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05-regression.html
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05-regression.html
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<!DOCTYPE html>
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<title>數值預測的任務 — 新手村逃脫！初心者的 Python 機器學習攻略 1.0.0 documentation</title>
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<div class="section" id="數值預測的任務">
<h1>數值預測的任務<a class="headerlink" href="#數值預測的任務" title="Permalink to this headline">¶</a></h1>
<p>我們先載入這個章節範例程式碼中會使用到的第三方套件、模組或者其中的部分類別、函式。</p>
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<span></span><span class="kn">from</span> <span class="nn">pyvizml</span> <span class="kn">import</span> <span class="n">CreateNBAData</span>
<span class="kn">import</span> <span class="nn">requests</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">MinMaxScaler</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
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<div class="section" id="關於數值預測的任務">
<h2>關於數值預測的任務<a class="headerlink" href="#關於數值預測的任務" title="Permalink to this headline">¶</a></h2>
<p>「數值預測」是「監督式學習」的其中一種應用類型，當預測的目標向量 <span class="math notranslate nohighlight">\(y\)</span> 屬於連續型的數值變數，那我們就能預期正在面對數值預測的任務，更廣泛被眾人知悉的名稱為「迴歸模型」。例如預測的目標向量 <span class="math notranslate nohighlight">\(y\)</span> 是 <code class="docutils literal notranslate"><span class="pre">players</span></code> 資料中的 <code class="docutils literal notranslate"><span class="pre">weightKilograms</span></code>，在資料類別中屬於連續型的數值類別 <code class="docutils literal notranslate"><span class="pre">float</span></code>；具體來說，迴歸模型想方設法將特徵矩陣 <span class="math notranslate nohighlight">\(X\)</span> 與目標向量 <span class="math notranslate nohighlight">\(y\)</span> 之間的關聯以一條迴歸線（Regression Line）描繪，而描繪迴歸線所依據的截距項和係數項，就是用來逼近 <span class="math notranslate nohighlight">\(f\)</span> 的 <span class="math notranslate nohighlight">\(h\)</span>。</p>
<p>我們也可依 <a class="reference external" href="https://en.wikipedia.org/wiki/Tom_M._Mitchell">Tom Mitchel</a> 對機器學習電腦程式的定義寫下數值預測的資料、任務、評估與但書，以預測 <code class="docutils literal notranslate"><span class="pre">players</span></code> 資料中的 <code class="docutils literal notranslate"><span class="pre">weightKilograms</span></code> 為例：</p>
<ul class="simple">
<li><p>資料（Experience）：一定數量的球員資料</p></li>
<li><p>任務（Task）：利用模型預測球員的體重</p></li>
<li><p>評估（Performance）：模型預測的體重與球員實際體重的誤差大小</p></li>
<li><p>但書（Condition）：隨著資料觀測值筆數增加，預測誤差應該要減少</p></li>
</ul>
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<span></span><span class="c1"># players 資料中的 weightKilograms</span>
<span class="n">cnd</span> <span class="o">=</span> <span class="n">CreateNBAData</span><span class="p">(</span><span class="mi">2019</span><span class="p">)</span>
<span class="n">players</span> <span class="o">=</span> <span class="n">cnd</span><span class="o">.</span><span class="n">create_players_df</span><span class="p">()</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">players</span><span class="p">[</span><span class="s1">'weightKilograms'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
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Creating players df...
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dtype('float64')
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<div class="section" id="以-Scikit-Learn-預測器完成數值預測任務">
<h2>以 Scikit-Learn 預測器完成數值預測任務<a class="headerlink" href="#以-Scikit-Learn-預測器完成數值預測任務" title="Permalink to this headline">¶</a></h2>
<p>將 <code class="docutils literal notranslate"><span class="pre">heightMeters</span></code> 當作特徵矩陣為例，特徵矩陣 <span class="math notranslate nohighlight">\(X\)</span> 與目標向量 <span class="math notranslate nohighlight">\(y\)</span> 之間的關聯可以這樣描述。</p>
<p><span class="math">\begin{equation}
\hat{y} = w_0 + w_1x_1
\end{equation}</span></p>
<p>以 Scikit-Learn 定義好的預測器類別 <code class="docutils literal notranslate"><span class="pre">LinearRegression</span></code> 可以快速找出描繪迴歸線所依據的截距項 <span class="math notranslate nohighlight">\(w_0\)</span> 和係數項 <span class="math notranslate nohighlight">\(w_1\)</span>。</p>
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<span></span><span class="n">X</span> <span class="o">=</span> <span class="n">players</span><span class="p">[</span><span class="s1">'heightMeters'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">players</span><span class="p">[</span><span class="s1">'weightKilograms'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_valid</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_valid</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.33</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
<span class="n">h</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
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<span></span><span class="nb">print</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">intercept_</span><span class="p">)</span> <span class="c1"># 截距項</span>
<span class="nb">print</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span> <span class="c1"># 係數項</span>
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-104.22092448587175
[101.82540151]
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<span></span><span class="c1"># 預測</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_valid</span><span class="p">)</span>
<span class="n">y_pred</span><span class="p">[:</span><span class="mi">10</span><span class="p">]</span>
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array([107.57591065, 82.11956027, 100.44813254, 105.53940262,
95.35686246, 112.66718072, 92.30210042, 92.30210042,
97.39337049, 95.35686246])
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<p>找出 <span class="math notranslate nohighlight">\(w_0\)</span> 與 <span class="math notranslate nohighlight">\(w_1\)</span> 就能夠描繪出一條迴歸線表達特徵矩陣 <span class="math notranslate nohighlight">\(X\)</span> 與目標向量 <span class="math notranslate nohighlight">\(y\)</span>。</p>
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<span></span><span class="c1"># 創建迴歸線的資料</span>
<span class="n">X1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">min</span><span class="p">()</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">X</span><span class="o">.</span><span class="n">max</span><span class="p">()</span><span class="o">+</span><span class="mf">0.1</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">y_hat</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X1</span><span class="p">)</span>
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<span></span><span class="c1"># 描繪迴歸線</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"training"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_valid</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">y_valid</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"valid"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X1</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">y_hat</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">"red"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"regression"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>使用 Scikit-Learn 預測器的最關鍵方法呼叫是 <code class="docutils literal notranslate"><span class="pre">fit()</span></code> 方法，究竟它是如何決定 <code class="docutils literal notranslate"><span class="pre">X_train</span></code> 與 <code class="docutils literal notranslate"><span class="pre">y_train</span></code> 之間的關聯 <span class="math notranslate nohighlight">\(w\)</span>？接下來我們試圖推導並理解它。</p>
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<div class="section" id="正規方程-Normal-Equation">
<h2>正規方程 Normal Equation<a class="headerlink" href="#正規方程-Normal-Equation" title="Permalink to this headline">¶</a></h2>
<p>使用機器學習解決數值預測的任務，顧名思義是能夠創建出一個 <span class="math notranslate nohighlight">\(h\)</span> 函式，這個函式可以將無標籤資料 <span class="math notranslate nohighlight">\(x\)</span> 作為輸入，並預測目標向量 <span class="math notranslate nohighlight">\(y\)</span> 的值作為其輸出。</p>
<p><span class="math">\begin{align}
\hat{y} &= h(x; w) \\
&= w_0 + w_1x_1 + ... + w_nx_n
\end{align}</span></p>
<p>為了寫作成向量相乘形式，為 <span class="math notranslate nohighlight">\(w_0\)</span> 補上 <span class="math notranslate nohighlight">\(x_0=1\)</span>：</p>
<p><span class="math">\begin{align}
\hat{y} &= w_0x_0 + w_1x_1 + ... + w_nx_n, \; where \; w_0 = 1 \\
&= w^Tx
\end{align}</span></p>
<p>其中 <span class="math notranslate nohighlight">\(\hat{y}\)</span> 是預測值、<span class="math notranslate nohighlight">\(n\)</span> 是特徵個數、<span class="math notranslate nohighlight">\(w\)</span> 是係數向量；並能夠進一步延展為 <code class="docutils literal notranslate"><span class="pre">m</span></code> 筆觀測值的外觀為：</p>
<p><span class="math">\begin{equation}
\hat{y} = h(X; w) =
\begin{bmatrix} x_{00}, x_{01}, ..., x_{0n} \\ x_{10}, x_{11}, ..., x_{1n} \\.\\.\\.\\ x_{m0}, x_{m1}, ..., x_{mn}
\end{bmatrix}
\begin{bmatrix} w_0 \\ w_1 \\.\\.\\.\\ w_n \end{bmatrix} = Xw
\end{equation}</span></p>
<p><span class="math notranslate nohighlight">\(h(X; w)\)</span> 是基於 <span class="math notranslate nohighlight">\(w\)</span> 的函式，如果第 <span class="math notranslate nohighlight">\(i\)</span> 個特徵 <span class="math notranslate nohighlight">\(x_i\)</span> 對應的係數 <span class="math notranslate nohighlight">\(w_i\)</span> 為正數，該特徵與 <span class="math notranslate nohighlight">\(\hat{y}\)</span> 的變動同向；如果第 <span class="math notranslate nohighlight">\(i\)</span> 個特徵 <span class="math notranslate nohighlight">\(x_i\)</span> 對應的係數 <span class="math notranslate nohighlight">\(w_i\)</span> 為負數，該特徵與 <span class="math notranslate nohighlight">\(\hat{y}\)</span> 的變動反向；如果第 <span class="math notranslate nohighlight">\(i\)</span> 個特徵 <span class="math notranslate nohighlight">\(x_i\)</span> 對應的係數 <span class="math notranslate nohighlight">\(w_i\)</span> 為零，該特徵對 <span class="math notranslate nohighlight">\(\hat{y}\)</span> 的變動沒有影響。</p>
<p>截至於此，資料（Experiment）與任務（Task）已經被定義妥善，特徵矩陣 <span class="math notranslate nohighlight">\(X\)</span> 外觀 <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n)</span></code>、目標向量 <span class="math notranslate nohighlight">\(y\)</span> 外觀 <code class="docutils literal notranslate"><span class="pre">(m,)</span></code>、係數向量 <span class="math notranslate nohighlight">\(w\)</span> 外觀 <code class="docutils literal notranslate"><span class="pre">(n,)</span></code>，通過將 <span class="math notranslate nohighlight">\(X\)</span> 輸入 <span class="math notranslate nohighlight">\(h\)</span> 來預測 <span class="math notranslate nohighlight">\(\hat{y}\)</span>，接下來還需要定義評估（Performance）。</p>
<p>評估 <span class="math notranslate nohighlight">\(h\)</span> 的方法是計算 <span class="math notranslate nohighlight">\(y^{(train)}\)</span> 與 <span class="math notranslate nohighlight">\(\hat{y}^{(train)}\)</span> 之間的均方誤差（Mean squared error）：</p>
<p><span class="math">\begin{equation}
MSE_{train} = \frac{1}{m}\sum_i(y^{(train)} - \hat{y}^{(train)})_i^2
\end{equation}</span></p>
<p>如果寫為向量運算的外觀：</p>
<p><span class="math">\begin{equation}
MSE_{train} = \frac{1}{m}\parallel y^{(train)} - \hat{y}^{(train)} \parallel^2
\end{equation}</span></p>
<p>電腦程式能夠通過觀察訓練資料藉此獲得一組能讓均方誤差最小化的係數向量 <span class="math notranslate nohighlight">\(w\)</span>，為了達成這個目的，將均方誤差表達為一個基於係數向量 <span class="math notranslate nohighlight">\(w\)</span> 的函式 <span class="math notranslate nohighlight">\(J(w)\)</span>：</p>
<p><span class="math">\begin{equation}
J(w) = MSE = \frac{1}{m} \parallel y - Xw \parallel^2
\end{equation}</span></p>
<p>整理一下函式 <span class="math notranslate nohighlight">\(J(w)\)</span> 的外觀：</p>
<p><span class="math">\begin{align}
J(w) &= \frac{1}{m}(Xw - y)^T(Xw - y) \\
&= \frac{1}{m}(w^TX^T - y^T)(Xw - y) \\
&= \frac{1}{m}(w^TX^TXw - w^TX^Ty - y^TXw + y^Ty) \\
&= \frac{1}{m}(w^TX^TXw - (Xw)^Ty - y^TXw + y^Ty) \\
&= \frac{1}{m}(w^TX^TXw - 2(Xw)^Ty + y^Ty)
\end{align}</span></p>
<p>求解 <span class="math notranslate nohighlight">\(J(w)\)</span> 斜率為零的位置：</p>
<p><span class="math">\begin{gather}
\frac{\partial}{\partial w} J(w) = 0 \\
2X^TXw - 2X^Ty = 0 \\
X^TXw = X^Ty \\
w^* = (X^TX)^{-1}X^Ty
\end{gather}</span></p>
<p>這個 <span class="math notranslate nohighlight">\(w^*\)</span> 求解亦被稱呼為「正規方程」（Normal equation）。</p>
</div>
<div class="section" id="自訂正規方程類別-NormalEquation">
<h2>自訂正規方程類別 NormalEquation<a class="headerlink" href="#自訂正規方程類別-NormalEquation" title="Permalink to this headline">¶</a></h2>
<p>我們可以依據正規方程自訂預測器類別，並與 Scikit-Learn 定義好的預測器類別 <code class="docutils literal notranslate"><span class="pre">LinearRegression</span></code> 比對係數向量是否一致。</p>
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<span></span><span class="k">class</span> <span class="nc">NormalEquation</span><span class="p">:</span>
<span class="sd">"""</span>
<span class="sd"> This class defines the Normal equation for linear regression.</span>
<span class="sd"> Args:</span>
<span class="sd"> fit_intercept (bool): Whether to add intercept for this model.</span>
<span class="sd"> """</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">fit_intercept</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_fit_intercept</span> <span class="o">=</span> <span class="n">fit_intercept</span>
<span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function uses Normal equation to solve for weights of this model.</span>
<span class="sd"> Args:</span>
<span class="sd"> X_train (ndarray): 2d-array for feature matrix of training data.</span>
<span class="sd"> y_train (ndarray): 1d-array for target vector of training data.</span>
<span class="sd"> """</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span> <span class="o">=</span> <span class="n">X_train</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_y_train</span> <span class="o">=</span> <span class="n">y_train</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_fit_intercept</span><span class="p">:</span>
<span class="n">X0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">m</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">float</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">X0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">X_train_T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span><span class="p">)</span>
<span class="n">left_matrix</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X_train_T</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span><span class="p">)</span>
<span class="n">right_matrix</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X_train_T</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_y_train</span><span class="p">)</span>
<span class="n">left_matrix_inv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">left_matrix</span><span class="p">)</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">left_matrix_inv</span><span class="p">,</span> <span class="n">right_matrix</span><span class="p">)</span>
<span class="n">w_ravel</span> <span class="o">=</span> <span class="n">w</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_w</span> <span class="o">=</span> <span class="n">w</span>
<span class="bp">self</span><span class="o">.</span><span class="n">intercept_</span> <span class="o">=</span> <span class="n">w_ravel</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">coef_</span> <span class="o">=</span> <span class="n">w_ravel</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X_test</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function returns predicted values with weights of this model.</span>
<span class="sd"> Args:</span>
<span class="sd"> X_test (ndarray): 2d-array for feature matrix of test data.</span>
<span class="sd"> """</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_X_test</span> <span class="o">=</span> <span class="n">X_test</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_fit_intercept</span><span class="p">:</span>
<span class="n">X0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">m</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">float</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_X_test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">X0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_X_test</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_X_test</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_w</span><span class="p">)</span>
<span class="k">return</span> <span class="n">y_pred</span>
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<span></span><span class="n">h</span> <span class="o">=</span> <span class="n">NormalEquation</span><span class="p">()</span>
<span class="n">h</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
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<span></span><span class="nb">print</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">intercept_</span><span class="p">)</span> <span class="c1"># 截距項</span>
<span class="nb">print</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span> <span class="c1"># 係數項</span>
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-104.22092448572948
[101.82540151]
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<span></span><span class="c1"># 預測</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_valid</span><span class="p">)</span>
<span class="n">y_pred</span><span class="p">[:</span><span class="mi">10</span><span class="p">]</span>
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array([107.57591065, 82.11956027, 100.44813254, 105.53940262,
95.35686246, 112.66718072, 92.30210042, 92.30210042,
97.39337049, 95.35686246])
</pre></div></div>
</div>
<p>比對 <span class="math notranslate nohighlight">\(w\)</span> 與前十筆預測值可以驗證自行定義的 <code class="docutils literal notranslate"><span class="pre">NormalEquation</span></code> 類別與 Scikit-Learn 求解 <span class="math notranslate nohighlight">\(w\)</span> 的邏輯相近。</p>
</div>
<div class="section" id="計算複雜性">
<h2>計算複雜性<a class="headerlink" href="#計算複雜性" title="Permalink to this headline">¶</a></h2>
<p>計算複雜性（Computational complexity）是電腦科學研究解決問題所需的資源，諸如時間（要通過多少步演算才能解決問題）和空間（在解決問題時需要多少記憶體），在演算法中常見到的大 O 符號就是表示演算所需時間的表達式。在正規方程中必須要透過計算 <span class="math notranslate nohighlight">\(X^TX\)</span> 的反矩陣 <span class="math notranslate nohighlight">\((X^TX)^{-1}\)</span> 求解 <span class="math notranslate nohighlight">\(w^*\)</span>，這是一個外觀 <code class="docutils literal notranslate"><span class="pre">(n+1,</span> <span class="pre">n+1)</span></code> 的二維數值陣列（<code class="docutils literal notranslate"><span class="pre">n</span></code> 為特徵個數），計算複雜性最多是 <span class="math notranslate nohighlight">\(O(n^3)\)</span>，這意味著如果特徵個數變為 2 倍，計算 <span class="math notranslate nohighlight">\((X^TX)^{-1}\)</span> 的時間最多會變為 8 倍。因此當面對的特徵矩陣
<code class="docutils literal notranslate"><span class="pre">n</span></code> 很大（約莫是大於 <span class="math notranslate nohighlight">\(10^4\)</span>），正規方程的計算複雜性問題就會浮現，這時讀者可能會好奇 <span class="math notranslate nohighlight">\(n \geq 10^4\)</span> 會很容易遭遇嗎？在特徵矩陣是圖像時很容易遭遇，例如低解析度 <span class="math notranslate nohighlight">\(100 \: px \times 100 \: px\)</span> 的灰階圖片。</p>
</div>
<div class="section" id="梯度遞減-Gradient-Descent">
<h2>梯度遞減 Gradient Descent<a class="headerlink" href="#梯度遞減-Gradient-Descent" title="Permalink to this headline">¶</a></h2>
<p>另外一種在機器學習、深度學習中更為廣泛使用的演算方法稱為「梯度遞減」（Gradient descent），基本概念是先隨機初始化一組係數向量，在基於降低 <span class="math notranslate nohighlight">\(y^{(train)}\)</span> 與 <span class="math notranslate nohighlight">\(\hat{y}^{(train)}\)</span> 之間誤差 <span class="math notranslate nohighlight">\(J(w)\)</span> 之目的標之下，以迭代方式更新該組係數向量，一直到 <span class="math notranslate nohighlight">\(J(w)\)</span> 收斂到局部最小值為止。</p>
<p>梯度遞減的精髓在於當演算方法更新係數向量時，並不是盲目亂槍打鳥地試誤（Trial and error），而是透過「有方向性」的依據進行更新，具體來說，就是根據誤差函式 <span class="math notranslate nohighlight">\(J(w)\)</span> 關於係數向量 <span class="math notranslate nohighlight">\(w\)</span> 的偏微分來決定更新的方向性，而更新的幅度大小則由一個大於零、稱為「學習速率」的常數 <span class="math notranslate nohighlight">\(\alpha\)</span> 決定：</p>
<p><span class="math">\begin{equation}
w := w - \alpha \frac{\partial J}{\partial w}
\end{equation}</span></p>
<p>讓我們用一個簡單的例子來看為什麼透過這個式子更新 <span class="math notranslate nohighlight">\(w\)</span> 是一種「有方向性」的依據，舉例來說如果給定一組 <span class="math notranslate nohighlight">\(X^{(train)}\)</span> 與 <span class="math notranslate nohighlight">\(y^{(train)}\)</span>：</p>
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<span></span><span class="n">X0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">X1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">X0</span><span class="p">,</span> <span class="n">X1</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">y_train</span><span class="p">)</span>
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[[ 1. 1.]
[ 1. 2.]
[ 1. 3.]
[ 1. 4.]
[ 1. 5.]
[ 1. 6.]
[ 1. 7.]
[ 1. 8.]
[ 1. 9.]
[ 1. 10.]]
[11. 17. 23. 29. 35. 41. 47. 53. 59. 65.]
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<p>從後見之明的視角來看，我們會知道係數向量 <span class="math notranslate nohighlight">\(w^*\)</span> 的組成 <span class="math notranslate nohighlight">\(w_0=5\)</span>、<span class="math notranslate nohighlight">\(w_1=6\)</span>：</p>
<p><span class="math">\begin{equation}
f(x) = y = 5x_0 + 6x_1
\end{equation}</span></p>
<p>亦即</p>
<p><span class="math">\begin{equation}
w^* = \begin{bmatrix} w_0^* \\ w_1^* \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}
\end{equation}</span></p>
<p>不過給定電腦程式一組 <span class="math notranslate nohighlight">\(X^{(train)}\)</span> 與 <span class="math notranslate nohighlight">\(y^{(train)}\)</span> 對於它來說像是拋出了一個大海撈針的問題，有無限多組的 <span class="math notranslate nohighlight">\(w\)</span> 等著要嘗試（它甚至不知道用 <span class="math notranslate nohighlight">\(w_0\)</span> 與 <span class="math notranslate nohighlight">\(w_1\)</span> 就可以找到跟 <span class="math notranslate nohighlight">\(f\)</span> 完全相同的 <span class="math notranslate nohighlight">\(h\)</span>），遑論找出 <span class="math notranslate nohighlight">\(w_0=5\)</span>、<span class="math notranslate nohighlight">\(w_1=6\)</span>；「梯度遞減」演算方法就是為電腦程式提供了一個尋找解題的方式，千里之行，始於足下，請先隨機初始化一組 <span class="math notranslate nohighlight">\(w\)</span>：</p>
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<span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">w</span>
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array([0.37454012, 0.95071431])
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<p>針對這組 <span class="math notranslate nohighlight">\(w\)</span> 可以得到一組 <span class="math notranslate nohighlight">\(\hat{y}^{(train)}\)</span>：</p>
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<span></span><span class="n">y_hat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>
<span class="n">y_hat</span>
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array([1.32525443, 2.27596873, 3.22668304, 4.17739734, 5.12811165,
6.07882596, 7.02954026, 7.98025457, 8.93096888, 9.88168318])
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<p>針對這組 <span class="math notranslate nohighlight">\(\hat{y}^{(train)}\)</span> 可以計算與 <span class="math notranslate nohighlight">\(y^{(train)}\)</span> 的均方誤差。</p>
<p><span class="math">\begin{equation}
J(w) = \frac{1}{m}\parallel y - Xw \parallel^2
\end{equation}</span></p>
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<span></span><span class="n">m</span> <span class="o">=</span> <span class="n">y_train</span><span class="o">.</span><span class="n">size</span>
<span class="n">j</span> <span class="o">=</span> <span class="p">((</span><span class="n">y_hat</span> <span class="o">-</span> <span class="n">y_train</span><span class="p">)</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">y_hat</span> <span class="o">-</span> <span class="n">y_train</span><span class="p">))</span> <span class="o">/</span> <span class="n">m</span>
<span class="n">j</span>
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1259.87134315462
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<p>那麼下一次的試誤該如何更新 <span class="math notranslate nohighlight">\(w\)</span> 才能確保離 <span class="math notranslate nohighlight">\(w^*\)</span> 更近，讓計算出來的均方誤差會更小一些？這時梯度遞減演算方法登場，它直截了當地說：請將目前的 <span class="math notranslate nohighlight">\(w_0\)</span> 減去學習速率 <span class="math notranslate nohighlight">\(\alpha\)</span> 乘上 <span class="math notranslate nohighlight">\(J(w)\)</span> 關於 <span class="math notranslate nohighlight">\(w_0\)</span> 的偏微分、將目前的 <span class="math notranslate nohighlight">\(w_1\)</span> 減去學習速率 <span class="math notranslate nohighlight">\(\alpha\)</span> 乘上 <span class="math notranslate nohighlight">\(J(w)\)</span> 關於 <span class="math notranslate nohighlight">\(w_1\)</span> 的偏微分：</p>
<p><span class="math">\begin{equation}
w_0 := w_0 - \alpha \frac{\partial J}{\partial w_0}
\end{equation}</span></p>
<p><span class="math">\begin{equation}
w_1 := w_1 - \alpha \frac{\partial J}{\partial w_1}
\end{equation}</span></p>
<p>以係數向量的外觀表示：</p>
<p><span class="math">\begin{equation}
w := w - \alpha \frac{\partial J}{\partial w}
\end{equation}</span></p>
<p>接著慢慢將 <span class="math notranslate nohighlight">\(J(w)\)</span> 關於 <span class="math notranslate nohighlight">\(w\)</span> 的偏微分式子展開：</p>
<p><span class="math">\begin{align}
\frac{\partial J}{\partial w} &= \frac{1}{m}\frac{\partial}{\partial w}(\parallel y - Xw \parallel^2) \\
&= \frac{1}{m}\frac{\partial}{\partial w}(Xw - y)^T(Xw-y) \\
&= \frac{1}{m}\frac{\partial}{\partial w}(w^TX^TXw - w^TX^Ty - y^TXw + y^Ty) \\
&= \frac{1}{m}\frac{\partial}{\partial w}(w^TX^TXw - (Xw)^Ty - (Xw)^Ty + y^Ty) \\
&= \frac{1}{m}\frac{\partial}{\partial w}(w^TX^TXw - 2(Xw)^Ty + y^Ty) \\
&= \frac{1}{m}(2X^TXw - 2X^Ty) \\
&= \frac{2}{m}(X^TXw - X^Ty) \\
&= \frac{2}{m}X^T(Xw - y) \\
&= \frac{2}{m}X^T(\hat{y} - y)
\end{align}</span></p>
<p><span class="math notranslate nohighlight">\(J(w)\)</span> 關於 <span class="math notranslate nohighlight">\(w\)</span> 的偏微分就是演算方法中所謂的「梯度」（Gradient），在迭代過程中 <span class="math notranslate nohighlight">\(w\)</span> 更新的方向性取決於梯度正負號，如果梯度為正，<span class="math notranslate nohighlight">\(w\)</span> 會向左更新（減小）；如果梯度為負，<span class="math notranslate nohighlight">\(w\)</span> 會向右更新（增大）。</p>
<p><span class="math">\begin{equation}
w := w - \alpha \frac{2}{m}X^T(\hat{y} - y)
\end{equation}</span></p>
<p>接著計算隨機初始化的 <span class="math notranslate nohighlight">\(w\)</span> 其梯度為何。</p>
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<span></span><span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">m</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">y_hat</span> <span class="o">-</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">gradients</span>
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array([ -64.79306239, -439.6750571 ])
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<p>當梯度為負，隨機初始化的 <span class="math notranslate nohighlight">\(w\)</span> 會向右更新（增大），離後見之明視角所知的 <span class="math notranslate nohighlight">\(w_0 = 5\)</span>、<span class="math notranslate nohighlight">\(w_1 = 6\)</span> 更加接近，在更新的方向性上是正確的。假設將學習速率設定為 0.001，更新的幅度就是：</p>
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<span></span><span class="n">learning_rate</span> <span class="o">=</span> <span class="mf">0.001</span>
<span class="o">-</span><span class="n">learning_rate</span> <span class="o">*</span> <span class="n">gradients</span>
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array([0.06479306, 0.43967506])
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<p>經過第一次迭代更新後的 <span class="math notranslate nohighlight">\(w\)</span>：</p>
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<span></span><span class="n">w</span> <span class="o">-=</span> <span class="n">learning_rate</span> <span class="o">*</span> <span class="n">gradients</span>
<span class="n">w</span>
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array([0.43933318, 1.39038936])
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<p>針對更新過一次的 <span class="math notranslate nohighlight">\(w\)</span> 可以得到一組 <span class="math notranslate nohighlight">\(\hat{y}^{(train)}\)</span>：</p>
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<span></span><span class="n">y_hat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>