inder: Interpolate derivatives
In chipPCR: Toolkit of Helper Functions to Pre-Process Amplification Data

Description Usage Arguments Details Value Author(s) References Examples

A function inder ("in" + "der" = interpolate derivatives) for interpolating first and second derivatives using the five-point stencil. Therefore this function can be used to estimate the Cq (cycle of quantification) of an amplification curve. First positive derivative also known as First Derivative Maximum (FDM) and the Second Derivative Maximum (SDM) are calculated this way (Ruijter et al. 2013). However, from the mathematical point of view it can also be used to calculate the melting point for melting curve analysis (compare Roediger et al. 2013) provided that the sign of the derivative is changed.

1	inder(x, y, Nip = 4, logy = FALSE, smooth.method = "spline")

`x`	is a `numeric` vector of independent variable or data frame/matrix containing abscissa and ordinate values.
`y`	is a vector of dependent variable. Omitted if `x` is data frame or matrix.
`Nip`	is a value which defines how often an interpolation takes place at n equally spaced points spanning the interval (default 4). `Nip` is a such a resolution. A high `Nip` may improve the precision. `Nip`s less than 2 and higher than 20 are not meaningful for conventional qPCR with up to 50 cycles. However, higher `Nip`s might be appropriate for isothermal amplification reactions.
`logy`	If `logy` is TRUE than a semi-decadic log scale graph (corresponds to the linear phase) to illustrate the exponential dynamic of the qPCR amplification is used.
`smooth.method`	a character vector of length 1 or `NULL` defining smoothing method used. Currently accepted character values: "spline", "supsmu". If `NULL`, smoothing is not performed.

The function y = f(x) is numerically derived using the five-point stencil. This method does not require any assumptions regarding the function f.

A smoothing procedure greatly enhances calculating derivative calculation. inder uses two smoothing algorithms best suited for this approach. A smoothing can be omitted by setting smooth.method to NULL, which is advisable in case of the already smoothed data.

An object of der class.

Stefan Roediger, Michal Burdukiewicz

Ruijter JM, Pfaffl MW, Zhao S, et al. (2013) Evaluation of qPCR curve analysis methods for reliable biomarker discovery: bias, resolution, precision, and implications. Methods San Diego Calif 59:32–46.

Surface Melting Curve Analysis with R. S. Roediger, A. Boehm and I. Schimke. The R Journal. 5(2):37–52, 2013. https://journal.r-project.org

# First example
# Derive sinus
x <- 1:100/10
y <- sin(x)
ders <- inder(x, y)
plot(x, y, type = "l")
lines(ders[, "x"], ders[, "d1y"], col = "red")
lines(ders[, "x"], ders[, "d2y"], col = "green")
legend("topright", c("f(x)", "f'(x)", "f''(x)"), lty = 1, col = c("black", 
	"red", "green"))

# Second example
# Determine the approximate second derivative maximum
# for a qPCR experiment. SDM, 
isPCR <- AmpSim(cyc = 1:40)

res <- inder(isPCR)

plot(isPCR, xlab = "Cycle", ylab = "RFU", ylim = c(-0.15,1),
     main = "Approximate Second Derivative Maximum (SDM)", 
     type = "b", pch = 20)

  lines(res[, "x"], res[, "d1y"], col = "blue")
  lines(res[, "x"], res[, "d2y"], col = "red")
  summ <- summary(res, print = FALSE)
  abline(v = c(summ["SDM"], summ["SDm"], summ["SDC"]), col = c(3,4,5))
  text(summ["SDM"], 0.5, paste0("SDM ~ ", round(summ["SDM"], 2)), 
       cex = 1.5, col = 3)
  text(summ["SDC"], 0.7, paste0("SDC ~ ", round(summ["SDC"], 2)), 
       cex = 1.5, col = 5)
  text(summ["SDm"], 0.9, paste0("SDm ~ ", round(summ["SDm"], 2)), 
       cex = 1.5, col = 4)
  text(summ["FDM"] + 10, 0.65, paste("FDM ~ ", round(summ["FDM"], 2)), 
       cex = 1.5, col = 1)
       
  legend(1, 1, c("raw", "first derivative", "second derivative"), 
	 col = c(1,4,2), lty = c(2,1,1), cex = 1.2)