在 R 中拟合累积分布后，根据拟合参数创建正态分布
After fitting the cumulative distribution in R creating the normal distribution from fitted parameters
用 Gompertz 函数成功拟合我的累积数据后，我需要从拟合函数创建正态分布。
这是目前的代码：
df <- data.frame(x = c(0.01,0.011482,0.013183,0.015136,0.017378,0.019953,0.022909,0.026303,0.0302,0.034674,0.039811,0.045709,0.052481,0.060256,0.069183,0.079433,0.091201,0.104713,0.120226,0.138038,0.158489,0.18197,0.20893,0.239883,0.275423,0.316228,0.363078,0.416869,0.47863,0.549541,0.630957,0.724436,0.831764,0.954993,1.096478,1.258925,1.44544,1.659587,1.905461,2.187762,2.511886,2.884031,3.311311,3.801894,4.365158,5.011872,5.754399,6.606934,7.585776,8.709636,10,11.481536,13.182567,15.135612,17.378008,19.952623,22.908677,26.30268,30.199517,34.673685,39.810717,45.708819,52.480746,60.255959,69.183097,79.432823,91.201084,104.712855,120.226443,138.038426,158.489319,181.970086,208.929613,239.883292,275.42287,316.227766,363.078055,416.869383,478.630092,549.540874,630.957344,724.43596,831.763771,954.992586,1096.478196),
y = c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.00044816,0.00127554,0.00221488,0.00324858,0.00438312,0.00559138,0.00686054,0.00817179,0.00950625,0.01085188,0.0122145,0.01362578,0.01514366,0.01684314,0.01880564,0.02109756,0.0237676,0.02683182,0.03030649,0.0342276,0.03874555,0.04418374,0.05119304,0.06076553,0.07437854,0.09380666,0.12115065,0.15836926,0.20712933,0.26822017,0.34131335,0.42465413,0.51503564,0.60810697,0.69886817,0.78237651,0.85461023,0.91287236,0.95616228,0.98569093,0.99869001,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999))
library(drc)
fm <- drm(y ~ x, data = df, fct = G.3())
options(scipen = 10) #to avoid scientific notation in x axis
plot(df$x, predict(fm),type = "l", log = "x",col = "blue",
main = "Cumulative function distribution",xlab = "x", ylab = "y")
points(df,col = "red")
legend("topleft", inset = .05,legend = c("exp","fit")
,lty = c(NA,1), col = c("red", "blue"), pch = c(1,NA), lwd=1, bty = "n")
summary(fm)
这是下面的情节：
我现在的想法是以某种方式将这种累积拟合转换为正态分布。有什么想法我该怎么做？
我在想 cumdiff（因为没有更好的术语）。 帮助很大。
编辑
plot(df$x[-1], Mod(df$y[-length(df$y)]-df$y[-1]), log = "x", type = "b",
main = "Normal distribution for original data",
xlab = "x", ylab = "y")
产量：
添加
为了从fitted函数得到高斯：
df$y_pred<-predict(fm)
plot(df$x[-1], Mod(df$y_pred[-length(df$y_pred)]-df$y_pred[-1]), log = "x",
type = "b", main="Normal distribution for fitted function",
xlab = "x", lab = "y")
产量：
虽然你的初衷可能是非参数的，但我建议使用参数估计方法：矩量法，这种方法广泛用于此类问题，因为你有一定的参数分布（正态分布）来拟合。思路很简单，从拟合的累积分布函数，你可以计算均值（我代码中的E1）和方差（我代码中SD的平方），然后问题就解决了，因为正态分布可以完全由均值和方差决定。
df <- data.frame(x=c(0.01,0.011482,0.013183,0.015136,0.017378,0.019953,0.022909,0.026303,0.0302,0.034674,0.039811,0.045709,0.052481,0.060256,0.069183,0.079433,0.091201,0.104713,0.120226,0.138038,0.158489,0.18197,0.20893,0.239883,0.275423,0.316228,0.363078,0.416869,0.47863,0.549541,0.630957,0.724436,0.831764,0.954993,1.096478,1.258925,1.44544,1.659587,1.905461,2.187762,2.511886,2.884031,3.311311,3.801894,4.365158,5.011872,5.754399,6.606934,7.585776,8.709636,10,11.481536,13.182567,15.135612,17.378008,19.952623,22.908677,26.30268,30.199517,34.673685,39.810717,45.708819,52.480746,60.255959,69.183097,79.432823,91.201084,104.712855,120.226443,138.038426,158.489319,181.970086,208.929613,239.883292,275.42287,316.227766,363.078055,416.869383,478.630092,549.540874,630.957344,724.43596,831.763771,954.992586,1096.478196),
y=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.00044816,0.00127554,0.00221488,0.00324858,0.00438312,0.00559138,0.00686054,0.00817179,0.00950625,0.01085188,0.0122145,0.01362578,0.01514366,0.01684314,0.01880564,0.02109756,0.0237676,0.02683182,0.03030649,0.0342276,0.03874555,0.04418374,0.05119304,0.06076553,0.07437854,0.09380666,0.12115065,0.15836926,0.20712933,0.26822017,0.34131335,0.42465413,0.51503564,0.60810697,0.69886817,0.78237651,0.85461023,0.91287236,0.95616228,0.98569093,0.99869001,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999))
library(drc)
fm <- drm(y ~ x, data = df, fct = G.3())
options(scipen = 10) #to avoid scientific notation in x axis
plot(df$x, predict(fm),type="l", log = "x",col="blue", main="Cumulative distribution function",xlab="x", ylab="y")
points(df,col="red")
E1 <- sum((df$x[-1] + df$x[-length(df$x)]) / 2 * diff(predict(fm)))
E2 <- sum((df$x[-1] + df$x[-length(df$x)]) ^ 2 / 4 * diff(predict(fm)))
SD <- sqrt(E2 - E1 ^ 2)
points(df$x, pnorm((df$x - E1) / SD), col = "green")
legend("topleft", inset = .05,legend= c("exp","fit","method of moment")
,lty = c(NA,1), col = c("red", "blue", "green"), pch = c(1,NA), lwd=1, bty="n")
summary(fm)
以及估算结果：
## > E1 (mean of fitted normal distribution)
## [1] 65.78474
## > E2 (second moment of fitted normal distribution)
##[1] 5792.767
## > SD (standard deviation of fitted normal distribution)
## [1] 38.27707
## > SD ^ 2 (variance of fitted normal distribution)
## [1] 1465.134
编辑：更新了根据 drc 拟合的 cdf 计算力矩的方法。下面定义的函数 moment 使用连续 r.v 的矩公式计算矩估计。 E(X ^ k) = k * \int x ^ {k - 1} (1 - cdf(x)) dx。这些是我可以从拟合的 cdf 得到的最佳估计。正如我在评论中讨论的那样，由于原始数据集中的原因，当 x 接近零时，拟合不是很好。
df <- data.frame(x=c(0.01,0.011482,0.013183,0.015136,0.017378,0.019953,0.022909,0.026303,0.0302,0.034674,0.039811,0.045709,0.052481,0.060256,0.069183,0.079433,0.091201,0.104713,0.120226,0.138038,0.158489,0.18197,0.20893,0.239883,0.275423,0.316228,0.363078,0.416869,0.47863,0.549541,0.630957,0.724436,0.831764,0.954993,1.096478,1.258925,1.44544,1.659587,1.905461,2.187762,2.511886,2.884031,3.311311,3.801894,4.365158,5.011872,5.754399,6.606934,7.585776,8.709636,10,11.481536,13.182567,15.135612,17.378008,19.952623,22.908677,26.30268,30.199517,34.673685,39.810717,45.708819,52.480746,60.255959,69.183097,79.432823,91.201084,104.712855,120.226443,138.038426,158.489319,181.970086,208.929613,239.883292,275.42287,316.227766,363.078055,416.869383,478.630092,549.540874,630.957344,724.43596,831.763771,954.992586,1096.478196),
y=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.00044816,0.00127554,0.00221488,0.00324858,0.00438312,0.00559138,0.00686054,0.00817179,0.00950625,0.01085188,0.0122145,0.01362578,0.01514366,0.01684314,0.01880564,0.02109756,0.0237676,0.02683182,0.03030649,0.0342276,0.03874555,0.04418374,0.05119304,0.06076553,0.07437854,0.09380666,0.12115065,0.15836926,0.20712933,0.26822017,0.34131335,0.42465413,0.51503564,0.60810697,0.69886817,0.78237651,0.85461023,0.91287236,0.95616228,0.98569093,0.99869001,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999))
library(drc)
fm <- drm(y ~ x, data = df, fct = G.3())
moment <- function(k){
f <- function(x){
x ^ (k - 1) * pmax(0, 1 - predict(fm, data.frame(x = x)))
}
k * integrate(f, lower = min(df$x), upper = max(df$x))$value
}
E1 <- moment(1)
E2 <- moment(2)
SD <- sqrt(E2 - E1 ^ 2)
用 Gompertz 函数成功拟合我的累积数据后，我需要从拟合函数创建正态分布。
这是目前的代码：
df <- data.frame(x = c(0.01,0.011482,0.013183,0.015136,0.017378,0.019953,0.022909,0.026303,0.0302,0.034674,0.039811,0.045709,0.052481,0.060256,0.069183,0.079433,0.091201,0.104713,0.120226,0.138038,0.158489,0.18197,0.20893,0.239883,0.275423,0.316228,0.363078,0.416869,0.47863,0.549541,0.630957,0.724436,0.831764,0.954993,1.096478,1.258925,1.44544,1.659587,1.905461,2.187762,2.511886,2.884031,3.311311,3.801894,4.365158,5.011872,5.754399,6.606934,7.585776,8.709636,10,11.481536,13.182567,15.135612,17.378008,19.952623,22.908677,26.30268,30.199517,34.673685,39.810717,45.708819,52.480746,60.255959,69.183097,79.432823,91.201084,104.712855,120.226443,138.038426,158.489319,181.970086,208.929613,239.883292,275.42287,316.227766,363.078055,416.869383,478.630092,549.540874,630.957344,724.43596,831.763771,954.992586,1096.478196),
y = c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.00044816,0.00127554,0.00221488,0.00324858,0.00438312,0.00559138,0.00686054,0.00817179,0.00950625,0.01085188,0.0122145,0.01362578,0.01514366,0.01684314,0.01880564,0.02109756,0.0237676,0.02683182,0.03030649,0.0342276,0.03874555,0.04418374,0.05119304,0.06076553,0.07437854,0.09380666,0.12115065,0.15836926,0.20712933,0.26822017,0.34131335,0.42465413,0.51503564,0.60810697,0.69886817,0.78237651,0.85461023,0.91287236,0.95616228,0.98569093,0.99869001,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999))
library(drc)
fm <- drm(y ~ x, data = df, fct = G.3())
options(scipen = 10) #to avoid scientific notation in x axis
plot(df$x, predict(fm),type = "l", log = "x",col = "blue",
main = "Cumulative function distribution",xlab = "x", ylab = "y")
points(df,col = "red")
legend("topleft", inset = .05,legend = c("exp","fit")
,lty = c(NA,1), col = c("red", "blue"), pch = c(1,NA), lwd=1, bty = "n")
summary(fm)
这是下面的情节：
我现在的想法是以某种方式将这种累积拟合转换为正态分布。有什么想法我该怎么做？
我在想 cumdiff（因为没有更好的术语）。
编辑
plot(df$x[-1], Mod(df$y[-length(df$y)]-df$y[-1]), log = "x", type = "b",
main = "Normal distribution for original data",
xlab = "x", ylab = "y")
产量：
添加
为了从fitted函数得到高斯：
df$y_pred<-predict(fm)
plot(df$x[-1], Mod(df$y_pred[-length(df$y_pred)]-df$y_pred[-1]), log = "x",
type = "b", main="Normal distribution for fitted function",
xlab = "x", lab = "y")
产量：
虽然你的初衷可能是非参数的，但我建议使用参数估计方法：矩量法，这种方法广泛用于此类问题，因为你有一定的参数分布（正态分布）来拟合。思路很简单，从拟合的累积分布函数，你可以计算均值（我代码中的E1）和方差（我代码中SD的平方），然后问题就解决了，因为正态分布可以完全由均值和方差决定。
df <- data.frame(x=c(0.01,0.011482,0.013183,0.015136,0.017378,0.019953,0.022909,0.026303,0.0302,0.034674,0.039811,0.045709,0.052481,0.060256,0.069183,0.079433,0.091201,0.104713,0.120226,0.138038,0.158489,0.18197,0.20893,0.239883,0.275423,0.316228,0.363078,0.416869,0.47863,0.549541,0.630957,0.724436,0.831764,0.954993,1.096478,1.258925,1.44544,1.659587,1.905461,2.187762,2.511886,2.884031,3.311311,3.801894,4.365158,5.011872,5.754399,6.606934,7.585776,8.709636,10,11.481536,13.182567,15.135612,17.378008,19.952623,22.908677,26.30268,30.199517,34.673685,39.810717,45.708819,52.480746,60.255959,69.183097,79.432823,91.201084,104.712855,120.226443,138.038426,158.489319,181.970086,208.929613,239.883292,275.42287,316.227766,363.078055,416.869383,478.630092,549.540874,630.957344,724.43596,831.763771,954.992586,1096.478196),
y=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.00044816,0.00127554,0.00221488,0.00324858,0.00438312,0.00559138,0.00686054,0.00817179,0.00950625,0.01085188,0.0122145,0.01362578,0.01514366,0.01684314,0.01880564,0.02109756,0.0237676,0.02683182,0.03030649,0.0342276,0.03874555,0.04418374,0.05119304,0.06076553,0.07437854,0.09380666,0.12115065,0.15836926,0.20712933,0.26822017,0.34131335,0.42465413,0.51503564,0.60810697,0.69886817,0.78237651,0.85461023,0.91287236,0.95616228,0.98569093,0.99869001,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999))
library(drc)
fm <- drm(y ~ x, data = df, fct = G.3())
options(scipen = 10) #to avoid scientific notation in x axis
plot(df$x, predict(fm),type="l", log = "x",col="blue", main="Cumulative distribution function",xlab="x", ylab="y")
points(df,col="red")
E1 <- sum((df$x[-1] + df$x[-length(df$x)]) / 2 * diff(predict(fm)))
E2 <- sum((df$x[-1] + df$x[-length(df$x)]) ^ 2 / 4 * diff(predict(fm)))
SD <- sqrt(E2 - E1 ^ 2)
points(df$x, pnorm((df$x - E1) / SD), col = "green")
legend("topleft", inset = .05,legend= c("exp","fit","method of moment")
,lty = c(NA,1), col = c("red", "blue", "green"), pch = c(1,NA), lwd=1, bty="n")
summary(fm)
以及估算结果：
## > E1 (mean of fitted normal distribution)
## [1] 65.78474
## > E2 (second moment of fitted normal distribution)
##[1] 5792.767
## > SD (standard deviation of fitted normal distribution)
## [1] 38.27707
## > SD ^ 2 (variance of fitted normal distribution)
## [1] 1465.134
编辑：更新了根据 drc 拟合的 cdf 计算力矩的方法。下面定义的函数 moment 使用连续 r.v 的矩公式计算矩估计。 E(X ^ k) = k * \int x ^ {k - 1} (1 - cdf(x)) dx。这些是我可以从拟合的 cdf 得到的最佳估计。正如我在评论中讨论的那样，由于原始数据集中的原因，当 x 接近零时，拟合不是很好。
df <- data.frame(x=c(0.01,0.011482,0.013183,0.015136,0.017378,0.019953,0.022909,0.026303,0.0302,0.034674,0.039811,0.045709,0.052481,0.060256,0.069183,0.079433,0.091201,0.104713,0.120226,0.138038,0.158489,0.18197,0.20893,0.239883,0.275423,0.316228,0.363078,0.416869,0.47863,0.549541,0.630957,0.724436,0.831764,0.954993,1.096478,1.258925,1.44544,1.659587,1.905461,2.187762,2.511886,2.884031,3.311311,3.801894,4.365158,5.011872,5.754399,6.606934,7.585776,8.709636,10,11.481536,13.182567,15.135612,17.378008,19.952623,22.908677,26.30268,30.199517,34.673685,39.810717,45.708819,52.480746,60.255959,69.183097,79.432823,91.201084,104.712855,120.226443,138.038426,158.489319,181.970086,208.929613,239.883292,275.42287,316.227766,363.078055,416.869383,478.630092,549.540874,630.957344,724.43596,831.763771,954.992586,1096.478196),
y=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.00044816,0.00127554,0.00221488,0.00324858,0.00438312,0.00559138,0.00686054,0.00817179,0.00950625,0.01085188,0.0122145,0.01362578,0.01514366,0.01684314,0.01880564,0.02109756,0.0237676,0.02683182,0.03030649,0.0342276,0.03874555,0.04418374,0.05119304,0.06076553,0.07437854,0.09380666,0.12115065,0.15836926,0.20712933,0.26822017,0.34131335,0.42465413,0.51503564,0.60810697,0.69886817,0.78237651,0.85461023,0.91287236,0.95616228,0.98569093,0.99869001,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999,0.99999999))
library(drc)
fm <- drm(y ~ x, data = df, fct = G.3())
moment <- function(k){
f <- function(x){
x ^ (k - 1) * pmax(0, 1 - predict(fm, data.frame(x = x)))
}
k * integrate(f, lower = min(df$x), upper = max(df$x))$value
}
E1 <- moment(1)
E2 <- moment(2)
SD <- sqrt(E2 - E1 ^ 2)