Behavioral Change Point Analysis

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:: <math> f(X_i|X_{i-1}) = {1\over  \sigma\sqrt{2 \pi (1-\rho^{2\tau_i})}} \exp {\left( \frac{\left(X_i - \rho^{\tau_i} (X_{i-1}-\mu)\right)^2}{2\sigma^2 (1 - \rho^{2\tau_i})}  \right)} </math>
 
:: <math> f(X_i|X_{i-1}) = {1\over  \sigma\sqrt{2 \pi (1-\rho^{2\tau_i})}} \exp {\left( \frac{\left(X_i - \rho^{\tau_i} (X_{i-1}-\mu)\right)^2}{2\sigma^2 (1 - \rho^{2\tau_i})}  \right)} </math>
  
: First, we identifying a likelihood function for <math> \rho </math>, given estimates of  <math> \mu = \overline{V_i} </math> and  <math>\sigma^2 = S_V^2 </math>, in a behaviorally homogenous region using the distribution function above:
+
* First, we identifying a likelihood function for <math> \rho </math>, given estimates of  <math> \mu = \overline{V_i} </math> and  <math>\sigma^2 = S_V^2 </math>, in a behaviorally homogenous region using the distribution function above:
 
:: <math> L(\rho|{X},{T}) = \prod_{i=1}^n f(X_i|X_{i-1},\tau_i,\rho) </math>
 
:: <math> L(\rho|{X},{T}) = \prod_{i=1}^n f(X_i|X_{i-1},\tau_i,\rho) </math>
  
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=== Likelihood of <math> \rho </math> parameter ===
 
=== Likelihood of <math> \rho </math> parameter ===
 
''Description'': This function works first by estimating the mean and standard deviation directly from x, using these to standardize the time series, and then optimizes for the likelihood of the value of rho. The equation for the likelihood is given above (and in equations 10 and 11 in the BCPA paper).
 
  
 
''Usage'':  GetRho(x, t)
 
''Usage'':  GetRho(x, t)
 +
 +
''Description'': This function works first by estimating the mean and standard deviation directly from x, using these to standardize the time series, and then optimizes for the likelihood of the value of rho. The equation for the likelihood is given above (and in equations 10 and 11 in the BCPA paper).
  
 
''Value'': Returns a vector with two values (again - not a list or dataframe for greater speed). The first value is "rho" estimate, the second is the log likelihood of the estimate.
 
''Value'': Returns a vector with two values (again - not a list or dataframe for greater speed). The first value is "rho" estimate, the second is the log likelihood of the estimate.
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=== Total likelihood within a behaviorally homogenous section ===
 
=== Total likelihood within a behaviorally homogenous section ===
''Description'': Returns log-likelihood of a given parameter set for a gappy time series.
 
  
 
''Usage'': GetLL(x, t, mu, s, rho)
 
''Usage'': GetLL(x, t, mu, s, rho)
 +
 +
''Description'': Returns log-likelihood of a given parameter set for a gappy time series.
  
 
''Value'': Returns value of the log likelihood
 
''Value'': Returns value of the log likelihood
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=== Likelihood of single change point ===
 
=== Likelihood of single change point ===
''Description'':  Takes a time series with values "x" obtained at time "t" and a time break "tbreak" and returns the estimates of "mu", "sigma" and "rho" as well as the negative log-likelihood of those estimates (given the data) both before and after the break.
 
  
 
''Usage'': GetDoubleL(x, t, tbreak)  
 
''Usage'': GetDoubleL(x, t, tbreak)  
 +
 +
''Description'':  Takes a time series with values "x" obtained at time "t" and a time break "tbreak" and returns the estimates of "mu", "sigma" and "rho" as well as the negative log-likelihood of those estimates (given the data) both before and after the break.
  
 
''Value'':  Returns a labeled matrix (more efficient than a data frame) with columns: "mu", "sigma", "rho" and "LL" corresponding to the estimates and 2 rows for each side of the break point.  
 
''Value'':  Returns a labeled matrix (more efficient than a data frame) with columns: "mu", "sigma", "rho" and "LL" corresponding to the estimates and 2 rows for each side of the break point.  
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=== Sweeping breaks ===
 
=== Sweeping breaks ===
''Description'': Finds a single change point within a time series.
 
  
 
''Usage'': SweepBreaks(x, t, range=0.6)
 
''Usage'': SweepBreaks(x, t, range=0.6)
 +
 +
''Description'': Finds a single change point within a time series.
  
 
''Arguments'':  
 
''Arguments'':  
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=== Choosing a model ===
 
=== Choosing a model ===
  
There are eight model functions, names M0 to M7, with redundant code which I have placed here: [[BCPA/Model Specification]].  Each of these function take data "x", at times "t", and breakpoint "tbreak" and return a named dataframe of parameters, log-likelihoods and BIC values: "data.frame(LL,bic,mu1,s1,rho1,mu2,s2,rho2)"
+
There are eight model functions, named M0 to M7, with redundant code which I have placed here: [[BCPA/Model Specification]].  Each of these function take data "x", at times "t", and breakpoint "tbreak" and return a named dataframe of parameters, log-likelihoods and BIC values: "data.frame(LL,bic,mu1,s1,rho1,mu2,s2,rho2)"
  
 
The ouput of all the models is obtained using this function, which returns a data.frame including columns: "Model", followed by the estimate output of each model function M0-M7 (i.e. data.frame(LL,bic,mu1,s1,rho1,mu2,s2,rho2)):
 
The ouput of all the models is obtained using this function, which returns a data.frame including columns: "Model", followed by the estimate output of each model function M0-M7 (i.e. data.frame(LL,bic,mu1,s1,rho1,mu2,s2,rho2)):
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=== Window sweeping ===
 
=== Window sweeping ===
  
This is the key mother-function which sweeps the analysis above using windows of size windowsize and selecting the best model according to BIC and recording the estimated parameters.  
+
This is the key mother-function which sweeps the analysis above using windows of size "windowsize", steps by "windowstep" and selecting the best model according to BIC and recording the estimated parameters.  
  
 
<pre>
 
<pre>
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Finally, the output of the windowsweep function (which we call "ws"), is converted to parameter estimates over the entire complex timeseries using the following function:
 
Finally, the output of the windowsweep function (which we call "ws"), is converted to parameter estimates over the entire complex timeseries using the following function:
  
 +
''Usage'': PartitionParameters(ws, t, windowsize = 50, windowstep = 1)
 +
 +
''Description'': Estimation of all parameters as a rolling average of the window-sweep output.
 +
 +
''Arguments'':
 +
* ws Output of WindowSweep
 +
* t Time values of time-series measurements.
 +
* windowsize Window size
 +
* windowstep Increment of window step
 +
 +
''Value'':  Returns a data frame with columns "mu.hat", "s.hat" and "rho.hat" for each location in the time-series (i.e., all of the time series minus the first and last range of windowsize/2).
 
<pre>
 
<pre>
 
PartitionParameters <- function(ws,t,windowsize=50,windowstep=1)
 
PartitionParameters <- function(ws,t,windowsize=50,windowstep=1)
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}
 
}
 
</pre>
 
</pre>
 
=== Plotting output ===
 
  
 
== Sample work flow ==
 
== Sample work flow ==
 +
 +
 +
Lets say your data is X,Y,T , and you convert the X,Y to step lengths (v) and turning angles (theta), and the "T" to a vector of midpoints, something like
 +
<pre>
 +
  t <- T[-1]+Т[-length(T)])/2
 +
</pre>
 +
 +
Vectors to apply the analysis:
 +
 +
<pre>
 +
  vc <- v*cos(theta)
 +
  vs <- v*sin(theta)
 +
</pre>
 +
 +
Once all the functions in the files are loaded, an analysis run would be as simple as:
 +
<pre>
 +
  vc.sweep <- WindowSweep(vc, t, windowsize = 50, windowstep = 1)
 +
  vc.output <- PartitionParameters(vc.sweep, t, windowsize=50, windowstep = 1)
 +
</pre>
 +
 +
The first function performs the windowsweep and returns all the possible breakpoints and their respective "model" (M0-M7) based on AIC, and the second function uses that output to produce estimates of the parameter values across the time series.

Revision as of 06:05, 19 July 2011

PLEASE NOTE: THIS PAGE IS UNDER CONSTRUCTION!

FurSealBCPA.png

The Behavioral Change Point Analysis (BCPA) is a method of identifying hidden shifts in the underlying parameters of a time series, developed specifically to be applied to animal movement data which is irregularly sampled.

The original paper on which it is based is: E. Gurarie, R. Andrews and K. Laidre A novel method for identifying behavioural changes in animal movement data (2009) Ecology Letters 12:5 395-408. Most of the material is also present in the Chapter 5 of my Pdf10.png PhD dissertation (click to access).

I have received numerous requests for the code behind the BCPA, so (after sending out more or less the same email with the same attachments several dozen times) have decided that it might be more efficient to post a sample work flow of using the BCPA on this wiki.

Please send all comments, questions, critiques, possible improvements, or (heaven forbid!) identification of errors to via e-mail at <text> eli.gurarie (at) noaa.gov </text>.

Contents

Brief Introduction

Likelihood of autocorrelation parameter for randomly subsampled process
Identifying MLCP from likelihood profile.

Briefly, there are several hierarchically layered parts to the method. For location and time data {Z,T}, the analysis is performed on velocity (estimated as V = ΔZ / ΔT components Vcos(θ) and Vsin(θ). These components are (importantly) assumed to be observations from a continuous time, Gaussian process, (i.e. an Ornstein-Uhlenbeck process), with mean μ(t), variance σ2(t) and autocorrelation ρ(t). The values of these parameters are assumed to change gradually or abruptly. The purpose of the BCPA is to identify the locations where changes are abrupt (assumed to correspond to discrete changes in an animal's behavior).

The distribution function of this process is given by:

 f(X_i|X_{i-1}) = {1\over  \sigma\sqrt{2 \pi (1-\rho^{2\tau_i})}} \exp {\left( \frac{\left(X_i - \rho^{\tau_i} (X_{i-1}-\mu)\right)^2}{2\sigma^2 (1 - \rho^{2\tau_i})}  \right)}
  • First, we identifying a likelihood function for ρ, given estimates of  \mu = \overline{V_i} and \sigma^2 = S_V^2 , in a behaviorally homogenous region using the distribution function above:
 L(\rho|{X},{T}) = \prod_{i=1}^n f(X_i|X_{i-1},\tau_i,\rho)
  • Second, we identify a "most likely change point" (MLCP) in a time series where the behavior may have changed by taking the product of the likelihoods of the estimates to the left and to the right of all possible change points in a time series. We identify which of the parameters (if any) have changed by comparing the BIC of eight possible models: M0 - no changes in any parameter, M1 - change in μ only, M2 - change in σ only,

M3 - change in ρ only ... etc ... M7 - chance in all three parameters.

  • Third, we sweep the MLCP changepoint across a complete data set, recording at every point what the parameter values are to the left and right of all MLCP's under the model with the highest BIC, and record the paraemters.
  • Fourth, we somehow present this mass of analysis.

I present here the code for all these steps, as they were applied in the original paper, and apply them to a simulated dataset. Naturally, implementation of this type of analysis should be specific to the relevant application.

Code pieces

Likelihood of ρ parameter

Usage: GetRho(x, t)

Description: This function works first by estimating the mean and standard deviation directly from x, using these to standardize the time series, and then optimizes for the likelihood of the value of rho. The equation for the likelihood is given above (and in equations 10 and 11 in the BCPA paper).

Value: Returns a vector with two values (again - not a list or dataframe for greater speed). The first value is "rho" estimate, the second is the log likelihood of the estimate.

Arguments

  • x Values of time series.
  • t Times of measurements associated with x.
 
GetRho <- function (x, t) 
{
    getL <- function(rho) {
        dt <- diff(t)
        s <- sd(x)
        mu <- mean(x)
        n <- length(x)
        x.plus <- x[-1]
        x.minus <- x[-length(x)]
        Likelihood <- dnorm(x.plus, mean = mu + (rho^dt) * (x.minus - 
            mu), sd = s * sqrt(1 - rho^(2 * dt)))
        logL <- sum(log(Likelihood))
        if (!is.finite(logL)) 
            logL <- -10^10
        return(-logL)
    }
    o <- optimize(getL, lower = 0, upper = 1, tol = 1e-04)
    return(c(o$minimum, o$objective))
}

Total likelihood within a behaviorally homogenous section

Usage: GetLL(x, t, mu, s, rho)

Description: Returns log-likelihood of a given parameter set for a gappy time series.

Value: Returns value of the log likelihood

Arguments:

  • x Time series data.
  • t Times at which data is obtained
  • mu Mean estimate
  • s Sigma (standard deviation) estimate (>0)
  • rho Rho estimate (between 0 and 1)
GetLL <- function (x, t, mu, s, rho) 
{
    dt <- diff(t)
    n <- length(x)
    x.plus <- x[-1]
    x.minus <- x[-length(x)]
    Likelihood <- dnorm(x.plus, mean = mu + (rho^dt) * (x.minus - 
        mu), sd = s * sqrt(1 - rho^(2 * dt)))
    LL <- -sum(log(Likelihood))
    return(LL)
}

Likelihood of single change point

Usage: GetDoubleL(x, t, tbreak)

Description: Takes a time series with values "x" obtained at time "t" and a time break "tbreak" and returns the estimates of "mu", "sigma" and "rho" as well as the negative log-likelihood of those estimates (given the data) both before and after the break.

Value: Returns a labeled matrix (more efficient than a data frame) with columns: "mu", "sigma", "rho" and "LL" corresponding to the estimates and 2 rows for each side of the break point.

Arguments:

  • x Values of time series.
  • t Times of measurements associated with x.
  • tbreak Breakpoint (in terms of the INDEX within "t" and "x", not actual time value).
GetDoubleL <- function(x,t,tbreak)
  {  
    x1 <- x[1:tbreak]  
    x2 <- x[tbreak:length(x)]
    
    t1 <- t[1:tbreak]
    t2 <- t[tbreak:length(t)]

    o1<-GetRho(x1,t1)
    o2<-GetRho(x2,t2)
    
    mu1 <- mean(x1)
    sigma1 <- sd(x1)
    rho1 <- o1[1]
    
    mu2 <- mean(x2)
    sigma2 <- sd(x2)
    rho2 <- o2[1]
  
    LL1 <- -o1[2]
    LL2 <- -o2[2]

    m <- matrix(c(mu1,mu2,sigma1,sigma2,rho1,rho2,LL1,LL2),2,4)
    colnames(m) <- c("mu","sigma","rho","LL")
    
    return(m)
  }

Sweeping breaks

Usage: SweepBreaks(x, t, range=0.6)

Description: Finds a single change point within a time series.

Arguments:

  • x Values of time series.
  • t Times of measurements associated with x.
  • range Range of possible breaks. Default (0.6) runs approximately from 1/5th to 4/5ths of the total length of the time series.

Value: Returns a matrix (not a data.frame for greater speed) with column headings: "breaks","tbreaks","mu1","sigma1","rho1","LL1","mu2","sigma2","rho2","LL2","LL". This is calculated for every possible break - which extends from 0.2l to 0.8l (where l is the length of the time series). The output of this function feeds WindowSweep.

SweepBreaks <- function(x,t,range=0.6)
    {
      n<-length(t)
      start <- (1-range)/2
      breaks<-round((start*n):((1-start)*n))
      Ls<-breaks*0
      
      l<-length(breaks)
      
      BreakParams <- matrix(NA,l,8)
      #BreakParams <- data.frame(mu1=NA,s1=NA,rho1=NA,LL1=NA,mu2=NA,s2=NA,rho2=NA,LL2=NA)
      
      for(i in 1:l)
      {
        myDoubleL <- GetDoubleL(x,t,breaks[i])
        BreakParams[i,] <- c(myDoubleL[1,],myDoubleL[2,])
      }
      
      # remember: LL1 and LL2 are columns 4 and 8
      BreakMatrix<- cbind(breaks,t[breaks], BreakParams, 
      BreakParams[,4]+BreakParams[,8])
      
      colnames(BreakMatrix) <- c("breaks","tbreaks","mu1","sigma1","rho1","LL1","mu2","sigma2","rho2","LL2","LL")
      
      return(BreakMatrix[2:nrow(BreakMatrix),])
    }

Choosing a model

There are eight model functions, named M0 to M7, with redundant code which I have placed here: BCPA/Model Specification. Each of these function take data "x", at times "t", and breakpoint "tbreak" and return a named dataframe of parameters, log-likelihoods and BIC values: "data.frame(LL,bic,mu1,s1,rho1,mu2,s2,rho2)"

The ouput of all the models is obtained using this function, which returns a data.frame including columns: "Model", followed by the estimate output of each model function M0-M7 (i.e. data.frame(LL,bic,mu1,s1,rho1,mu2,s2,rho2)):

 
GetModels <- function(x,t,tbreak,K=2)
{
  for(i in 0:7)
  {
    f<-get(paste("M",i,sep=""))
    myr<-data.frame(Model=i,f(x,t,tbreak,K))
    ifelse(i==0,
      r<-myr,
      r<-rbind(r,myr))
  }
  return(r)
}

Window sweeping

This is the key mother-function which sweeps the analysis above using windows of size "windowsize", steps by "windowstep" and selecting the best model according to BIC and recording the estimated parameters.

WindowSweep <- function (x, t, windowsize = 50, windowstep = 1, sine = 0, K = 2, 
    plotme = 1) 
{
    low <- seq(1, (length(t) - windowsize), windowstep)
    hi <- low + windowsize
    for (i in 1:length(low)) {
        myx <- x[low[i]:hi[i]]
        myt <- t[low[i]:hi[i]]
        bp <- SweepBreaks(myx, myt)
        myestimate <- bp[bp[, 11] == max(bp[, 11]), ]
        breakpoint <- myestimate[1]
        tbreak <- myestimate[2]
        ifelse(sine, allmodels <- GetModelsSin(myx, myt, breakpoint, 
            K), allmodels <- GetModels(myx, myt, breakpoint, 
            K))
        mymodel <- allmodels[allmodels$bic == min(allmodels$bic), 
            ]
        mymodel <- data.frame(mymodel, Break = tbreak)
        ifelse(i == 1, estimates <- mymodel, estimates <- rbind(estimates, 
            mymodel))
        if (plotme) {
            plot.ts(t, x, type = "l", col = "grey")
            lines(t, x, type = "l")
            lines(myt, myx, col = "green")
            abline(v = tbreak)
            print(estimates[i, ])
        }
    }
    return(data.frame(estimates))
}

Partitioning parameters

Finally, the output of the windowsweep function (which we call "ws"), is converted to parameter estimates over the entire complex timeseries using the following function:

Usage: PartitionParameters(ws, t, windowsize = 50, windowstep = 1)

Description: Estimation of all parameters as a rolling average of the window-sweep output.

Arguments:

  • ws Output of WindowSweep
  • t Time values of time-series measurements.
  • windowsize Window size
  • windowstep Increment of window step

Value: Returns a data frame with columns "mu.hat", "s.hat" and "rho.hat" for each location in the time-series (i.e., all of the time series minus the first and last range of windowsize/2).

PartitionParameters <- function(ws,t,windowsize=50,windowstep=1)
{
	n.col<-length(t)
	n.row<-dim(ws)[1]

	mu.M <- matrix(NA,n.row,n.col)
	s.M <- matrix(NA,n.row,n.col)
	rho.M <- matrix(NA,n.row,n.col)

	for(i in 1:n.row)
	{
		myws<-ws[i,]
		dts <- abs(t-myws$Break)
		tbreak <- match(min(dts),dts)

		max <- min(n.col,i+windowsize)

		mu.M[i,i:tbreak] <- myws$mu1
		mu.M[i,(tbreak+1):max] <- myws$mu2
		s.M[i,i:tbreak] <- myws$s1
		s.M[i,(tbreak+1):max] <- myws$s2
		rho.M[i,i:tbreak] <- myws$rho1
		rho.M[i,(tbreak+1):max] <- myws$rho2
		  }

		adjust <- colSums(!is.na(mu.M))


		mu.hat<-colSums(mu.M,na.rm=1)/adjust
		s.hat<-colSums(s.M,na.rm=1)/adjust
		rho.hat<-colSums(rho.M,na.rm=1)/adjust

		return(data.frame(mu.hat,s.hat,rho.hat))
	}
}

Sample work flow

Lets say your data is X,Y,T , and you convert the X,Y to step lengths (v) and turning angles (theta), and the "T" to a vector of midpoints, something like
  t <- T[-1]+Т[-length(T)])/2

Vectors to apply the analysis:

  vc <- v*cos(theta)
  vs <- v*sin(theta)

Once all the functions in the files are loaded, an analysis run would be as simple as:

  vc.sweep <- WindowSweep(vc, t, windowsize = 50, windowstep = 1)
  vc.output <- PartitionParameters(vc.sweep, t, windowsize=50, windowstep = 1)
The first function performs the windowsweep and returns all the possible breakpoints and their respective "model" (M0-M7) based on AIC, and the second function uses that output to produce estimates of the parameter values across the time series.
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