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统计学
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基础
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统计量
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抽样
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</span>
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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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</label>
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<span class="md-ellipsis">
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计数
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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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<span class="md-ellipsis">
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概率概念
|
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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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<span class="md-ellipsis">
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分布
|
||
|
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|
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|
||
</span>
|
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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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|
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贝叶斯
|
||
|
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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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|
||
|
||
</span>
|
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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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|
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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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|
||
机器学习
|
||
|
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|
||
|
||
</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_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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|
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<span class="md-ellipsis">
|
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|
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|
||
经典机器学习
|
||
|
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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%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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|
||
梯度机器学习
|
||
|
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|
||
|
||
</span>
|
||
|
||
|
||
|
||
</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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|
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<span class="md-ellipsis">
|
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|
||
|
||
深度学习
|
||
|
||
|
||
|
||
</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%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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|
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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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|
||
|
||
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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</ul>
|
||
</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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|
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<li class="md-nav__item md-nav__item--nested">
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|
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|
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|
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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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|
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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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|
||
</span>
|
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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>
|
||
</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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<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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|
||
|
||
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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|
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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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|
||
|
||
|
||
<span class="md-ellipsis">
|
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|
||
|
||
Transformer 与语言模型
|
||
|
||
|
||
|
||
</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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|
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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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|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
高级文本生成
|
||
|
||
|
||
|
||
</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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</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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|
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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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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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</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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自主系统
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感知
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机器人学习
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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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ARM 与 NEON
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x86 与 AVX
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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">统计学基础<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>统计学提供了描述数据和量化不确定性的语言。本节涵盖分布、随机变量、PMF、PDF、CDF、期望、方差、矩以及中心极限定理——这些概念支撑着每一个机器学习评估指标和损失函数。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>统计学是从数据中学习的科学。你收集观测值，对其进行汇总，并得出结论——通常针对那些无法直接测量的事物。</p>
|
||
</li>
|
||
<li>
|
||
<p>假设你想知道某个国家所有成年人的平均身高。你不可能测量每一个人，因此你测量一个<strong>样本</strong>，并利用统计学对整个<strong>总体</strong>做出有根据的推测。</p>
|
||
</li>
|
||
<li>
|
||
<p>统计学有两个主要分支：</p>
|
||
<ul>
|
||
<li><strong>描述性统计</strong>：对已有数据进行汇总（平均值、图表、表格）</li>
|
||
<li><strong>推断性统计</strong>：利用样本对更大群体做出推断</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>统计学的基本构件是<strong>分布</strong>——一种描述数值如何分布的方式。其他一切——平均值、检验、预测——都源于对分布的理解。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>频率分布</strong>统计数据中每个值（或值区间）出现的次数。想象一下把考试成绩分到不同的区间，然后统计每个区间中有多少学生。结果就是直方图。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>概率分布</strong>用概率代替原始计数。它不说"12 名学生的分数在 70 到 80 之间"，而是说"分数在 70 到 80 之间的概率为 0.24"。当数据连续时，直方图的柱状会变成一条平滑曲线。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="频率分布直方图与概率分布平滑曲线对比" src="../../images/distribution_types.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>左侧的直方图基于你实际收集的数据构建。右侧的平滑曲线是一个数学模型，描述了数据背后的模式。一个是经验性的，另一个是理论性的。</p>
|
||
</li>
|
||
<li>
|
||
<p>为了从数学上处理分布，我们需要一种将结果赋予数值的方法。这正是<strong>随机变量</strong>所做的。</p>
|
||
</li>
|
||
<li>
|
||
<p>随机变量是一个将每次试验的结果映射到实数的函数。抛一枚硬币：结果是"正面"或"反面"，但随机变量 <span class="arithmatex">\(X\)</span> 将其转换为 <span class="arithmatex">\(X(正面) = 1\)</span> 和 <span class="arithmatex">\(X(反面) = 0\)</span>。现在我们就可以进行算术运算了。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="随机变量将结果（硬币、骰子）映射到数轴" src="../../images/random_variable.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>离散</strong>随机变量取值为可数集：10 次抛掷中的正面次数、骰子的点数、一小时内收到的电子邮件数量。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>连续</strong>随机变量可以在一个区间内取任意值：你的精确身高、下一班公交车到达的时间、中午的温度。</p>
|
||
</li>
|
||
<li>
|
||
<p>这种区别很重要，因为它改变了我们计算概率的方式。对于离散变量，我们求和。对于连续变量，我们积分（回顾第 3 章的积分内容）。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于离散随机变量，<strong>概率质量函数（PMF）</strong>给出每个具体值的概率：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[P(X = x) = p(x), \quad \text{其中 } \sum_{x} p(x) = 1\]</div>
|
||
<ul>
|
||
<li>对于连续随机变量，<strong>概率密度函数（PDF）</strong>给出落在某个区间内的概率。任何单个精确值的概率为零；只有区间才具有正概率：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[P(a \le X \le b) = \int_a^b f(x)\, dx, \quad \text{其中 } \int_{-\infty}^{\infty} f(x)\, dx = 1\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>既然我们可以将结果赋予数值，最自然的问题就是：平均而言我们期望得到什么值？</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>期望</strong>（或期望值）是所有可能值的加权平均值，权重即为概率。可以将其视为分布的"重心"。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果你多次掷一个公平的骰子，你的平均掷点数会收敛到 3.5。这就是期望值，尽管你实际上永远掷不出 3.5。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于离散随机变量：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[E[X] = \sum_{x} x \cdot p(x)\]</div>
|
||
<ul>
|
||
<li>对于连续随机变量（使用第 3 章的积分）：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[E[X] = \int_{-\infty}^{\infty} x \cdot f(x)\, dx\]</div>
|
||
<ul>
|
||
<li>示例：一个公平的六面骰子，对于 <span class="arithmatex">\(x = 1, 2, 3, 4, 5, 6\)</span>，有 <span class="arithmatex">\(p(x) = 1/6\)</span>。</li>
|
||
</ul>
|
||
<div class="arithmatex">\[E[X] = 1 \cdot \tfrac{1}{6} + 2 \cdot \tfrac{1}{6} + 3 \cdot \tfrac{1}{6} + 4 \cdot \tfrac{1}{6} + 5 \cdot \tfrac{1}{6} + 6 \cdot \tfrac{1}{6} = \frac{21}{6} = 3.5\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>期望具有线性性质，即 <span class="arithmatex">\(E[aX + b] = aE[X] + b\)</span>。这一性质极其有用，在机器学习损失函数中频繁出现。</p>
|
||
</li>
|
||
<li>
|
||
<p>期望告诉我们中心位置，但完全没有说明数值的分散程度。为了描述分布的完整形状，我们需要<strong>矩</strong>。</p>
|
||
</li>
|
||
<li>
|
||
<p>矩是 <span class="arithmatex">\(X\)</span> 的某次幂的期望。第 <span class="arithmatex">\(k\)</span> 阶<strong>原点矩</strong>为：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mu_k' = E[X^k]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>一阶原点矩（<span class="arithmatex">\(k = 1\)</span>）就是均值：<span class="arithmatex">\(\mu_1' = E[X] = \mu\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>原点矩是从零点开始度量的。通常我们更关心相对于均值的偏差。第 <span class="arithmatex">\(k\)</span> 阶<strong>中心矩</strong>将测量中心化：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mu_k = E[(X - \mu)^k]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>一阶中心矩始终为零（均值上下方的偏差相互抵消）。二阶中心矩就是<strong>方差</strong>。</p>
|
||
</li>
|
||
<li>
|
||
<p>为了比较不同尺度上的分布，我们通过除以标准差 <span class="arithmatex">\(\sigma\)</span> 的适当幂次来进行<strong>标准化</strong>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\tilde{\mu}_k = \frac{\mu_k}{\sigma^k}\]</div>
|
||
<ul>
|
||
<li>每个矩捕捉分布形状的不同方面：</li>
|
||
</ul>
|
||
<p><img alt="钟形曲线标注了每个矩所捕捉的特征：均值（中心）、方差（分散程度）、偏度（不对称性）、峰度（尾部重量）" src="../../images/moments_shape.svg" /></p>
|
||
<ul>
|
||
<li><strong>1 阶矩（均值）</strong>：分布的中心位置。平衡点。</li>
|
||
<li><strong>2 阶矩（方差）</strong>：数值围绕均值的分散程度。方差越大，分布越宽。</li>
|
||
<li><strong>3 阶矩（偏度）</strong>：分布向左还是向右倾斜。偏度为零表示对称。</li>
|
||
<li>
|
||
<p><strong>4 阶矩（峰度）</strong>：尾部的重量。峰度越高，极端异常值越多。</p>
|
||
</li>
|
||
<li>
|
||
<p>让我们对具体数据集 <span class="arithmatex">\(X = \{2, 4, 4, 4, 5, 5, 7, 9\}\)</span> 计算全部四个矩。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>步骤 1：均值</strong>（一阶原点矩）</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mu = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = \frac{40}{8} = 5\]</div>
|
||
<ul>
|
||
<li><strong>步骤 2：方差</strong>（二阶中心矩）。从每个值中减去均值，平方，然后取平均：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\sigma^2 = \frac{(2{-}5)^2 + (4{-}5)^2 + (4{-}5)^2 + (4{-}5)^2 + (5{-}5)^2 + (5{-}5)^2 + (7{-}5)^2 + (9{-}5)^2}{8}\]</div>
|
||
<div class="arithmatex">\[= \frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8} = \frac{32}{8} = 4\]</div>
|
||
<ul>
|
||
<li>
|
||
<p><strong>标准差</strong>为 <span class="arithmatex">\(\sigma = \sqrt{4} = 2\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>步骤 3：偏度</strong>（标准化三阶中心矩）。偏差取三次方，求平均，再除以 <span class="arithmatex">\(\sigma^3\)</span>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\tilde{\mu}_3 = \frac{1}{8} \cdot \frac{(-3)^3 + (-1)^3 + (-1)^3 + (-1)^3 + 0^3 + 0^3 + 2^3 + 4^3}{2^3}\]</div>
|
||
<div class="arithmatex">\[= \frac{1}{8} \cdot \frac{-27 -1 -1 -1 + 0 + 0 + 8 + 64}{8} = \frac{42}{64} = 0.656\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>正偏度表示右尾更长，这很合理，因为 9 远高于均值。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>步骤 4：峰度</strong>（标准化四阶中心矩）。偏差取四次方：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\tilde{\mu}_4 = \frac{1}{8} \cdot \frac{(-3)^4 + (-1)^4 + (-1)^4 + (-1)^4 + 0^4 + 0^4 + 2^4 + 4^4}{2^4}\]</div>
|
||
<div class="arithmatex">\[= \frac{1}{8} \cdot \frac{81 + 1 + 1 + 1 + 0 + 0 + 16 + 256}{16} = \frac{356}{128} = 2.781\]</div>
|
||
<ul>
|
||
<li>正态分布的峰度为 3（称为"常峰态"）。我们的 2.781 很接近，表明尾部大致呈正态。大于 3 的值（"尖峰态"）表示尾部更重；小于 3（"低峰态"）表示尾部更轻。某些公式会报告<strong>超值峰度</strong>（减去 3），因此我们的超值峰度为 <span class="arithmatex">\(-0.219\)</span>。</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>计算一个加载骰子的期望值，其中面 6 的概率为 0.3，其余面均分剩余概率。通过模拟 100,000 次投掷进行验证。
|
||
<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</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></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-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># 加载骰子：面 6 的 p=0.3，其余面均分 0.7</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">probs</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="mf">0.14</span><span class="p">,</span> <span class="mf">0.14</span><span class="p">,</span> <span class="mf">0.14</span><span class="p">,</span> <span class="mf">0.14</span><span class="p">,</span> <span class="mf">0.14</span><span class="p">,</span> <span class="mf">0.30</span><span class="p">])</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">faces</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>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="c1"># 解析法计算期望值</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">ev</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">faces</span> <span class="o">*</span> <span class="n">probs</span><span class="p">)</span>
|
||
<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="sa">f</span><span class="s2">"期望值（公式法）: </span><span class="si">{</span><span class="n">ev</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-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="c1"># 模拟</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jax</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">42</span><span class="p">)</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="n">rolls</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">faces</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">100_000</span><span class="p">,),</span> <span class="n">p</span><span class="o">=</span><span class="n">probs</span><span class="p">)</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"期望值（模拟法）: </span><span class="si">{</span><span class="n">rolls</span><span class="o">.</span><span class="n">mean</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>计算示例数据集的所有四个矩（均值、方差、偏度、峰度），然后修改数据并观察每个矩如何变化。
|
||
<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.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></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">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</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">7</span><span class="p">,</span> <span class="mi">9</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-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></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">x</span><span class="p">)</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">variance</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">x</span> <span class="o">-</span> <span class="n">mean</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">std</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">variance</span><span class="p">)</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">skewness</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">x</span> <span class="o">-</span> <span class="n">mean</span><span class="p">)</span> <span class="o">/</span> <span class="n">std</span><span class="p">)</span> <span class="o">**</span> <span class="mi">3</span><span class="p">)</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">kurtosis</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">x</span> <span class="o">-</span> <span class="n">mean</span><span class="p">)</span> <span class="o">/</span> <span class="n">std</span><span class="p">)</span> <span class="o">**</span> <span class="mi">4</span><span class="p">)</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"均值: </span><span class="si">{</span><span class="n">mean</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-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></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">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</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">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"偏度: </span><span class="si">{</span><span class="n">skewness</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-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"峰度: </span><span class="si">{</span><span class="n">kurtosis</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-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"超值峰度: </span><span class="si">{</span><span class="n">kurtosis</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">3</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>并排可视化公平骰子的 PMF 和 CDF。尝试修改概率以观察形状如何变化。
|
||
<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><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-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="n">faces</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>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">pmf</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span> <span class="o">/</span> <span class="mi">6</span> <span class="c1"># 公平骰子；试试修改这些值！</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">cdf</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">pmf</span><span class="p">)</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span><span class="p">)</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="mi">1</span><span class="p">,</span> <span class="mi">2</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-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="n">ax1</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">faces</span><span class="p">,</span> <span class="n">pmf</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#3498db"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"PMF"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"面值"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"P(X = x)"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="n">ax1</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
|
||
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a>
|
||
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="n">ax2</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">faces</span><span class="p">,</span> <span class="n">cdf</span><span class="p">,</span> <span class="n">where</span><span class="o">=</span><span class="s2">"mid"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#e74c3c"</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-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"CDF"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"面值"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"P(X ≤ x)"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="n">ax2</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">)</span>
|
||
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a>
|
||
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
|
||
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
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