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滞后数据对回归的影响
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<p>最近几天才开始着手研究ML量化，有点晚，个人的预期还是很看好ML在Quant买方策略里的效果。</p>
<p>此篇文章非结论性的文章，只是记录一下学习过程中的想法，大家有兴趣可以探讨。</p>
<p>在看一个kaggle竞赛kernel的时候，发现一个延迟相关性的问题。</p>
<div class="highlight"><pre><span></span><code><span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">({</span><span class="s1">'C'</span><span class="p">:</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">]})</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"C"</span><span class="p">]</span><span class="o">.</span><span class="n">corr</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"C"</span><span class="p">]))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"C"</span><span class="p">]</span><span class="o">.</span><span class="n">corr</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"C"</span><span class="p">]</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="mi">1</span><span class="p">)))</span>
</code></pre></div>
<p>结果是：</p>
<blockquote>
<p>1.0</p>
<p>0.07459885356228374</p>
</blockquote>
<p>当时我还是挺震惊的，自己和过去的自己完全不相关了吗？</p>
<p>我再回想一下现在的一些回归方法大部分(<strong>除了RNN这种含有历史信息的模型</strong>)都是 f(X_i)=y_i ，这个i是啥呢？我们做股价数据预测时，i就是时间。 X_i 就是i时刻对应的价格(Price)，成交量(Vol)，特征1(Feature1)，特征2(Feature2) ......组成的向量。这应该是横截面回归，计算时不含有历史信息的。我预测明天的股价时，只用到了今天的信息，与昨天无关。</p>
<p>假如<strong>特征1</strong>的预测性很好，那么我们一般会检测一下特征1和y的相关性，结果应该是不错的。但是万一这个特征1的效果有点滞后，今天的新闻对明天本来没什么影响，经过某些专家解读后对后天的股价产生了影响。实际环境中有没有这样的特征呢？这个我也不太清楚。才入坑ML，这些对我来讲一切都是新的......</p>
<p>如果<strong>特征1</strong>的效果滞后了两天(今天->后天)，而且我们一般把y会设置成明天的股价变化，那特征1和y的相关性就会发生前面代码中的情况（低相关性）。</p>
<p>来代码测试一下：</p>
<div class="highlight"><pre><span></span><code><span class="c1"># 用的某篇论文中用GBM产生的股价方法，论文中为stock B，故起名为B</span>
<span class="n">stock_price</span> <span class="o">=</span> <span class="n">B</span><span class="p">(</span><span class="mf">12.0</span><span class="p">)</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">stock_price</span>
<span class="c1"># 计算 log return</span>
<span class="n">df</span><span class="p">[</span><span class="s2">"log_return"</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">'close'</span><span class="p">])</span><span class="o">.</span><span class="n">diff</span><span class="p">()</span>
<span class="c1"># 把y设置成明天的 log return</span>
<span class="n">df</span><span class="p">[</span><span class="s2">"y"</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s2">"log_return"</span><span class="p">]</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">OFFSET</span> <span class="o">=</span> <span class="mi">2</span>
<span class="c1"># 我自己造的一个未来数据特征，其实就是后天的 log return 加了高斯噪声，我假如是个前天的新闻一定会对今天的股价产生影响</span>
<span class="n">df</span><span class="p">[</span><span class="s2">"Feature1"</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s2">"log_return"</span><span class="p">]</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="o">-</span><span class="n">OFFSET</span><span class="p">)</span><span class="o">*</span><span class="mi">100</span> <span class="o">+</span> <span class="n">RS</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"log_return"</span><span class="p">]))</span>
<span class="n">df</span><span class="o">.</span><span class="n">dropna</span><span class="p">(</span><span class="n">inplace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</code></pre></div>
<p>有未来数据在，按理来讲应该是接近1的相关性才对，但是有偏移，按前面的结论应该相关性很低。</p>
<p>来看一下特征1和y相关性如何</p>
<div class="highlight"><pre><span></span><code><span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"y"</span><span class="p">]</span><span class="o">.</span><span class="n">corr</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"Feature1"</span><span class="p">]))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"log_return"</span><span class="p">]</span><span class="o">.</span><span class="n">corr</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"Feature1"</span><span class="p">]))</span>
<span class="o">--------</span> <span class="n">output</span>
<span class="o">-</span><span class="mf">0.02142933034920817</span>
<span class="o">-</span><span class="mf">0.0007564299402125546</span>
</code></pre></div>
<p>在意料之中，和y基本不相关，和过去的自己也没有关系。直接回归试试看呢？我这里用了<strong>线性回归</strong>和<strong>树模型</strong>回归来进行测试。</p>
<div class="highlight"><pre><span></span><code><span class="n">make_regress</span><span class="p">(</span><span class="n">df</span><span class="p">)</span>
<span class="o">--------</span> <span class="n">output</span>
<span class="n">linear</span> <span class="n">regression</span> <span class="n">result</span><span class="p">:</span>
<span class="ow">in</span> <span class="n">sample</span> <span class="n">R</span><span class="o">^</span><span class="mi">2</span><span class="p">:</span> <span class="mf">0.01</span> <span class="n">out</span> <span class="n">sample</span> <span class="n">R</span><span class="o">^</span><span class="mi">2</span><span class="p">:</span> <span class="o">-</span><span class="mf">0.06</span>
<span class="n">ExtraTrees</span> <span class="n">regression</span> <span class="n">result</span><span class="p">:</span>
<span class="ow">in</span> <span class="n">sample</span> <span class="n">R</span><span class="o">^</span><span class="mi">2</span><span class="p">:</span> <span class="mf">0.05</span> <span class="n">out</span> <span class="n">sample</span> <span class="n">R</span><span class="o">^</span><span class="mi">2</span><span class="p">:</span> <span class="o">-</span><span class="mf">0.15</span>
</code></pre></div>
<p>样本内外的R2是负的，那是基本没有预测作用。如果我做如下一个变换呢？从特征1再衍生出一个特征</p>
<div class="highlight"><pre><span></span><code><span class="c1"># 新增加一个特征</span>
<span class="n">df</span><span class="p">[</span><span class="s2">"Feature1_shift1"</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s2">"Feature1"</span><span class="p">]</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">df</span><span class="o">.</span><span class="n">dropna</span><span class="p">(</span><span class="n">inplace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</code></pre></div>
<p>再来进行回归测试</p>
<div class="highlight"><pre><span></span><code><span class="n">make_regress</span><span class="p">(</span><span class="n">df</span><span class="p">)</span>
<span class="o">--------</span> <span class="n">output</span>
<span class="n">linear</span> <span class="n">regression</span> <span class="n">result</span><span class="p">:</span>
<span class="ow">in</span> <span class="n">sample</span> <span class="n">R</span><span class="o">^</span><span class="mi">2</span><span class="p">:</span> <span class="mf">0.90</span> <span class="n">out</span> <span class="n">sample</span> <span class="n">R</span><span class="o">^</span><span class="mi">2</span><span class="p">:</span> <span class="mf">0.90</span>
<span class="n">ExtraTrees</span> <span class="n">regression</span> <span class="n">result</span><span class="p">:</span>
<span class="ow">in</span> <span class="n">sample</span> <span class="n">R</span><span class="o">^</span><span class="mi">2</span><span class="p">:</span> <span class="mf">0.87</span> <span class="n">out</span> <span class="n">sample</span> <span class="n">R</span><span class="o">^</span><span class="mi">2</span><span class="p">:</span> <span class="mf">0.86</span>
</code></pre></div>
<p>结果显而易见，回到了用未来数据进行预测的效果。</p>
<p>当然这个例子有点特殊，实际生活中是不是这样还得去进行测试。如果通过简单的<strong>特征工程</strong>（shift）能增加预测效果，那么哪些特征需要做shift呢？shift多少个周期呢？如果新增出来的特征太多，怎么去评价哪些shift是有用的呢？</p>
<p>欢迎大家一起讨论。最后附上所有测试代码(python3)</p>
<div class="highlight"><pre><span></span><code><span class="kn">import</span> <span class="nn">math</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">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">ExtraTreesRegressor</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="n">RS</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">RandomState</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
<span class="n">LENGTH</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">OFFSET</span> <span class="o">=</span> <span class="mi">2</span>
<span class="k">def</span> <span class="nf">split</span><span class="p">(</span><span class="n">df</span><span class="p">):</span>
<span class="n">sticker</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">columns</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
<span class="n">sticker</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="s2">"y"</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="n">sticker</span><span class="p">]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s2">"y"</span><span class="p">]</span>
<span class="n">split_idx</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">df</span><span class="p">)</span><span class="o">*</span><span class="mf">0.7</span><span class="p">)</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:</span><span class="n">split_idx</span><span class="p">]</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:</span><span class="n">split_idx</span><span class="p">]</span>
<span class="n">X_test</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">split_idx</span><span class="p">:]</span>
<span class="n">y_test</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="n">split_idx</span><span class="p">:]</span>
<span class="k">return</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span>
<span class="k">def</span> <span class="nf">make_regress</span><span class="p">(</span><span class="n">df</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="n">X_test</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">split</span><span class="p">(</span><span class="n">df</span><span class="p">)</span>
<span class="n">lr_model</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
<span class="n">lr_model</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>
<span class="n">lr_in_sample_r2</span> <span class="o">=</span> <span class="n">lr_model</span><span class="o">.</span><span class="n">score</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="n">lr_out_sample_r2</span> <span class="o">=</span> <span class="n">lr_model</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"linear regression result:"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"in sample R^2: </span><span class="si">%.2f</span><span class="s2"> out sample R^2: </span><span class="si">%.2f</span><span class="s2">"</span> <span class="o">%</span> <span class="p">(</span><span class="n">lr_in_sample_r2</span><span class="p">,</span> <span class="n">lr_out_sample_r2</span><span class="p">))</span>
<span class="n">etr_model</span> <span class="o">=</span> <span class="n">ExtraTreesRegressor</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
<span class="n">etr_model</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>
<span class="n">etr_in_sample_r2</span> <span class="o">=</span> <span class="n">etr_model</span><span class="o">.</span><span class="n">score</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="n">etr_out_sample_r2</span> <span class="o">=</span> <span class="n">etr_model</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"ExtraTrees regression result:"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"in sample R^2: </span><span class="si">%.2f</span><span class="s2"> out sample R^2: </span><span class="si">%.2f</span><span class="s2">"</span> <span class="o">%</span> <span class="p">(</span><span class="n">etr_in_sample_r2</span><span class="p">,</span> <span class="n">etr_out_sample_r2</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">B</span><span class="p">(</span><span class="n">b0</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""</span>
<span class="sd"> gen B price Done</span>
<span class="sd"> param bo: initial price</span>
<span class="sd"> """</span>
<span class="k">global</span> <span class="n">LENGTH</span>
<span class="n">mu</span> <span class="o">=</span> <span class="mf">0.003</span>
<span class="n">delta_t</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="mf">251.0</span>
<span class="n">delta</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="n">b_close</span> <span class="o">=</span> <span class="p">[</span><span class="n">b0</span><span class="p">]</span> <span class="c1"># first price</span>
<span class="n">n_rand</span> <span class="o">=</span> <span class="n">RS</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">LENGTH</span><span class="p">)</span> <span class="c1"># N~(0,1) random numbers</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">LENGTH</span><span class="p">):</span>
<span class="n">b</span> <span class="o">=</span> <span class="nb">pow</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">e</span><span class="p">,</span> <span class="n">mu</span><span class="o">*</span><span class="n">delta_t</span><span class="o">+</span><span class="n">delta</span><span class="o">*</span><span class="n">n_rand</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">delta_t</span><span class="p">))</span> <span class="o">*</span> <span class="n">b_close</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">b_close</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">dates_idx</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="s1">'20130101'</span><span class="p">,</span> <span class="n">periods</span><span class="o">=</span><span class="n">LENGTH</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">freq</span><span class="o">=</span><span class="s1">'D'</span><span class="p">)</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">({</span><span class="s2">"close"</span><span class="p">:</span> <span class="n">b_close</span><span class="p">},</span> <span class="n">index</span><span class="o">=</span><span class="n">dates_idx</span><span class="p">)</span>
<span class="k">return</span> <span class="n">df</span>
<span class="c1"># 用的某篇论文中用GBM产生的股价方法，论文中为stock B，故起名为B</span>
<span class="n">stock_price</span> <span class="o">=</span> <span class="n">B</span><span class="p">(</span><span class="mf">12.0</span><span class="p">)</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">stock_price</span>
<span class="c1"># 计算 log return</span>
<span class="n">df</span><span class="p">[</span><span class="s2">"log_return"</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">'close'</span><span class="p">])</span><span class="o">.</span><span class="n">diff</span><span class="p">()</span>
<span class="c1"># 把y设置成明天的 log return</span>
<span class="n">df</span><span class="p">[</span><span class="s2">"y"</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s2">"log_return"</span><span class="p">]</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="c1"># 我自己造的一个未来数据特征，其实就是后天的 log return 加了高斯噪声，我假如是个前天的新闻一定会对今天的股价产生影响</span>
<span class="n">df</span><span class="p">[</span><span class="s2">"Feature1"</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s2">"log_return"</span><span class="p">]</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="o">-</span><span class="n">OFFSET</span><span class="p">)</span><span class="o">*</span><span class="mi">100</span> <span class="o">+</span> <span class="n">RS</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"log_return"</span><span class="p">]))</span>
<span class="n">df</span><span class="o">.</span><span class="n">dropna</span><span class="p">(</span><span class="n">inplace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"y"</span><span class="p">]</span><span class="o">.</span><span class="n">corr</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"Feature1"</span><span class="p">]))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"log_return"</span><span class="p">]</span><span class="o">.</span><span class="n">corr</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">"Feature1"</span><span class="p">]))</span>
<span class="n">make_regress</span><span class="p">(</span><span class="n">df</span><span class="p">)</span>
<span class="c1"># 新增加一个特征</span>
<span class="n">df</span><span class="p">[</span><span class="s2">"Feature1_shift1"</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s2">"Feature1"</span><span class="p">]</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">df</span><span class="o">.</span><span class="n">dropna</span><span class="p">(</span><span class="n">inplace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">make_regress</span><span class="p">(</span><span class="n">df</span><span class="p">)</span>
</code></pre></div>
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