News

Entertainment

Science & Technology

Life

Culture & Art

Hobbies

News

Entertainment

Science & Technology

Culture & Art

Hobbies

Fortran is a programming language that has been around since the 1950s. It is one of the oldest programming languages still in use today and is the primary language used in scientific computing and engineering applications. The language has seen a resurgence in popularity in recent years due to its ability to integrate with high-level languages such as Python and C++. This makes it an ideal choice for data-intensive projects and applications. Here you will find news, articles and videos related to the language.

Python is widely considered the gold standard language for Data Science, and the entire range of packages, literature, and resources related to Data Science is always available in Python. This is not…

Abstract / TLDR R-universe now provides MacOS arm64 binaries for all R packages. This means that MacOS users on Apple Silicon hardware (aka M1/M2/M3) can install the very latest builds of any R package without the need for any compilation: install.packages('arrow', repos = c('https://apache.r-universe.dev', 'https://cloud.r-project.org')) R-universe uses cross-compiling for arm64 binaries, though this should not make much of a difference for package authors and R users. Packages with C/C++/Fortran/Rust code are all supported. Why cross compiling Because GitHub Actions currently does not offer arm64 runners for OSS projects, the arm64 binaries are cross-compiled on the MacOS intel runners. The cross build environment is set up to mimic a native arm64 machine, such that most R packages do not need any modification to work. We found only a small number of packages with a buggy configure script that may need a fix to allow cross-compilation. The r-universe workflow only builds arm64 binaries when needed, i.e. for packages that include compiled code (C/C++/Fortran/Rust). Packages that do not include any compiled code are portable by design, so for these packages the binaries built for MacOS on intel are served both in the x86_64 and arm64 cranlike repositories, without the need for an additional cross compile step. Just like CRAN, the r-universe package homepage shows a link to the r-4.3-x86_64 and r-4.3-arm64 binaries. Packages without compiled code have a r-4.3-any binary which is used for either architecture. To have a look at the build logs, click on the little apple icon next to these links. Alternatively, you can use the /buildlog shortlink, for example https://apache.r-universe.dev/arrow/buildlog will take you to the latest arrow builds. On this page you can find the arm64 build log specifically by expanding the r-release-macos job and then under the section “Cross Compile for MacOS ARM64”. If this section is disabled, it means it was skipped because this package does not have compiled code, and does not need cross compilation. Some technical details For those interested how the cross compilation is set up, here are the main ingredients: The standard MacOS Xcode toolchains are used to cross compile C/C++ code by passing the -arch arm64 flag to clang and clang++. The universal gfortran 12.2 version from CRAN (thanks to Simon Urbanek) is used to cross compile fortran code, also by passing gfortran -arch arm64. The same collection of system libs used by CRAN is preinstalled in the build environment. R packages with a configure script are built with --configure-args="--build=x86_64-apple-darwin20 --host=aarch64-apple-darwin20". These flags are needed by autoconf scripts, but other packages can use them as well. The r-universe macos-cross workflow overrides some more files and variables to target arm64. We put some shell shims on the PATH to help packages that shell out to uname or arch to determine the architecture. A clever cargo shim is used to override the default cargo build target to aarch64-apple-darwin and copy outputs to the expected directory after the build. Packages are built with strict linking (without the -undefined dynamic_lookup flag). With this setup, almost any R package can be built in the cross environment exactly the same way they do on normal arm64 hardware. But if your package does not work and you need some help fixing it, please feel free to open an issue. Universal binaries Finally, some R packages download precompiled binaries for libraries too big or complicated to build on the fly. It is safest to distribute such libraries in the form of universal binaries which contain both the x86_64 and arm64 libs. This way your download script does not need to make any guesses about the target architecture it is building for: the same libs can be linked on either target. Creating a universal binary can be done for both static and dynamic libraries and is really easy. If you have a x86_64 and arm64 version of libfoo.a you can glue them together with lipo: lipo -create x86_64/libfoo.a arm64/libfoo.a -output universal/libfoo.a And this new libfoo.a can be used when building either for intel or arm. Again if you need any help with this feel free to reach out.

The foundational importance of arrays in computer science cannot be overstated. Arrays and lists are the bedrock of data structures, often the first concepts introduced to budding programmers. Since their inception back to Fortran in 1957 and continuing to hold prominence in contemporary languages like Python, arrays maintain a consistent and universal presence across the programming landscape. Their enduring popularity can be attributed to their simplicity and versatility, providing a straightforward means of organizing data in multidimensional grids. While dense arrays excel in performance, they do not encapsulate the entirety of real-world data scenarios. The challenge lies in structured data,

The Mistral AI Team has announced the release of its groundbreaking code generation model, Codestral-22B. This contributes toward a new direction and benchmark of AI for software development. Codestral empowers developers by enhancing their coding capabilities and streamlining the development process. Codestral is an open-weight generative AI model explicitly crafted for code generation tasks. It supports over 80 programming languages, including popular ones like Python, Java, C, C++, JavaScript, and Bash, as well as more specialized languages like Swift and Fortran. This extensive language base ensures that Codestral can be an invaluable tool across diverse coding environments and projects. The

Introduction When fitting a nonlinear regression model in R with nls(), the first step is to select an appropriate regression model to fit the observed data, the second step is to find reasonable starting values for the model parameters in order to initialize the nonlinear least-squares (NLS) algorithm. In some cases, choosing starting values is straightforward, for instance when there is some physical interpretation to the model or the least squares objective function is highly regular and easy to optimize. In other cases, this can be more challenging, especially when the least squares objective consists of large plateaus with local minima located in small narrow canyons (see e.g. Transtrum, Machta, and Sethna (2011)), or when the parameters live on very different scales. To illustrate, consider the following Hobbs’ weed infestation example dataset from (Nash 1979) and (Nash 2014): ## Hobbs' weed infestation data hobbs_weed 34094.85 (3.11e+00): par = (100 10 1) #> 11186.49 (3.86e+00): par = (78.18911 3.290385 0.4759647) #> 5112.607 (4.69e+01): par = (80.9356 3.413601 0.1187296) #> 5032.864 (3.53e+01): par = (114.9669 5.478666 0.1037978) #> 4937.315 (2.78e+01): par = (184.1488 9.704171 0.09525315) #> 4798.508 (2.41e+01): par = (346.9847 19.72996 0.09116909) #> 4624.151 (2.16e+01): par = (1325.61 80.41635 0.08932604) #> 3859.425 (1.91e+01): par = (23585.2 1465.277 0.09589796) #> 1605.353 (1.32e+01): par = (7709100 479913.4 0.1272658) #> Error in nls(formula = y ~ b1/(1 + b2 * exp(-b3 * x)), data = hobbs_weed, : singular gradient This is in fact an example where the Gauss-Newton direction initially points to the boundary of the parameter space (see also Figure 7 in Transtrum, Machta, and Sethna (2011)). Instead, using a damped least squares algorithm (Levenberg-Marquardt) that combines both the Gauss-Newton and gradient descent directions, we expect the parameters to evaporate less quickly: library(gslnls) ## initial attempt gsl_nls gsl_nls( fn = y ~ b1 / (1 + b2 * exp(-b3 * x)), data = hobbs_weed, start = c(b1 = 100, b2 = 10, b3 = 1), algorithm = "lm" ) #> Nonlinear regression model #> model: y ~ b1/(1 + b2 * exp(-b3 * x)) #> data: hobbs_weed #> b1 b2 b3 #> 196.1863 49.0916 0.3136 #> residual sum-of-squares: 2.587 #> #> Algorithm: multifit/levenberg-marquardt, (scaling: more, solver: qr) #> #> Number of iterations to convergence: 15 #> Achieved convergence tolerance: 1.554e-14 In an attempt to reduce the dependence of the NLS algorithm on a single set of (poorly selected) starting values, this post demonstrates a new multistart procedure available1 in R-package gslnls, which can be useful when we only have limited knowledge regarding the expected parameter values or when we wish to automate nonlinear model fits across multiple different datasets. A common approach in R to avoid the need for user-supplied NLS starting values is to make use of so-called selfStart models (SSasymp(), SSfpl(), SSmicmen(), etc.) that include an initialization function to return reasonable starting values for the nonlinear model given the available data. The initialization function typically considers a simpler approximate or linearized version of the nonlinear model for which parameters can be estimated without starting values, using e.g. lm(). This is the model fitting approach that is utilized by drc and nlraa among other packages. If we intend to fit a nonlinear model for which a selfStart implementation is already available (or straightforward to implement), this is definitely the recommended approach, since the starting values obtained from a selfStart model are usually well-informed and most NLS routines will not have trouble obtaining the correct parameter estimates from these starting values. If no selfStart model is available, another approach is to repeatedly call nls() using different starting values drawn as random or fixed points from a pre-defined grid. This is implemented by nls2::nls2() using one of the methods "brute-force", "grid-search", or "random-search". The new multistart procedure in gsl_nls() tries to improve on naive random or grid-based multistart optimization, which can be a very time consuming process, especially if the number of parameters is large (curse of dimensionality) or the scale of the parameter ranges to evaluate is quite broad. As in any multistart global optimization procedure, the ideal approach would be to start a local NLS optimizer within each basin of attraction2 exactly once to reach all existing local minima with minimal computational effort. In practice, we try to avoid running too many local optimizers in the same basin of attraction (resulting in the same local minimum) by exploring the parameter space for promising starting values that might converge to an unseen (local) optimum. The multistart algorithm implemented in gsl_nls() is a modified version of the algorithm in (Hickernell and Yuan 1997) that works both with or without a pre-defined grid of starting ranges. Before describing the details of the multistart procedure, below are several examples illustrating its usage with gsl_nls(). NLS examples Hobbs’ weed infestation example Revisiting the Hobbs’ weed infestation example above, we fit the nonlinear model with gsl_nls() using a (fixed) set of starting ranges for the parameters instead of individual starting values: ## multistart attempt gsl_nls gsl_nls( fn = y ~ b1 / (1 + b2 * exp(-b3 * x)), data = hobbs_weed, start = list(b1 = c(0, 1000), b2 = c(0, 1000), b3 = c(0, 10)) ) #> Nonlinear regression model #> model: y ~ b1/(1 + b2 * exp(-b3 * x)) #> data: hobbs_weed #> b1 b2 b3 #> 196.1863 49.0916 0.3136 #> residual sum-of-squares: 2.587 #> #> Algorithm: multifit/levenberg-marquardt, (scaling: more, solver: qr) #> #> Number of iterations to convergence: 3 #> Achieved convergence tolerance: 2.842e-14 Alternatively, we can leave one or more starting ranges undefined, in which case they are updated dynamically during the multistart optimization: ## multistart attempt 2 gsl_nls gsl_nls( fn = y ~ b1 / (1 + b2 * exp(-b3 * x)), data = hobbs_weed, start = list(b1 = NA, b2 = NA, b3 = NA) ) #> Nonlinear regression model #> model: y ~ b1/(1 + b2 * exp(-b3 * x)) #> data: hobbs_weed #> b1 b2 b3 #> 196.1863 49.0916 0.3136 #> residual sum-of-squares: 2.587 #> #> Algorithm: multifit/levenberg-marquardt, (scaling: more, solver: qr) #> #> Number of iterations to convergence: 4 #> Achieved convergence tolerance: 5.329e-15 NIST StRD Gauss1 example Second, we consider the Gauss1 regression test problem as listed in the NIST StRD Nonlinear Regression archive. The observed data takes the shape of a camel’s back consisting of two Gaussians on a decaying exponential baseline subject to additive Gaussian noise. The data, nonlinear model and target parameter values are included in the gslnls package and available through the function nls_test_problem(): gauss1 'data.frame': 250 obs. of 2 variables: #> $ y: num 97.6 97.8 96.6 92.6 91.2 ... #> $ x: num 1 2 3 4 5 6 7 8 9 10 ... ## model + target parameters gauss1[c("fn", "target")] #> $fn #> y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)^2/b5^2) + b6 * exp(-(x - #> b7)^2/b8^2) #> #> #> $target #> b1 b2 b3 b4 b5 b6 #> 98.77821087 0.01049728 100.48990633 67.48111128 23.12977336 71.99450300 #> b7 b8 #> 178.99805021 18.38938902 Due to the large number of parameters in the model y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)**2 / b5**2) + b6 * exp(-(x - b7)**2 / b8**2), finding starting values from which nls() is able to correctly fit the model is a tedious task. Below is an initial attempt at solving the NLS problem with the NL2SOL algorithm (algorithm = "port") and all parameters bounded from below by zero: ## initial attempt nls nls( formula = gauss1$fn, data = gauss1$data, start = c(b1 = 100, b2 = 0, b3 = 100, b4 = 75, b5 = 25, b6 = 75, b7 = 125, b8 = 25), lower = 0, algorithm = "port" ) #> Error in nls(formula = gauss1$fn, data = gauss1$data, start = c(b1 = 100, : Convergence failure: singular convergence (7) As seen from this example, most parameter starting values –except for b7– are not that far from their target values. However, nls() still fails to converge due to a single poorly selected starting value for the parameter b7. Using the multistart procedure in gsl_nls(), we can combine fixed starting values or ranges when good initial guesses for the parameters are available together with missing starting values for the parameters that are lacking such information. This avoids the need to select poor starting values for certain parameters, which may cause the NLS optimization to fail as in our previous attempt to fit the model. ## multistart attempt gsl_nls gsl_nls( fn = gauss1$fn, data = gauss1$data, start = list(b1 = 100, b2 = c(0, 1), b3 = NA, b4 = NA, b5 = NA, b6 = NA, b7 = NA, b8 = NA), lower = 0 ) #> Nonlinear regression model #> model: y ~ b1 * exp(-b2 * x) + b3 * exp(-(x - b4)^2/b5^2) + b6 * exp(-(x - b7)^2/b8^2) #> data: gauss1$data #> b1 b2 b3 b4 b5 b6 b7 b8 #> 98.7782 0.0105 100.4899 67.4811 23.1298 71.9945 178.9981 18.3894 #> residual sum-of-squares: 1316 #> #> Algorithm: multifit/levenberg-marquardt, (scaling: more, solver: qr) #> #> Number of iterations to convergence: 4 #> Achieved convergence tolerance: 2.274e-13 Lubricant dataset (Bates & Watts) As a final example, consider the Lubricant dataset from (Bates and Watts 1988, Appendix 1, A1.8), which measures the kinematic viscosity of a lubricant as a function of temperature (°C) and pressure (atm). The Lubricant data, model and target parameter are included as an NLS test problem in gslnls and can be retrieved with nls_test_problem() similar to the previous example. Here, x1 encodes the temperature predictor (in °C) and x2 is the pressure (atm) predictor. lubricant 'data.frame': 53 obs. of 3 variables: #> $ y : num 5.11 6.39 7.39 5.79 5.11 ... #> $ x1: num 0 0 0 0 0 0 0 0 0 0 ... #> $ x2: num 0.001 0.741 1.407 0.363 0.001 ... ## model + target parameters lubricant[c("fn", "target")] #> $fn #> y ~ b1/(b2 + x1) + b3 * x2 + b4 * x2^2 + b5 * x2^3 + (b6 * x2 + #> b7 * x2^3) * exp(-x1/(b8 + b9 * x2^2)) #> #> #> $target #> b1 b2 b3 b4 b5 #> 1054.54053186 206.54577890 1.46031479 -0.25965483 0.02257371 #> b6 b7 b8 b9 #> 0.40138428 0.03528405 57.40463143 -0.47672110 The nonlinear model contains a relatively large number of parameters, similar to the Gauss1 example, and the model fitting is further complicated by the large differences in magnitude of the target parameters. Without a more systematic approach, e.g. by linearizing parts of the model to initialize parameters as in (Bates and Watts 1988, Ch 3.6), it is difficult to obtain a good model fit with nls() by naively trying different sets of starting values. Here, we again use the NL2SOL algorithm (algorithm = "port"), selecting starting values that are all close to the target parameters, but which still cause nls() to fail. ## initial attempt nls nls( formula = lubricant$fn, data = lubricant$data, start = c(b1 = 1000, b2 = 200, b3 = 1, b4 = -0.01, b5 = 0.01, b6 = 0.01, b7 = 0.01, b8 = 50, b9 = -0.01), algorithm = "port" ) #> Error in nls(formula = lubricant$fn, data = lubricant$data, start = c(b1 = 1000, : Convergence failure: false convergence (8) Attempting to solve the NLS problem with the multistart procedure in gsl_nls(), we are able to obtain correct NLS convergence without any knowledge of the target parameters by leaving all starting values unspecified: ## multistart attempt gsl_nls gsl_nls( fn = lubricant$fn, data = lubricant$data, start = c(b1 = NA, b2 = NA, b3 = NA, b4 = NA, b5 = NA, b6 = NA, b7 = NA, b8 = NA, b9 = NA) ) #> Nonlinear regression model #> model: y ~ b1/(b2 + x1) + b3 * x2 + b4 * x2^2 + b5 * x2^3 + (b6 * x2 + b7 * x2^3) * exp(-x1/(b8 + b9 * x2^2)) #> data: lubricant$data #> b1 b2 b3 b4 b5 b6 b7 #> 1054.54080 206.54584 1.46031 -0.25965 0.02257 0.40138 0.03528 #> b8 b9 #> 57.40467 -0.47672 #> residual sum-of-squares: 0.08702 #> #> Algorithm: multifit/levenberg-marquardt, (scaling: more, solver: qr) #> #> Number of iterations to convergence: 6 #> Achieved convergence tolerance: 5.69e-16 Multistart algorithm details Before evaluating more NLS test problems, this section provides a more comprehensive overview of the implemented multistart algorithm. The implementation is primarily based on the global optimization algorithm in (Hickernell and Yuan 1997), but slightly modified to the context of trust-region nonlinear least squares. The multistart algorithm consists of multiple major iterations. At the start of major iteration \(M\), a set of pseudo-random starting points is sampled inside the grid of starting ranges. Starting points with an almost singular (approximate) Hessian matrix are discarded immediately, as these are unlikely to converge to a local optimum. For the remaining points, a few iterations of the NLS optimizer are applied in order to distinguish promising from unpromising starting points. At each major iteration, only the \(q\) most promising starting points are kept, and if a starting point is not discarded from this set for at least \(s\) major iterations, a full NLS optimization routine is executed using this starting point. If a new optimal solution is found, the number of optimal stationary points NOSP is incremented and the number of found worse stationary points NWSP is reset to zero. If the obtained solution has already been observed before, or if it converges to a local minimum that is worse than the current optimal solution, the number of worse stationary points NSWP is incremented. If the number of found worse stationary points (NWSP) becomes much larger than the number of optimal starting points (NOSP), then it is likely that we have exhausted the search space and are unable to further improve the current optimal solution. The following pseudo-code provides a high-level description of the multistart procedure. For simplicity, we first consider the scenario in which all \(p\) parameter starting ranges \([l_1, u_1] \times \ldots \times [l_{p}, u_{p}]\) are pre-defined and fixed. INITIALIZE Choose parameters \(N \geq q \geq 1\), \(L \geq \ell \geq 1\), \(s \geq 1\), \(r > 1\), maxstart \(\geq 1\), minsp \(\geq 1\). Set \(M = 0\), NOSP = NWSP = 0, and initialize a scaling vector \(\boldsymbol{D} = (0.75, \dots, 0.75) \in [0, 1]^p\). SAMPLE Sample \(N\) initial \(p\)-dimensional points \(\boldsymbol{\theta}_1,\ldots, \boldsymbol{\theta}_N\) inside the grid of starting ranges \([l_1, u_1] \times \ldots \times [l_{p}, u_{p}]\) from a pseudo-random Sobol sequence. For \(i = 1,\ldots,N\) and \(k = 1,\ldots,p\), rescale the sampled points inside the grid of starting ranges favoring points near zero according to3: \(\theta^*_{i,k} = \frac{\text{sgn}(\theta_{i,k})}{D_k} \cdot ((|\theta_{i,k}| - l_k^+ - u_k^- + 1)^{D_k} - 1) + l_k^+ - u_k^-\). REDUCE 1 AND CONCENTRATE Discard all points for which \(\det\left(J(\boldsymbol{\theta}^*_i)^TJ(\boldsymbol{\theta}^*_i)\right) < \epsilon\), with \(\epsilon > 0\) small and \(J(\boldsymbol{\theta}^*_i)\) the Jacobian evaluated at \(\boldsymbol{\theta}^*_i\). For each remaining point apply (a small number) \(\ell\) iterations of the NLS optimizer, obtaining concentrated points \(\boldsymbol{\theta}_1^{**},\ldots,\boldsymbol{\theta}_{N_1}^{**}\) REDUCE 2 Identify the first \(q\) order statistics of \(F(\boldsymbol{\theta}_1^{**}),\ldots,F(\boldsymbol{\theta}_{N_1}^{**})\), with \(F(\boldsymbol{\theta}_i^{**})\) the NLS objective evaluated at \(\boldsymbol{\theta}_i^{**}\). Denote \(I_q\) for the set of indices (of size \(q\)). Update counter \(S(i) = S(i) + 1\) if \(i \in I_q\), otherwise set \(S(i) = 0\). OPTIMIZE For each \(i\) with \(S(i) \geq s\), if \(F(\boldsymbol{\theta}_i^{**}) < (1 + \delta) \cdot F(\boldsymbol{\theta}^{opt})\), apply \(L\) iterations of the NLS optimizer, obtaining \(\boldsymbol{\theta}_i^{***}\), and set \(S(i) = 0\). If \(F(\boldsymbol{\theta}_i^{***}) < F(\boldsymbol{\theta}^{opt})\), set \(\boldsymbol{\theta}^{opt} = \boldsymbol{\theta}_i^{***}\), and update the scaling vector \(\boldsymbol{D}\) by \(D_k = \left(\frac{\min_\kappa \Delta_\kappa}{\Delta_k}\right)^\alpha\), with exponent \(\alpha \in [0, 1]\) and \(\boldsymbol{\Delta} = \text{diag}(J(\boldsymbol{\theta}^{***}_i)^TJ(\boldsymbol{\theta}^{***}_i))\), i.e. the diagonal damping matrix as used by Marquardt’s scaling method. Update the number of found optimal stationary points NOSP = NOSP + 1, and reset the number of found worse stationary points NWSP = 0. If \(F(\boldsymbol{\theta}_i^{***}) \geq F(\boldsymbol{\theta}^{opt})\), set NWSP = NWSP + 1. REPEAT If NWSP \(\geq\) \(r \cdot\) NSP and NSP \(\geq\) minsp (minimum required number of stationary points) then stop with success. Otherwise, if \(M 0\). It is important to point out that there is no guarantee that the NLS objective \(F(\boldsymbol{\theta}^{opt})\) at the solution returned by the multistart procedure indeed evaluates to the global minimum inside the grid of starting ranges. This may for instance be due to the rescaling function in step 2., which is a type of inverse logistic function scaled to the starting range \([l_k, u_k]\). The scaling exponent \(D_k\) is calculated from the damping matrix in Marquardt’s scaling method, thereby rescaling each parameter differently based on an approximate measure of its order of magnitude. If the damping matrix that is used to calculate the \(D_k\)’s is highly –but incorrectly– confident about the order of magnitudes of certain parameters, we may not explore the parameter starting ranges \([l_k, u_k]\) sufficiently broadly in the subsequent major iterations. Increasing the number of sampled points \(N\) at the start of each major iteration may help to overcome such issues. If we suspect that the returned optimal solution \(\boldsymbol{\theta}^{opt}\) is only a local optimizing solution, we can always force the multistart procedure to continue searching until a better optimum is found by increasing minsp (minimum required number of stationary points). Missing starting ranges If no fixed starting values or ranges are defined for certain parameters, as demonstrated in the above examples, then the missing ranges \([l_k, u_k]\) are initialized to the unit interval and dynamically increased or decreased in each major iteration of the multistart algorithm. The decision to increase or decrease the limits of a parameter’s starting range is driven by the minimum and maximum parameter values obtained from the \(q\) best-performing concentrated points (step 3. and 4.) with indices included in \(I_q\). These typically provide a rough indication of the order of magnitude of the parameter range in which to search for the optimal solution. If the dynamic parameter ranges fail to grow sufficiently large to include the global optimizing solution, it may help to increase the values of \(N\), \(r\), maxstart or minsp to avoid early termination of the algorithm at the cost of increased computation effort. NLS test problems At the moment of writing this post, 59 NLS test problems are included in gslnls originating primarily from the NIST StRD Nonlinear Regression archive, (Bates and Watts 1988) and (Moré, Garbow, and Hillstrom 1981). This collection of test problems contains 33 regression problems, with nonlinear models defined as a formula and the number of parameters and observations fixed (p, n fixed). The other 26 problems are NLS optimization problems, ported from the Fortran library TEST_NLS. For these problems the nonlinear models are defined as a function and some of the models allow for the number of parameters and observations to be freely varied, only requiring that the number of parameters does not exceed the number of observations/residuals (p 7 40.02 332.8 #> 8 44.82 378.4 #> 9 50.76 434.8 #> 10 55.05 477.3 #> 11 61.01 536.8 #> 12 66.40 593.1 #> 13 75.47 689.1 #> 14 81.78 760.0 #> #> $fn #> y ~ b1 * (1 - exp(-b2 * x)) #> #> #> $start #> b1 b2 #> 5e+02 1e-04 #> #> $target #> b1 b2 #> 2.389421e+02 5.501564e-04 #> #> attr(,"class") #> [1] "nls_test_formula" ## nls model fit with(misra1a, gsl_nls( fn = fn, data = data, start = start ) ) #> Nonlinear regression model #> model: y ~ b1 * (1 - exp(-b2 * x)) #> data: data #> b1 b2 #> 2.389e+02 5.502e-04 #> residual sum-of-squares: 0.1246 #> #> Algorithm: multifit/levenberg-marquardt, (scaling: more, solver: qr) #> #> Number of iterations to convergence: 16 #> Achieved convergence tolerance: 6.745e-15 Example optimization problem (Rosenbrock) ## nls model fit with(nls_test_problem("Rosenbrock"), gsl_nls( fn = fn, y = y, start = start, jac = jac ) ) #> Nonlinear regression model #> model: y ~ fn(x) #> x1 x2 #> 1 1 #> residual sum-of-squares: 9.762e-19 #> #> Algorithm: multifit/levenberg-marquardt, (scaling: more, solver: qr) #> #> Number of iterations to convergence: 16 #> Achieved convergence tolerance: 7.696e-14 Benchmark NLS fits To conclude, we benchmark the performance of the multistart algorithm by computing NLS model fits for each of the 59 test problems using the multistart algorithm with no starting values provided, i.e. all starting values are set to NA. As trust region method we choose respectively: the default Levenberg-Marquardt algorithm (algorithm = "lm"); the double dogleg algorithm (algorithm = "ddogleg"); and the Levenberg-Marquardt algorithm with geodesic acceleration (algorithm = "lmaccel"). The maximum number of allowed iterations maxiter is set to \(10\ 000\), all other tuning parameters in the control argument are kept at their default values according to gsl_nls_control(). For comparison, we also compute single-start NLS model fits using the default Levenberg Marquardt algorithm (algorithm = "lm"), with as naive choice of starting values a vector of all ones \((1, \ldots, 1)\), similar to nls() when argument start is missing. The table below displays the NLS model fit results for each individual test problem using the following status colors: success ; the NLS routine converged successfully and the residual sum-of-squares (SSR) under the fitted parameters coincides (up to a small numeric tolerance) with the SSR under the target parameter values4. The total runtime is displayed in seconds, timed on a modern laptop computer (Intel i7-8550U CPU, 1.80GHz) using a single core. false convergence ; the NLS routine converged successfully, but the residual SSR under the fitted parameters is larger than the SSR under the target parameter values. max. iterations ; the NLS routine failed to converge within the maximum number of allowed iterations. failed/non-zero exit ; the NLS routine failed to converge and returns an error or an NLS object with a non-zero exit code. We observe that the naive single-start model fits manage to correctly fit about half of the test problems (27 out of 59), suggesting that these test problems are straightforward to optimize and do not require well-informed starting values. The multistart model fits using the double dogleg method improve upon the naive single-start model fits achieving correct convergence for 51 out of 59 test problems. The multistart Levenberg-Marquardt model fits correctly converge for a few more test problems (56 out of 59). Finally, the most robust results are obtained with the multistart model fits using the Levenberg-Marquardt algorithm with geodesic acceleration, which correctly fit all 59 test problems without initializing proper starting values or starting ranges! 🎉 lm/single-startlm/multi-startddogleg/multi-startlmaccel/multi-startMisra1a0.4s0.2s0.2sChwirut20.4s0.3s0.4sChwirut10.5s0.4s0.7sLanczos30.02s2s2s2sGauss14s2sGauss24s2sDanWood0.003s0.3s0.3s0.3sMisra1b0.4s0.6s0.6sKirby20.01s0.3s0.3s0.4sHahn12sNelson0.3s0.6s0.2sMGH170.9s0.3s1sLanczos10.02s3s2s1sLanczos20.02s2s2s2sGauss34s2sMisra1c0.8sMisra1d2s2s0.5sRoszman14s5sENSO6s5s5sMGH090.004s0.4s0.4s0.5sThurber1s1s2sBoxBOD0.3s0.2s0.4sRatkowsky20.1s0.1s0.3sMGH102s0.7s1sEckerle40.4s0.4s0.5sRatkowsky30.4s0.3s0.4sBennett51sIsomerization0.008s0.6s0.6s0.6sLubricant0.01s2s2s3sSulfisoxazole0.3s0.4s0.5sLeaves1s0.9s0.3sChloride0.3s0.4s0.6sTetracycline0.005s0.5s0.3s0.4sLinear, full rank0.002s0.2s0.2s0.2sLinear, rank 10.002s0.5s0.3s0.4sLinear, rank 1, zero columns and rows0.002s0.4s0.3s0.3sRosenbrock0.002s0.3s0.1s0.3sHelical valley0.002s0.5s0.2s0.3sPowell singular0.002s0.3s0.2s0.3sFreudenstein/Roth0.4s0.2s0.4sBard0.001s0.3s0.2s0.3sKowalik and Osborne0.002s0.3s0.2s0.2sMeyer1s0.5s1sWatson0.004s0.9s0.4s0.5sBox 3-dimensional0.002s0.1s0.2s0.4sJennrich and Sampson0.003s0.2s0.2s0.3sBrown and Dennis0.008s0.6s0.5s0.7sChebyquad0.009s1s0.5s1sBrown almost-linear0.001s0.7s0.6s0.9sOsborne 10.4s0.2s0.5sOsborne 21s1sHanson 10.2s0.2s0.4sHanson 20.2s0.2s0.2sMcKeown 10.002s0.2s0.2s0.3sMcKeown 20.003s0.3s0.2s0.4sMcKeown 30.004s0.4s0.3s0.4sDevilliers and Glasser 10.4s0.4s0.6sDevilliers and Glasser 20.6s0.7s1sMadsen example0.004s0.2s0.2s0.3s# Successful fits27/5956/5951/5959/59 References Bates, D. M., and D. G. Watts. 1988. “Nonlinear Regression Analysis and Its Applications.” Wiley. Hickernell, F. J., and Y. Yuan. 1997. “A Simple Multistart Algorithm for Global Optimization.” OR Transactions 1 (2): 1–11. Madsen, K. 1988. “A Combined Gauss-Newton and Quasi-Newton Method for Non-Linear Least Squares.” Institute for Numerical Analysis, DTU. McKeown, J. J. 1975. “Specialised Versus General-Purpose Algorithms for Minimising Functions That Are Sums of Squared Terms.” Mathematical Programming 9: 57–68. Moré, J. J., B. S. Garbow, and K. E. Hillstrom. 1981. “Testing Unconstrained Optimization Software.” ACM Transactions on Mathematical Software (TOMS) 7 (1): 17–41. Nash, J. C. 1979. Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation. Bristol, UK: Adam Hilger. ———. 2014. Nonlinear Parameter Optimization Using r Tools. UK: Wiley. NIST StRD. 1978. “NIST Statistical Reference Datasets (StRD) Archive.” Online source. https://www.itl.nist.gov/div898/strd/nls/nls_main.shtml. Salane, D. E. 1987. “A Continuation Approach for Solving Large-Residual Nonlinear Least Squares Problems.” SIAM Journal on Scientific and Statistical Computing 8 (4): 655–71. Transtrum, M. K., B. B. Machta, and J. P. Sethna. 2011. “Geometry of Nonlinear Least Squares with Applications to Sloppy Models and Optimization.” Physical Review E 83 (3). multistart optimization requires gslnls version >= 1.3.0.↩︎ a basin of attraction refers to the basin surrounding a local minimum of the objective function, such that starting from any point inside the basin the local optimizer converges to the same local minimum.↩︎ \(f^+ = f \vee 0\) and \(f^- =-f \vee 0\) denote the positive and negative part of \(f\).↩︎ The reason for using the residual sum-of-squares (SSR) to check for correct convergence of the NLS model fits is that several problems have multiple distinct parameter solutions that result in the same (optimal) SSR.↩︎