# Easy error propagation in R

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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:

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

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:

- the formula operator:
`~`

- the left-hand side (LHS) of the formula:
`z`

- 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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