normalizeFragmentLength: Normalizes signals for PCR fragment-length effects
In aroma.light: Light-Weight Methods for Normalization and Visualization of Microarray Data using Only Basic R Data Types

Description Usage Arguments Value Multi-enzyme normalization Target functions Author(s) References Examples

Normalizes signals for PCR fragment-length effects. Some or all signals are used to estimated the normalization function. All signals are normalized.

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## Default S3 method:
normalizeFragmentLength(y, fragmentLengths, targetFcns=NULL, subsetToFit=NULL,
  onMissing=c("ignore", "median"), .isLogged=TRUE, ..., .returnFit=FALSE)

`y`	A `numeric` `vector` of length K of signals to be normalized across E enzymes.
`fragmentLengths`	An `integer` KxE `matrix` of fragment lengths.
`targetFcns`	An optional `list` of E `function`s; one per enzyme. If `NULL`, the data is normalized to have constant fragment-length effects (all equal to zero on the log-scale).
`subsetToFit`	The subset of data points used to fit the normalization function. If `NULL`, all data points are considered.
`onMissing`	Specifies how data points for which there is no fragment length is normalized. If `"ignore"`, the values are not modified. If `"median"`, the values are updated to have the same robust average as the other data points.
`.isLogged`	A `logical`.
`...`	Additional arguments passed to `lowess`.
`.returnFit`	A `logical`.

Returns a numeric vector of the normalized signals.

It is assumed that the fragment-length effects from multiple enzymes added (with equal weights) on the intensity scale. The fragment-length effects are fitted for each enzyme separately based on units that are exclusively for that enzyme. If there are no or very such units for an enzyme, the assumptions of the model are not met and the fit will fail with an error. Then, from the above single-enzyme fits the average effect across enzymes is the calculated for each unit that is on multiple enzymes.

It is possible to specify custom target function effects for each enzyme via argument targetFcns. This argument has to be a list containing one function per enzyme and ordered in the same order as the enzyme are in the columns of argument fragmentLengths. For instance, if one wish to normalize the signals such that their mean signal as a function of fragment length effect is constantly equal to 2200 (or the intensity scale), the use targetFcns=function(fl, ...) log2(2200) which completely ignores fragment-length argument 'fl' and always returns a constant. If two enzymes are used, then use targetFcns=rep(list(function(fl, ...) log2(2200)), 2).

Note, if targetFcns is NULL, this corresponds to targetFcns=rep(list(function(fl, ...) 0), ncol(fragmentLengths)).

Alternatively, if one wants to only apply minimal corrections to the signals, then one can normalize toward target functions that correspond to the fragment-length effect of the average array.

Henrik Bengtsson

[1] H. Bengtsson, R. Irizarry, B. Carvalho, and T. Speed, Estimation and assessment of raw copy numbers at the single locus level, Bioinformatics, 2008.

  # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
# Example 1: Single-enzyme fragment-length normalization of 6 arrays
# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
# Number samples
I <- 9

# Number of loci
J <- 1000

# Fragment lengths
fl <- seq(from=100, to=1000, length.out=J)

# Simulate data points with unknown fragment lengths
hasUnknownFL <- seq(from=1, to=J, by=50)
fl[hasUnknownFL] <- NA

# Simulate data
y <- matrix(0, nrow=J, ncol=I)
maxY <- 12
for (kk in 1:I) {
  k <- runif(n=1, min=3, max=5)
  mu <- function(fl) {
    mu <- rep(maxY, length(fl))
    ok <- !is.na(fl)
    mu[ok] <- mu[ok] - fl[ok]^{1/k}
    mu
  }
  eps <- rnorm(J, mean=0, sd=1)
  y[,kk] <- mu(fl) + eps
}

# Normalize data (to a zero baseline)
yN <- apply(y, MARGIN=2, FUN=function(y) {
  normalizeFragmentLength(y, fragmentLengths=fl, onMissing="median")
})

# The correction factors
rho <- y-yN
print(summary(rho))
# The correction for units with unknown fragment lengths
# equals the median correction factor of all other units
print(summary(rho[hasUnknownFL,]))

# Plot raw data
layout(matrix(1:9, ncol=3))
xlim <- c(0,max(fl, na.rm=TRUE))
ylim <- c(0,max(y, na.rm=TRUE))
xlab <- "Fragment length"
ylab <- expression(log2(theta))
for (kk in 1:I) {
  plot(fl, y[,kk], xlim=xlim, ylim=ylim, xlab=xlab, ylab=ylab)
  ok <- (is.finite(fl) & is.finite(y[,kk]))
  lines(lowess(fl[ok], y[ok,kk]), col="red", lwd=2)
}

# Plot normalized data
layout(matrix(1:9, ncol=3))
ylim <- c(-1,1)*max(y, na.rm=TRUE)/2
for (kk in 1:I) {
  plot(fl, yN[,kk], xlim=xlim, ylim=ylim, xlab=xlab, ylab=ylab)
  ok <- (is.finite(fl) & is.finite(y[,kk]))
  lines(lowess(fl[ok], yN[ok,kk]), col="blue", lwd=2)
}


  # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
# Example 2: Two-enzyme fragment-length normalization of 6 arrays
# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
set.seed(0xbeef)

# Number samples
I <- 5

# Number of loci
J <- 3000

# Fragment lengths (two enzymes)
fl <- matrix(0, nrow=J, ncol=2)
fl[,1] <- seq(from=100, to=1000, length.out=J)
fl[,2] <- seq(from=1000, to=100, length.out=J)

# Let 1/2 of the units be on both enzymes
fl[seq(from=1, to=J, by=4),1] <- NA
fl[seq(from=2, to=J, by=4),2] <- NA

# Let some have unknown fragment lengths
hasUnknownFL <- seq(from=1, to=J, by=15)
fl[hasUnknownFL,] <- NA

# Sty/Nsp mixing proportions:
rho <- rep(1, I)
rho[1] <- 1/3;  # Less Sty in 1st sample
rho[3] <- 3/2;  # More Sty in 3rd sample


# Simulate data
z <- array(0, dim=c(J,2,I))
maxLog2Theta <- 12
for (ii in 1:I) {
  # Common effect for both enzymes
  mu <- function(fl) {
    k <- runif(n=1, min=3, max=5)
    mu <- rep(maxLog2Theta, length(fl))
    ok <- is.finite(fl)
    mu[ok] <- mu[ok] - fl[ok]^{1/k}
    mu
  }

  # Calculate the effect for each data point
  for (ee in 1:2) {
    z[,ee,ii] <- mu(fl[,ee])
  }

  # Update the Sty/Nsp mixing proportions
  ee <- 2
  z[,ee,ii] <- rho[ii]*z[,ee,ii]

  # Add random errors
  for (ee in 1:2) {
    eps <- rnorm(J, mean=0, sd=1/sqrt(2))
    z[,ee,ii] <- z[,ee,ii] + eps
  }
}


hasFl <- is.finite(fl)

unitSets <- list(
  nsp  = which( hasFl[,1] & !hasFl[,2]),
  sty  = which(!hasFl[,1] &  hasFl[,2]),
  both = which( hasFl[,1] &  hasFl[,2]),
  none = which(!hasFl[,1] & !hasFl[,2])
)

# The observed data is a mix of two enzymes
theta <- matrix(NA_real_, nrow=J, ncol=I)

# Single-enzyme units
for (ee in 1:2) {
  uu <- unitSets[[ee]]
  theta[uu,] <- 2^z[uu,ee,]
}

# Both-enzyme units (sum on intensity scale)
uu <- unitSets$both
theta[uu,] <- (2^z[uu,1,]+2^z[uu,2,])/2

# Missing units (sample from the others)
uu <- unitSets$none
theta[uu,] <- apply(theta, MARGIN=2, sample, size=length(uu))

# Calculate target array
thetaT <- rowMeans(theta, na.rm=TRUE)
targetFcns <- list()
for (ee in 1:2) {
  uu <- unitSets[[ee]]
  fit <- lowess(fl[uu,ee], log2(thetaT[uu]))
  class(fit) <- "lowess"
  targetFcns[[ee]] <- function(fl, ...) {
    predict(fit, newdata=fl)
  }
}


# Fit model only to a subset of the data
subsetToFit <- setdiff(1:J, seq(from=1, to=J, by=10))

# Normalize data (to a target baseline)
thetaN <- matrix(NA_real_, nrow=J, ncol=I)
fits <- vector("list", I)
for (ii in 1:I) {
  lthetaNi <- normalizeFragmentLength(log2(theta[,ii]), targetFcns=targetFcns,
                     fragmentLengths=fl, onMissing="median",
                     subsetToFit=subsetToFit, .returnFit=TRUE)
  fits[[ii]] <- attr(lthetaNi, "modelFit")
  thetaN[,ii] <- 2^lthetaNi
}


# Plot raw data
xlim <- c(0, max(fl, na.rm=TRUE))
ylim <- c(0, max(log2(theta), na.rm=TRUE))
Mlim <- c(-1,1)*4
xlab <- "Fragment length"
ylab <- expression(log2(theta))
Mlab <- expression(M==log[2](theta/theta[R]))

layout(matrix(1:(3*I), ncol=I, byrow=TRUE))
for (ii in 1:I) {
  plot(NA, xlim=xlim, ylim=ylim, xlab=xlab, ylab=ylab, main="raw")

  # Single-enzyme units
  for (ee in 1:2) {
    # The raw data
    uu <- unitSets[[ee]]
    points(fl[uu,ee], log2(theta[uu,ii]), col=ee+1)
  }

  # Both-enzyme units (use fragment-length for enzyme #1)
  uu <- unitSets$both
  points(fl[uu,1], log2(theta[uu,ii]), col=3+1)

  for (ee in 1:2) {
    # The true effects
    uu <- unitSets[[ee]]
    lines(lowess(fl[uu,ee], log2(theta[uu,ii])), col="black", lwd=4, lty=3)

    # The estimated effects
    fit <- fits[[ii]][[ee]]$fit
    lines(fit, col="orange", lwd=3)

    muT <- targetFcns[[ee]](fl[uu,ee])
    lines(fl[uu,ee], muT, col="cyan", lwd=1)
  }
}

# Calculate log-ratios
thetaR <- rowMeans(thetaN, na.rm=TRUE)
M <- log2(thetaN/thetaR)

# Plot normalized data
for (ii in 1:I) {
  plot(NA, xlim=xlim, ylim=Mlim, xlab=xlab, ylab=Mlab, main="normalized")
  # Single-enzyme units
  for (ee in 1:2) {
    # The normalized data
    uu <- unitSets[[ee]]
    points(fl[uu,ee], M[uu,ii], col=ee+1)
  }
  # Both-enzyme units (use fragment-length for enzyme #1)
  uu <- unitSets$both
  points(fl[uu,1], M[uu,ii], col=3+1)
}

ylim <- c(0,1.5)
for (ii in 1:I) {
  data <- list()
  for (ee in 1:2) {
    # The normalized data
    uu <- unitSets[[ee]]
    data[[ee]] <- M[uu,ii]
  }
  uu <- unitSets$both
  if (length(uu) > 0)
    data[[3]] <- M[uu,ii]

  uu <- unitSets$none
  if (length(uu) > 0)
    data[[4]] <- M[uu,ii]

  cols <- seq_along(data)+1
  plotDensity(data, col=cols, xlim=Mlim, xlab=Mlab, main="normalized")

  abline(v=0, lty=2)
}