用r查找双峰分布中的局部最小值 [英] Find local minimum in bimodal distribution with r
问题描述
我的数据是经过预处理的图像数据，我想将两个类分开.在理论上(希望在实践中)，最佳阈值是双峰分布数据中两个峰之间的局部最小值.
我的测试数据是: http://www.file-upload .net/download-9365389/data.txt.html
有几种方法可以做到这一点.
首先，按照问题中的说明使用d作为密度，d$x和d$y包含密度图的x和y值.最小值在导数dy/dx = 0时发生.由于x值的间距相等，我们可以使用diff(d$y)估算dy，并在d$x最小化abs(diff(d$y))的情况下寻找d$x:
d$x[which.min(abs(diff(d$y)))]
# [1] 2.415785
问题在于，当dy/dx = 0时，密度曲线也可能最大化.在这种情况下，最小值最小，但最大值已达到峰值，因此可以使用，但是您不能依靠这一点.
因此，第二种方法使用optimize(...)，它在给定的时间间隔内寻找局部最小值. optimize(...)需要一个函数作为参数，因此我们使用approxfun(d$x,d$y)创建一个插值函数.
optimize(approxfun(d$x,d$y),interval=c(1,4))$minimum
# [1] 2.415791
最后，我们证明这确实是最低要求:
hist(data,prob=TRUE)
lines(d, col="red", lty=2)
v <- optimize(approxfun(d$x,d$y),interval=c(1,4))$minimum
abline(v=v, col="blue")
实际上是首选的另一种方法是使用k-均值聚类.
df <- read.csv(header=F,"data.txt")
colnames(df) = "X"
# bimodal
km <- kmeans(df,centers=2)
df$clust <- as.factor(km$cluster)
library(ggplot2)
ggplot(df, aes(x=X)) +
geom_histogram(aes(fill=clust,y=..count../sum(..count..)),
binwidth=0.5, color="grey50")+
stat_density(geom="line", color="red")
实际上，数据看起来比双峰更具三峰性.
# trimodal
km <- kmeans(df,centers=3)
df$clust <- as.factor(km$cluster)
library(ggplot2)
ggplot(df, aes(x=X)) +
geom_histogram(aes(fill=clust,y=..count../sum(..count..)),
binwidth=0.5, color="grey50")+
stat_density(geom="line", color="red")
My data are pre-processed image data and I want to seperate two classes. In therory (and hopefully in practice) the best threshold is the local minimum between the two peaks in the bimodal distributed data.
My testdata is: http://www.file-upload.net/download-9365389/data.txt.html
I tried to follow this thread: I plotted the histogram and calculated the kernel density function:
datafile <- read.table("....txt")
data <- data$V1
hist(data)
d <- density(data) # returns the density data with defaults
hist(data,prob=TRUE)
lines(d) # plots the results
But how to continue?
I would calculate the first and second derivates of the density function to find the local extrema, specifically the local minimum. However I have no idea how to do this in R and density(test) seems not to be a normal function. Thus please help me: how can I calculate the derivates and find the local minimum of the pit between the two peaks in the density function density(test)?
There are a few ways to do this.
First, using d for the density as in your question, d$x and d$y contain the x and y values for the density plot. The minimum occurs when the derivative dy/dx = 0. Since the x-values are equally spaced, we can estimate dy using diff(d$y), and seek d$x where abs(diff(d$y)) is minimized:
d$x[which.min(abs(diff(d$y)))]
# [1] 2.415785
The problem is that the density curve could also be maximized when dy/dx = 0. In this case the minimum is shallow but the maxima are peaked, so it works, but you can't count on that.
So a second way uses optimize(...) which seeks a local minimum in a given interval. optimize(...) needs a function as argument, so we use approxfun(d$x,d$y) to create an interpolation function.
optimize(approxfun(d$x,d$y),interval=c(1,4))$minimum
# [1] 2.415791
Finally, we show that this is indeed the minimum:
hist(data,prob=TRUE)
lines(d, col="red", lty=2)
v <- optimize(approxfun(d$x,d$y),interval=c(1,4))$minimum
abline(v=v, col="blue")
Another approach, which is preferred actually, uses k-means clustering.
df <- read.csv(header=F,"data.txt")
colnames(df) = "X"
# bimodal
km <- kmeans(df,centers=2)
df$clust <- as.factor(km$cluster)
library(ggplot2)
ggplot(df, aes(x=X)) +
geom_histogram(aes(fill=clust,y=..count../sum(..count..)),
binwidth=0.5, color="grey50")+
stat_density(geom="line", color="red")
The data actually looks more trimodal than bimodal.
# trimodal
km <- kmeans(df,centers=3)
df$clust <- as.factor(km$cluster)
library(ggplot2)
ggplot(df, aes(x=X)) +
geom_histogram(aes(fill=clust,y=..count../sum(..count..)),
binwidth=0.5, color="grey50")+
stat_density(geom="line", color="red")
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