计算曲线下的面积 [英] Calculate the Area under a Curve

问题描述

我想计算曲线下的面积以进行积分，而无需定义诸如 integrate() 之类的函数.

I would like to calculate the area under a curve to do integration without defining a function such as in integrate().

我的数据如下所示:

Date          Strike     Volatility
2003-01-01    20         0.2
2003-01-01    30         0.3
2003-01-01    40         0.4
etc.

我绘制了 plot(strike,volatile) 以查看波动率微笑.有没有办法整合这个绘制的曲线"?

I plotted plot(strike, volatility) to look at the volatility smile. Is there a way to integrate this plotted "curve"?

推荐答案

AUC 通过查看大量梯形图很容易近似，每次都在 x_i, x_{ 之间i+1}、y{i+1} 和 y_i.使用zoo包的rollmean，可以做到:

The AUC is approximated pretty easily by looking at a lot of trapezium figures, each time bound between x_i, x_{i+1}, y{i+1} and y_i. Using the rollmean of the zoo package, you can do:

library(zoo)

x <- 1:10
y <- 3*x+25
id <- order(x)

AUC <- sum(diff(x[id])*rollmean(y[id],2))

确保您对 x 值进行排序，否则您的结果将毫无意义.如果沿 y 轴的某处有负值，则必须弄清楚如何准确定义曲线下的区域，并进行相应调整(例如使用 abs() )

Make sure you order the x values, or your outcome won't make sense. If you have negative values somewhere along the y axis, you'd have to figure out how exactly you want to define the area under the curve, and adjust accordingly (e.g. using abs() )

关于你的后续:如果你没有正式的函数，你会如何绘制它?因此，如果您只有值，那么您唯一可以近似的就是定积分.即使你有 R 中的函数，你也只能使用 integrate() 计算定积分.绘制形式函数只有在您也可以定义它的情况下才有可能.

Regarding your follow-up : if you don't have a formal function, how would you plot it? So if you only have values, the only thing you can approximate is a definite integral. Even if you have the function in R, you can only calculate definite integrals using integrate(). Plotting the formal function is only possible if you can also define it.

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