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<!DOCTYPE html>
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<title>苹果的味道</title>
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<a href="." class="brand">苹果的味道</a>
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<div class="span3" id="blog-nav">
<ul class="nav nav-pills pull-right">
<li><a href="./pages/about.html">About</a></li>
<li >
<a href="./category/distributed-system.html ">Distributed System</a>
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<a href="./category/life.html ">Life</a>
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<a href="./category/machine-learning.html ">Machine Learning</a>
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<a href="./category/notes.html ">Notes</a>
<li class="active" >
<a href="./category/pearls.html ">Pearls</a>
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<a href="./category/viewpoint.html ">Viewpoint</a>
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<h1>小小收藏夹(II)</h1b>
</div>
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<div class="row-fluid">
<div class="span2">
<p>日 15 六月 2014 </p>
<p style="text-align: left;">
Filed under <a href="./category/pearls.html">Pearls</a>
</p>
<p style="text-align: left;">
Tags <a href="./tag/suan-fa.html">算法</a> <a href="./tag/fun.html">Fun</a> <a href="./tag/staff.html">Staff</a> <a href="./tag/shou-cang-jia.html">收藏夹</a> <a href="./tag/data-structure.html">Data Structure</a> <a href="./tag/algorithm.html">Algorithm</a> <a href="./tag/pgm.html">PGM</a> </p>
<p>
</p>
</div>
<div class="span8">
<h1 id="_1">傅里叶变换及应用(下)</h1>
<hr />
<blockquote>
<p>本部分为傅里叶变换及应用(下),我们接着<a href="http://www.qingyuanxingsi.com/xiao-xiao-shou-cang-jia-i.html">小小收藏夹(I)</a>一文中<strong>傅里叶变换及应用(上)</strong>部分继续我们关于傅里叶变换及其应用的学习。</p>
</blockquote>
<ul>
<li>
<p><a href="http://v.163.com/movie/2008/2/3/6/M7Q4BLENR_M7QBSAA36.html">时延性，尺度变化，卷积</a>;主要介绍傅里叶变换的若干性质。</p>
<ul>
<li>Delay;<em>A shift in time corresponds to a phase shift in frequency.</em></li>
</ul>
<p>\begin{equation}
\begin{split}
\mathscr{F} f(t-b) &=\int_{-\infty}^{\infty} e^{-2 \pi ist} f(t-b)dt \\
&= \int_{-\infty}^{\infty} e^{-2\pi is(u+b)}f(u)du \\
&= e^{-2\pi isb} \int_{-\infty}^{\infty} e^{-2\pi isu}f(u)du \\
&= e^{-2\pi isb} \mathscr{F} f(t)
\end{split}
\end{equation}</p>
<ul>
<li>Stretch;(以下仅给出当$a>0$时的情形,$a$的正负会影响积分上下限)</li>
</ul>
<p>\begin{equation}
\begin{split}
\mathscr{F} f(at) &= \int_{-\infty}^{\infty} e^{-2 \pi ist} f(at)dt\\
&= \frac{1}{a} \int_{-\infty}^{\infty} e^{-2 \pi i(s/a)u} f(u)du \\
&= \frac{1}{a} \mathscr{F} f(\frac{s}{a})
\end{split}
\end{equation}</p>
<ul>
<li>Convolution;
\begin{equation}
\begin{split}
\mathscr{F} f(t) \mathscr{F} g(x) &= (\int_{-\infty}^{\infty} e^{-2 \pi ist} f(t)dt)(\int_{-\infty}^{\infty} e^{-2 \pi isx} g(x)dx) \\
&= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-2 \pi ist} e^{-2 \pi isx} f(t)g(x)dtdx \\
&= \int_{-\infty}^{\infty}(\int_{-\infty}^{\infty} e^{-2 \pi is(t+x)}f(t)dt)g(x)dx \\
&= \int_{-\infty}^{\infty}(\int_{-\infty}^{\infty} e^{-2 \pi isu}f(u-x)du)g(x)dx \\
&= \int_{-\infty}^{\infty}(\int_{-\infty}^{\infty} f(u-x)g(x)dx)e^{-2 \pi isu}du \\
&= \mathscr{F} h(u) \\
&= \mathscr{F}(\int_{-\infty}^{\infty} f(u-x)g(x)dx)
\end{split}
\end{equation}</li>
</ul>
</li>
</ul>
<h1 id="_2">访谈</h1>
<hr />
<ul>
<li><a href="http://www.52ml.net/14374.html">LinkedIn高级分析师王益：大数据时代的理想主义和现实主义(图灵访谈)</a>;个人觉得读过之后觉得很受用的一篇文章,从事机器学习相关学习和工作的朋友可以看看。</li>
</ul>
<h1 id="_3">算法</h1>
<hr />
<ul>
<li><a href="http://nlp.stanford.edu/IR-book/html/htmledition/rocchio-classification-1.html">Rocchio_classification</a>;一种非常经典的文本分类算法，能够在用户查询时根据用户相关度反馈对Query向量进行调整,以使得搜索引擎返回的文档与搜索词间相关度更强。(<em>Relevance Feedback</em>)</li>
</ul>
<h1 id="_4">深入了解机器学习</h1>
<hr />
<p>硕士生涯马上就要开始了,在硕士阶段开始之前,想对机器学习相关知识有一个更为深入的了解。</p>
<ul>
<li>
<p>Beta/Dirichlet分布。贝叶斯方法为了简化计算量，在计算Posterier时均采用了Conjugate Prior,而Beta/Dirichlet分布则是最为常见的Conjugate Prior.以下几个文章都灰常不错,建议阅读;</p>
<ul>
<li><a href="http://www.52nlp.cn/lda-math-%E8%AE%A4%E8%AF%86betadirichlet%E5%88%86%E5%B8%831">LDA-math-认识Beta/Dirichlet分布(1)</a></li>
<li><a href="http://www.52nlp.cn/lda-math-%E8%AE%A4%E8%AF%86betadirichlet%E5%88%86%E5%B8%833">LDA-math-认识Beta/Dirichlet分布(3)</a></li>
</ul>
</li>
<li>
<p>Gamma函数递归性质的证明;</p>
</li>
</ul>
<p>\begin{equation}
\begin{split}
\Gamma(x+1) &= \int_{0}^{\infty} t^x e^{-t} dt \\
&= -\int_{0}^{\infty} t^x d e^{-t} \\
&= [-t^x e^{-t}]_0^{\infty} + \int_0^{\infty} e^{-t} x t^{x-1} dt \\
&= x \Gamma(x)
\end{split}
\end{equation}</p><script type= "text/javascript">
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