Multifractals in ecology using R - Day 2

Posgraduate course, University of Maringá, 2013

Cumulative distributions and ranks

We want to make a plot of the cumulative distribution of a function $P(x)$ the frequency of words in a text.
The cumulative distribution of the frequency is defined such that $P(x)$ is the fraction of words with frequency greater than or equal to $x$.
If $x$ is the frequency of the most frequent word, usually “the”, then there is exactly one word with frequency greater than or equal to $x$.
Similarly for the second most frequent word, usually “of”, there are two words with frequency greater than or equal: “of” and “the”.

Cumulative distributions and ranks 1

In general if we rank the words in descending order then by definition there are n words with frequency greater than or equal than that of the nth most commond word.
Thus the cumulative distribution $P(x)$ is proportional to the rank n of a word.
Then to plot $P(x)$ we only need to plot the ranks as a function of the frequency.

Cumulative distributions and fractal dimension

We can analyze the data from Metzler (1992)

      rm(list=ls())

      ps <- read.table("patch1968.dat",header=T)

      ps$r <- rank(-ps$pSize)

      plot(r ~ pSize,data=ps)

      plot(log(r)~log(pSize),data=ps)		

      lm0 <- lm(log(r)~log(pSize),data=ps)

      summary(lm0)

      abline(lm0)

Cumulative distributions and fractal dimension 1

If $B$ is the exponent then $H = 2 - 2B$
```
      slope0 <- coef(lm0)[2]
		
      2+slope0*2
```
The patches are persistent because H=1.18 > 0.5
We need to install the package “car” to test for autocorrelation with the Durbin-Watson statistic. We can do this using the RStudio menu Tools/Install Packages.
```
      require(car)

      dwt(lm0)
```
we can draw a grid to determine the break point.
```
      grid()
```

Exercise 1

Split the data in two to obtain two fractal dimensions without correlation

There is a shorcut for doing this: the package “segmented” fits a broken line and finds the break point.

      require(segmented)

      ps$logr <- log(ps$r)
      ps$logpSize <- log(ps$pSize)
      lm0 <- lm(logr~logpSize,data=ps)
      seg <- segmented(lm0, seg.Z = ~logpSize, psi=4)
      summary(seg)
      slope(seg)

Exercise 1 (Cont.)

Let’s do a function to calculate H

small patches are persistent

      calcH <- function(B) { 2-2*abs(B)}

      calcH(.1550) # H = 1.69

Big patches are anti-persistent

      calcH(.9036) # H = 0.19

What is the breakpoint value in ha? Let’s do another function.

A possible answer:

      calcBreak <- function(B) { 0.1*exp(B)*0.65 } 

      calcBreak(3.35) # 1.85 ha

Conclusion

small patches: if they are growing they keep growing, if they are reducing they vanish.
big patches: if they are growing they will reduce, if they are reducing they will grow.
Thus big patches are more stable, small patches appear and disappear.

Exercise 2

Let’s do the same thing using segmented with the 1985 data: “patch1985.dat”

We can do a plot with the segmented object

      plot(seg,col="green",xlab="Log Patch Size"
              ,ylab="Acum Freq")
		
      points(log(r)~log(pSize),data=ps,pch=2,cex=.5)

We can use the functions:

      summary(seg)
      slope(seg)

      calcH(0.2676)
      calcH(1.26)

      calcBreak(2.708) # 0.97 ha

A different graphic analysis

Using graphics package “ggplot2”. We need to add both datasets in one data frame

      ps <- read.table("patch1985.dat",header=T)
      ps$r <- rank(-ps$pSize)
      ps$Year <- "1985"

      ps1 <- read.table("patch1968.dat",header=T)
      ps1$r <- rank(-ps1$pSize)
      ps1$Year <- "1968"

      ps <- rbind(ps,ps1)

Gramar of graphics ggplot2

		require(ggplot2)
		ggplot(data=ps,aes(x=pSize,y=r,color=Year))
					+geom_point()

		p <- ggplot(data=ps,aes(x=log(pSize),y=log(r),
				color=Year))+geom_point(aes(shape=Year))

Leonardo A. Saravia