数据框中变量之间的快速成对简单线性回归
李哲源*_*李哲源 11 performance regression r linear-regression lm
我已经在Stack Overflow上多次看到成对或一般配对的简单线性回归.这是一个针对此类问题的玩具数据集.
set.seed(0)
X <- matrix(runif(100), 100, 5, dimnames = list(1:100, LETTERS[1:5]))
b <- c(1, 0.7, 1.3, 2.9, -2)
dat <- X * b[col(X)] + matrix(rnorm(100 * 5, 0, 0.1), 100, 5)
dat <- as.data.frame(dat)
pairs(dat)
所以基本上我们想要计算5*4 = 20个回归线:
----- A ~ B A ~ C A ~ D A ~ E
B ~ A ----- B ~ C B ~ D B ~ E
C ~ A C ~ B ----- C ~ D C ~ E
D ~ A D ~ B D ~ C ----- D ~ E
E ~ A E ~ B E ~ C E ~ D -----
这是一个穷人的策略:
poor <- function (dat) {
n <- nrow(dat)
p <- ncol(dat)
## all formulae
LHS <- rep(colnames(dat), p)
RHS <- rep(colnames(dat), each = p)
## function to fit and summarize a single model
fitmodel <- function (LHS, RHS) {
if (RHS == LHS) {
z <- data.frame("LHS" = LHS, "RHS" = RHS,
"alpha" = 0,
"beta" = 1,
"beta.se" = 0,
"beta.tv" = Inf,
"beta.pv" = 0,
"sig" = 0,
"R2" = 1,
"F.fv" = Inf,
"F.pv" = 0,
stringsAsFactors = FALSE)
} else {
result <- summary(lm(reformulate(RHS, LHS), data = dat))
z <- data.frame("LHS" = LHS, "RHS" = RHS,
"alpha" = result$coefficients[1, 1],
"beta" = result$coefficients[2, 1],
"beta.se" = result$coefficients[2, 2],
"beta.tv" = result$coefficients[2, 3],
"beta.pv" = result$coefficients[2, 4],
"sig" = result$sigma,
"R2" = result$r.squared,
"F.fv" = result$fstatistic[[1]],
"F.pv" = pf(result$fstatistic[[1]], 1, n - 2, lower.tail = FALSE),
stringsAsFactors = FALSE)
}
z
}
## loop through all models
do.call("rbind.data.frame", c(Map(fitmodel, LHS, RHS),
list(make.row.names = FALSE,
stringsAsFactors = FALSE)))
}
逻辑很明确:获取所有对,构造模型公式(reformulate),拟合回归(lm),进行汇总summary,返回所有统计信息,rbind并将它们作为数据框.
好的,很好,但如果有p变量怎么办？然后我们需要做p * (p - 1)回归!
我能想到的立竿见影的改进是使用多个LHS拟合线性模型.例如,该公式矩阵的第一列合并到
cbind(B, C, D, E) ~ A
这减少了从回归的数量p * (p - 1)来p.
但是我们绝对可以做得更好而不使用lm和summary.这是我之前的尝试:是否有简单回归的快速估计(仅具有截距和斜率的回归线)？.它很快,因为它使用变量之间的协方差进行估计,就像求解正规方程一样.但simpleLM那里的功能非常有限:
- 它需要计算残差向量来估计残差标准误差,这是一个性能瓶颈;
- 它不支持多个LHS,因此需要
p * (p - 1)在成对回归设置中调用时间).
我们可以通过编写函数来推广快速成对回归pairwise_simpleLM吗？
一般配对简单线性回归
上述成对回归的更有用的变化是一组LHS变量和一组RHS变量之间的一般配对回归.
例1
适合配对回归LHS变量之间A,B,C和RHS变量D,E即,配合6个简单的线性回归线:
A ~ D A ~ E
B ~ D B ~ E
C ~ D C ~ E
例2
将具有多个LHS变量的简单线性回归拟合到特定的RHS变量,例如:cbind(A, B, C, D) ~ E.
例3
使用特定LHS变量和一组RHS变量一次拟合一个简单的线性回归,例如:
A ~ B A ~ C A ~ D A ~ E
我们还能general_paired_simpleLM为此快速发挥作用吗？
警告
- 所有变量必须是数字; 不允许因素或成对回归没有意义.
- 没有讨论加权回归,因为在这种情况下方差 - 协方差方法是不合理的.
一些统计结果/背景
(图中链接:R中计算R2(R平方)的函数)
计算细节
这里涉及的计算基本上是方差 - 协方差矩阵的计算.一旦我们得到它,所有成对回归的结果就是元素方法矩阵算法.
方差 - 协方差矩阵可以通过R函数获得cov,但下面的函数使用手动计算crossprod.优点是如果你有它,它显然可以从优化的BLAS库中受益.请注意,以这种方式进行了大量的简化.R函数cov有use允许处理的参数NA,但crossprod没有.我假设你dat没有任何遗漏的价值!如果确实有缺失值,请自行删除它们na.omit(dat).
as.matrix将数据帧转换为矩阵的初始值可能是开销.原则上,如果我用C/C++编写所有代码,我可以消除这种强制.事实上,许多元素矩阵矩阵算法可以合并为单个循环嵌套.但是,我现在真的很麻烦(因为我没有时间).
有些人可能会争辩说最终回归的格式不方便.可能还有其他格式:
- 数据框列表,每个数据框给出特定LHS变量的回归结果;
- 数据框列表,每个数据框给出特定RHS变量的回归结果.
这真的是基于意见的.无论如何,您可以随时split.data.frame在我返回的数据框上执行"LHS"列或"RHS"列.
R功能 pairwise_simpleLM
pairwise_simpleLM <- function (dat) {
## matrix and its dimension (n: numbeta.ser of data; p: numbeta.ser of variables)
dat <- as.matrix(dat)
n <- nrow(dat)
p <- ncol(dat)
## variable summary: mean, (unscaled) covariance and (unscaled) variance
m <- colMeans(dat)
V <- crossprod(dat) - tcrossprod(m * sqrt(n))
d <- diag(V)
## R-squared (explained variance) and its complement
R2 <- (V ^ 2) * tcrossprod(1 / d)
R2_complement <- 1 - R2
R2_complement[seq.int(from = 1, by = p + 1, length = p)] <- 0
## slope and intercept
beta <- V * rep(1 / d, each = p)
alpha <- m - beta * rep(m, each = p)
## residual sum of squares and standard error
RSS <- R2_complement * d
sig <- sqrt(RSS * (1 / (n - 2)))
## statistics for slope
beta.se <- sig * rep(1 / sqrt(d), each = p)
beta.tv <- beta / beta.se
beta.pv <- 2 * pt(abs(beta.tv), n - 2, lower.tail = FALSE)
## F-statistic and p-value
F.fv <- (n - 2) * R2 / R2_complement
F.pv <- pf(F.fv, 1, n - 2, lower.tail = FALSE)
## export
data.frame(LHS = rep(colnames(dat), times = p),
RHS = rep(colnames(dat), each = p),
alpha = c(alpha),
beta = c(beta),
beta.se = c(beta.se),
beta.tv = c(beta.tv),
beta.pv = c(beta.pv),
sig = c(sig),
R2 = c(R2),
F.fv = c(F.fv),
F.pv = c(F.pv),
stringsAsFactors = FALSE)
}
让我们比较问题中玩具数据集的结果.
oo <- poor(dat)
rr <- pairwise_simpleLM(dat)
all.equal(oo, rr)
#[1] TRUE
让我们看看它的输出:
rr[1:3, ]
# LHS RHS alpha beta beta.se beta.tv beta.pv sig
#1 A A 0.00000000 1.0000000 0.00000000 Inf 0.000000e+00 0.0000000
#2 B A 0.05550367 0.6206434 0.04456744 13.92594 5.796437e-25 0.1252402
#3 C A 0.05809455 1.2215173 0.04790027 25.50126 4.731618e-45 0.1346059
# R2 F.fv F.pv
#1 1.0000000 Inf 0.000000e+00
#2 0.6643051 193.9317 5.796437e-25
#3 0.8690390 650.3142 4.731618e-45
当我们具有相同的LHS和RHS时,回归是没有意义的,因此截距为0,斜率为1,等等.
速度怎么样？还在使用这个玩具示例:
library(microbenchmark)
microbenchmark("poor_man's" = poor(dat), "fast" = pairwise_simpleLM(dat))
#Unit: milliseconds
# expr min lq mean median uq max
# poor_man's 127.270928 129.060515 137.813875 133.390722 139.029912 216.24995
# fast 2.732184 3.025217 3.381613 3.134832 3.313079 10.48108
由于我们有更多变数,差距将越来越大.例如,我们有10个变量:
set.seed(0)
X <- matrix(runif(100), 100, 10, dimnames = list(1:100, LETTERS[1:10]))
b <- runif(10)
DAT <- X * b[col(X)] + matrix(rnorm(100 * 10, 0, 0.1), 100, 10)
DAT <- as.data.frame(DAT)
microbenchmark("poor_man's" = poor(DAT), "fast" = pairwise_simpleLM(DAT))
#Unit: milliseconds
# expr min lq mean median uq max
# poor_man's 548.949161 551.746631 573.009665 556.307448 564.28355 801.645501
# fast 3.365772 3.578448 3.721131 3.621229 3.77749 6.791786
R功能 general_paired_simpleLM
general_paired_simpleLM <- function (dat_LHS, dat_RHS) {
## matrix and its dimension (n: numbeta.ser of data; p: numbeta.ser of variables)
dat_LHS <- as.matrix(dat_LHS)
dat_RHS <- as.matrix(dat_RHS)
if (nrow(dat_LHS) != nrow(dat_RHS)) stop("'dat_LHS' and 'dat_RHS' don't have same number of rows!")
n <- nrow(dat_LHS)
pl <- ncol(dat_LHS)
pr <- ncol(dat_RHS)
## variable summary: mean, (unscaled) covariance and (unscaled) variance
ml <- colMeans(dat_LHS)
mr <- colMeans(dat_RHS)
vl <- colSums(dat_LHS ^ 2) - ml * ml * n
vr <- colSums(dat_RHS ^ 2) - mr * mr * n
##V <- crossprod(dat - rep(m, each = n)) ## cov(u, v) = E[(u - E[u])(v - E[v])]
V <- crossprod(dat_LHS, dat_RHS) - tcrossprod(ml * sqrt(n), mr * sqrt(n)) ## cov(u, v) = E[uv] - E{u]E[v]
## R-squared (explained variance) and its complement
R2 <- (V ^ 2) * tcrossprod(1 / vl, 1 / vr)
R2_complement <- 1 - R2
## slope and intercept
beta <- V * rep(1 / vr, each = pl)
alpha <- ml - beta * rep(mr, each = pl)
## residual sum of squares and standard error
RSS <- R2_complement * vl
sig <- sqrt(RSS * (1 / (n - 2)))
## statistics for slope
beta.se <- sig * rep(1 / sqrt(vr), each = pl)
beta.tv <- beta / beta.se
beta.pv <- 2 * pt(abs(beta.tv), n - 2, lower.tail = FALSE)
## F-statistic and p-value
F.fv <- (n - 2) * R2 / R2_complement
F.pv <- pf(F.fv, 1, n - 2, lower.tail = FALSE)
## export
data.frame(LHS = rep(colnames(dat_LHS), times = pr),
RHS = rep(colnames(dat_RHS), each = pl),
alpha = c(alpha),
beta = c(beta),
beta.se = c(beta.se),
beta.tv = c(beta.tv),
beta.pv = c(beta.pv),
sig = c(sig),
R2 = c(R2),
F.fv = c(F.fv),
F.pv = c(F.pv),
stringsAsFactors = FALSE)
}
将此应用于问题中的示例1.
general_paired_simpleLM(dat[1:3], dat[4:5])
# LHS RHS alpha beta beta.se beta.tv beta.pv sig
#1 A D -0.009212582 0.3450939 0.01171768 29.45071 1.772671e-50 0.09044509
#2 B D 0.012474593 0.2389177 0.01420516 16.81908 1.201421e-30 0.10964516
#3 C D -0.005958236 0.4565443 0.01397619 32.66585 1.749650e-54 0.10787785
#4 A E 0.008650812 -0.4798639 0.01963404 -24.44040 1.738263e-43 0.10656866
#5 B E 0.012738403 -0.3437776 0.01949488 -17.63426 3.636655e-32 0.10581331
#6 C E 0.009068106 -0.6430553 0.02183128 -29.45569 1.746439e-50 0.11849472
# R2 F.fv F.pv
#1 0.8984818 867.3441 1.772671e-50
#2 0.7427021 282.8815 1.201421e-30
#3 0.9158840 1067.0579 1.749650e-54
#4 0.8590604 597.3333 1.738263e-43
#5 0.7603718 310.9670 3.636655e-32
#6 0.8985126 867.6375 1.746439e-50
将此应用于问题中的示例2.
general_paired_simpleLM(dat[1:4], dat[5])
# LHS RHS alpha beta beta.se beta.tv beta.pv sig
#1 A E 0.008650812 -0.4798639 0.01963404 -24.44040 1.738263e-43 0.1065687
#2 B E 0.012738403 -0.3437776 0.01949488 -17.63426 3.636655e-32 0.1058133
#3 C E 0.009068106 -0.6430553 0.02183128 -29.45569 1.746439e-50 0.1184947
#4 D E 0.066190196 -1.3767586 0.03597657 -38.26820 9.828853e-61 0.1952718
# R2 F.fv F.pv
#1 0.8590604 597.3333 1.738263e-43
#2 0.7603718 310.9670 3.636655e-32
#3 0.8985126 867.6375 1.746439e-50
#4 0.9372782 1464.4551 9.828853e-61
将此应用于问题中的示例3.
general_paired_simpleLM(dat[1], dat[2:5])
# LHS RHS alpha beta beta.se beta.tv beta.pv sig
#1 A B 0.112229318 1.0703491 0.07686011 13.92594 5.796437e-25 0.16446951
#2 A C 0.025628210 0.7114422 0.02789832 25.50126 4.731618e-45 0.10272687
#3 A D -0.009212582 0.3450939 0.01171768 29.45071 1.772671e-50 0.09044509
#4 A E 0.008650812 -0.4798639 0.01963404 -24.44040 1.738263e-43 0.10656866
# R2 F.fv F.pv
#1 0.6643051 193.9317 5.796437e-25
#2 0.8690390 650.3142 4.731618e-45
#3 0.8984818 867.3441 1.772671e-50
#4 0.8590604 597.3333 1.738263e-43
我们甚至可以在两个变量之间进行简单的线性回归:
general_paired_simpleLM(dat[1], dat[2])
# LHS RHS alpha beta beta.se beta.tv beta.pv sig
#1 A B 0.1122293 1.070349 0.07686011 13.92594 5.796437e-25 0.1644695
# R2 F.fv F.pv
#1 0.6643051 193.9317 5.796437e-25
这意味着simpleLM函数现在已经过时了.
附录:Markdown(需要MathJax支持)照片
Denote our variables by $x_1$, $x_2$, etc, a pairwise simple linear regression takes the form $$x_i = \alpha_{ij} + \beta_{ij}x_j$$ where $\alpha_{ij}$ and $\beta_{ij}$ is the intercept and the slope of $x_i \sim x_j$, respectively. We also denote $m_i$ and $v_i$ as the sample mean and **unscaled** sample variance of $x_i$. Here, the unscaled variance is just the sum of squares without dividing by sample size, that is $v_i = \sum_{k = 1}^n(x_{ik} - m_i)^2 = (\sum_{k = 1}^nx_{ik}^2) - n m_i^2$. We also denote $V_{ij}$ as the **unscaled** covariance between $x_i$ and $x_j$: $V_{ij} = \sum_{k = 1}^n(x_{ik} - m_i)(x_{jk} - m_j)$ = $(\sum_{k = 1}^nx_{ik}x_{jk}) - nm_im_j$.
Using the results for a simple linear regression given in [Function to calculate R2 (R-squared) in R](https://stackoverflow.com/a/40901487/4891738), we have $$\beta_{ij} = V_{ij} \ / \ v_j,\quad \alpha_{ij} = m_i - \beta_{ij}m_j,\quad r_{ij}^2 = V_{ij}^2 \ / \ (v_iv_j),$$ where $r_{ij}^2$ is the R-squared. Knowing $r_{ij}^2 = RSS_{ij} \ / \ TSS_{ij}$ where $RSS_{ij}$ and $TSS_{ij} = v_i$ are residual sum of squares and total sum of squares of $x_i \sim x_j$, we can derive $RSS_{ij}$ and residual standard error $\sigma_{ij}$ **without actually computing residuals**: $$RSS_{ij} = (1 - r_{ij}^2)v_i,\quad \sigma_{ij} = \sqrt{RSS_{ij} \ / \ (n - 2)}.$$
F-statistic $F_{ij}$ and associated p-value $p_{ij}^F$ can also be obtained from sum of squares: $$F_{ij} = \tfrac{(TSS_{ij} - RSS_{ij}) \ / \ 1}{RSS_{ij} \ / \ (n - 2)} = (n - 2) r_{ij}^2 \ / \ (1 - r_{ij}^2),\quad p_{ij}^F = 1 - \texttt{CDF_F}(F_{ij};\ 1,\ n - 2),$$ where $\texttt{CDF_F}$ denotes the CDF of F-distribution.
The only thing left is the standard error $e_{ij}$, t-statistic $t_{ij}$ and associated p-value $p_{ij}^t$ for $\beta_{ij}$, which are $$e_{ij} = \sigma_{ij} \ / \ \sqrt{v_i},\quad t_{ij} = \beta_{ij} \ / \ e_{ij},\quad p_{ij}^t = 2 * \texttt{CDF_t}(-|t_{ij}|; \ n - 2),$$ where $\texttt{CDF_t}$ denotes the CDF of t-distribution.
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