## ADALINE
パーセプトロンとの違いは、重みの更新方法
* パーセプトロン：単位ステップ関数
* ADALINE：線形活性化関数 \\(\phi(z)\\)

目的関数(Objective function) ... 学習過程で最適化される関数。多くの場合は ** コスト関数 ** (cost function)

## コスト関数
誤差平方和(Sum of Squared Error：SSE)
$$ J(w) = \frac{1}{2}\sum_i(y^{(i)}-\phi(z^{(i)}))^2 $$

利点
* 微分可能である
* 凸関数であるため勾配降下法(gradient descent)を用いてコスト関数を最小化する重みを見つけ出すことができる。

### 勾配降下法を使った重み更新
コスト関数\\( J(w) \\)の勾配\\( \nabla J(w) \\)に沿って1ステップ進む
$$ w := w + \Delta w $$
重みの変化である\\( \Delta w \\)は、負の勾配に学習率\\( \eta \\)を掛けたもの
$$ \Delta w = -\eta\nabla J(w) $$

### 勾配計算(偏微分係数)
$$ \begin{align} \frac{\delta J}{\delta w_j} = \frac{\delta}{\delta w_j}\frac{1}{2}\sum_i(y^{(i)}-\phi(z^{(i)}))^2 \\
= \frac{1}{2}\frac{\delta}{\delta w_j}\sum_i(y^{(i)}-\phi(z^{(i)}))^2 \\
= \frac{1}{2}\sum_i2(y^{(i)}-\phi(z^{(i)}))\frac{\delta}{\delta w_j}(y^{(i)}-\phi(z^{(i)})) \\
= \sum_i(y^{(i)}-\phi(z^{(i)}))\frac{\delta}{\delta w_j}\Bigl( y^{(i)}-\sum_k(w_kx_k^{(i)})\Bigr) \\
= \sum_i(y^{(i)}-\phi(z^{(i)}))(-x_j^{(i)}) \\
= -\sum_i(y^{(i)}-\phi(z^{(i)}))x_j^{(i)} \\
\end{align} $$


```python
class AdalineGD(object):
    """ADAptive LInear NEuron classifier.

    Parameters
    ------------
    eta : float
        Learning rate (between 0.0 and 1.0)
    n_iter : int
        Passes over the training dataset.

    Attributes
    -----------
    w_ : 1d-array
        Weights after fitting.
    cost_ : list
        Sum-of-squares cost function value in each epoch.

    """
    def __init__(self, eta=0.01, n_iter=50):
        self.eta = eta
        self.n_iter = n_iter

    def fit(self, X, y):
        """ Fit training data.

        Parameters
        ----------
        X : {array-like}, shape = [n_samples, n_features]
            Training vectors, where n_samples is the number of samples and
            n_features is the number of features.
        y : array-like, shape = [n_samples]
            Target values.

        Returns
        -------
        self : object

        """
        self.w_ = np.zeros(1 + X.shape[1])
        self.cost_ = []

        for i in range(self.n_iter):
            net_input = self.net_input(X)
            # Please note that the "activation" method has no effect
            # in the code since it is simply an identity function. We
            # could write `output = self.net_input(X)` directly instead.
            # The purpose of the activation is more conceptual, i.e.,  
            # in the case of logistic regression, we could change it to
            # a sigmoid function to implement a logistic regression classifier.
            output = self.activation(X)
            errors = (y - output)
            self.w_[1:] += self.eta * X.T.dot(errors)
            self.w_[0] += self.eta * errors.sum()
            cost = (errors**2).sum() / 2.0
            self.cost_.append(cost)
        return self

    def net_input(self, X):
        """Calculate net input"""
        return np.dot(X, self.w_[1:]) + self.w_[0]

    def activation(self, X):
        """Compute linear activation"""
        return self.net_input(X)

    def predict(self, X):
        """Return class label after unit step"""
        return np.where(self.activation(X) >= 0.0, 1, -1)
```


```python
import pandas as pd

df = pd.read_csv('https://archive.ics.uci.edu/ml/'
        'machine-learning-databases/iris/iris.data', header=None)
import matplotlib.pyplot as plt
import numpy as np

# select setosa and versicolor
y = df.iloc[0:100, 4].values
y = np.where(y == 'Iris-setosa', -1, 1)

# extract sepal length and petal length
X = df.iloc[0:100, [0, 2]].values

fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(8, 4))

ada1 = AdalineGD(n_iter=10, eta=0.01).fit(X, y)
ax[0].plot(range(1, len(ada1.cost_) + 1), np.log10(ada1.cost_), marker='o')
ax[0].set_xlabel('Epochs')
ax[0].set_ylabel('log(Sum-squared-error)')
ax[0].set_title('Adaline - Learning rate 0.01')

ada2 = AdalineGD(n_iter=10, eta=0.0001).fit(X, y)
ax[1].plot(range(1, len(ada2.cost_) + 1), ada2.cost_, marker='o')
ax[1].set_xlabel('Epochs')
ax[1].set_ylabel('Sum-squared-error')
ax[1].set_title('Adaline - Learning rate 0.0001')

plt.tight_layout()
# plt.savefig('./adaline_1.png', dpi=300)
plt.show()
```


```python
from matplotlib.colors import ListedColormap


def plot_decision_regions(X, y, classifier, resolution=0.02):

    # setup marker generator and color map
    markers = ('s', 'x', 'o', '^', 'v')
    colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')
    cmap = ListedColormap(colors[:len(np.unique(y))])

    # plot the decision surface
    x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1
    x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1
    xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution),
                           np.arange(x2_min, x2_max, resolution))
    Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)
    Z = Z.reshape(xx1.shape)
    plt.contourf(xx1, xx2, Z, alpha=0.4, cmap=cmap)
    plt.xlim(xx1.min(), xx1.max())
    plt.ylim(xx2.min(), xx2.max())

    # plot class samples
    for idx, cl in enumerate(np.unique(y)):
        plt.scatter(x=X[y == cl, 0], y=X[y == cl, 1],
                    alpha=0.8, c=cmap(idx),
                    edgecolor='black',
                    marker=markers[idx], 
                    label=cl)
```


```python
# standardize features
X_std = np.copy(X)
X_std[:, 0] = (X[:, 0] - X[:, 0].mean()) / X[:, 0].std()
X_std[:, 1] = (X[:, 1] - X[:, 1].mean()) / X[:, 1].std()

ada = AdalineGD(n_iter=15, eta=0.01)
ada.fit(X_std, y)

plot_decision_regions(X_std, y, classifier=ada)
plt.title('Adaline - Gradient Descent')
plt.xlabel('sepal length [standardized]')
plt.ylabel('petal length [standardized]')
plt.legend(loc='upper left')
plt.tight_layout()
# plt.savefig('./adaline_2.png', dpi=300)
plt.show()

plt.plot(range(1, len(ada.cost_) + 1), ada.cost_, marker='o')
plt.xlabel('Epochs')
plt.ylabel('Sum-squared-error')

plt.tight_layout()
# plt.savefig('./adaline_3.png', dpi=300)
plt.show()
```


```python

```

36a3fc0d9f7f9a08d3abf29f71817425cdd7b91a
ipynb
Jupyter Notebook
python-machine-learning/ch02/ADALINE.ipynb
hide-tono/python-training
eacbb4954b50e3b02d2a8031825f28a4ad8c624c
Apache-2.0
python-machine-learning/ch02/ADALINE.ipynb
hide-tono/python-training
eacbb4954b50e3b02d2a8031825f28a4ad8c624c
Apache-2.0
python-machine-learning/ch02/ADALINE.ipynb
hide-tono/python-training
eacbb4954b50e3b02d2a8031825f28a4ad8c624c
Apache-2.0
Qwen/Qwen-72B
1. YES
2. YES