r - 意外的卷积结果

Question

我正在尝试在 R 中实现以下卷积，但没有得到预期的结果:

$$
C_{\sigma}[i]=\sum\limits_{k=-P}^P SDL_{\sigma}[i-k,i]\centerdot S[i]
$$


其中 $S[i]$ 是光谱强度向量(洛伦兹信号/NMR 光谱)，$i\in [1,N]$ 其中 $N$ 是数据点的数量(在实际示例中，可能是 32K值)。这是 Jacob, Deborde and Moing, Analytical Bioanalytical Chemistry (2013) 405:5049-5061 (DOI 10.1007/s00216-013-6852-y) 中的方程 1。

$SDL_{\sigma}$ 是计算洛伦兹曲线二阶导数的函数，我实现如下(基于论文中的方程 2):

SDL <- function(x, x0, sigma = 0.0005){
    if (!sigma > 0) stop("sigma must be greater than zero.")
    num <- 16 * sigma * ((12 * (x-x0)^2) - sigma^2)
    denom <- pi * ((4 * (x - x0)^2) + sigma^2)^3
    sdl <-  num/denom
    return(sdl)
    }

sigma是半峰宽， x0是洛伦兹信号的中心。

我相信 SDL工作正常(因为返回值的形状类似于经验 Savitzky-Golay 二阶导数)。我的问题是实现 $C_{\sigma}$，我写成:

CP <- function(S = NULL, X = NULL, method = "SDL", W = 2000, sigma = 0.0005) {
    # S is the spectrum, X is the frequencies, W is the window size (2*P in the eqn above)
    # Compute the requested 2nd derivative
    if (method == "SDL") {

        P <- floor(W/2)
        sdl <- rep(NA_real_, length(X)) # initialize a vector to store the final answer

        for(i in 1:length(X)) {
            # Shrink window if necessary at each extreme
            if ((i + P) > length(X)) P <- (length(X) - i + 1)
            if (i < P) P <- i
            # Assemble the indices corresponding to the window
            idx <- seq(i - P + 1, i + P - 1, 1)
            # Now compute the sdl
            sdl[i] <- sum(SDL(X[idx], X[i], sigma = sigma))
            P <- floor(W/2) # need to reset at the end of each iteration
            }
        }

    if (method == "SG") {
        sdl <- sgolayfilt(S, m = 2)     
        }

    # Now convolve!  There is a built-in function for this!
    cp <- convolve(S, sdl, type = "open")
    # The convolution has length 2*(length(S)) - 1 due to zero padding
    # so we need rescale back to the scale of S
    # Not sure if this is the right approach, but it doesn't affect the shape
    cp <- c(cp, 0.0)
    cp <- colMeans(matrix(cp, ncol = length(cp)/2)) # stackoverflow.com/q/32746842/633251
    return(cp)
    }

根据引用，二阶导数的计算仅限于大约 2000 个数据点的窗口以节省时间。我认为这部分工作正常。它应该只产生微不足道的扭曲。

下面演示一下整个过程和问题:

require("SpecHelpers")
require("signal")
# Create a Lorentzian curve
loren <- data.frame(x0 = 0, area = 1, gamma = 0.5)
lorentz1 <- makeSpec(loren, plot = FALSE, type = "lorentz", dd = 100, x.range = c(-10, 10))
#
# Compute convolution
x <- lorentz1[1,] # Frequency values
y <- lorentz1[2,] # Intensity values
sig <- 100 * 0.0005 # per the reference
cpSDL <- CP(S = y, X = x, sigma = sig)
sdl <- sgolayfilt(y, m = 2)
cpSG <- CP(S = y, method = "SG")
#
# Plot the original data, compare to convolution product
ylabel <- "data (black), Conv. Prod. SDL (blue), Conv. Prod. SG (red)"
plot(x, y, type = "l", ylab = ylabel, ylim = c(-0.75, 0.75))
lines(x, cpSG*100, col = "red")
lines(x, cpSDL/2e5, col = "blue")

如您所见，来自 CP 的卷积乘积使用 SDL (蓝色)与 CP 的卷积乘积不同使用 SG方法(红色，这是正确的，除了比例)。我期待使用 SDL 的结果方法应该具有相似的形状，但规模不同。

如果到目前为止你一直坚持我，a) 谢谢，b) 你能看出哪里出了问题吗？毫无疑问，我有一个根本的误解。

Answer 1

您正在执行的手动卷积存在一些问题。如果您查看维基百科页面上为“Savitzky–Golay filter”定义的卷积函数here ，您会看到 y[j+i]求和中与 S[i] 冲突的项您引用的等式中的术语。我相信您引用的公式可能不正确/打错了。

我按如下方式修改了您的函数，现在似乎可以产生与 sgolayfilt() 相同的形状。版本，虽然我不确定我的实现是完全正确的。注意选择sigma很重要并且确实会影响最终的形状。如果最初没有得到相同的形状，请尝试显着调整 sigma范围。

CP <- function(S = NULL, X = NULL, method = "SDL", W = 2000, sigma = 0.0005) {
    # S is the spectrum, X is the frequencies, W is the window size (2*P in the eqn above)
    # Compute the requested 2nd derivative
    if (method == "SDL") {
        sdl <- rep(NA_real_, length(X)) # initialize a vector to store the final answer

        for(i in 1:length(X)) {
            bound1 <- 2*i - 1
            bound2 <- 2*length(X) - 2*i + 1
            P <- min(bound1, bound2)
            # Assemble the indices corresponding to the window
            idx <- seq(i-(P-1)/2, i+(P-1)/2, 1)
            # Now compute the sdl
            sdl[i] <- sum(SDL(X[idx], X[i], sigma = sigma) * S[idx])
            }
        }

    if (method == "SG") {
        sdl <- sgolayfilt(S, m = 2)     
        }

    # Now convolve!  There is a built-in function for this!
    cp <- convolve(S, sdl, type = "open")
    # The convolution has length 2*(length(S)) - 1 due to zero padding
    # so we need rescale back to the scale of S
    # Not sure if this is the right approach, but it doesn't affect the shape
    cp <- c(cp, 0.0)
    cp <- colMeans(matrix(cp, ncol = length(cp)/2)) # stackoverflow.com/q/32746842/633251
    return(cp)
}