fda (version 6.2.0)

cor.fd: Correlation matrix from functional data object(s)

Description

Compute a correlation matrix for one or two functional data objects.

Usage

cor.fd(evalarg1, fdobj1, evalarg2=evalarg1, fdobj2=fdobj1)

Value

A matrix or array:

With one or two functional data objects, fdobj1 and possibly fdobj2, the value is a matrix of dimensions length(evalarg1) by length(evalarg2) giving the correlations at those points of fdobj1 if missing(fdobj2) or of correlations between eval.fd(evalarg1, fdobj1) and eval.fd(evalarg2, fdobj2).

With a single multivariate data object with k variables, the value is a 4-dimensional array of dim = c(nPts, nPts, 1, choose(k+1, 2)), where nPts = length(evalarg1).

Arguments

evalarg1

a vector of argument values for fdobj1.

evalarg2

a vector of argument values for fdobj2.

fdobj1, fdobj2

functional data objects

Details

1. var1 <- var.fd(fdobj1) 2. evalVar1 <- eval.bifd(evalarg1, evalarg1, var1) 3. if(missing(fdobj2)) Convert evalVar1 to correlations 4. else: 4.1. var2 <- var.fd(fdobj2) 4.2. evalVar2 <- eval.bifd(evalarg2, evalarg2, var2) 4.3. var12 <- var.df(fdobj1, fdobj2) 4.4. evalVar12 <- eval.bifd(evalarg1, evalarg2, var12) 4.5. Convert evalVar12 to correlations

References

Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009), Functional data analysis with R and Matlab, Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2005), Functional Data Analysis, 2nd ed., Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York.

See Also

mean.fd, sd.fd, std.fd stdev.fd var.fd

Examples

daybasis3 <- create.fourier.basis(c(0, 365))
daybasis5 <- create.fourier.basis(c(0, 365), 5)
tempfd3 <- with(CanadianWeather, smooth.basis(
       day.5, dailyAv[,,"Temperature.C"], 
       daybasis3, fdnames=list("Day", "Station", "Deg C"))$fd )
precfd5 <- with(CanadianWeather, smooth.basis(
       day.5, dailyAv[,,"log10precip"], 
       daybasis5, fdnames=list("Day", "Station", "Deg C"))$fd )

# Correlation matrix for a single functional data object
(tempCor3 <- cor.fd(seq(0, 356, length=4), tempfd3))

# Cross correlation matrix between two functional data objects 
# Compare with structure described above under 'value':
(tempPrecCor3.5 <- cor.fd(seq(0, 365, length=4), tempfd3,
                          seq(0, 356, length=6), precfd5))

# The following produces contour and perspective plots

daybasis65 <- create.fourier.basis(rangeval=c(0, 365), nbasis=65)
daytempfd <- with(CanadianWeather, smooth.basis(
       day.5, dailyAv[,,"Temperature.C"], 
       daybasis65, fdnames=list("Day", "Station", "Deg C"))$fd )
dayprecfd <- with(CanadianWeather, smooth.basis(
       day.5, dailyAv[,,"log10precip"], 
       daybasis65, fdnames=list("Day", "Station", "log10(mm)"))$fd )

str(tempPrecCor <- cor.fd(weeks, daytempfd, weeks, dayprecfd))
# dim(tempPrecCor)= c(53, 53)

op <- par(mfrow=c(1,2), pty="s")
contour(weeks, weeks, tempPrecCor, 
        xlab="Average Daily Temperature",
        ylab="Average Daily log10(precipitation)",
        main=paste("Correlation function across locations\n",
          "for Canadian Anual Temperature Cycle"),
        cex.main=0.8, axes=FALSE)
axisIntervals(1, atTick1=seq(0, 365, length=5), atTick2=NA, 
            atLabels=seq(1/8, 1, 1/4)*365,
            labels=paste("Q", 1:4) )
axisIntervals(2, atTick1=seq(0, 365, length=5), atTick2=NA, 
            atLabels=seq(1/8, 1, 1/4)*365,
            labels=paste("Q", 1:4) )
persp(weeks, weeks, tempPrecCor,
      xlab="Days", ylab="Days", zlab="Correlation")
mtext("Temperature-Precipitation Correlations", line=-4, outer=TRUE)
par(op)

# Correlations and cross correlations
# in a bivariate functional data object
gaittime   <- (1:20)/21
gaitbasis5 <- create.fourier.basis(c(0,1),nbasis=5)
gaitfd5    <- smooth.basis(gaittime, gait, gaitbasis5)$fd

gait.t3 <- (0:2)/2
(gaitCor3.5 <- cor.fd(gait.t3, gaitfd5))
# Check the answers with manual computations
gait3.5 <- eval.fd(gait.t3, gaitfd5)
all.equal(cor(t(gait3.5[,,1])), gaitCor3.5[,,,1])
# TRUE
all.equal(cor(t(gait3.5[,,2])), gaitCor3.5[,,,3])
# TRUE
all.equal(cor(t(gait3.5[,,2]), t(gait3.5[,,1])),
               gaitCor3.5[,,,2])
# TRUE

# NOTE:
dimnames(gaitCor3.5)[[4]]
# [1] Hip-Hip
# [2] Knee-Hip 
# [3] Knee-Knee
# If [2] were "Hip-Knee", then
# gaitCor3.5[,,,2] would match 
# cor(t(gait3.5[,,1]), t(gait3.5[,,2]))
# *** It does NOT.  Instead, it matches:  
# cor(t(gait3.5[,,2]), t(gait3.5[,,1]))