概述
1. R中的方差和标准差
方差:var
,是样本方差
var(y) instructs R to calculate the sample variance of Y. In other words it uses n-1 ‘degrees of freedom’, where n is the number of observations in Y.
标准差:sd
,是样本标准差
var(y) instructs R to calculate the sample variance of Y. In other words it uses n-1 ‘degrees of freedom’, where n is the number of observations in Y.
var
和sd
的关系
sd = sqrt(var),两者都是样本的结果。
sd(y) = sqrt(var(y)). In other words, this is the uncorrected sample standard deviation.
2. 手动计算方差和标准差
2.1 生成数据
生成数据:
library(tidyverse)
y = c(0.1,0.2,0.2,0.1,0.3,0.8,0.5)
> y
[1] 0.1 0.2 0.2 0.1 0.3 0.8 0.5
2.2 样本方差
公式计算:
> var(y)
[1] 0.0647619
手动计算:注意,分母是n-1
> sum((y - mean(y))^2)/(length(y)-1)
[1] 0.0647619
2.3 样本标准差
公式计算:
> ## 样本标准差
> sd(y)
[1] 0.2544836
手动计算:
> sqrt(var(y))
[1] 0.2544836
或者:
> sqrt(sum((y - mean(y))^2)/(length(y)-1))
[1] 0.2544836
2.4 总体方差
> ## 总体方差
> sum((y - mean(y))^2)/length(y)
[1] 0.0555102
或者:
> mean((y-mean(y))^2)
[1] 0.0555102
2.5 总体标准差
> ## 总体标准差
> sqrt(sum((y - mean(y))^2)/length(y))
[1] 0.235606
或者:
> sqrt(mean((y-mean(y))^2))
[1] 0.235606
3. 中心化和标准化
中心化和标准化意义一样,都是消除量纲的影响
- 中心化:数据-均值
- 标准化:(数据-均值)/标准差
3.1 R中的scale函数中心化和标准化
scale用法:
Description
scale centers and/or scales the columns of a numeric table.
Usage
## S4 method for signature 'db.obj'
scale(x, center = TRUE, scale = TRUE)
Arguments
x
A db.obj object. It represents a table/view in the database if it is an db.data.frame object, or a series of operations applied on an existing db.data.frame object if it is a db.Rquery object.
center
either a logical value or a numeric vector of length equal to the number of columns of 'x'.
scale
either a logical value or a numeric vector of length equal to the number of columns of 'x'.
参数解释:
- center 为TRUE时,中心化,为FALSE不中心化
- sacle 为TRUE时,为标准化,为FALSE不标准化
3.2 center=T,scale=T
scale默认的参数是center=T,scale=T
公式计算:
> ## 标准化: center =T, scale =T
> scale(y,center = T,scale = T)
[,1]
[1,] -0.84204134
[2,] -0.44908871
[3,] -0.44908871
[4,] -0.84204134
[5,] -0.05613609
[6,] 1.90862703
[7,] 0.72976916
attr(,"scaled:center")
[1] 0.3142857
attr(,"scaled:scale")
[1] 0.2544836
手动计算:
> (y - mean(y))/sd(y)
[1] -0.84204134 -0.44908871 -0.44908871 -0.84204134 -0.05613609 1.90862703 0.72976916
3.3 center=T,scale=F
公式计算:
> ## 标准化: center =T, scale =F
> scale(y,center = T,scale = F)
[,1]
[1,] -0.21428571
[2,] -0.11428571
[3,] -0.11428571
[4,] -0.21428571
[5,] -0.01428571
[6,] 0.48571429
[7,] 0.18571429
attr(,"scaled:center")
[1] 0.3142857
手动计算:
> y - mean(y)
[1] -0.21428571 -0.11428571 -0.11428571 -0.21428571 -0.01428571 0.48571429 0.18571429
3.4 center=F,scale=F
公式计算:
> ## 标准化: center =F, scale =F
> scale(y,center = F,scale = F)
[,1]
[1,] 0.1
[2,] 0.2
[3,] 0.2
[4,] 0.1
[5,] 0.3
[6,] 0.8
[7,] 0.5
手动计算:
> y
[1] 0.1 0.2 0.2 0.1 0.3 0.8 0.5
3.5 center=F,scale=T
公式计算:
> ## 标准化: center =F, scale =T
> scale(y,center = F,scale = T)
[,1]
[1,] 0.2357023
[2,] 0.4714045
[3,] 0.4714045
[4,] 0.2357023
[5,] 0.7071068
[6,] 1.8856181
[7,] 1.1785113
attr(,"scaled:scale")
[1] 0.4242641
手动计算:
手动计算,不仅仅是分子变化了,分母也变化了,注意!
> y/sqrt(sum(y^2)/(length(y)-1))
[1] 0.2357023 0.4714045 0.4714045 0.2357023 0.7071068 1.8856181 1.1785113
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