# Lecture 16: Functions as Objects

36-350
20 October 2014

### Previously...

• Writing our own functions
• Dividng labor with multiple functions
• Refactoring to create higher-level operations
• Using apply, sapply, etc., to avoid iteration

### Agenda

• Functions are objects, and can be arguments to other functions
• Functions are objects, and can be returned by other functions
• Example: surface

Reading: Sections 7.5, 7.11 and 7.13 of Matloff

Optional Recommended Reading: Chapter 3 of Chambers

### Functions as Objects

• In R, functions are objects, just like everything else

• This means that they can be passed to functions as arguments
and returned by functions as outputs as well

### Functions of Functions: Computationally

• We often want to do very similar things to many different functions

• The procedure is the same, only the function we're working with changes

• $$\therefore$$ Write one function to do the job, and pass the function as an argument

• Because R treats a function like any other object, we can do this simply: invoke the function by its argument name in the body

• We have already seen examples

### R Functions That Take Functions as Arguments

• apply(), sapply(), etc.: Take this function and use it on all of these objects
• nlm(): Take this function and try to make it small, starting from here
• ks.test(): Compare these data to this cumulative distribution function
• curve(): Evaluate this function over that range, and plot the results

### Some R Syntax Facts About Functions

• Typing a function's name, without parentheses, in the terminal gives you its source code:
sample

function (x, size, replace = FALSE, prob = NULL)
{
if (length(x) == 1L && is.numeric(x) && x >= 1) {
if (missing(size))
size <- x
sample.int(x, size, replace, prob)
}
else {
if (missing(size))
size <- length(x)
x[sample.int(length(x), size, replace, prob)]
}
}
<bytecode: 0x102d004a0>
<environment: namespace:base>


### Some R Syntax Facts About Functions

• Functions are their own class in R:
class(sin)

 "function"

class(sample)

 "function"

resample <- function(x) { sample(x, size=length(x), replace=TRUE) }
class(resample)

 "function"


### Some R Syntax Facts About Functions

• Functions can be put into lists or even arrays
• A call to function returns a function object
• body executed; access with body(foo)
• arguments required: access with formals(foo)
gives argument list of foo: names are argument names, values are expressions for defaults (if any)
• parent environment: access with environment(foo)

### Some R Syntax Facts About Functions

• R has separate types for built-in functions and for those written in R:
typeof(resample)

 "closure"

typeof(sample)

 "closure"

typeof(sin)

 "builtin"


Why closure for written-in-R functions? Because expressions are “closed” by referring to the parent environment

There's also a 2nd class of built-in functions called primitive

### Anonymous Functions

• function() returns an object of class function
• So far we've assigned that object to a name
• If we don't have an assignment, we get an anonymous function
• Usually part of some larger expression:
sapply((-2):2,function(log.ratio){exp(log.ratio)/(1+exp(log.ratio))})

 0.1192 0.2689 0.5000 0.7311 0.8808


### Anonymous Functions

• Often handy when connecting other pieces of code
• especially in things like apply and sapply
• Won't cluttering the workspace
• Can't be examined or re-used later

• Problems in stats. come down to optimization
So do lots of problems in econ., physics, CS, bio, …

• Lots of optimization problems require the gradient of the objective function

• Gradient of $$f$$ at $$x$$: $\nabla f(x) = \left[\left.\frac{\partial f}{\partial x_1}\right|_{x} \ldots \left.\frac{\partial f}{\partial x_p}\right|_{x}\right]$

• We do the same thing to get the gradient of $$f$$ at $$x$$ no matter what $$f$$ is:
  find the partial derivative of f with respect to each component of x
return the vector of partial derivatives

• It makes no sense to re-write this every time we change $$f$$!

• $$\therefore$$ write code to calculate the gradient of an arbitrary function

• We could write our own, but there are lots of tricky issues

• Best way to calculate partial derivative
• What if $$x$$ is at the edge of the domain of $$f$$?
• Fortunately, someone has already done this

From the package numDeriv

grad(func, x, ...) # Plus other arguments

• Assumes func is a function which returns a single floating-point value
• Assumes x is a vector of arguments to func
• If x is a vector and func(x) is also a vector, then it's assumed func is vectorized and we get a vector of derivatives
• Extra arguments in ... get passed along to func
• Other functions in the package for the Jacobian of a vector-valued function, and the matrix of 2nd partials (Hessian)

• Does it work as advertized?
require("numDeriv")
just_a_phase <- runif(n=1,min=-pi,max=pi)

 TRUE

phases <- runif(n=10,min=-pi,max=pi)

 TRUE

grad(func=function(x){x^2+x^3}, x=c(1,-1))

 2 3


Note: grad is perfectly happy with func being an anonymous function!

Now we can use this as a piece of a larger machine:

gradient.descent <- function(f,x,max.iterations,step.scale,
stopping.deriv,...) {
for (iteration in 1:max.iterations) {
if(all(abs(gradient) < stopping.deriv)) { break() }
}
iterations=iteration)
return(fit)
}

• Works equally well whether f is mean squared error of a regression, $$\psi$$ error of a regression, (negative log) likelihood, cost of a production plan, …

### Cautions

• Scoping f takes values for all names which aren't its arguments from the environment where it was defined, not the one where it is called (e.g., not from inside grad or gradient.descent)
• Debugging If f and g are both complicated, avoid debugging g(f) as a block; divide the work by writing very simple f.dummy to debug/test g, and debug/test the real f separately

### Returning Functions: A trivial example

Functions can be return values like anything else

make.noneuclidean <- function(ratio.to.diameter=pi) {
circumference <- function(d) { return(ratio.to.diameter*d) }
return(circumference)
}


### Returning Functions: A trivial example (cont'd.)

try(circumference(10))
kings.i <- make.noneuclidean(3)
try(kings.i(10))

 30

formals(kings.i)

$d  body(kings.i)  { return(ratio.to.diameter * d) }  environment(kings.i)  <environment: 0x100b588d8>  try(circumference(10))  ### A Less Trivial Example Create a linear predictor, based on sample values of two variables make.linear.predictor <- function(x,y) { linear.fit <- lm(y~x) predictor <- function(x) { return(predict(object=linear.fit,newdata=data.frame(x=x))) } return(predictor) }  The predictor function persists and works, even when the data we used to create it is gone ### A Less Trivial Example library(MASS); data(cats) vet_predictor <- make.linear.predictor(x=cats$Bwt,y=cats$Hwt) rm(cats) # Data set goes away vet_predictor(3.5) # My cat's body mass in kilograms   1 13.76  ### A more mathematical example • Instead of finding $$\nabla f(x)$$, find the function $$\nabla f$$: nabla <- function(f,...) { require("numDeriv") g <- function(x,...) { grad(func=f,x=x,...) } return(g) }  Exercise: Write a test case! ### Example: curve() • You learned to use curve in the first week (because you did all of the assigned reading, including section 2.3.3 of the textbook) • A call to curve looks like this: curve(expr, from = a, to = b, ...)  expr is some expression involving a variable called x which is swept from the value a to the value b ... are other plot-control arguments • curve feeds the expression a vector x and expects a numeric vector back, e.g.  curve(x^2 * sin(x))  is fine ### Using curve() with our own functions • If we have defined a function already, we can use it in curve: psi <- function(x,c=1) {ifelse(abs(x)>c,2*c*abs(x)-c^2,x^2)} curve(psi(x,c=10),from=-20,to=20)  Try this! Also try curve(psi(x=10,c=x),from=-20,to=20)  and explain it to yourself ### Using curve() with our own functions • If our function doesn't take vectors to vectors, curve becomes unhappy mse <- function(y0,a,Y=gmp$pcgmp,N=gmp\$pop) {
mean((Y - y0*(N^a))^2)
}

> curve(mse(a=x,y0=6611),from=0.10,to=0.15)
Error in curve(mse(a = x, y0 = 6611), from = 0.1, to = 0.15) :
'expr' did not evaluate to an object of length 'n'
In N^a : longer object length is not a multiple of shorter object length


How do we solve this?

### Using curve() with our own functions

• Define a new, vectorized function, say with sapply:
sapply(seq(from=0.10,to=0.15,by=0.01),mse,y0=6611)

 154701953 102322974  68755654  64529166 104079527 207057513

mse(6611,0.10)

 154701953

mse.plottable <- function(a,...){ return(sapply(a,mse,...)) }
mse.plottable(seq(from=0.10,to=0.15,by=0.01),y0=6611)

 154701953 102322974  68755654  64529166 104079527 207057513


### Using curve() with our own functions

curve(mse.plottable(a=x,y0=6611),from=0.10,to=0.20,xlab="a",ylab="MSE") ### Using curve() with our own functions

• Alternate strategy: Vectorize() returns a new, vectorized function
mse.vec <- Vectorize(mse, vectorize.args=c("y0","a"))
mse.vec(a=seq(from=0.10,to=0.15,by=0.01),y0=6611)

 154701953 102322974  68755654  64529166 104079527 207057513

mse.vec(a=1/8,y0=c(5000,6000,7000))

 134617132  74693733  63732256


### Using curve() with our own functions

curve(mse.vec(a=x,y0=6611),from=0.10,to=0.20,xlab="a",ylab="MSE") ### Example: surface()

• curve takes an expression and, as a side-effect, plots a 1-D curve by sweeping over x

• Suppose we want something like that but sweeping over two variables

• Built-in plotting function contour:

contour(x,y,z, [[other stuff]])


x and y are vectors of coordinates, z is a matrix of the corresponding shape
(see help(contour) for graphical options)

• Strategy: surface should make x and y sequences, evaluate the expression at each combination to get z, and then call contour

### First attempt at surface()

• Only works with vector-to-number functions:
surface.1 <- function(f,from.x=0,to.x=1,from.y=0,to.y=1,n.x=101,
n.y=101,...) {
x.seq <- seq(from=from.x,to=to.x,length.out=n.x)
y.seq <- seq(from=from.y,to=to.y,length.out=n.y)
plot.grid <- expand.grid(x=x.seq,y=y.seq)
z.values <- apply(plot.grid,1,f)
z.matrix <- matrix(z.values,nrow=n.x)
contour(x=x.seq,y=y.seq,z=z.matrix,...)
invisible(list(x=x.seq,y=y.seq,z=z.matrix))
}


### First attempt at surface()

surface.1(function(p){return(sum(p^3))},from.x=-1,from.y=-1) ### Expressions and Evaluation

• curve doesn't require us to write a function every time — what's it's trick?

• Expressions are just another class of R object, so they can be created and manipulated

• One manipulation is evaluation

eval(expr,envir)


evaluates the expression expr in the environment envir, which can be a data frame or even just a list

• When we type something like x^2+y^2 as an argument to surface.1, R tries to evaluate it prematurely

• substitute returns the unevaluted expression

• curve uses first substitute(expr) and then eval(expr,envir), having made the right envir

### Second attempt at surface()

surface.2 <- function(expr,from.x=0,to.x=1,from.y=0,to.y=1,n.x=101,
n.y=101,...) {
x.seq <- seq(from=from.x,to=to.x,length.out=n.x)
y.seq <- seq(from=from.y,to=to.y,length.out=n.y)
plot.grid <- expand.grid(x=x.seq,y=y.seq)
unevaluated.expression <- substitute(expr)
z.values <- eval(unevaluated.expression,envir=plot.grid)
z.matrix <- matrix(z.values,nrow=n.x)
contour(x=x.seq,y=y.seq,z=z.matrix,...)
invisible(list(x=x.seq,y=y.seq,z=z.matrix))
}


### Second attempt at surface()

surface.2(abs(x^3)+abs(y^3),from.x=-1,from.y=-1) ### Evaluating at Combinations

• Evaluating a function at every combination of two arguments is a really common task

• There is a function to do it for us: outer

### Third attempt at surface()

surface.3 <- function(expr,from.x=0,to.x=1,from.y=0,to.y=1,n.x=101,
n.y=101,...) {
x.seq <- seq(from=from.x,to=to.x,length.out=n.x)
y.seq <- seq(from=from.y,to=to.y,length.out=n.y)
unevaluated.expression <- substitute(expr)
z <- function(x,y) {
return(eval(unevaluated.expression,envir=list(x=x,y=y)))
}
z.values <- outer(X=x.seq,Y=y.seq,FUN=z)
z.matrix <- matrix(z.values,nrow=n.x)
contour(x=x.seq,y=y.seq,z=z.matrix,...)
invisible(list(x=x.seq,y=y.seq,z=z.matrix, func=z))
}


### Third attempt at surface()

surface.3(x^4-y^4,from.x=-1,from.y=-1)