在R中拟合正态分布

Question

我正在使用以下代码来适应正态分布."b"(太大而不能直接发布)的数据集链接是:

b的链接

setwd("xxxxxx")
library(fitdistrplus)

require(MASS)
tazur <-read.csv("b", header= TRUE, sep=",")
claims<-tazur$b
a<-log(claims)
plot(hist(a))

绘制直方图后,似乎正态分布应该很好.

f1n <- fitdistr(claims,"normal")
summary(f1n)

#Length Class  Mode   
#estimate 2      -none- numeric
#sd       2      -none- numeric
#vcov     4      -none- numeric
#n        1      -none- numeric
#loglik   1      -none- numeric

plot(f1n)

xy.coords(x,y,xlabel,ylabel,log)中的错误:

'x'是一个列表,但没有组件'x'和'y'

当我尝试绘制拟合分布时,我得到上述错误,甚至f1n的摘要统计信息都没有.

非常感谢任何帮助.

Answer 1

看起来你MASS::fitdistr和之间的混淆fitdistrplus::fitdist.

• MASS::fitdistr返回类"fitdistr"的对象,并且没有用于此的绘图方法.因此,您需要提取估计的参数并自己绘制估计的密度曲线.
• 我不知道为什么你加载包fitdistrplus,因为你的函数调用清楚地表明你正在使用MASS.无论如何,fitdistrplus具有fitdist返回类"fitdist"对象的功能.这个类有plot方法,但是对于返回的"fitdistr"不起作用MASS.

我将向您展示如何使用这两个包.

## reproducible example
set.seed(0); x <- rnorm(500)

运用 MASS::fitdistr

没有绘图方法,所以自己做.

library(MASS)
fit <- fitdistr(x, "normal")
class(fit)
# [1] "fitdistr"

para <- fit$estimate
#         mean            sd 
#-0.0002000485  0.9886248515 

hist(x, prob = TRUE)
curve(dnorm(x, para[1], para[2]), col = 2, add = TRUE)

运用 fitdistrplus::fitdist

library(fitdistrplus)
FIT <- fitdist(x, "norm")    ## note: it is "norm" not "normal"
class(FIT)
# [1] "fitdist"

plot(FIT)    ## use method `plot.fitdist`

Answer 2

复习先前的答案

在前面的答案中，我没有提到两种方法之间的区别。通常，如果我们选择最大似然推断，我建议使用MASS::fitdistr，因为对于许多基本分布，它执行精确推断而不是数值优化。Doc ?fitdistr明确指出了这一点：

For the Normal, log-Normal, geometric, exponential and Poisson
distributions the closed-form MLEs (and exact standard errors) are
used, and ‘start’ should not be supplied.

For all other distributions, direct optimization of the
log-likelihood is performed using ‘optim’.  The estimated standard
errors are taken from the observed information matrix, calculated
by a numerical approximation.  For one-dimensional problems the
Nelder-Mead method is used and for multi-dimensional problems the
BFGS method, unless arguments named ‘lower’ or ‘upper’ are
supplied (when ‘L-BFGS-B’ is used) or ‘method’ is supplied
explicitly.

另一方面，fitdistrplus::fitdist即使存在精确推断，也始终以数字方式执行推断。当然，其优点fitdist是可以使用更多的推理原理：

Fit of univariate distributions to non-censored data by maximum
likelihood (mle), moment matching (mme), quantile matching (qme)
or maximizing goodness-of-fit estimation (mge).

该答案的目的

这个答案将探索正态分布的确切推论。它具有理论上的味道，但是没有似然原理的证明。仅给出结果。基于这些结果，我们编写了自己的R函数以进行精确推断，可以将其与进行比较MASS::fitdistr。另一方面，为了与进行比较fitdistrplus::fitdist，我们optim在数值上使用了最小对数似然函数。

这是学习统计学和相对高级使用的绝佳机会optim。为方便起见，我将估算比例参数：方差，而不是标准误差。

正态分布的精确推断

自己编写推理功能

以下代码得到了很好的注释。有个开关exact。如果设置FALSE，则选择数值解。

## fitting a normal distribution
fitnormal <- function (x, exact = TRUE) {
  if (exact) {
    ################################################
    ## Exact inference based on likelihood theory ##
    ################################################
    ## minimum negative log-likelihood (maximum log-likelihood) estimator of `mu` and `phi = sigma ^ 2`
    n <- length(x)
    mu <- sum(x) / n
    phi <- crossprod(x - mu)[1L] / n  # (a bised estimator, though)
    ## inverse of Fisher information matrix evaluated at MLE
    invI <- matrix(c(phi, 0, 0, phi * phi), 2L,
                   dimnames = list(c("mu", "sigma2"), c("mu", "sigma2")))
    ## log-likelihood at MLE
    loglik <- -(n / 2) * (log(2 * pi * phi) + 1)
    ## return
    return(list(theta = c(mu = mu, sigma2 = phi), vcov = invI, loglik = loglik, n = n))
    }
  else {
    ##################################################################
    ## Numerical optimization by minimizing negative log-likelihood ##
    ##################################################################
    ## negative log-likelihood function
    ## define `theta = c(mu, phi)` in order to use `optim`
    nllik <- function (theta, x) {
      (length(x) / 2) * log(2 * pi * theta[2]) + crossprod(x - theta[1])[1] / (2 * theta[2])
      }
    ## gradient function (remember to flip the sign when using partial derivative result of log-likelihood)
    ## define `theta = c(mu, phi)` in order to use `optim`
    gradient <- function (theta, x) {
      pl2pmu <- -sum(x - theta[1]) / theta[2]
      pl2pphi <- -crossprod(x - theta[1])[1] / (2 * theta[2] ^ 2) + length(x) / (2 * theta[2])
      c(pl2pmu, pl2pphi)
      }
    ## ask `optim` to return Hessian matrix by `hessian = TRUE`
    ## use "..." part to pass `x` as additional / further argument to "fn" and "gn"
    ## note, we want `phi` as positive so box constraint is used, with "L-BFGS-B" method chosen
    init <- c(sample(x, 1), sample(abs(x) + 0.1, 1))  ## arbitrary valid starting values
    z <- optim(par = init, fn = nllik, gr = gradient, x = x, lower = c(-Inf, 0), method = "L-BFGS-B", hessian = TRUE)
    ## post processing ##
    theta <- z$par
    loglik <- -z$value  ## flip the sign to get log-likelihood
    n <- length(x)
    ## Fisher information matrix (don't flip the sign as this is the Hessian for negative log-likelihood)
    I <- z$hessian / n  ## remember to take average to get mean
    invI <- solve(I, diag(2L))  ## numerical inverse
    dimnames(invI) <- list(c("mu", "sigma2"), c("mu", "sigma2"))
    ## return
    return(list(theta = theta, vcov = invI, loglik = loglik, n = n))
    }
  }

我们仍然使用以前的数据进行测试：

set.seed(0); x <- rnorm(500)

## exact inference
fit <- fitnormal(x)

#$theta
#           mu        sigma2 
#-0.0002000485  0.9773790969 
#
#$vcov
#              mu    sigma2
#mu     0.9773791 0.0000000
#sigma2 0.0000000 0.9552699
#
#$loglik
#[1] -703.7491
#
#$n
#[1] 500

hist(x, prob = TRUE)
curve(dnorm(x, fit$theta[1], sqrt(fit$theta[2])), add = TRUE, col = 2)

数值方法也相当准确，除了方差协方差在对角线处没有精确的0：

fitnormal(x, FALSE)

#$theta
#[1] -0.0002235315  0.9773732277
#
#$vcov
#                 mu       sigma2
#mu     9.773826e-01 5.359978e-06
#sigma2 5.359978e-06 1.910561e+00
#
#$loglik
#[1] -703.7491
#
#$n
#[1] 500