意外的卷积结果

Unexpected Convolution Results

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,可能有 32K 个值)。这是 Jacob、Deborde 和 Moing 中的等式 1,Analytical Bioanalytical Chemistry (2013) 405:5049-5061 (DOI 10.1007/s00216-013-6852-y).

$SDL_{\sigma}$ 是计算洛伦兹曲线二阶导数的函数,我实现如下(基于论文中的等式 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 $C_{\sigma}$,我将其写为:

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)) # whosebug.com/q/32746842/633251
    return(cp)
    }

根据参考文献,二阶导数的计算限于 window 约 2000 个数据点以节省时间。我认为这部分工作正常。它应该只会产生微不足道的扭曲。

下面是整个过程和问题的演示:

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,CP 使用 SDL 的卷积乘积(蓝色)与 CP 使用 SG 方法的卷积乘积不同(红色,正确,比例除外)。我希望使用 SDL 方法的结果应该具有相似的形状但比例不同。

如果到目前为止您一直支持我,a) 谢谢,b) 您能看出哪里出了问题吗?毫无疑问,我有一个根本性的误解。

您正在执行的手动卷积存在一些问题。如果您查看 "Savitzky–Golay filter" here 维基百科页面上定义的卷积函数,您会看到求和中的 y[j+i] 项与 S[i] 项冲突您引用的方程式。我相信您引用的公式可能不正确/打字错误。

我按如下方式修改了你的函数,它现在似乎可以生成与 sgolayfilt() 版本相同的形状,尽管我不确定我的实现是否完全正确。请注意 sigma 的选择很重要并且会影响最终的形状。如果您最初没有得到相同的形状,请尝试显着调整 sigma 参数。

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)) # whosebug.com/q/32746842/633251
    return(cp)
}

您的代码有几个问题。在 CP 中,当您计算 SDL 时,您似乎正在尝试在方程中对 $C_{\sigma}$ 求和,但这个求和是卷积的定义。

当你实际计算 SDL 时,你正在改变 x0 的值,但这个值是洛伦兹的平均值,应该是常数(在本例中为 0)。

最后可以计算出卷积的bounds,拉出具有原始bounds的信号

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)) {
        sdl[i] <- SDL(X[i], 0, sigma = sigma)
        }
    }

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

# Now convolve!  There is a built-in function for this!
cp <- convolve(S, sdl, type = "open")
shift <- floor(length(S)/2) #both signals are the same length and symmetric around zero
                             #so the fist point of the convolution is half the signal 
                             #before the first point of the signal   
print(shift)      
cp <- cp[shift:(length(cp)-shift)]
return (cp)
}

运行本次测试。

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)

#
# Plot the original data, compare to convolution product

plot(x, cpSDL)

结果符合预期形状:

cpSDL

我也不完全确定您的 SDL 定义是否正确。 This article 对于洛伦兹的二阶导数有一个更复杂的公式。