These notes are inspired by

a talk by Stuart Sierra on Design Patterns in Functional Programming and

some thoughts I found on F# for fun an profit

and are reflection on how I use different strategies to solve things in R. Design Pattern seems to be a big word, especially because of its use in object-oriented programming. But in the end I think it is simply the correct label for reoccurring strategies to design software.

The Fixed Point Algorithm

I keep my notes using R and as an ongoing example I use a fixed point algorithm to compute the square root of a positive real number. The algorithm is defined as:

and the fixed point function to find the square root is given by:

where \(p\) is a given (positive) number for which we want to find the square root. To represent this in R I define the algorithm as:

fp <- function ( f , x , converged , ... ) { value <- f ( x , ... ) if ( converged ( x , value )) value else Recall ( f , value , converged , ... ) }

\(x\) is the value of the last iteration or the starting value. converged is a function with two arguments and ... can be used for currying in R. Let the fixed point function be defined as:

fpsqrt <- function ( x , p ) p / x

and

converged <- function ( x , y ) all ( abs ( x - y ) < 0.001 )

So let’s compute that value:

fp ( fpsqrt , 2 , converged , p = 2 )

## Error: evaluation nested too deeply: infinite recursion / options(expressions=)?

And as so often nothing really works the first time. In the current implementation it is also kind of hard to find out what is going wrong but we will get there. In the following I will apply different patterns to modify the above framework to get to a solution.

Wrapper Pattern

This pattern is one I saw in the talk by Stuart Sierra. The Wrapper Pattern is something I use to add some functionality to a function, without actually changing it. One of the things I do is adding a logger to a function or taking care that a function preserves attributes, which is not the case for many functions in R. Something else would be to try to call a function and retry every two minutes because of a failing connection to a database or a file system that is not responding.

A function has a single, well defined purpose; logging and writing to a database are two things. Computing the next iteration and keeping track of the number of iterations are also two things. That one feature, with an additional argument for logging yes/no, will not be alone for long.

The problem in my example is that the fixed point function oscillates between two values instead of converging against the square root. A trick to overcome this is to use average damping. This means that instead of directly taking the value \(x_{n}\) for computing \(x_{n + 1}\) we use \(\frac{x_{n - 1} + x_n}{2}\). And this is actually not part of the logic of the fixed point function so it should not be polluted with it:

averageDamp <- function ( fun ) { function ( x , ... ) ( x + fun ( x , ... )) / 2 } fp ( averageDamp ( fpsqrt ), 2 , converged , p = 2 )

## [1] 1.414214

# and to compare: sqrt ( 2 )

## [1] 1.414214

Okay, great, now it seems to work. An additional wrapper I want to have is for printing the value in each iteration to the console:

printValue <- function ( fun ) { function ( x , ... ) { cat ( x , "

" ) fun ( x , ... ) } } fp ( printValue ( averageDamp ( fpsqrt )), 2 , converged , p = 2 )

## 2 ## 1.5 ## 1.416667 ## 1.414216

## [1] 1.414214

The problem now becomes that if we add to many wrappers it gets complicated. Actually try to figure out which wrapper is called first, maybe that is not obvious to you yet.

The wrapper pattern can be applied to add features before or after (or both) the actual function. printValue adds the printing before and averageDamp the correction after the original function. If we look at a different formulation of averageDamp the pattern becomes more obvious:

averageDamp <- function ( fun ) { function ( x , ... ) { value <- fun ( x , ... ) ( x + value ) / 2 } }

Interface Patterns

Currying

The value of this technique - from my perspective - is that you can build interfaces more easily (with sufficient support of the language). For example the fixed point function for the square root needs two arguments. The algorithm, however, actually knows only about one argument. Currying in this case simply means to make the two-argument function fpsqrt into a one-argument function. We can do this by setting \(p = 2\) which is what I accomplished using the ... so far.

In R you have two native options to model currying. The one you see most of the time is to use the dots argument to allow to pass additional arguments to the function. However, this puts an additional burden to every implementation in my framework because I need to take care that I allow for dots in every wrapper function I define. The other option is to use an anonymous function to wrap the original version in an one-argument function which would look like this:

fp ( averageDamp ( function ( x ) fpsqrt ( x , p = 2 )), 2 , converged )

## [1] 1.414214

If I rely on this interface (one-argument function) I can get rid of all the dots. However the syntactical support of this technique is limited in R which is why packages like purrr and rlist try to improve the situation; The packages functional and pryr provide dedicated functions for currying.

Closures

Every function in R is a closure (except for primitive functions). A closure is a function which has an environment associated to it. E.g. a function in a R-package has access to the packages namespace or a method in a class as in object orientation has access to the scope of the class. But typically the term is used when functions are returned from other functions (except Rs error message whenever you try to subset a closure). If you don’t know about them, but want to, you can read this article or the chapter in Advanced R.

For my example I use a closure to redefine the fixed point function for the square root for a given value of \(p\). Where I think that the given-value-of-p part is only emphasized by the following implementation:

fpsqrt <- function ( p ) { function ( x ) p / x }

And this actually makes the call of the algorithm a bit more concise:

fp ( averageDamp ( fpsqrt ( 2 )), 2 , converged )

## [1] 1.414214

Cache Pattern

In various situations I want to cache some results instead of recomputing them. This I do for performance reasons because no matter which library you use, the time to compute an inverse is not linear in the sample size. Once you have 10.000 observations and compute the inverse of a \((10.000 \times 10.000)\) variance-covariance matrix combined with a bootstrap in a Monte-Carlo simulation study you have to wait. To illustrate this I want to compute a linear estimator. And although the estimator can be identified analytically I use the fixed point algorithm. The fixed point function, in this case I use Newton-Raphson algorithm, is defined as:

with

being the first and second order derivatives for \(\beta\) of the likelihood under a normal distribution. If this does not mean anything to you, take it for granted and focus on the pattern. Consider the following implementation in R where I already applied the interface pattern using a closure:

nr <- function ( X , y ) { function ( beta ) beta - solve ( - crossprod ( X )) %*% crossprod ( X , y - X %*% beta ) } # Some data to test the function: set.seed ( 1 ) X <- cbind ( 1 , 1 : 10 ) y <- X %*% c ( 1 , 2 ) + rnorm ( 10 ) # Average damping in this case will make the convergence a bit slower: fp ( printValue ( averageDamp ( nr ( X , y ))), c ( 1 , 2 ), converged )

## 1 2 ## 0.9155882 2.027366 ## 0.8733823 2.041049 ## 0.8522794 2.047891 ## 0.8417279 2.051311 ## 0.8364522 2.053022 ## 0.8338143 2.053877 ## 0.8324954 2.054304

## [,1] ## [1,] 0.8318359 ## [2,] 2.0545183

# And to have a comparison: stats :: lm.fit ( X , y ) $ coefficients

## x1 x2 ## 0.8311764 2.0547321

The results look promising. We should choose a different tolerance level, which at the moment is chosen very liberal, to get closer to Rs implementation; but that is not the focus here. What I want to do now is to make the returned function of nr rely on pre-calculated values to avoid that they are recomputed in every iteration. In this example I combine this with the definition of local functions:

nr <- function ( X , y ) { # f1 relies on values in its scope: f 1 <- function ( beta ) Xy - XX %*% beta Xy <- crossprod ( X , y ) XX <- crossprod ( X ) f 2i nv <- solve ( - XX ) function ( beta ) beta - f 2i nv %*% f 1 ( beta ) } fp ( averageDamp ( nr ( X , y )), c ( 1 , 2 ), converged )

## [,1] ## [1,] 0.8318359 ## [2,] 2.0545183

Some remarks:

local function definitions like f1 in the above example, where I am sure to control the environment, are the only places where I rely on free variables. I.e. f1 relies on Xy and XX which are values defined in the enclosing environment; which I avoid at all cost in a top-level function.

in the above example, where I am sure to control the environment, are the only places where I rely on free variables. I.e. relies on and which are values defined in the enclosing environment; which I avoid at all cost in a top-level function. I like this representation because I keep the logic of the fixed point function local to nr ; that function knows everything there is to know about how to compute the next iteration, given the data. A different approach is to define nr such that it expects XX , Xy and f2inv as arguments, which means some other part of my code has to know about the implementation in nr and I have to look at different places to understand how the next iteration is computed.

Counter Pattern

So far the fixed point framework does not allow to restrict the number of iterations. This is of course something you always want to control. This time I use closures to model mutable state. Consider the two implementations of a counter:

# Option 1: counterConst <- function () { # like a constructor function count <- 0 function () { count <<- count + 1 count } } counter <- counterConst () counter ()

## [1] 1

counter ()

## [1] 2

# Option 2: counter <- local ({ count <- 0 function () { count <<- count + 1 count } }) counter ()

## [1] 1

counter ()

## [1] 2

I remember that closures are hard to understand because as in the above example they can be used to model mutable state when almost everything in R is immutable. I probably banged my head against the wall for a couple of hours over one of Hadleys examples to get the idea.

To implement a maximum number of iterations in the fixed point framework I combine the wrapper pattern and the counter pattern to modify the convergence criterion such that the algorithm will terminate after a given number of iterations:

addMaxIter <- function ( converged , maxIter ) { count <- 0 function ( ... ) { count <<- count + 1 if ( count >= maxIter ) TRUE else converged ( ... ) } }

This allows us to explore the error which occurred in the initial example:

fp ( printValue ( fpsqrt ( 2 )), 2 , addMaxIter ( converged , 4 ))

## 2 ## 1 ## 2 ## 1

## [1] 2

Now we can see that the initial version of the algorithm oscillates between 1 and 2. You may argue that in this case you can also see the number of iterations as logic of the algorithm (as a responsibility of fp ). In that case I would argue that the code does not reflect the previously introduced formula of the algorithm any more. But let’s compare a different implementation:

fpImp <- function ( f , x , convCrit , maxIter = 100 ) { converged <- function () { convCrit ( x , value ) | count >= maxIter } count <- 0 value <- NULL repeat { count <- count + 1 value <- f ( x ) if ( converged ()) break else { x <- value next } } list ( result = value , iter = count ) } fpImp ( averageDamp ( fpsqrt ( 2 )), 2 , converged )

## $result ## [1] 1.414214 ## ## $iter ## [1] 4

And now let’s add the number of iterations to the return value of fp :

addIter <- function ( fun ) { count <- 0 function ( x ) { count <<- count + 1 value <- fun ( x ) attr ( value , "count" ) <- count value } } fp ( addIter ( averageDamp ( fpsqrt ( 2 ))), 2 , converged )

## [1] 1.414214 ## attr(,"count") ## [1] 4

Maybe this implementation deserves an own name, but you can still see the wrapper and counter pattern and it is similar to the averageDamp function. Compared to fpImp the logic around a maximum number of iterations has been separated from the concrete implementation of the algorithm. Especially if I think about adding more features, the imperative implementation has to cope with more and more things. Instead I can plug-in new features around my fixed point framework. So I’d argue it is open for extensions and closed for modifications which is not only a good thing if you like object-orientation.

The counter pattern is of course more general. It simply reflects one strategy to model mutable state. There are just very few situations in which I really need to do that and counting is a reoccurring theme.

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