Linear Regression, GLMs and GAMs with R demonstrates how to use R to extend the basic assumptions and constraints of linear regression to specify, model, and interpret the results of generalized linear (GLMs) and generalized additive (GAMs) models. The course demonstrates the estimation of GLMs and GAMs by working through a series of practical. I have to get GAMS to find the maximum element of a set. This should result in some linear regression model, where the objective is not the least squares but the least maximum deviation.
Many data in the environmental sciences do not fit simple linear models and are best described by “wiggly models”, also known as Generalised Additive Models (GAMs).Let’s start with a famous by one Gavin Simpson, which amounts to:1. GAMs are just GLMs2. GAMs fit wiggly terms3. Use + s(x) not x in your syntax4. Use method = 'REML'5. Always look at gam.checkThis is basically all there is too it – an extension of with a smoothing function.
Of course, there may be many sophisticated things going on when you fit a model with smooth terms, but you only need to understand the rationale and some basic theory. There are also lots of what would be apparently magic things happening when we try to understand what is under the hood of say lmer or glmer, but we use them all the time without reservation! GAMs in a nutshellLet’s start with an equation for a Gaussian: y = beta0 + x1beta1 + varepsilon, quad varepsilon sim N(0, sigma^2) What changes in a GAM is the presence of a smoothing term: y = beta0 + f(x1) + varepsilon, quad varepsilon sim N(0, sigma^2) This simply means that the contribution to the linear predictor is now some function (f). This is not that dissimilar conceptually to using a quadratic ( (x1^2)) or cubic term ( (x1^3)) as your predictor.The function (f) can be something more funky or kinky – here, we’re going to focus on splines.
In the old days, it might have been something like piecewise linear functions.You can have combinations of linear and smooth terms in your model, for example y = beta0 + x1beta1 + f(x2) + varepsilon, quad varepsilon sim N(0, sigma^2) or we can fit generalised distributions and random effects, for example ln(y) = beta0 + f(x1) + varepsilon, quad varepsilon sim Poisson(lambda) ln(y) = beta0 + f(x1) + z1gamma + varepsilon, quad varepsilon sim Poisson(lambda), quad gamma sim N(0,Sigma). A simple exampleLets try a simple example. First, let’s create a data frame and fill it with some simulated data with an obvious non-linear trend and compare how well some models fit to that data.
IntroductionMathematical functions play an important role in the GAMS language, especially for nonlinear models. Like other programming languages, GAMS provides a number of built-in. GAMS is used in an extremely diverse set of application areas and this creates frequent requests for the addition of new and often sophisticated and specialized functions.
There is a trade-off between satisfying these requests and avoiding complexity not needed by most users. The GAMS Function Library Facility provides the means for managing this trade-off, since it allows users to import functions from an external library into a GAMS model. However, these external libraries can currently only provide functionality for the evaluation of functions (incl. Their first and second derivatives) in a point. Solvers that need to analyze the algebraic structure of the model instance are therefore not able to work with extrinsic functions. This includes the class of deterministic global solvers, see column 'Global' in, while, for example, stochastic global solvers can work with extrinsic functions.In this chapter we will demonstrate how to access functions from an extrinsic function library in a GAMS model and we will describe the extrinsic function libraries that are included in the GAMS distribution. In addition, we will provide some pointers for users who wish to build their own extrinsic function library.Using Function LibrariesFunction libraries are made available to a model with the following compiler directive: $FuncLibIn Here InternalLibName is a handle that will be used to refer to the library inside the model source code, ExternalLibName is the file name of the shared library that implements the extrinsic functions.
To access libraries included with the GAMS distribution, the name of the library may be used with no path prefix. GAMS will look for the library in the GAMS system directory. To access a library that does not reside in this standard place, the external name should include a relative or absolute path to the location of the library.
GAMS will then search for the specified library using the mechanisms specific to the host operating system. When processing the directive, GAMS will validate the library, make the included functions available for use and add a table of the included functions to the listing file. Note The function library facility gives users complete control over naming so that potential name conflicts between libraries can be avoided.Before the individual functions may be used, they have to be declared in the following way: Function /./;Here InternalFuncName is the name of the individual function that will be used in the GAMS code.
The user may choose this internal name freely and thus avoid potential naming conflicts. InternalLibName is the name of the function library as defined by the directive and Func.Name is the name of the individual function in the external function library. Once functions have been declared in this way they may be used like intrinsic functions.Consider the following simple example: $FuncLibIn myLib tricclibFunction myCos /myLib.Cosine/mySin /myLib.Sine/myPi /myLib.Pi/;Scalar d;d = myCos(myPi/3);display d;Note that in the first line the external tricclib is activated and the internal name myLib is specified for it. Then the functions are declared.
Observe that Cosine, Sine and Pi are functions in the trigonometric library. After the library has been loaded and the functions have been declared, the functions may be used as usual.
The trigonometric library is discussed in section below.Libraries that are included in the GAMS DistributionIn this section we will present the libraries that are included in the GAMS distribution: the, the and the. In addition, we will provide details on the. However, note that the LINDO Sampling Library is only available for LINDO license holders.In the tables that follow, the 'Endogenous Classification' (last column) specifies in which models the function may legally appear. In order of least to most restrictive, the choices are any, DNLP, NLP, none. See section for details on model types in GAMS. Note well that functions classified as any are only permitted with exogenous (constant) arguments.A word on the notation in the tables below: for function arguments, lower case indicates that an endogenous variable is allowed.
For details on endogeneous variables, see section. Upper case function arguments indicate that a constant is required. Arguments in square brackets may be omitted: the default values used in such cases are specified in the function description provided.The Fitpack Library is a Fortran-based library for one-dimensional and two-dimensional spline interpolations.
This library has been repackaged to work with the GAMS Function Library Facility. The model from the GAMS Test Library is an example of the use of the library FITPACK inside GAMS.Note that the supporting points to which the function will be fit need to be stored in a three-dimensional parameter, fitdata, in a fit.gdx. The first dimension is a function index, the second dimension is the index of the supporting point and the third dimension takes one of the following four values: 'w' (weight), 'x' (x-value), 'y' (y-value) or 'z' (z-value).The FITPACK library is made available with the following directive: $FuncLibIn fitfclibIt provides the following functions: FunctionDescriptionEnd. Classif.fitFunc(FUNIND,x,y)Evaluate splineDNLPfitParam(FUNIND,PARAM,VALUE)Read or set parametersnoneThe function FitParam may be used to change certain parameters that are used for the evaluation.
The following values are defined:. 1: Smoothing factor (S). 2: Degree of spline in direction x (Kx).
3: Degree of spline in direction y (Ky). 4: Lower bound of function in direction x (LOx).
5: Lower bound of function in direction y (LOy). 6: Upper bound of function in direction x (UPx). 7: Upper bound of function in direction y (UPy)The Piecewise Polynomial LibraryThe Piecewise Polynomial Library may be used to evaluate piecewise polynomial functions. An example is given in the model in the GAMS Test Library. Note that the functions that are to be evaluated need to be defined and stored in a.