---
category: literaturenote
citekey: eddelbuettelrcpparmadilloacceleratinghighperformance2014
title: "RcppArmadillo: Accelerating R with high-performance C++ linear algebra"
authors: "Eddelbuettel, Dirk; Sanderson, Conrad"
year: 2014
date: 2014-03-00 March 2014
doi: 10.1016/j.csda.2013.02.005
publication: "Computational Statistics & Data Analysis"
url: "http://www.sciencedirect.com/science/article/pii/S0167947313000492"
zotero_key: 83J6PKAH
zotero_storage: B37Z5SZX
collections: R / kalman
folder: Liiklussageduse kaudne hindamine/04_Tehniline
firstAuthor: "Eddelbuettel, Dirk"
status: converted
---
# RcppArmadillo: Accelerating R with High-Performance C++ Linear Algebra

Dirk Eddelbuettela,<sup>∗</sup> , Conrad Sandersonb,c

*<sup>a</sup>Debian Project,* http: // www. debian. org *<sup>b</sup>NICTA, PO Box 6020, St Lucia, QLD 4067, Australia <sup>c</sup>Queensland University of Technology (QUT), Brisbane, QLD 4000, Australia*

## Abstract

The R statistical environment and language has demonstrated particular strengths for interactive development of statistical algorithms, as well as data modelling and visualisation. Its current implementation has an interpreter at its core which may result in a performance penalty in comparison to directly executing user algorithms in the native machine code of the host CPU. In contrast, the C++ language has no built-in visualisation capabilities, handling of linear algebra or even basic statistical algorithms; however, user programs are converted to high-performance machine code, ahead of execution. A new method avoids possible speed penalties in R by using the Rcpp extension package in conjunction with the Armadillo C++ matrix library. In addition to the inherent performance advantages of compiled code, Armadillo provides an easy-touse template-based meta-programming framework, allowing the automatic pooling of several linear algebra operations into one, which in turn can lead to further speedups. With the aid of Rcpp and Armadillo, conversion of linear algebra centered algorithms from R to C++ becomes straightforward. The algorithms retains the overall structure as well as readability, all while maintaining a bidirectional link with the host R environment. Empirical timing comparisons of R and C++ implementations of a Kalman filtering algorithm indicate a speedup of several orders of magnitude.

*Keywords:* Software, R, C++, linear algebra

#### 1. Overview

Linear algebra is a cornerstone of statistical computing and statistical software systems. Various matrix decompositions, linear program solvers, and eigenvalue / eigenvector computations are key to many estimation and analysis routines. As generally useful procedures, these are often abstracted and regrouped in specific libraries for linear algebra which statistical programmers have provided for various programming languages and environments.

<sup>∗</sup>Corresponding author. Email: edd@debian.org. Postal address: 711 Monroe Avenue, River Forest, IL 60305, USA. The RcppArmadillo package is available with the electronic version of this article, as well as via every CRAN mirror.

One such environment and statistical programming language is R (R Development Core Team, 2012). It has become a tool of choice for data analysis and applied statistics (Morandat et al., 2012). While R has particular strengths at fast prototyping and easy visualisation of data, its implementation has an interpreter at its core. In comparison to running user algorithms in the native machine code of the host CPU, the use of an interpreter often results in a performance penalty for non-trivial algorithms that perform elaborate data manipulation (Morandat et al., 2012). With user algorithms becoming more complex and increasing in functionality, as well as with data sets which continue to increase in size, the issue of execution speed becomes more important.

The C++ language offers a complementary set of attributes: while it has no built-in visualisation capabilities nor handling of linear algebra or statistical methods, user programs are converted to high-performance machine code ahead of execution. It is also inherently flexible. One key feature is *operator overloading* which allows the programmer to define custom behaviour for mathematical operators such as +, −, ∗ (Meyers, 2005). C++ also provides language constructs known as *templates*, originally intended to easily allow the reuse of algorithms for various object types, and later extended to a programming construct in its own right called *template meta-programming* (Vandevoorde and Josuttis, 2002; Abrahams and Gurtovoy, 2004)

Operator overloading allows mathematical operations to be extended to user-defined objects, such as matrices. This in turn allows linear algebra expressions to be written in a more natural manner (eg. *<sup>X</sup>* <sup>=</sup> <sup>0</sup>.<sup>1</sup> <sup>∗</sup> *<sup>A</sup>* <sup>+</sup> <sup>0</sup>.<sup>2</sup> <sup>∗</sup> *<sup>B</sup>*), rather than the far less readable traditional function call syntax, eg. *<sup>X</sup>* <sup>=</sup> *add*(*multiply*(0.1, *<sup>A</sup>*), *multiply*(0.2, *<sup>B</sup>*)).

Template meta-programming is the process of inducing the C++ compiler to execute, at compile time, Turing-complete programs written in a somewhat opaque subset of the C++ language (Vandevoorde and Josuttis, 2002; Abrahams and Gurtovoy, 2004). These meta-programs in effect generate further C++ code (often specialised for particular object types), which is finally converted into machine code.

An early and influential example of exploiting both meta-programming and overloading of mathematical operators was provided by the Blitz++ library (Veldhuizen, 1998), targeted for efficient processing of arrays. Blitz++ employed elaborate metaprogramming to avoid the generation of temporary array objects during the evaluation of mathematical expressions. However, the library's capabilities and usage were held back at the time by the limited availability of compilers correctly implementing all the necessary features and nuances of the C++ language.

We present a new method of avoiding the speed penalty in R by using the Rcpp extension package (Eddelbuettel and Franc¸ois, 2011, 2012; Eddelbuettel, 2013) in conjunction with the Armadillo C++ linear algebra library (Sanderson, 2010). Similar to Blitz++, Armadillo uses operator overloading and various template meta-programming techniques to attain efficiency. However, it has been written to target modern C++ compilers as well as providing a much larger set of linear algebra operations than Blitz++. R programs augmented to use Armadillo retain the overall structure as well as readability, all while retaining a bidirectional link with the host R environment.

Section 2 provides an overview of Armadillo, followed by its integration with the Rcpp extension package. Section 4 shows an example of an R program and its conversion to C++ via Rcpp and Armadillo. Section 5 discusses an empirical timing comparison between the R and C++ versions before Section 6 concludes.

### 2. Armadillo

The Armadillo C++ library provides vector, matrix and cube types (supporting integer, floating point and complex numbers) as well as a subset of trigonometric and statistics functions (Sanderson, 2010). In addition to elementary operations such as addition and matrix multiplication, various matrix factorisations and submatrix manipulation operations are provided. The corresponding application programming interface (syntax) enables the programmer to write code which is both concise and easy-to-read to those familiar with scripting languages such as Matlab and R. Table 1 lists a few common Armadillo functions.

Matrix multiplication and factorisations are accomplished through integration with the underlying operations stemming from standard numerical libraries such as BLAS and LAPACK (Demmel, 1997). Similar to how environments such as R are implemented, these underlying libraries can be replaced in a transparent manner with variants that are optimised to the specific hardware platform and/or multi-threaded to automatically take advantage of the now-common multi-core platforms (Kurzak et al., 2010).

Armadillo uses a delayed evaluation approach to combine several operations into one and reduce (or eliminate) the need for temporary objects. In contrast to brute-force evaluations, delayed evaluation can provide considerable performance improvements as well as reduced memory usage. The delayed evaluation machinery is accomplished through template meta-programming (Vandevoorde and Josuttis, 2002; Abrahams and Gurtovoy, 2004), where the C++ compiler is induced to reason about mathematical expressions at *compile time*. Where possible, the C++ compiler can generate machine code that is tailored for each expression.

As an example of the possible efficiency gains, let us consider the expression

| Armadillo function                | Description                                              |  |
|-----------------------------------|----------------------------------------------------------|--|
| X(1,2) = 3                        | Assign value 3 to element at location (1,2) of matrix X  |  |
| X = A + B                         | Add matrices A and B                                     |  |
| X( span(1,2), span(3,4) )         | Provide read/write access to submatrix of X              |  |
| zeros(rows [, cols [, slices]))   | Generate vector (or matrix or cube) of zeros             |  |
| ones(rows [, cols [, slices]))    | Generate vector (or matrix or cube) of ones              |  |
| eye(rows, cols)                   | Matrix diagonal set to 1, off-diagonal elements set to 0 |  |
| repmat(X, row_copies, col_copies) | Replicate matrix X in block-like manner                  |  |
| det(X)                            | Returns the determinant of matrix X                      |  |
| norm(X, p)                        | Compute the p-norm of matrix or vector X                 |  |
| rank(X)                           | Compute the rank of matrix X                             |  |
| min(X, dim=0); max(X, dim=0)      | Extremum value of each column of X (row if dim=1)        |  |
| trans(X) or X.t()                 | Return transpose of X                                    |  |
| R = chol(X)                       | TR = X<br>Cholesky decomposition of X such that R        |  |
| inv(X) or X.i()                   | Returns the inverse of square matrix X                   |  |
| pinv(X)                           | Returns the pseudo-inverse of matrix X                   |  |
| lu(L, U, P, X)                    | LU decomp. with partial pivoting; also lu(L, U, X)       |  |
| qr(Q, R, X)                       | QR decomp. into orthogonal Q and right-triangular R      |  |
| X = solve(A, B)                   | Solve system AX = B for X                                |  |
| s = svd(X); svd(U, s, V, X)       | Singular-value decomposition of X                        |  |

Table 1: Selected Armadillo functions with brief descriptions; see http://arma.sf.net/docs.html for more complete documentation. Several optional additional arguments have been omitted here for brevity.

*X* = *A* − *B* + *C*, where *A*, *B* and *C* are matrices. A brute-force implementation would evaluate *A* − *B* first and store the result in a temporary matrix *T*. The next operation would be *T* + *C*, with the result finally stored in *X*. The creation of the temporary matrix, and using two separate loops for the subtraction and addition of matrix elements is suboptimal from an efficiency point of view.

Through the overloading of mathematical operators, Armadillo avoids the generation of the temporary matrix by first converting the expression into a set of lightweight Glue objects, which only store references to the matrices and Armadillo's representations of mathematical expressions (eg. other Glue objects). To indicate that an operation comprised of subtraction and addition is required, the exact type of the Glue objects is automatically inferred from the given expression through template metaprogramming. More specifically, given the expression *X* = *A* − *B* + *C*, Armadillo automatically induces the compiler to generate an instance of the lightweight Glue storage object with the following C++ *type*:

```
Glue< Glue<Mat, Mat, glue minus>, Mat, glue plus>
```

where Glue<...> indicates that Glue is a C++ template class, with the items between '<' and '>' specifying template parameters; the outer Glue<..., Mat, glue plus> is the Glue object indicating an addition operation, storing a reference to a matrix as well as a reference to another Glue object; the inner Glue<Mat, Mat, glue minus> stores references to two matrices and indicates a subtraction operation. In both the inner and outer Glue, the type Mat specifies that a reference to a matrix object is to be held.

The expression evaluator in Armadillo is then automatically invoked through the "=" operation, which interprets (at compile time) the template parameters of the compound Glue object and generates C++ code equivalent to:

```
for(int i=0; i<N; i++) { X[i] = (A[i] - B[i]) + C[i]; }
```

where *N* is the number of elements in *A*, *B* and *C*, with A[i] indicating the *i*-th element in *A*. As such, apart from the lightweight Glue objects (for which memory is preallocated at compile time), no other temporary object is generated, and only one loop is required instead of two. Given a sufficiently advanced C++ compiler, the lightweight Glue objects can be optimised away, as they are automatically generated by the compiler and only contain compile-time generated references; the resultant machine code can appear as if the Glue objects never existed in the first place.

Note that due to the ability of the Glue object to hold references to other Glue objects, far longer and more complicated operations can be easily accommodated. Further discussion of template meta-programming is beyond the scope of this paper; for more details, the interested reader is referred to Vandevoorde and Josuttis (2002) as well as Abrahams and Gurtovoy (2004). Reddy et al. (2013) provide a recent application of Armadillo in computer vision and pattern recognition.

#### 3. RcppArmadillo

The RcppArmadillo package (Franc¸ois et al., 2012) employs the Rcpp package (Eddelbuettel and Franc¸ois, 2011, 2012; Eddelbuettel, 2013) to provide a bidirectional interface between R and C++ at the object level. Using templates, R objects such as vectors and matrices can be mapped directly to the corresponding Armadillo objects.

Consider the simple example in Listing 1. Given a vector, the g() function returns both the outer and inner products. We load the inline package (Sklyar et al., 2012), which provides cxxfunction() that we use to compile, link and load the C++ code which is passed as the body argument. We declare the function signature to contain a single argument named 'vs'. On line five, this argument is used to instantiate an Armadillo column vector object named 'v' (using the templated conversion function as() from Rcpp). In lines six and seven, the outer and inner product of the column vector are calculated by appropriately multiplying the vector with its transpose. This shows how the \* operator for multiplication has been overloaded to provide the appropriate operation for the types implemented by Armadillo. The inner product creates a scalar variable, and in contrast to R where each object is a vector type (even if of length one), we have to explicitly convert using as\_scalar() to assign the value to a variable of type double.

Finally, the last line creates an R named list type containing both results. As a result of calling cxxfunction(), a new function is created. It contains a reference to the native code, compiled on the fly based on the C++ code provided to cxxfunction()

```
R> library(inline)
2 R>
   R> g <- cxxfunction(signature(vs="numeric" ),
4 + plugin="RcppArmadillo" , body='
   + arma::vec v = Rcpp::as<arma::vec>(vs);
6 + arma::mat op = v * v.t();
   + double ip = arma::as_scalar(v.t() * v);
8 + return Rcpp::List::create(Rcpp::Named("outer")=op,
   + Rcpp::Named("inner")=ip);
10 +' )
   R> g(7:11)
12 $outer
       [,1] [,2] [,3] [,4] [,5]
14 [1,] 49 56 63 70 77
   [2,] 56 64 72 80 88
16 [3,] 63 72 81 90 99
   [4,] 70 80 90 100 110
18 [5,] 77 88 99 110 121
20 $inner
   [1] 415
```

Listing 1: Integrating Armadillo-based C++ code via the RcppArmadillo package.

and makes it available directly from R under a user-assigned function name, here g(). The listing also shows how the Rcpp and arma namespaces are used to disambiguate symbols from the two libraries; the :: operator is already familiar to R programmers who use the NAMESPACE directive in R in a similar fashion.

The listing also demonstrates how the new function g() can be called with a suitable argument. Here we create a vector of five elements, containing values ranging from 7 to 11. The function's output, here the list containing both outer and inner product, is then displayed as it is not assigned to a variable.

This simple example illustrates how R objects can be transferred directly into corresponding Armadillo objects using the interface code provided by Rcpp. It also shows how deployment of RcppArmadillo is straightforward, even for interactive work where functions can be compiled on the fly. Similarly, usage in packages is also uncomplicated and follows the documentation provided with Rcpp (Eddelbuettel and Franc¸ois, 2012; Eddelbuettel, 2013).

#### 4. Kalman Filtering Example

The Kalman filter is ubiquitous in many engineering disciplines as well as in statistics and econometrics (Tusell, 2011). A recent example of an application is volatility extraction in a diffusion option pricing model (Li, 2013). Even in its simplest linear form, the Kalman filter can provide simple estimates by recursively applying linear updates which are robust to noise and can cope with missing data. Moreover, the estimation process is lightweight and fast, and consumes only minimal amounts of memory as few state variables are required. We discuss a standard example below. The (twodimensional) position of an object is estimated based on past values. A 6 × 1 state vector includes *X* and *Y* coordinates determining the position, two variables for speed (or velocity) *V<sup>X</sup>* and *V<sup>Y</sup>* relative to the two coordinates, as well as two acceleration variables *A<sup>X</sup>* and *A<sup>Y</sup>* .

We have the positions being updated as a function of the velocity

$$X = X_0 + V_X dt$$
 and  $Y = Y_0 + V_Y dt$ ,

and the velocity being updated as a function of the (unobserved) acceleration:

$$V_x = V_{X,0} + A_X dt$$
 and  $V_y = V_{Y,0} + A_Y dt$ .

With covariance matrices *Q* and *R* for (Gaussian) error terms, the standard Kalman filter estimation involves a linear prediction step resulting in a new predicted state vector, and a new covariance estimate. This leads to a residuals vector and a covariance matrix for residuals which are used to determine the (optimal) Kalman gain, which is then used to update the state estimate and covariance matrix.

All of these steps involve only matrix multiplication and inversions, making the algorithm very suitable for an implementation in any language which can use matrix

```
% Copyright 2010 The MathWorks, Inc.
2 function y = kalmanfilter(z)
    dt=1;
4 % Initialize state transition matrix
    A=[ 1 0 dt 0 0 0; 0 1 0 dt 0 0;... % [x ], [y ]
6 0 0 1 0 dt 0; 0 0 0 1 0 dt;... % [Vx], [Vy]
        0 0 0 0 1 0 ; 0 0 0 0 0 1 ]; % [Ax], [Ay]
8 H = [ 1 0 0 0 0 0; 0 1 0 0 0 0 ]; % Init. measuremnt mat
    Q = eye(6);
10 R = 1000 * eye(2);
    persistent x_est p_est % Init. state cond.
12 if isempty(x_est)
      x_est = zeros(6, 1); % x_est=[x,y,Vx,Vy,Ax,Ay]'
14 p_est = zeros(6, 6);
    end
    x_prd = A * x_est; % Predicted state and covariance
18 p_prd = A * p_est * A' + Q;
20 S = H * p_prd' * H' + R; % Estimation
    B = H * p_prd';
22 klm_gain = (S \ B)';
24 % Estimated state and covariance
    x_est = x_prd + klm_gain * (z - H * x_prd);
26 p_est = p_prd - klm_gain * H * p_prd;
    y = H * x_est; % Compute the estimated measurements
28 end % of the function
```

Listing 2: Basic Kalman Filter in Matlab.

expressions. An example for Matlab is provided on the Mathworks website<sup>1</sup> and shown in Listing 2.

A straightforward R implementation can be written as a close transcription of the Matlab version; we refer to this version as FirstKalmanR. It is shown in Listing 3. A slightly improved version (where several invariant statements are moved out of the repeatedly-called function) is provided in Listing 4 on page 9 showing the function KalmanR. The estimates of the state vector and its covariance matrix are updated iteratively. The Matlab implementation uses two variables declared 'persistent' for this. In R, which does not have such an attribute for variables, we store them in the enclosing environment of the outer function KalmanR, which contains an inner function kalmanfilter that is called for each observation.

Armadillo provides efficient vector and matrix classes to implement the Kalman

<sup>1</sup>See http://www.mathworks.com/products/matlab-coder/demos.html?file=/products/ demos/shipping/coder/coderdemo\_kalman\_filter.html.

filter. In Listing 5 on page 10, we show a simple C++ class containing a basic constructor as well as one additional member function. The constructor can be used to initialise all variables as we are guaranteed that the code in the class constructor will be executed exactly once when this class is instantiated. A class also makes it easy to add 'persistent' local variables, which is a feature we need here. Given such a class, the estimation can be accessed from R via a short and simple routine such as the one shown in Listing 6.

```
FirstKalmanR <- function(pos) {
      kalmanfilter <- function(z) {
4 dt <- 1
          A <- matrix(c( 1, 0, dt, 0, 0, 0, 0, 1, 0, dt, 0, 0, # x, y
6 0, 0, 1, 0, dt, 0, 0, 0, 0, 1, 0, dt, # Vx, Vy
                       0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1), # Ax, Ay
8 6, 6, byrow=TRUE)
          H <- matrix( c(1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0),
10 2, 6, byrow=TRUE)
          Q <- diag(6)
12 R <- 1000 * diag(2)
14 xprd <- A %*% xest # predicted state and covriance
          pprd <- A %*% pest %*% t(A) + Q
          S <- H %*% t(pprd) %*% t(H) + R # estimation
18 B <- H %*% t(pprd)
          kalmangain <- t(solve(S, B))
          ## estimated state and covariance, assign to vars in parent env
22 xest <<- xprd + kalmangain %*% (z - H %*% xprd)
          pest <<- pprd - kalmangain %*% H %*% pprd
          ## compute the estimated measurements
26 y <- H %*% xest
      }
28 xest <- matrix(0, 6, 1)
      pest <- matrix(0, 6, 6)
      N <- nrow(pos)
32 y <- matrix(NA, N, 2)
      for (i in 1:N) {
34 y[i,] <- kalmanfilter(t(pos[i,,drop=FALSE]))
      }
36 invisible(y)
   }
```

Listing 3: Basic Kalman filter in R (referred to as *FirstKalmanR*).

```
KalmanR <- function(pos) {
      kalmanfilter <- function(z) {
4 ## predicted state and covariance
          xprd <- A %*% xest
6 pprd <- A %*% pest %*% t(A) + Q
8 ## estimation
          S <- H %*% t(pprd) %*% t(H) + R
10 B <- H %*% t(pprd)
12 kalmangain <- t(solve(S, B))
14 ## estimated state and covariance
          ## assigned to vars in parent env
16 xest <<- xprd + kalmangain %*% (z - H %*% xprd)
          pest <<- pprd - kalmangain %*% H %*% pprd
          ## compute the estimated measurements
20 y <- H %*% xest
      }
      dt <- 1
24 A <- matrix( c( 1, 0, dt, 0, 0, 0, # x
                   0, 1, 0, dt, 0, 0, # y
26 0, 0, 1, 0, dt, 0, # Vx
                   0, 0, 0, 1, 0, dt, # Vy
28 0, 0, 0, 0, 1, 0, # Ax
                   0, 0, 0, 0, 0, 1), # Ay
30 6, 6, byrow=TRUE)
      H <- matrix( c(1, 0, 0, 0, 0, 0,
32 0, 1, 0, 0, 0, 0),
                2, 6, byrow=TRUE)
34 Q <- diag(6)
      R <- 1000 * diag(2)
36 N <- nrow(pos)
      Y <- matrix(NA, N, 2)
      xest <- matrix(0, 6, 1)
40 pest <- matrix(0, 6, 6)
42 for (i in 1:N) {
          Y[i,] <- kalmanfilter(t(pos[i,,drop=FALSE]))
44 }
      invisible(Y)
46 }
```

Listing 4: An improved Kalman filter implemented in R (referred to as *KalmanR*).

```
using namespace arma;
   class Kalman {
4 private:
      mat A, H, Q, R, xest, pest;
6 double dt;
8 public:
      // constructor, sets up data structures
10 Kalman() : dt(1.0) {
          A.eye(6,6);
12 A(0,2) = A(1,3) = A(2,4) = A(3,5) = dt;
          H.zeros(2,6);
14 H(0,0) = H(1,1) = 1.0;
          Q.eye(6,6);
16 R = 1000 * eye(2,2);
          xest.zeros(6,1);
18 pest.zeros(6,6);
      }
      // sole member function: estimate model
22 mat estimate(const mat & Z) {
         unsigned int n = Z.n_rows, k = Z.n_cols;
24 mat Y = zeros(n, k);
         mat xprd, pprd, S, B, kalmangain;
26 colvec z, y;
28 for (unsigned int i = 0; i<n; i++) {
            z = Z.row(i).t();
30 // predicted state and covariance
            xprd = A * xest;
32 pprd = A * pest * A.t() + Q;
            // estimation
34 S = H * pprd.t() * H.t() + R;
            B = H * pprd.t();
36 kalmangain = (solve(S, B)).t();
            // estimated state and covariance
38 xest = xprd + kalmangain * (z - H * xprd);
            pest = pprd - kalmangain * H * pprd;
40 // compute the estimated measurements
            y = H * xest;
42 Y.row(i) = y.t();
         }
44 return Y;
      }
46 };
```

Listing 5: A Kalman filter class in C++, using Armadillo classes.

```
R> kalmanSrc <- '
2 + mat Z = as<mat>(ZS); // passed from R
  + Kalman K;
4 + mat Y = K.estimate(Z);
  + return wrap(Y);'
6 R> KalmanCpp <- cxxfunction(signature(ZS="numeric" ),
  + body=kalmanSrc, include=kalmanClass,
8 + plugin="RcppArmadillo" )
```

Listing 6: A Kalman filter function implemented in a mixture of R and C++ code, using the RcppArmadillo package to embed Armadillo based C++ code (using the *Kalman* class from Listing 5) within R code. The resulting program is referred to as *KalmanCpp*.

The content of Listing 5 is assigned to a variable kalmanClass which (on line seven) is passed to the include= argument. This provides the required class declaration and definition. The four lines of code in lines two to five, assigned to kalmanSrc, provide the function body required by cxxfunction(). From both these elements and the function signature argument, cxxfunction() creates a very simple yet efficient C++ implementation of the Kalman filter which we can access from R. Given a vector of observations *Z*, it estimates a vector of position estimates *Y*.

![](_page_10_Figure_3.jpeg)

Figure 1: An example of object trajectory and the corresponding Kalman filter estimate.

```
R> require(rbenchmark)
2 R> require(compiler)
  R>
4 R> FirstKalmanRC <- cmpfun(FirstKalmanR)
  R> KalmanRC <- cmpfun(KalmanR)
6 R>
  R> ## Read data, ensure identical results
8 R> pos <- as.matrix(read.table("pos.txt" , header=FALSE,
  + col.names=c("x" ,"y" )))
10 R> stopifnot(all.equal(KalmanR(pos), KalmanRC(pos)),
  + all.equal(KalmanR(pos), KalmanCpp(pos)),
12 + all.equal(FirstKalmanR(pos), FirstKalmanRC(pos)),
  + all.equal(KalmanR(pos), FirstKalmanR(pos)))
14 R>
  R> res <- benchmark(KalmanR(pos), KalmanRC(pos),
16 + FirstKalmanR(pos), FirstKalmanRC(pos),
  + KalmanCpp(pos),
18 + columns = c("test" , "replications" ,
  + "elapsed" , "relative" ),
20 + order="relative" ,
  + replications=100)
```

Listing 7: R code for timing comparison of Kalman filter implementations.

This is illustrated in Figure 1 which displays the original object trajectory (using light-coloured square symbols) as well as the position estimates provided by the Kalman filter (using dark-coloured circles). This uses the same dataset provided by the Mathworks for their example; the data is believed to be simulated.

We note that this example is meant to be illustrative and does not attempt to provide a reference implementation of a Kalman filter. R contains several packages providing various implementations, as discussed in the survey provided by Tusell (2011).

#### 5. Empirical Speed Comparison

Listing 7 contains the code for creating a simple benchmarking exercise. It compares several functions for implementing the Kalman filter, all executed within the R environment. Specifically, we examine the initial R version FirstKalmanR shown in Listing 3, a refactored version KalmanR shown in Listing 4, an improved version<sup>2</sup> due to an anonymous referee (not shown, but included in the package), as well as byte-compiled versions (designated with a trailing 'C') created by using the byte-code compiler introduced with R version 2.13.0 (Tierney, 2012). Finally, the C++ version shown in Listings 5 and 6 is used. Also shown are the R statements for creating the byte-compiled variants via calls to cmpfun(). This is followed by a test to ensure that

<sup>2</sup>The improved version replaces explicit transpose and multiplication with the crossprod function.

| Implementation | Time in seconds | Relative to best solution |
|----------------|-----------------|---------------------------|
| KalmanCpp      | 0.73            | 1.0                       |
| KalmanRimpC    | 21.10           | 29.0                      |
| KalmanRimp     | 22.01           | 30.2                      |
| KalmanRC       | 28.64           | 39.3                      |
| KalmanR        | 31.30           | 43.0                      |
| FirstKalmanRC  | 49.40           | 67.9                      |
| FirstKalmanR   | 64.00           | 88.0                      |

Table 2: Performance comparison of various implementations of a Kalman filter. KalmanCpp is the RcppArmadillo based implementation in C++ shown in Listings 5 and 6. KalmanRimp is an improved version supplied by an anonymous referee which uses the crossprod function instead of explicit transpose and multiplication. KalmanR is the R implementation shown in Listing 4; FirstKalmanR is a direct translation of the original Matlab implementation shown in Listing 3. In all cases, the trailing 'C' denotes a byte-compiled variant of the corresponding R code. Timings are averaged over 500 replications. The comparison was made using R version 2.15.2, Rcpp version 0.10.2 and RcppArmadillo version 0.3.6.1 on Ubuntu 12.10 running in 64-bit mode on a 2.67 GHz Intel i7 processor.

all variants provide the same results. Next, the actual benchmark is executed before the result is displayed.

The results are shown in Table 2. Optimising and improving the R code has merits: we notice a steady improvement from the slowest R version to the fastest R version. Byte-compiling R code provides an additional performance gain which is more pronounced for the slower variant than the fastest implementation in R. However, the KalmanCpp function created using RcppArmadillo clearly outperforms all other variants, underscoring the principal point of this paper.

These results are consistent with the empirical observations made by Morandat et al. (2012), who also discuss several reasons for the slow speed of R compared to the C language, a close relative of C++.

# 6. Conclusion

This paper introduced the RcppArmadillo package for use within the R statistical environment. By using the Rcpp interface package, RcppArmadillo brings the speed of C++ along with the highly expressive Armadillo linear algebra library to the R language. A small example implementing a Kalman filter illustrated two key aspects. First, orders of magnitude of performance gains can be obtained by deploying C++ code along with R. Second, the ease of use and readability of the corresponding C++ code is similar to the R code from which it was derived.

This combination makes RcppArmadillo a compelling tool in the arsenal of applied researchers deploying computational methods in statistical computing and data analysis. As of early-2013, about 30 R packages on CRAN deploy RcppArmadillo<sup>3</sup> , showing both the usefulness of Armadillo and its acceptance by the R community.

<sup>3</sup>See http://cran.r-project.org/package=RcppArmadillo for more details.

#### Acknowledgements

NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy, as well as the Australian Research Council through the ICT Centre of Excellence program.

Adam M. Johansen provided helpful comments on an earlier draft. Comments by two anonymous referees and one editor further improved the paper and are gratefully acknowledged.

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