《Practical Statistics for Data Scientists》Chapter 2：Data and Sampling distributions

《Practical Statistics for Data Scientists》Chapter 2：Data and Sampling distributions

本章主要总结几个概念性的东西：

• target shuffle：discern between signal and noise
• confidence intervals | the level of confidence
• QQ-plots：利用该图判定数据是否服从正态分布
• long-tailed distribution
• t-distribution
• binomial distribution
• possion distribution | exponential distribution | weibull distribution

1、target shuffle：当你训练了一个model后，想要验证model的预测效果，可以首先用model预测若干sample的label，然后将这些label重新shuffle，得到label*，将这2种label与sample实际的label_y进行比较，如果label与label_y大部分相同，而label*与label_y仅有几例是相同的，说明，model能够有效预测sample。target shuffle这种方法可以convince client to believe the model。
参考文章：target shuffle

2、confidence interval（置信区间） && the level of confidence（置信度）
置信区间 一般与 置信度 相对应，如下图所示：90%的置信度 对应的 置信区间为：[53610，62279]。

对于上述置信度和置信区间，我们可以这样理解：我们有90%的把握说，总体的真值在[53610，62279]区间内。
一般，置信度越小，置信区间越窄。

3、在真实世界，鲜少有raw data服从normal distribution，要判断一个raw data是否服从normal distribution(Guassian)，我们可以通过如下步骤进行判断：
step1：将raw data中的value进行z-score转换，记为Z；（z-score就是将value进行如下转化：(value - mean)/std ）
step2：将Z进行升序排序；
step3：plots each value’s z-score on the y-axis; the x-axis is the corresponding quantile of a normal distribution for that value’s rank.
step4：If the points roughly fall on the diagonal line, then the sample distribution can be considered close to normal.

在R中的代码实现如下：

norm_samp <- rnorm(100)
qqnorm(norm_samp)
abline(a=0, b=1, col='grey')

4、介绍几种分布

• long-tailed distribution
  Most data is not normally distributed.
  Assuming a normal distribution can lead to underestimation of extreme events，下图为QQ-plot，图中展现了long-tailed distribution 与 normal distribution的区别，从图中可以看出，在long-tailed distribution中，极值部分 都与 normal dsitribution(diagonal line)不重叠，long-tailed distribution更能代表“实际数据的分布”。
  除此以外，从图中还可以看出，在均值的1个标准差范围内，long-tailed distribution接近line。
  
  在实际中，对于raw data的distribution要根据“领域经验”和“统计知识”一起判断。比如，拟合“events per time period”，可以采用Possion distribution。
• Student’s t-Distribution

1. The t-distribution is actually a family of distributions resembling the normal distribution, but with thicker tails.
2. It is widely used as a reference basis for the distribution of sample means, differerences between two sample means, regression parameters, and other sample statistics.
3. sample数量越大，其拟合的t-distribution越接近于normal distribution.

• Binomial distribution

1. Binomial outcomes are important to model, since they represent, among other things, fundamental decisions (buy or don’t buy, click or don’t click, survive or die, etc.).
2. A binomial trial is an experiment with two possible outcomes: one with probability p and the other with probability 1 – p.
3. With large n, and provided p is not too close to 0 or 1, the binomial distribution can be approximated by the normal distribution.

• Possion and related distribution