In a previous post I demonstrated how to use R’s simple built-in symbolic engine to generate Jacobian and (pseudo)-Hessian matrices that make non-linear optimization perform much more efficiently. Another related application is Gaussian error propagation.

Say you have data from a set of measurements in variables x and y where you know the corresponding measurement errors (dx and dy, typically the standard deviation or error from a set of replicates or a calibration curve). Next you want to create a derived value defined by an arbitrary function z = f(x,y). What would the corresponding error in value of z, i.e. dz = df, be?

If the function f(x,y) is a simple sum or product, their are simple equations for determining df. However, if f(x,y) is something more complex, like:

$$z = f(x,y) = frac{x-y}{(x+y)^2}$$

you’ll need to use a bit of calculus, specifically the chain rule:

$$df = sqrt{left(dx frac{partial f}{partial x}right)^2 + left(dy frac{partial f}{partial y}right)^2 + ...}$$

Applying the above equation allows for the derivation of Gaussian error propagation for any arbitrary function. So how does one do this in R? Again, the D() function and R expression() objects come to our rescue.

Say the definition of z (ergo f(x,y)) is defined in an R formula:

> f = z ~ (x-y)/(x+y)^2

If you probe the structure of a formula object you get:

> str(f)
Class 'formula' length 3 z ~ (x - y)/(x + y)^2
  ..- attr(*, ".Environment")=<environment: R_GlobalEnv>

What’s key is the “length 3” bit:

> f[[1]]; f[[2]]; f[[3]]
`~`
z
(x - y)/(x + y)^2

The code above shows us that a formula object can be subsetted into its constituent parts:

1. the formula operator: ~
2. the left-hand side (LHS) of the formula: z
3. the right-hand side (RHS) of the formula: (x - y)/(x + y)^2

The class() of the RHS is a call, which is close enough to an R expression that both all.vars() and D() work as expected to generate the mathematical expressions for the partial derivatives with respect to each variable:

> all.vars(f[[3]])
[1] "x" "y"

> lapply(all.vars(f[[3]]), function(v) D(f[[3]], v))
[[1]]
1/(x + y)^2 - (x - y) * (2 * (x + y))/((x + y)^2)^2

[[2]]
-(1/(x + y)^2 + (x - y) * (2 * (x + y))/((x + y)^2)^2)

These expressions need to be modified a bit – i.e. in this case they need to be multiplied by dx and dy, respectively and then squared. What’s returned from D() is a call object, so the elements above need to be converted to character to manipulate them accordingly. This is done with deparse().

> lapply(all.vars(f[[3]]), function(v) deparse(D(f[[3]], v)))
[[1]]
[1] "1/(x + y)^2 - (x - y) * (2 * (x + y))/((x + y)^2)^2"

[[2]]
[1] "-(1/(x + y)^2 + (x - y) * (2 * (x + y))/((x + y)^2)^2)"

The final error propagation expression is created with a bit of string manipulation:

> sprintf('sqrt(%s)', 
    paste(
        sapply(all.vars(f[[3]]), function(v) {
            sprintf('(d%s*(%s))^2', v, deparse(D(f[[3]], v)))
        }), 
        collapse='+'
    )
  )
[1] "sqrt((dx*(1/(x + y)^2 - (x - y) * (2 * (x + y))/((x + y)^2)^2))^2+(dy*(-(1/(x + y)^2 + (x - y) * (2 * (x + y))/((x + y)^2)^2)))^2)"

Now that we’ve got the basics down, let’s test this out with some data …

> set.seed(0)
> data = data.frame(
    x  = runif(5), 
    y  = runif(5), 
    dx = runif(5)/10, 
    dy = runif(5)/10
  )
> data
          x         y          dx         dy
1 0.8966972 0.2016819 0.006178627 0.07698414
2 0.2655087 0.8983897 0.020597457 0.04976992
3 0.3721239 0.9446753 0.017655675 0.07176185
4 0.5728534 0.6607978 0.068702285 0.09919061
5 0.9082078 0.6291140 0.038410372 0.03800352

and with a little help from dplyr:

> library(dplyr)
> data %>%
+   mutate_(.dots=list(
+     # compute derived value
+     z  = deparse(f[[3]]),
+     
+     # generates a mathematical expression to compute dz
+     # as a character string
+     dz = sapply(all.vars(f[[3]]), function(v) {
+             dfdp = deparse(D(f[[3]], v))
+             sprintf('(d%s*(%s))^2', v, dfdp)
+           }) %>%
+           paste(collapse='+') %>%
+           sprintf('sqrt(%s)', .)
+       ))
          x         y          dx         dy           z         dz
1 0.8966972 0.2016819 0.006178627 0.07698414  0.57608929 0.14457245
2 0.2655087 0.8983897 0.020597457 0.04976992 -0.46718831 0.03190297
3 0.3721239 0.9446753 0.017655675 0.07176185 -0.33019871 0.01978697
4 0.5728534 0.6607978 0.068702285 0.09919061 -0.05778613 0.07604809
5 0.9082078 0.6291140 0.038410372 0.03800352  0.11809201 0.02424023

Taking this a step further, this method can be wrapped in a chainable function that determines the name of new variables from the LHS of a formula argument:

mutate_with_error = function(.data, f) {
  exprs = list(
      # expression to compute new variable values
      deparse(f[[3]]),

      # expression to compute new variable errors
      sapply(all.vars(f[[3]]), function(v) {
        dfdp = deparse(D(f[[3]], v))
        sprintf('(d%s*(%s))^2', v, dfdp)
      }) %>%
        paste(collapse='+') %>%
        sprintf('sqrt(%s)', .)
  )
  names(exprs) = c(
    deparse(f[[2]]),
    sprintf('d%s', deparse(f[[2]]))
  )

  .data %>%
    # the standard evaluation alternative of mutate()
    mutate_(.dots=exprs)
}

Thus, adding new derived variables and propagating errors accordingly becomes relatively easy:

> set.seed(0)
> data = data.frame(x=runif(5), y=runif(5), dx=runif(5)/10, dy=runif(5)/10)
> data
          x         y          dx         dy
1 0.8966972 0.2016819 0.006178627 0.07698414
2 0.2655087 0.8983897 0.020597457 0.04976992
3 0.3721239 0.9446753 0.017655675 0.07176185
4 0.5728534 0.6607978 0.068702285 0.09919061
5 0.9082078 0.6291140 0.038410372 0.03800352

> data %>% mutate_with_error(z ~ (x-y)/(x+y)^2)
          x         y          dx         dy           z         dz
1 0.8966972 0.2016819 0.006178627 0.07698414  0.57608929 0.14457245
2 0.2655087 0.8983897 0.020597457 0.04976992 -0.46718831 0.03190297
3 0.3721239 0.9446753 0.017655675 0.07176185 -0.33019871 0.01978697
4 0.5728534 0.6607978 0.068702285 0.09919061 -0.05778613 0.07604809
5 0.9082078 0.6291140 0.038410372 0.03800352  0.11809201 0.02424023

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