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统计学
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基础
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统计量
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统计量
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抽样
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</a>
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假设检验
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</span>
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</a>
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推断
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<span class="md-ellipsis">
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概率论
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</span>
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<span class="md-nav__icon md-icon"></span>
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概率论
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计数
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</span>
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|
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</a>
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</li>
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概率概念
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<span class="md-ellipsis">
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分布
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|
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|
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|
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</span>
|
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|
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|
||
|
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</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter%2005%3A%20probability/04.%20bayesian/" class="md-nav__link">
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<span class="md-ellipsis">
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贝叶斯
|
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|
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|
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|
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</span>
|
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|
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|
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|
||
</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter%2005%3A%20probability/05.%20information%20theory/" class="md-nav__link">
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<span class="md-ellipsis">
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信息论
|
||
|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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</ul>
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</nav>
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</li>
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|
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<li class="md-nav__item md-nav__item--nested">
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<span class="md-ellipsis">
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|
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机器学习
|
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|
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|
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|
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</span>
|
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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机器学习
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter%2006%3A%20machine%20learning/01.%20classical%20machine%20learning/" class="md-nav__link">
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|
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<span class="md-ellipsis">
|
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|
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|
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经典机器学习
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter%2006%3A%20machine%20learning/02.%20gradient%20machine%20learning/" class="md-nav__link">
|
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|
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<span class="md-ellipsis">
|
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|
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|
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梯度机器学习
|
||
|
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|
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|
||
</span>
|
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|
||
|
||
|
||
</a>
|
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</li>
|
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<li class="md-nav__item">
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<a href="../../chapter%2006%3A%20machine%20learning/03.%20deep%20learning/" class="md-nav__link">
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|
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<span class="md-ellipsis">
|
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|
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|
||
深度学习
|
||
|
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|
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|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2006%3A%20machine%20learning/04.%20reinforcement%20learning/" class="md-nav__link">
|
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|
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|
||
<span class="md-ellipsis">
|
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|
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|
||
强化学习
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2006%3A%20machine%20learning/05.%20distributed%20deep%20learning/" class="md-nav__link">
|
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|
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|
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|
||
<span class="md-ellipsis">
|
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|
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|
||
分布式深度学习
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
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</li>
|
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|
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|
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|
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|
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</ul>
|
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</nav>
|
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|
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</li>
|
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|
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|
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|
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|
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|
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|
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|
||
|
||
|
||
|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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|
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<span class="md-ellipsis">
|
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|
||
|
||
计算语言学
|
||
|
||
|
||
|
||
</span>
|
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|
||
|
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|
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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计算语言学
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|
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|
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter%2007%3A%20computational%20linguistics/01.%20linguistic%20foundations/" class="md-nav__link">
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|
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|
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|
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<span class="md-ellipsis">
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|
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|
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语言学基础
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="../../chapter%2007%3A%20computational%20linguistics/02.%20text%20processing%20and%20classic%20NLP/" class="md-nav__link">
|
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|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
||
文本处理与经典 NLP
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2007%3A%20computational%20linguistics/03.%20embeddings%20and%20sequence%20models/" class="md-nav__link">
|
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|
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|
||
|
||
<span class="md-ellipsis">
|
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|
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|
||
嵌入与序列模型
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2007%3A%20computational%20linguistics/04.%20transformers%20and%20language%20models/" class="md-nav__link">
|
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|
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|
||
|
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<span class="md-ellipsis">
|
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|
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|
||
Transformer 与语言模型
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2007%3A%20computational%20linguistics/05.%20advanced%20text%20generation/" class="md-nav__link">
|
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|
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|
||
|
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<span class="md-ellipsis">
|
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|
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|
||
高级文本生成
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
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</a>
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</li>
|
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|
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|
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</ul>
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</nav>
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</li>
|
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|
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|
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|
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|
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|
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|
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|
||
|
||
|
||
|
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|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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计算机视觉
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图像基础
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目标检测与分割
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音频与语音
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数字信号处理
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自动语音识别
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</span>
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语音合成
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</span>
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</a>
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<span class="md-ellipsis">
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说话人与音频分析
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</span>
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|
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|
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</a>
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</li>
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源分离与降噪
|
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</span>
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|
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<span class="md-ellipsis">
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多模态学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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多模态学习
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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多模态表征
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</span>
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</a>
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视觉语言模型
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</span>
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</a>
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图像与视频 Token 化
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</span>
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|
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|
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</a>
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跨模态生成
|
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</span>
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2010%3A%20multimodal%20learning/05.%20unified%20multimodal%20architectures/" class="md-nav__link">
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<span class="md-ellipsis">
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统一多模态架构
|
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|
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|
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</span>
|
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|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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自主系统
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</span>
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<span class="md-nav__icon md-icon"></span>
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感知
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机器人学习
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</span>
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</a>
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视觉-语言-动作模型
|
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</span>
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|
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|
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|
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</a>
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自动驾驶
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</span>
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|
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|
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|
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</a>
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<a href="../../chapter%2011%3A%20autonomous%20systems/05.%20space%20and%20extreme%20robotics/" class="md-nav__link">
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<span class="md-ellipsis">
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太空与极端机器人
|
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|
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|
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|
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</span>
|
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|
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|
||
|
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图神经网络
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</span>
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<span class="md-nav__icon md-icon"></span>
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图神经网络
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几何深度学习
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</span>
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|
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ARM 与 NEON
|
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x86 与 AVX
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Triton、TPU 与 Pallas
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边缘推理
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ML 系统设计
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应用 AI
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前沿 AI
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神经形态计算
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太空数据中心
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去中心化 AI
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编程练习（使用 CoLab 或 notebook）
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<h1 id="_1">统计量<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>统计量用单个数值概括数据，捕捉其离散程度、位置、形状和关联。本节涵盖方差、标准差、四分位数、偏度、峰度、协方差、相关和 z 分数——这是探索性数据分析和机器学习特征工程的基础工具集。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>在上一节中，我们介绍了矩作为一组概括性统计量家族。在此，我们展开讨论从矩中衍生出的实用工具：度量离散程度、位置、形状和关联的统计量。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>离散程度</strong>回答了这样一个问题：数据的分布有多分散？两个班级的平均考试成绩可能相同，但其分散程度却可能大相径庭。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="均值相同但离散程度不同的两个分布" src="../../images/variance_spread.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>窄（蓝色）分布的方差较小：大部分数值紧密聚集在均值周围。宽（红色）分布的方差较大：数值散布得更远。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>方差</strong>是距均值距离的平方的平均值。取平方是为了避免正负偏差相互抵消。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2\]</div>
|
||
<ul>
|
||
<li>当处理样本（而非整个总体）时，我们用 <span class="arithmatex">\(N - 1\)</span> 而不是 <span class="arithmatex">\(N\)</span> 来除。这种修正（称为<strong>贝塞尔校正</strong>）是因为样本往往会低估真实的变异性：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[s^2 = \frac{1}{N-1} \sum_{i=1}^{N} (x_i - \bar{x})^2\]</div>
|
||
<ul>
|
||
<li>
|
||
<p><strong>标准差</strong>是方差的平方根：<span class="arithmatex">\(\sigma = \sqrt{\sigma^2}\)</span>。它将度量单位恢复为原始单位。如果数据的单位是厘米，方差的单位是 cm<span class="arithmatex">\(^2\)</span>，而标准差的单位又回到了 cm。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>平均绝对偏差（MAD）</strong>是一个更简单的替代方案。它不取平方，而是取每个偏差的绝对值：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\text{MAD} = \frac{1}{N} \sum_{i=1}^{N} |x_i - \mu|\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>MAD 对方差而言对异常值更稳健，因为它不会通过平方来放大大的偏差。然而，方差在数学上更便利（在证明和机器学习优化中更容易分解）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>位置</strong>回答了一个不同的问题：特定数值相对于其余数据的位置在哪里？</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>四分位数</strong>将排序后的数据分成四个相等的部分。Q1（第 25 百分位数）是低于该值的数据占 25% 的值。Q2 是中位数（第 50 百分位数）。Q3 是第 75 百分位数。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>四分位距（IQR）</strong>是 <span class="arithmatex">\(Q3 - Q1\)</span>。它捕捉了中间 50% 数据的离散程度，排除了极端值。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="显示 Q1、中位数、Q3、IQR、须线和异常值的箱线图" src="../../images/quartiles_boxplot.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>箱线图</strong>是统计学中最有用的可视化工具之一。箱体从 Q1 延伸到 Q3，中间的线为中位数，须线延伸到最远的非异常值，而须线之外的点则为异常值。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>百分位数</strong>是四分位数的推广。第 <span class="arithmatex">\(p\)</span> 百分位数是低于该值的观测值占 <span class="arithmatex">\(p\%\)</span> 的值。Q1 是第 25 百分位数，中位数是第 50 百分位数，Q3 是第 75 百分位数。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>z 分数</strong>告诉你一个值距均值有多少个标准差：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[z = \frac{x - \mu}{\sigma}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>z 分数为 2 表示该值高于均值 2 个标准差。z 分数为 <span class="arithmatex">\(-1.5\)</span> 表示低于均值 1.5 个标准差。这也称为<strong>标准化</strong>，在机器学习中广泛用于特征缩放，因为它将任何分布变换为均值为 0、标准差为 1。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>形状</strong>描述了分布超出其中心和离散程度之外的几何特征。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>偏度</strong>（上一节中的标准化三阶矩）衡量不对称性。像正态曲线这样完全对称的分布，其偏度为零。正偏度表示右尾较长（如收入分布）。负偏度表示左尾较长（如退休年龄分布）。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\text{偏度} = \frac{1}{N} \sum_{i=1}^{N} \left(\frac{x_i - \mu}{\sigma}\right)^3\]</div>
|
||
<ul>
|
||
<li><strong>峰度</strong>（标准化四阶矩）衡量尾部厚度。正态分布的峰度为 3。尾部更厚（更容易出现异常值）的分布的峰度大于 3。</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\text{峰度} = \frac{1}{N} \sum_{i=1}^{N} \left(\frac{x_i - \mu}{\sigma}\right)^4\]</div>
|
||
<ul>
|
||
<li><strong>相关</strong>衡量两个变量之间关系的强度和方向。它回答了：当一个变量上升时，另一个变量倾向于上升、下降，还是基本不变？</li>
|
||
</ul>
|
||
<p><img alt="三个散点图，分别显示正相关、无相关和负相关" src="../../images/correlation_scatter.svg" /></p>
|
||
<ul>
|
||
<li><strong>皮尔森相关</strong>（<span class="arithmatex">\(r\)</span>）衡量<em>线性</em>关联。其取值范围从 <span class="arithmatex">\(-1\)</span>（完全负相关）到 <span class="arithmatex">\(0\)</span>（无相关）再到 <span class="arithmatex">\(+1\)</span>（完全正相关）。</li>
|
||
</ul>
|
||
<div class="arithmatex">\[r = \frac{\sum_{i=1}^{N} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2} \cdot \sqrt{\sum (y_i - \bar{y})^2}}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>如果你还记得第 1 章中的点积，皮尔森相关本质上就是 <span class="arithmatex">\(\mathbf{x}\)</span> 和 <span class="arithmatex">\(\mathbf{y}\)</span> 均值中心化之后的余弦相似度。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>斯皮尔曼相关</strong>（<span class="arithmatex">\(\rho\)</span>）衡量<em>单调</em>关联。它不使用原始值，而是先对它们进行排序，然后在排序上计算皮尔森相关。这使得它对异常值稳健，并且即使关系是非线性的，只要是一致递增或递减的，也能正常工作。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>几何平均数</strong>是当数值相互乘除时（如增长率）合适的平均值。如果你的投资分别增长了 10%、20% 和 30%，那么平均增长因子并不是这些增长率的算术平均数。而是：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\bar{x}_{\text{geo}} = \left(\prod_{i=1}^{N} x_i\right)^{1/N}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>具体到增长率，先将百分比转换为因子（1.10、1.20、1.30），计算几何平均数，再减去 1。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>指数移动平均（EMA）</strong>赋予最近观测值更高的权重。与简单移动平均中窗口内所有点权重相等不同，EMA 呈指数衰减：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\text{EMA}_t = \alpha \cdot x_t + (1 - \alpha) \cdot \text{EMA}_{t-1}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>平滑因子 <span class="arithmatex">\(\alpha\)</span>（介于 0 和 1 之间）控制旧观测值失去影响的速度。<span class="arithmatex">\(\alpha\)</span> 越大，对近期变化的响应越灵敏；<span class="arithmatex">\(\alpha\)</span> 越小，曲线越平滑。在机器学习中，EMA 被用于 Adam 等优化器以及批归一化的运行统计中。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>异常值检测</strong>识别出与其余数据异常遥远的数点。两种常用方法：</p>
|
||
<ul>
|
||
<li><strong>IQR 法</strong>：如果一个点低于 <span class="arithmatex">\(Q1 - 1.5 \times \text{IQR}\)</span> 或高于 <span class="arithmatex">\(Q3 + 1.5 \times \text{IQR}\)</span>，则为异常值</li>
|
||
<li><strong>Z 分数法</strong>：如果 <span class="arithmatex">\(|z| > 3\)</span>（距均值超过 3 个标准差），则为异常值</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>IQR 法更稳健，因为它不假设正态分布。Z 分数法在数据近似正态时效果良好，但当分布高度偏斜时可能失效。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colab-notebook">编程练习（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>计算数据集的方差、标准差和 MAD，并进行比较。观察添加极端异常值时发生的变化。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">data</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">9</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="mi">7</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">jnp</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">mean</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">variance</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">std</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">mad</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">data</span> <span class="o">-</span> <span class="n">mean</span><span class="p">))</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="nb">print</span><span class="p">(</span><span class="s2">"原始数据："</span><span class="p">)</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">" 方差：</span><span class="si">{</span><span class="n">variance</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">，标准差：</span><span class="si">{</span><span class="n">std</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">，MAD：</span><span class="si">{</span><span class="n">mad</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="c1"># 添加一个异常值并重新计算</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="n">data_outlier</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="mf">100.0</span><span class="p">)</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="n">mean2</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">data_outlier</span><span class="p">)</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="se">\n</span><span class="s2">添加异常值（100）后："</span><span class="p">)</span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">" 方差：</span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">data_outlier</span><span class="p">)</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">，标准差：</span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">data_outlier</span><span class="p">)</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">，MAD：</span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">data_outlier</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">mean2</span><span class="p">))</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>计算两个变量之间的皮尔森相关和斯皮尔曼相关。尝试不同的关系。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="c1"># 完全线性关系</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</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="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">jnp</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">y</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span> <span class="c1"># 试试修改这个！</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">pearson</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a> <span class="n">a_c</span> <span class="o">=</span> <span class="n">a</span> <span class="o">-</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a> <span class="n">b_c</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a> <span class="k">return</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">a_c</span> <span class="o">*</span> <span class="n">b_c</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">a_c</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">b_c</span><span class="o">**</span><span class="mi">2</span><span class="p">)))</span>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="k">def</span><span class="w"> </span><span class="nf">spearman</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a> <span class="n">rank_a</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">a</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a> <span class="n">rank_b</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a> <span class="k">return</span> <span class="n">pearson</span><span class="p">(</span><span class="n">rank_a</span><span class="p">,</span> <span class="n">rank_b</span><span class="p">)</span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"皮尔森 r： </span><span class="si">{</span><span class="n">pearson</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"斯皮尔曼 ρ：</span><span class="si">{</span><span class="n">spearman</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>分别使用 IQR 和 Z 分数方法实现异常值检测，然后比较它们在偏斜数据上的结果。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="n">data</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</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="mi">5</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="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">50</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">jnp</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="c1"># IQR 方法</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">q1</span><span class="p">,</span> <span class="n">q3</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">percentile</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="mi">25</span><span class="p">),</span> <span class="n">jnp</span><span class="o">.</span><span class="n">percentile</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="mi">75</span><span class="p">)</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">iqr</span> <span class="o">=</span> <span class="n">q3</span> <span class="o">-</span> <span class="n">q1</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">lower</span><span class="p">,</span> <span class="n">upper</span> <span class="o">=</span> <span class="n">q1</span> <span class="o">-</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">iqr</span><span class="p">,</span> <span class="n">q3</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">iqr</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">iqr_outliers</span> <span class="o">=</span> <span class="n">data</span><span class="p">[(</span><span class="n">data</span> <span class="o"><</span> <span class="n">lower</span><span class="p">)</span> <span class="o">|</span> <span class="p">(</span><span class="n">data</span> <span class="o">></span> <span class="n">upper</span><span class="p">)]</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"IQR 边界：[</span><span class="si">{</span><span class="n">lower</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2">, </span><span class="si">{</span><span class="n">upper</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2">]"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"IQR 异常值：</span><span class="si">{</span><span class="n">iqr_outliers</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="c1"># Z 分数方法</span>
|
||
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="n">z_scores</span> <span class="o">=</span> <span class="p">(</span><span class="n">data</span> <span class="o">-</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">))</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="n">z_outliers</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">z_scores</span><span class="p">)</span> <span class="o">></span> <span class="mi">3</span><span class="p">]</span>
|
||
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="se">\n</span><span class="s2">Z 分数：</span><span class="si">{</span><span class="n">z_scores</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Z 分数异常值（|z| > 3）：</span><span class="si">{</span><span class="n">z_outliers</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>在不同平滑因子下计算并绘制带噪声数据的指数移动平均。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a>
|
||
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="c1"># 生成带噪声的数据</span>
|
||
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="n">key</span> <span class="o">=</span> <span class="nb">__import__</span><span class="p">(</span><span class="s2">"jax"</span><span class="p">)</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
||
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="n">noise</span> <span class="o">=</span> <span class="nb">__import__</span><span class="p">(</span><span class="s2">"jax"</span><span class="p">)</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">50</span><span class="p">,))</span>
|
||
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="n">signal</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span> <span class="o">+</span> <span class="n">noise</span>
|
||
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a>
|
||
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="k">def</span><span class="w"> </span><span class="nf">ema</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">alpha</span><span class="p">):</span>
|
||
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a> <span class="n">result</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
|
||
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a> <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">data</span><span class="p">)):</span>
|
||
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a> <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">alpha</span> <span class="o">*</span> <span class="n">data</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">alpha</span><span class="p">)</span> <span class="o">*</span> <span class="n">result</span><span class="p">[</span><span class="n">t</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
|
||
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a> <span class="k">return</span> <span class="n">result</span>
|
||
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a>
|
||
<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">signal</span><span class="p">,</span> <span class="s2">"o"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"原始数据"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#999"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="k">for</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="p">[(</span><span class="mf">0.1</span><span class="p">,</span> <span class="s2">"#e74c3c"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="s2">"#3498db"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.7</span><span class="p">,</span> <span class="s2">"#27ae60"</span><span class="p">)]:</span>
|
||
<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ema</span><span class="p">(</span><span class="n">signal</span><span class="p">,</span> <span class="n">alpha</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">"α=</span><span class="si">{</span><span class="n">alpha</span><span class="si">}</span><span class="s2">"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"不同平滑因子下的 EMA"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
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