip(X, Y):
        n1 += 1
        n2 = 0
        for x2, y2 in zip(X, Y):
            n2 += 1
            if n1 > n2:
                dist = ((x1[0] - x2[0])**2 + (x1[1] - x2[1])**2) + 1e-53
                if y1 != y2:
                    sum += 1 / dist
                else:
                    pass
                    
    return sum / (len(Y) * (len(Y) - 1) / 2)

それ以外は、回帰問題と同じですね。

乳がんデータセット

乳がんデータセットを用いた２値分類の例です。

from sklearn.datasets import load_breast_cancer

dataset = load_breast_cancer()

import optuna

objective = SupervisedIsomap(dataset.data, dataset.target, classification_scorer)
study = optuna.create_study(direction="minimize")
study.optimize(objective, n_trials=100)

Isomap(eigen_solver='dense', n_neighbors=47, neighbors_algorithm='ball_tree',
       path_method='D')

Isomap(eigen_solver='arpack', n_neighbors=47, neighbors_algorithm='brute',
       path_method='FW')

Isomap(eigen_solver='arpack', n_neighbors=47, neighbors_algorithm='brute',
       path_method='FW')

Isomap(eigen_solver='arpack', n_neighbors=47, neighbors_algorithm='brute',
       path_method='FW')

Isomap(eigen_solver='arpack', n_neighbors=47, neighbors_algorithm='brute',
       path_method='FW')

Isomap(eigen_solver='arpack', n_neighbors=47, neighbors_algorithm='brute',
       path_method='FW')

ワインデータセット

ワインデータセットを用いた例です。このように、ラベルが３種類以上の場合でも実行可能です。

from sklearn.datasets import load_wine

dataset = load_wine()

import optuna

objective = SupervisedIsomap(dataset.data, dataset.target, classification_scorer)
study = optuna.create_study(direction="minimize")
study.optimize(objective, n_trials=100)

Isomap(eigen_solver='arpack', n_neighbors=8, neighbors_algorithm='ball_tree',
       path_method='D')

Isomap(eigen_solver='arpack', n_neighbors=42, neighbors_algorithm='ball_tree',
       path_method='FW')

Isomap(eigen_solver='dense', n_neighbors=42, neighbors_algorithm='brute',
       path_method='FW')

Isomap(eigen_solver='arpack', n_neighbors=45, neighbors_algorithm='brute',
       path_method='D')

Isomap(eigen_solver='arpack', n_neighbors=45, neighbors_algorithm='ball_tree',
       path_method='D')

手書き数字データセット

手書き数字データセットを用いた例です。この例ではラベルが１０種類あります。

from sklearn.datasets import load_digits

dataset = load_digits()

import optuna

objective = SupervisedIsomap(dataset.data, dataset.target, classification_scorer)
study = optuna.create_study(direction="minimize")
study.optimize(objective, n_trials=100)

Isomap(eigen_solver='arpack', n_neighbors=32, neighbors_algorithm='kd_tree',
       path_method='D')

Isomap(eigen_solver='arpack', n_neighbors=21, neighbors_algorithm='kd_tree',
       path_method='FW')

Isomap(eigen_solver='dense', n_neighbors=45, neighbors_algorithm='brute',
       path_method='FW')

Isomap(eigen_solver='dense', n_neighbors=50, neighbors_algorithm='kd_tree',
       path_method='FW')

Isomap(eigen_solver='dense', n_neighbors=44, neighbors_algorithm='kd_tree',
       path_method='FW')

Isomap(eigen_solver='arpack', n_neighbors=29, neighbors_algorithm='kd_tree',
       path_method='D')

Isomap(eigen_solver='arpack', n_neighbors=29, neighbors_algorithm='kd_tree',
       path_method='D')

おわりに

Isomapは、データにもよりますがパラメータ依存が小さいようですね。t-SNEと比べてみると面白いです。