在 R 中拟合零膨胀泊松的问题答案

【问题标题】：Issue with fitting Zero Inflated Poisson in R在 R 中拟合零膨胀泊松的问题
【发布时间】：2020-05-05 16:57:01
【问题描述】：

我有 755 行数据和约 87% 的零。我很难将零膨胀泊松或负二项式（或任何）回归拟合到该数据。我尝试了 4 种不同的方法，但无法让它发挥作用。我什至不确定这些是否是我应该使用的回归。任何帮助将非常感激。我也不擅长编码，我相信这很明显。

我知道这很长，但这是我的实际数据...

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, 0.404749055585537, 0, 0.134916351861846, 
0, 0.149907057624273, 0, 0, 0, 0, 0, 0, 0, 0, 0.269832703723691, 
0.134916351861846, 0.134916351861846, 0, 0, 0.269832703723691, 
0, 0.269832703723691, 0, 0, 0, 0, 0, 0.134916351861846, 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, 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, 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, 0, 0.112430293218205, 0, 0, 0, 
0, 0, 0, 0.367953686895943, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0.122651228965314, 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, 0.337290879654614, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0.578212936550767, 0, 0, 0.404749055585537, 0, 0, 0, 0.269832703723691, 
0.269832703723691, 0, 0, 0.299814115248546, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0.404749055585537, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.192737645516922, 
0.192737645516922, 0, 0, 0, 0.134916351861846, 0, 0.134916351861846, 
0, 0, 0, 0, 0, 0.404749055585537, 0.134916351861846, 0.134916351861846, 
0.337290879654614, 0, 0, 0, 0, 0.674581759309228, 0, 0.134916351861846, 
0, 0.299814115248546, 0.168645439827307, 0.449721172872819, 0, 
0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0.134916351861846, 
0, 0, 0, 0.122651228965314, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 
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, 0.134916351861846, 0, 0.134916351861846, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0.122651228965314, 0, 0, 0.134916351861846, 
0, 0, 0.149907057624273, 0, 0, 0, 0, 0.269832703723691, 0, 0, 
0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0.134916351861846, 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, 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, 0.134916351861846, 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, 0, 0, 0, 0, 0.449721172872819, 0, 0, 0, 0, 0, 0, 0, 0.112430293218205, 
0, 0, 0.134916351861846, 0.539665407447383, 0.134916351861846, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0.134916351861846, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0.134916351861846, 0.134916351861846, 0, 0, 0, 0, 0, 0.134916351861846, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 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, 0.134916351861846, 
0, 0, 0, 0, 0, 0, 0, 0, 0.404749055585537, 0, 0, 0, 0.404749055585537, 
0, 0, 0.674581759309228, 0.269832703723691, 0, 0, 0, 0, 0, 0, 
0.134916351861846, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0.269832703723691, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.404749055585537, 
0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0.269832703723691, 
0.269832703723691, 0.134916351861846, 0, 0.404749055585537, 0.809498111171074, 
0, 0.134916351861846, 0.134916351861846, 1.07933081489477, 0.134916351861846, 
0, 0.269832703723691, 0, 0.94441446303292, 0.245302457930628, 
0, 0, 0, 0, 0, 0.245302457930628, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

这是我今天尝试的 4 种方法。

> hog.cpue <- hogA$hog.cpue
> fitg <- fitdist(hog.cpue, "ZIP")
Error in computing default starting values.
Error in manageparam(start.arg = start, fix.arg = fix.arg, obs = data,  : 
  Error in start.arg.default(obs, distname) : 
  Unknown starting values for distribution ZIP.

> fit_zip2 <- fitdist(hogA$hog.cpue, 'nbinom', start = list(mu = 0.293, size = 0.1)) 
<simpleError in optim(par = vstart, fn = fnobj, fix.arg = fix.arg, obs = data,     gr = gradient, ddistnam = ddistname, hessian = TRUE, method = meth,     lower = lower, upper = upper, ...): function cannot be evaluated at initial parameters>
Error in fitdist(hogA$hog.cpue, "nbinom", start = list(mu = 0.293, size = 0.1)) : 
  the function mle failed to estimate the parameters, 
                with the error code 100

> fitzip <- fitdist(hogA$hog.cpue, "ZIP", start = list(mu = 0.293, sigma = 0.1), discrete = TRUE,
+                   optim.method = "L-BFGS-B", lower = c(0, 0), upper = c(Inf, 1))
<simpleError in dZIP(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, 0.404749055585537, 0, 0.134916351861846, 0, 0.149907057624273, 0, 0, 0, 0, 0, 0, 0, 0, 0.269832703723691, 0.134916351861846, 0.134916351861846, 0, 0, 0.269832703723691, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0.134916351861846, 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, 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, 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, 0, 0.112430293218205, 0, 0, 0, 0, 0, 0, 0.367953686895943, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.122651228965314, 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, 0.337290879654614, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.578212936550767, 0, 0, 0.404749055585537, 0, 0, 0, 0.269832703723691, 0.269832703723691, 0, 0, 0.299814115248546, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.404749055585537, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.192737645516922, 0.192737645516922, 0, 0, 0, 0.134916351861846, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0.404749055585537, 0.134916351861846, 0.134916351861846, 0.337290879654614, 0, 0, 0, 0, 0.674581759309228, 0, 0.134916351861846, 0, 0.299814115248546, 0.168645439827307, 0.449721172872819, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0.122651228965314, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 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, 0.134916351861846, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.122651228965314, 0, 0, 0.134916351861846, 0, 0, 0.149907057624273, 0, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 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, 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, 0.134916351861846, 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, 0, 0, 0, 0, 0.449721172872819, 0, 0, 0, 0, 0, 0, 0, 0.112430293218205, 0, 0, 0.134916351861846, 0.539665407447383, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0.134916351861846, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 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, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0.404749055585537, 0, 0, 0, 0.404749055585537, 0, 0, 0.674581759309228, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.404749055585537, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0.269832703723691, 0.269832703723691, 0.134916351861846, 0, 0.404749055585537, 0.809498111171074, 0, 0.134916351861846, 0.134916351861846, 1.07933081489477, 0.134916351861846, 0, 0.269832703723691, 0, 0.94441446303292, 0.245302457930628, 0, 0, 0, 0, 0, 0.245302457930628, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),     mu = 0, sigma = 1, log = TRUE): mu must be greater than 0 
 >
Error in fitdist(hogA$hog.cpue, "ZIP", start = list(mu = 0.293, sigma = 0.1),  : 
  the function mle failed to estimate the parameters, 
                with the error code 100
In addition: Warning messages:
1: In fitdist(hogA$hog.cpue, "ZIP", start = list(mu = 0.293, sigma = 0.1),  :
  The dZIP function should return a zero-length vector when input has length zero
2: In fitdist(hogA$hog.cpue, "ZIP", start = list(mu = 0.293, sigma = 0.1),  :
  The pZIP function should return a zero-length vector when input has length zero

> fpoisZI <- fitdist(hogA$hog.cpue, "ZIP", start=list(sigma=sum(hogA$hog.cpue == 0)/length(hogA$hog.cpue), mu=mean(hogA$hog.cpue)))
<simpleError in dZIP(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, 0.404749055585537, 0, 0.134916351861846, 0, 0.149907057624273, 0, 0, 0, 0, 0, 0, 0, 0, 0.269832703723691, 0.134916351861846, 0.134916351861846, 0, 0, 0.269832703723691, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0.134916351861846, 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, 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, 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, 0, 0.112430293218205, 0, 0, 0, 0, 0, 0, 0.367953686895943, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.122651228965314, 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, 0.337290879654614, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.578212936550767, 0, 0, 0.404749055585537, 0, 0, 0, 0.269832703723691, 0.269832703723691, 0, 0, 0.299814115248546, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.404749055585537, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.192737645516922, 0.192737645516922, 0, 0, 0, 0.134916351861846, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0.404749055585537, 0.134916351861846, 0.134916351861846, 0.337290879654614, 0, 0, 0, 0, 0.674581759309228, 0, 0.134916351861846, 0, 0.299814115248546, 0.168645439827307, 0.449721172872819, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0.122651228965314, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 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, 0.134916351861846, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.122651228965314, 0, 0, 0.134916351861846, 0, 0, 0.149907057624273, 0, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 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, 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, 0.134916351861846, 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, 0, 0, 0, 0, 0.449721172872819, 0, 0, 0, 0, 0, 0, 0, 0.112430293218205, 0, 0, 0.134916351861846, 0.539665407447383, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0.134916351861846, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 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, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0.404749055585537, 0, 0, 0, 0.404749055585537, 0, 0, 0.674581759309228, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0.269832703723691, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.404749055585537, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0.269832703723691, 0.269832703723691, 0.134916351861846, 0, 0.404749055585537, 0.809498111171074, 0, 0.134916351861846, 0.134916351861846, 1.07933081489477, 0.134916351861846, 0, 0.269832703723691, 0, 0.94441446303292, 0.245302457930628, 0, 0, 0, 0, 0, 0.245302457930628, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.134916351861846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),     sigma = 0.426547699594046, mu = -0.020557328452897, log = TRUE): mu must be greater than 0 
 >
Error in fitdist(hogA$hog.cpue, "ZIP", start = list(sigma = sum(hogA$hog.cpue ==  : 
  the function mle failed to estimate the parameters, 
                with the error code 100
In addition: Warning messages:
1: In fitdist(hogA$hog.cpue, "ZIP", start = list(sigma = sum(hogA$hog.cpue ==  :
  The dZIP function should return a zero-length vector when input has length zero
2: In fitdist(hogA$hog.cpue, "ZIP", start = list(sigma = sum(hogA$hog.cpue ==  :
  The pZIP function should return a zero-length vector when input has length zero

【问题讨论】：

标签： r statistics regression poisson

【解决方案1】：

对于泊松分布，无论是否膨胀为零，这些值都必须是正数和离散的，即整数，没有小数。我不知道您所拥有的值是否是预期的。

整理好数值后，可以考虑使用基于 glm 的方法：

library(pscl)

x = rpois(1000,20)
x[sample(length(x),200)] = 0
# fits intercept only model
fit = zeroinfl(x ~ 1,dist="poisson")

estimated_mean = exp(coefficients(fit)["count_(Intercept)"])
count_(Intercept) 
         20.14875

estimated_missing = coefficients(fit)["zero_(Intercept)"]
# it's a logit you need to convert to prob
estimated_missing = exp(estimated_missing)/(1+exp(estimated_missing))
zero_(Intercept) 
             0.2

如果你有小数，因为它是一个比率，你需要的是一个offset，所以假设基础平均比率为 0.5，缺少 0.2：

n = rep(1000 * 1:5 , each=100)
x = rpois(length(n),0.5*n)
x[sample(length(x),0.2*length(n))] = 0
# fits intercept only model
fit = zeroinfl(x ~ 1,dist="poisson",offset=log(n))

然后你重复上面的操作，得到 0.5 作为系数和 0.2 作为缺失率。

【讨论】：

我发布的值至关重要，不能四舍五入。任何想法如何处理？
如果可能的话，您能否详细说明您是如何得出这些值的？您是否将计数除以某物以获得比率？
是的，它是鱼的数量除以采样的面积，从而为您提供每单位采样的鱼的标准化数量。
对不起，这本来是一个结果..就像如果你输入estimated_missing你得到那个zero_(...)
是的，你需要的是一个偏移量，stats.stackexchange.com/questions/11182/…