Fitting Curves and Surfaces Using SAS Software - ISS - University of ...

Information Systems Services 

Fitting Curves and Surfaces Using 

SAS Software 

TUT125 

Version 1.1 (February 2007)

Fitting Curves and Surfaces Using SAS Software 

Version 1.1 (February 2007) 

Contents 

1. Introduction ............................................................................................................................................ 3 

1.1 Documentation ........................................................................................................................... 3 

1.2 Getting Started ........................................................................................................................... 4 

2. Using Interpolating Splines in SAS/GRAPH ........................................................................................... 4 

3. SAS/STAT Modelling Procedures .......................................................................................................... 5 

3.1 Parametric Modelling using REG and NLIN ............................................................................... 6 

3.2 Non-Parametric Modelling using Loess Smoothing .................................................................. 10 

3.3 Semi-Parametric Modelling using Thin-Plate Splines ............................................................... 13 

3.4 Generalised Additive Models ................................................................................................... 17 

References ..................................................................................................................................................... 19 

Annex ............................................................................................................................................................. 20 

Format Conventions 

In this document the following format conventions are used: 

Menu items and commands that you must type Name 

in are shown in bold 

Keys that you press and options that you or 

select are enclosed in angle brackets. 

Feedback 

If you notice any mistakes in this document please contact the Information Officer. Email should be 

sent to the address info-officer@leeds.ac.uk 

Copyright 

This document is copyright University of Leeds. Permission to use material in this document should be 

obtained from the Information Officer (email should be sent to the address info-officer@leeds.ac.uk) 

Information Systems Services Page 2 of 22 

Version 1.1 (Feb 2007) tut125.doc



1. Introduction 

In studies involving continuous valued variables, a common requirement is the ability to predict the value of 

one variable given the value of another variable. As a first step, the relationship between a pair of variables 

can be examined using a scatter plot. It may be sufficient to obtain a visual impression of the relationship 

between the variables by smoothing the data - for example, by using a smoothing spline, specified on a 

SYMBOL statement. Alternatively, it may be required, to obtain a deeper understanding about how changes in 

one variable (referred to as the independent or explanatory variable) influence values of another variable, 

referred to as the dependent variable. For values of the independent variable within the range of the current 

data, this is referred to as interpolation. For values of the independent variable outside the range of the current 

data, the problem is referred to as extrapolation. 

Whilst the use of the SYMBOL spline facility may be adequate for visual interpolation, it does not provide 

numerical representations of interpolated values and cannot be relied upon for extrapolation. Consider the two 

graphs below. 

Both curves appear to fit the data equally well. In the first graph, y increases exponentially as x increases – in 

the second graph y tails off asymptotically as x increases. Which curve represents ‘the best fit’? The answer is, 

of course, that there is no such thing as an absolute best fit. This demonstrates an idea central to curve fitting 

– that an understanding of the nature of the variables is essential if meaningful predictions are to be made. If it 

is known that y settles down to a finite value as x increases, a functional form expressing that relationship is 

essential if extrapolations are to be made. Thus, an alternative to the simple smoothing approach is the 

modelling approach in which a mathematical function, based upon knowledge of the data, is sought to 

describe the relationship between the two variables. 

This document looks at a range of methods available in SAS for smoothing and modelling data. A familiarity 

with the basics of SAS/GRAPH is assumed. 

1.1 Documentation 

The main reference documentation for the facilities described in this document are as follows: 

• SAS/GRAPH Reference Manual Vols. 1 & 2 

• SAS/STAT Reference Manual Vols. 1 & 2 

The Reference Manuals are also available in the on-line Help system. 

The SAS Sample Library, accessible from within a SAS session, also contains the numerous examples found 

in the above volumes. 





1.2 Getting Started 

A variety of SAS programs and data sets will be used for these exercises. These files are stored in a zip file 

named sascurve.zip on the ISS website. Before starting, it is advised that you download the zip file to your 

hard disk. To download the file, open a web browser and go to 

http://iss.leeds.ac.uk/software. 

Click on ‘PDF files (suitable for printing)’ and scroll down to ‘SAS example files’. Right-click on ‘sascurve.zip’. 

Select Save Link As… (Save Target As… in Internet Explorer). When the Save As dialog box appears, 

select a suitable directory (see below) and click Save. Go into Windows Explorer and double click the zip file. 

Click File/Extract All to unzip the files. 

A directory such as C:\courses\sas\graph is recommended as the location in which to save the data files. If 

you are working in a public cluster, you may find it necessary to use the Temp directory instead (eg use 

C:\temp\sas\graph). This is the path used in the example programs. If you store your files in a different 

directory, you will need to change the Filename and Libname statements used in the example files 

accordingly. 

To start SAS, locate SAS from the Statistics menu and double-click to open SAS. After a short period, the SAS 

data editor window will be displayed. 

2. Using Interpolating Splines in SAS/GRAPH 

The INTERPOL= option (or I=) of the SYMBOL statement offers a number of spline interpolation algorithms for 

smoothing data. 

The first method uses I=SPLINE, to fit a cubic spline with continuous second derivatives. The second method, 

I=SMnn, is more appropriate for noisy data and allows the degree of smoothing to be varied by specifying a 

parameter nn as a number in the range 0 to 99. Larger values of nn produce more smoothing. The methods 

are compared in the following example. 

Example 1: Comparison of Spline Interpolation Methods 

The effect of using SPLINE and SM with varying degrees of smoothing is illustrated in the graphs below. (The 

program to produce these graphs is in file splines.sas). 





For this data set, a value of nn of about 30 or less gives a result from using SMnn equivalent to that obtained 

from using the SPLINE method. 

3. SAS/STAT Modelling Procedures 

SAS/STAT has a number of procedures that can be used for curve fitting. Some use a purely modelling 

approach, others adopt the smoothing approach and one is a hybrid method using a mixture of modelling and 

smoothing. Amongst the procedures currently available, the following are the ones most useful for curve fitting: 

PROC REG - multiple linear regression modelling 

PROC NLIN - non-linear regression modelling 

PROC LOESS - median smoothing 

PROC TPSPLINE - thin-plate spline smoothing 

PROC GAM - generalised additive modelling 

The REG procedure is a general-purpose procedure for multiple linear regression and is the simplest of all 

these procedures to use. It is appropriate when a functional form, linear in the parameters, can be specified for 

the model. 

The NLIN procedure fits models which are non-linear in the parameters. NLIN requires more information to be 

specified than REG, including initial estimates of parameter values and mathematical expressions for the 

functional form and its derivatives with respect to each parameter. In some cases, NLIN may fail to produce a 

globally optimal solution. Repeated alterations to the starting values may be required before a successful fit 

can be achieved. 

The table below contains some functions commonly used for curve fitting. Some of these will be used in the 

examples that follow to illustrate the different procedures and to indicate some of the pitfalls that can arise in 

fitting curves to data. 

Some Functions Commonly Used in Curve Fitting 

Description Functional form 

Straight line Y = β0 + β1X 

Polynomial Y = β0 + β1X + β2X 2 + … βkX k 

Exponential Y = β0 e β 1 X 

Negative exponential Y = β0(1 – e -β 1 X ) 

Logistic Y = β2/(1 + e -(β 0 + β 1 X) ) 

Inverse linear polynomial Y = 1/(β0 + β1/X) 





Inverse quadratic polynomial Y= 1/( β0 + β1/X + β2X) 

Both REG and NLIN are useful when you know the functional form of the model you want to fit. However, 

when you have no prior knowledge about the model, or when you know that the data cannot be represented 

by a model with a fixed number of parameters, two other procedures can be used – LOESS and TPSPLINE. 

PROC LOESS uses a non-parametric method called Loess Smoothing for estimating regression surfaces. 

This approach makes no assumptions about the parametric form of the regression surface and as a result 

offers greater flexibility for model fitting. In particular, the LOESS procedure is suitable when there are outliers 

in the data and a robust fitting method is necessary. 

The TPSPLINE procedure uses a method called penalised least squares to fit non-parametric and semiparametric 

models. TPSPLINE uses thin-plate smoothing splines to approximate smooth multivariate functions 

observed with noise. The degree of smoothing can be controlled using the generalised cross validation 

function (GCV). 

The GAM procedure fits Generalised Additive Models as defined by Hastie and Tibshirani (1990). These 

models are extensions of the generalised linear model, allowing functions to take the place of individual 

explanatory variables in the model. 

A full description of these procedures is beyond the scope of this text. The user is referred to the SAS/STAT 

User Guide (available within the online Help system within SAS) for details. The examples below illustrate their 

use on the sample data sets described in the Annex. 

3.1 Parametric Modelling using REG and NLIN 

Example 2: A Comparison of REG, NLIN and I=SPLINE 

PROC REG and PROC NLIN are the main procedures for curve fitting when a functional form for a suitable 

curve can be found. This example shows how to use both REG and NLIN and compares a variety of curves 

fitted to the same data. 

(i) Open the file curve.sas. The program, shown below, creates a small data set used for the example. 

data curve1; 

input x y @@; 

xsq=x*x; 

datalines; 

0.3 0.2 0.5 0.33 1.0 0.5 1.5 0.6 2.0 0.67 2.5 0.72 3.0 0.75 4.0 0.8 

5.0 0.85 6.0 0.87 7.0 0.89 8.0 0.9 9.0 0.91 12.0 0.92 13.0 . 14.0 . 15.0 . 

; 

run; 

(ii) Submit the program by pressing F8. 

The last three observations are included with missing values for Y. It will be instructive to see how well 

each model performs in estimating values of Y for these observations. 

A very simple approach would be to use the SYMBOL statement to fit a simple polynomial to the data. 

(iii) Open the file splineq.sas. This program uses I=RQ to fit a quadratic to the data. 

(iv) Submit the program by pressing F8. The results should correspond to the display shown below. 





title ‘Quadratic using SYMBOL 

I=RQ’; 

proc gplot; 

plot y*x y*x/overlay; 

symbol1 v=star c=blue I=none; 

symbol2 v=none c=red I=RQ; 

run; 

A better way to fit polynomials is to use PROC REG. Note that the data step above created a variable 

called XSQ as the square of X. The next solution uses PROC REG to fit a quadratic polynomial to the 

data. 

(v) Open the file regq.sas. 

(vi) Submit the program by pressing F8. 

The model statement includes both X and XSQ in the model. The fitted values estimated by the model 

are saved in an output data set fitquad specified on the OUTPUT statement. The fitted values are 

stored in a variable yfit. PROC GPLOT takes the output data set saved by PROC REG, which contains 

both the original data values and the fitted values, and plots the raw data points and the fitted values 

overlaid. Note that since the fitted values already lie on a smooth curve, the use of I=SPLINE merely 

serves to deliver a curve connecting the fitted values and does not contribute any further smoothing. 

proc reg data=curve1; 

model y=x xsq; 

output out=fitquad p=yfit r=resid; 

run; 

title ‘Quadratic using PROC REG’; 

proc gplot data=fitquad; 

plot y*x yfit*x/overlay; 


symbol2 v=none c=red I=spline; 

run; 

Both of these examples show the limitations of using polynomials. Although they may offer a 

reasonable approximation to the shape of data, they may be useless for prediction beyond the range of 

the data or for interpolation within the range of the data. 

The next two solutions use PROC NLIN to fit non-linear functions. Both functions have been chosen to 

model the behaviour of the data at extreme values of X, something which a polynomial is unable to do 

with this data. 

PROC NLIN requires the form of the curve to be fitted to be specified as a mathematical equation using 

a model statement. The derivatives of the curve with respect to each parameter must also be 

specified. Initial guesses at the values of the parameters must be specified using a parms statement. 

Single or multiple values may be specified. 





(vii) Open the file negexp.sas. This program fits a negative exponential (see the table above for a 

description of the functional form of this, and other curves). 

(viii) Submit the program by pressing F8. The program and resulting fit are shown below. 





proc nlin data=curve1; 

parms b0=1 b1=0.2 to 0.4 by 0.1; 

model y=b0*(1-exp(-b1*x)); 

der.b0=1-exp(-b1*x); 

der.b1=b0*x*exp(-b1*x); 

output out=fitexp p=yfit r=resid; 

run; 

title ‘Negative Exponential using PROC 

NLIN’; 

proc gplot data=fitexp; 




run; 

The fitted values are saved in the data set fitexp and plotted using the technique described in the 

previous example. 

(ix) Open the file invpoly.sas. This program fits an inverse polynomial to the data. 

Inverse polynomials are an under-used class of functions that are highly suited to small data sets by 

virtue of their having very few parameters. 

(x) Submit the program by pressing F8. The program and resulting fit are shown below. 

proc nlin data=curve1; 

parms b0=1 b1=1; 

model y=1/(b0+b1/x); 

der.b0 = -1/((b0+b1/x)**2); 

der.b1 = -1/(x*((b0+b1/x)**2)); 

output out=fitinvp p=yfit r=resid; 

run; 

title ‘Inverse Polynomial using PROC 

NLIN’; 

proc gplot data=fitinvp; 




run; 

The fitted values are saved in the data set fitinvp. The fit is an improvement upon the negative 

exponential. In fact, the data has been generated to conform to an inverse polynomial for the purpose 

of illustration. However, this does not detract from the message that consideration of the nature of the 

data is essential in guiding the user to an appropriate choice of model. 





3.2 Non-Parametric Modelling using Loess Smoothing 

The technique of locally weighted scatterplot smoothing, or Loess (alt. Lowess), was originally proposed by 

Cleveland (1979) and further developed by Cleveland and Devlin (1988). Loess smoothing uses weighted 

least squares to fit linear or quadratic functions of the explanatory variables at the centres of neighbourhoods. 

The radius of each neighbourhood is chosen so that the neighbourhood contains a specified percentage of the 

data points. The fraction of the data in each local neighbourhood, referred to as the smoothing parameter, 

controls the smoothness of the estimated surface. Data points in a given local neighbourhood are weighted by 

a smooth decreasing function of their distance from the centre of the neighbourhood. 

The technique is implemented in the SAS/STAT procedure LOESS. The Loess technique also allows 

confidence intervals to be computed for the smoothed estimates, subject to assumptions of Normality in the 

data. (In fact, by using iterative re-weighting, the LOESS procedure can also provide statistical inference when 

the error distribution is symmetric but not necessarily Normal). 

Example 3: Non-parametric Smoothing using PROC LOESS 

The following example is taken from the SAS Sample Library and illustrates the application of LOESS to the 

melanoma data listed in Annex 1. The data is taken from the Connecticut Tumour Registry, and represents 

age-adjusted numbers of melanoma incidences per 100,000 people for 37 years from 1936 to 1972. 

(Houghton, Flannery and Viola, 1980). 

(i) Open the file melanoma.sas. (This is described in Annex 1). 


The data set sasgraph.melanoma will be created. 

(iii) Open the file loess1.sas. 

As a first step, this program contains the following code to execute the LOESS procedure using default 

choices for all parameters. 

proc loess data=sasgraph.melanoma; 

model Incidences=Year; 

ods output OutputStatistics=Results; 

run; 

(iv) Submit the program by pressing F8. 

The fitted values are captured using the Output Delivery System in a data set called Results. 

The plot below shows the raw data with the loess fit superimposed. 





goptions reset=all; 

symbol1 c=blue v=dot; 

symbol2 c=red i=join v=none; 

%let opts=vaxis=axis1 hm=3 vm=3 

overlay; 

axis1 label=(f=zapf h=2 a=90 r=0); 

title1 'Melanoma Data with Default 

LOESS Fit'; 

proc gplot data=Results; 

plot depvar*year 

pred*year/&opts; 

run; 

title; 

Melanoma data with default LOESS fit 

The degree of smoothing can be controlled using the smooth option. In the next step, four different fits 

are produced, each with a different value of the smoothing parameter. 

(v) Open the file loess2.sas. (The PROC LOESS step uses 4 different values for the smoothing parameter 

as shown below). 


model Incidences=Year/ 

smooth=0.1 0.2 0.3 0.4 residual; 

ods output OutputStatistics=Results; 

run; 

(vi) Submit the program by pressing F8. 

The results from all four models are stored in the output data set Results. 

(vii) Open the file loess2g.sas. This program plots each set of results and replays all four graphs in a 

single display using PROC GREPLAY. The code and results are shown below. 

goptions nodisplay; 


by SmoothingParameter; 

plot depvar*year pred*year/ 

&opts name='fit'; 

run; 

goptions display; 

proc greplay nofs tc=sashelp.templt 

template=l2r2; 

igout gseg; 

treplay 1:fit 2:fit2 3:fit1 4:fit3; 

run; 

LOESS fits with different values of the 

smoothing parameter 

Examination of the residuals associated with each model can help in the selection of the best value of 

the smoothing parameter. The step below applies PROC LOESS to the residuals from each model. 





(viii) Open the file loess3.sas. (The code is shown below). 

proc loess data=Results; 


ods output OutputStatistics=residout; 

model Residual=Year/smooth=0.3; 

run; 

(ix) Submit the program by pressing F8. 

The results are stored in the output data set residout. 

(x) Open the file loess3g.sas. (The program and the graphs produced are shown below). 

(xi) Submit the program by pressing F8. The program plots the results in the same fashion as before using 

PROC GREPLAY. 

axis1 label = (angle=90 rotate=0) 

order = (-0.8 to 0.8 by 0.4); 

goptions nodisplay; 

proc gplot data=residout; 


plot DepVar*Year 

Pred*Year/&opts vref=0 lv=2 

vm=1 name='resids'; 

run; 

goptions display ; 

proc greplay nofs tc=sashelp.templt 

template=l2r2; 

igout gseg; 

treplay 1:resids 2:resids2 

3:resids1 4:resids3; 

run; quit; 

title; 

Residual plots for the four models 

An inspection of these plots suggests that a value of 0.2 for the smoothing parameter provides a good 

compromise between representing the general trend in the data and tracking the detailed fluctuations 

about that trend. 

If the observations are independently and identically distributed and follow the Normal distribution, we 

can obtain confidence limits for the model predictions by adding the clm option in the model 

statement. By default, 95% limits are produced, but this can be changed by using the alpha= option in 

the model statement. 

(xii) To complete this sequence, open the file loess4.sas. 

(xiii) This final step specifies alpha=0.1, causing 90% confidence limits to be included in the Results 

data set and produces the fitted curve with confidence limits superimposed. The program and the 

graph are shown below. 






model Incidences=Year/smooth=0.2 

residual clm alpha=0.1; 

ods output 

OutputStatistics=Results; 

run; 

symbol3 c=green i=join value=none; 

symbol4 c=green i=join v=none; 

axis1 label = (a=90 r=0) order = (0 to 6); 

title1 'Age-adjusted Melanoma 

Incidences for 37 Years'; 


plot DepVar*Year Pred*Year 

LowerCl*Year 

UpperCL*Year/&opts; 

run; 

Age-adjusted Melanoma Incidences for 37 

Years 

A facility for performing LOESS smoothing is also provided in SAS/INSIGHT. 

3.3 Semi-Parametric Modelling using Thin-Plate Splines 

PROC TPSPLINE offers a hybrid approach to curve fitting by allowing both a parametric component and a 

non-parametric component to be included in a model. The general form of the model estimated by TPSPLINE 

is 

yi = f( xi) + ziβ + εI 

where yi is the ith observation, xi is a d-dimensional vector of covariates and zi is a p-dimensional vector of 

covariates. The term f( xi) is the non-parametric component, where f is an unknown function that is assumed to 

be reasonably smooth. The xi are the smoothing variables. The parametric component, ziβ, models a linear 

regression relationship with the yi. The zi are the regression variables. (If the non-parametric component is 

absent, the model reduces to a fully parametric linear model for which PROC REG would be a more suitable 

tool). 

The TPSPLINE procedure estimates f( xi) and β using penalised least squares. This is 

an extension of the ordinary least squares technique in which a penalty function is added to the least squares 

objective function. The function to be minimised has the form 

n 

S i 

1 

( λ ) = ( ( x ) ) 

2 

∑ 

yi 

− f i − z β + λP( 

f ) 

where P( f ) is the penalty on the roughness of f. The λ term is the smoothing parameter which controls the 

balance between smoothness and goodness of fit. 

The choice of the smoothing parameter is critical. The simplest way is to try a range of values for λ and choose 

the value that gives the best result. Provided that sufficient data is available (n>25 at least), and that the 

observations are not highly correlated, a more objective way to proceed is to use the Generalised Cross 

Validation (GCV) function proposed by Wahba (1990). (For full details of this function and the criteria 

governing its use, see the section on PROC TPSPLINE in SAS OnlineDoc). 





Example 4: Surface Fitting using Thin-Plate Smoothing Splines 

The following example is taken from the SAS Sample Library and illustrates the use of TPSPLINE to fit a 

smooth surface to the sulphate data listed in Annex 1. The data represent the deposition of sulphate (SO4) at 

179 sites in 48 contiguous states of the United States in 1990. Each observation records the latitude and 

longitude of the site as well as the SO4 deposition at the site measured in gram per square meter (g/m 2 ). 

(i) Open the file tpsplin1.sas. 


The DATA step (shown below) creates a data set pred containing values of latitude and longitude over 

a finely divided lattice. This data set will be offered to TPSPLINE as a scoring data set upon which the 

estimated surface will be based. 

data pred; 

do latitude = 25 to 47 by 1; 

do longitude = 68 to 124 by 1; 

output; 

end; 

end; 

run; 

The TPSPLINE step specifies a range of values for the smoothing parameter. 

proc tpspline data=sasgraph.so4; 

ods output GCVFunction=gcv; 

model so4 = (latitude longitude) /lognlambda=(-6 to 1 by 0.1); 

score data=pred out=prediction1; 

run; 

The ods statement saves values of the generalised cross validation function, gcv, for the different 

values of λ (or more precisely lognlambda (=log10(n*λ))) in an output data set called gcv. A plot of 

gcv against lognlambda is presented below. 

goptions reset=all; 

axis1 minor=none; 

axis2 minor=none; 

title 'GCV Function'; 

proc gplot data=gcv; 

plot gcv*lognlambda/ cframe=white 

vaxis=axis1 haxis=axis2 ; 

format gcv lognlambda d5.2; 

symbol1 c=blue i=join v=none w=2; 

run; 

The gcv plot reveals a local minimum at around –2.56 and a global minimum around 0.277. The fitted 

estimates saved in the data set prediction1 are based upon the global minimum. 

It is instructive to compare the results with estimates based upon the local minimum. 





(iii) Open the file tpsplin2.sas. This program executes TPSPLINE specifying a single value of lognlambda 

of –2.56 (shown below). The results are stored in prediction2. 

proc tpspline data=sasgraph.so4; 

model so4 = (latitude longitude) /lognlambda0 = -2.56; 


run; 

(iv) Submit the program by pressing F8. 

A convenient way to inspect the results is to use PROC GCONTOUR (or G3D). 

(iv) Open file tpsplin3.sas. It contains two steps of the form 

legend1 frame cborder=black position=center ; 

proc gcontour data=; 

plot latitude*longitude = P_so4/ 

legend=legend1 hreverse 

vaxis=axis1 haxis=axis2 

caxis=blue cframe=white; 

run; 

producing contour plots for each set of predictions, as shown below. 

(v) Submit the program by pressing F8. The results are shown below. 

TPSPLINE fit with lognlambda = 0.277 TPSPLINE fit with lognlambda = -.56 

An alternative way to view the results is to use the ACTIVEX device driver (see Chapter 4 of ‘An 

Introduction to SAS/GRAPH’). This allows the predicted surface and the contour lines to viewed 

simultaneously in a Web browser. 

(vi) Open the file tpsplin4.sas. 

(vii) Submit the program by pressing F8. 

This program plots the two sets of predictions using PROC G3D and uses the Output Delivery System 

to save HTML output in a file. The graph will be displayed automatically in the Web browser. The 

resulting surface for predictions1 should resemble that shown alongside the code below. 





ods html 

file=”C:\temp\sas\graph\s04.html”; 

goptions device=activex; 

proc g3d data=prediction1; 

plot latitude*longitude = P_so4; 

run; 

ods html close; 

ods listing; 

The activex driver allows further modifications to the display to be carried out. 

(viii) Click the right mouse button anywhere within the frame of the graph and from the pop-up menu 

displayed select Surface Options. 

(ix) Under Appearance, specify Banded Gradient for Surface and Below for Contour. Click OK. 

(x) Click the right mouse button again and select Options … Colors from the pop-up menu. Change the 

colours of the walls and the floor to a colour of your choice. Click OK. 

The ACTIVEX device driver allows keyboard controls to be used to shift, rotate and zoom the graph, 

allowing the fitted surface to be inspected more thoroughly. Table 1, below, describes these controls. 

Keyboard Action Effect 

Ctrl + LMB Rotate 

Alt + LMB Shift 

Shift + LMB Zoom 

Table 1: Activex Keyboard Modifiers 

(xi) To rotate the graph, keep the Ctrl key and left mouse button depressed and move the mouse. 

(xii) To zoom-in, keep the Alt key and left mouse button depressed and move the mouse. 

The graphs below show the results for each of the two data sets predictions1 and predictions2. 





TPSPLINE fit with lognlambda = 0.277 

TPSPLINE fit with lognlambda = -2.56 

The contour bands and the projected contour lines were added using the procedure described in (v) above. It 

is clear from these displays that the two values of lognlambda correspond to very different models. The value 

of 0.277 generates a model which represents the overall trend in the data. The lower value of –2.56 results in 

a model which has a greater goodness of fit but which is more dominated by noise. 

3.4 Generalised Additive Models 

The non-parametric methods described above, Loess smoothing and thin-plate spline smoothing, tend not to 

perform well when the number of variables in the model is large. The increasing dimensionality of the 

multivariate space defined by the variables has the effect of increasing the variance of the fitted values. The 

idea of additive models was introduced by Stone (1985) in an attempt to address this problem. 

In contrast with ordinary multiple linear regression, in which the expected value of the dependent variable Y is 

expressed as a linear function of the dependent variables of the form 

E(Y) = β0 + β1X1 + … + βpXp 

the additive model expresses the expected value of Y as 

E(Y) = S0 + S1(X1) + … + Sp(Xp) 

where the functions Si(Xi) are smooth functions to be estimated non-parametrically. 

The concept was subsequently extended by Hastie and Tibshirani (1990) to allow the dependent variable to 

have a distribution from the exponential family of distributions rather than being restricted to the Normal 

distribution. The extended model is called the generalised additive model. PROC GAM implements the 

generalised additive model and recognises data from Normal, Binomial, Poisson and Gamma distributions. 

The methodology of implementing this model is beyond the scope of this document. The reader is referred to 

the SAS/STAT User Guide for full details. The following example illustrates an application of the technique to 

the sulphate data used earlier in Example 4 to illustrate the thin-plate spline smoothing method. 





Example 5: Fitting a Generalised Additive Model using PROC GAM 

(i) Open the file gam.sas and execute the program by pressing F8. 

ods html 

file="c:\temp\sas\graph\so4b.htm"; 

goptions reset=all device=activex; 

proc gam data=sasgraph.so4; 

model so4=spline(longitude) 

spline(latitude) 


run; 

ods html close; 

ods listing; 

The surface is much smoother than that produced by TPSPLINE. However, GAM uses much less cpu 

time than TPSPLINE. In this example, GAM ran in less than a fifth of the time taken by TPSPLINE. The 

contour bands and the projected contour lines were added using the technique described in 1.2.3. 





References 

Hastie, T.J. and Tibshirani, R.J. (1990), Generalized Additive Models, New York: Chapman and Hall 

Cleveland, W.S., Devlin, S.J., and Grosse, E. (1988), "Regression By Local Fitting," Journal of Econometrics, 

37, 87 -114. 

Houghton, A.N., Flannery, J., and Viola, M.V. (1980), "Malignant Melanoma in Connecticut and Denmark," 

International Journal of Cancer, 25, 95 -104. 

Wahba, G., (1990), Spline Models for Observational Data, Philadelphia: Society for Industrial and Applied 

Mathematics. 





Annex 

This annex describes the data sets used in the examples and the DATA step code required to read the data. 

In some cases, the raw data is also stored separately in a file of type .dat. (All files are assumed to be held in 

the directory C:\temp\sas\graph.) 

Data set name: melanoma 

Description: The data is taken from the Connecticut Tumour Registry, and represents age-adjusted numbers 

of melanoma incidences per 100,000 people for 37 years from 1936 to 1972. (Houghton, Flannery and Viola, 

1980). 

Program name: melanoma.sas 

libname sasgraph ‘c:\temp\sas\graph’; 

data sasgraph.melanoma; 

input Year Incidences @@; 

format year d4.0; 

format incidences d4.1; 

cards; 

1936 0.9 1937 0.8 1938 0.8 1939 1.3 

1940 1.4 1941 1.2 1942 1.7 1943 1.8 

1944 1.6 1945 1.5 1946 1.5 1947 2.0 

1948 2.5 1949 2.7 1950 2.9 1951 2.5 

1952 3.1 1953 2.4 1954 2.2 1955 2.9 

1956 2.5 1957 2.6 1958 3.2 1959 3.8 

1960 4.2 1961 3.9 1962 3.7 1963 3.3 

1964 3.7 1965 3.9 1966 4.1 1967 3.8 

1968 4.7 1969 4.4 1970 4.8 1971 4.8 

1972 4.8 

run; 





Data set name: so4 

Description: The data represent the deposition of sulphate (SO4) at 179 sites in 48 contiguous states of the 

United States in 1990. Each observation records the latitude and longitude of the site as well as the SO4 

deposition at the site measured in gram per square meter (g/m 2 ). 

Program name: so4.sas 

libname sasgraph ‘c:\temp\sas\graph’; 

data sasgraph.so4; 

input latitude longitude so4 @@; 

cards; 

32.45833 87.24222 1.403 34.28778 85.96889 2.103 

33.07139 109.86472 0.299 36.07167 112.15500 0.304 

31.95056 112.80000 0.263 33.60500 92.09722 1.950 

34.17944 93.09861 2.168 36.08389 92.58694 1.578 

36.10056 94.17333 1.708 39.00472 123.08472 0.096 

36.56694 118.77722 0.259 41.76583 122.47833 0.065 

34.80611 119.01139 0.053 38.53528 121.77500 0.135 

37.44139 105.86528 0.247 38.11778 103.31611 0.326 

39.99389 105.48000 0.687 39.40306 107.34111 0.225 

39.42722 107.37972 0.339 37.19806 108.49028 0.559 

40.50750 107.70194 0.250 40.36417 105.58194 0.307 

40.53472 106.78000 0.564 37.75139 107.68528 0.557 

39.10111 105.09194 0.371 40.80639 104.75472 0.286 

29.97472 82.19806 1.279 28.54278 80.64444 1.564 

25.39000 80.68000 0.912 30.54806 84.60083 1.243 

27.38000 82.28389 0.991 32.14111 81.97139 1.225 

33.17778 84.40611 1.580 31.47306 83.53306 0.851 

43.46139 113.55472 0.180 43.20528 116.74917 0.103 

44.29778 116.06361 0.161 40.05333 88.37194 2.940 

41.84139 88.85111 2.090 41.70111 87.99528 3.171 

37.71000 89.26889 2.523 38.71000 88.74917 2.317 

37.43556 88.67194 3.077 40.93333 90.72306 2.006 

40.84000 85.46389 2.725 38.74083 87.48556 3.158 

41.63250 87.08778 3.443 40.47528 86.99222 2.775 

42.90972 91.47000 1.154 40.96306 93.39250 1.423 

37.65111 94.80361 1.863 39.10222 96.60917 0.898 

38.67167 100.91639 0.536 37.70472 85.04889 2.693 

37.07778 82.99361 2.195 38.11833 83.54694 2.762 

36.79056 88.06722 2.377 29.92972 91.71528 1.276 

30.81139 90.18083 1.393 44.37389 68.26056 2.268 

46.86889 68.01472 1.551 44.10750 70.72889 1.631 

45.48917 69.66528 1.369 39.40889 76.99528 2.535 

38.91306 76.15250 2.477 41.97583 70.02472 1.619 

42.39250 72.34444 2.156 42.38389 71.21472 2.417 

45.56083 84.67833 1.701 46.37417 84.74139 1.539 

47.10472 88.55139 1.048 42.41028 85.39278 3.107 

44.22389 85.81806 2.258 47.53111 93.46861 0.550 

47.94639 91.49611 0.563 46.24944 94.49722 0.591 

44.23722 95.30056 0.604 32.30667 90.31833 1.614 

32.33472 89.16583 1.135 34.00250 89.80000 1.503 

38.75361 92.19889 1.814 36.91083 90.31861 2.435 

45.56861 107.43750 0.217 48.51028 113.99583 0.387 

48.49917 109.79750 0.100 46.48500 112.06472 0.209 

41.15306 96.49278 0.743 41.05917 100.74639 0.391 

36.13583 115.42556 0.139 41.28528 115.85222 0.075 

38.79917 119.25667 0.053 39.00500 114.21583 0.273 

43.94306 71.70333 2.391 40.31500 74.85472 2.593 

33.22028 108.23472 0.377 35.78167 106.26750 0.315 

32.90944 105.47056 0.355 36.04083 106.97139 0.376 

36.77889 103.98139 0.326 42.73389 76.65972 3.249 

42.29944 79.39639 3.344 43.97306 74.22306 2.322 





44.39333 73.85944 2.111 41.35083 74.04861 3.306 

43.52611 75.94722 3.948 42.10639 77.53583 2.231 

41.99361 74.50361 3.022 36.13250 77.17139 1.857 

35.06056 83.43056 2.393 35.69694 80.62250 2.082 

35.02583 78.27833 1.729 34.97083 79.52833 1.959 

35.72833 78.68028 1.780 47.60139 103.26417 0.354 

48.78250 97.75417 0.306 47.12556 99.23694 0.273 

39.53139 84.72417 3.828 40.35528 83.06611 3.401 

39.79278 81.53111 3.961 40.78222 81.92000 3.349 

36.80528 98.20056 0.603 34.98000 97.52139 0.994 

36.59083 101.61750 0.444 44.38694 123.62306 0.629 

44.63472 123.19000 0.329 45.44917 122.15333 0.716 

43.12167 121.05778 0.050 44.21333 122.25333 0.423 

43.89944 117.42694 0.071 45.22444 118.51139 0.109 

40.78833 77.94583 3.275 41.59778 78.76750 4.336 

40.65750 77.93972 3.352 41.32750 74.82028 3.081 

33.53944 80.43500 1.456 44.35500 98.29083 0.372 

43.94917 101.85833 0.224 35.96139 84.28722 3.579 

35.18250 87.19639 2.148 35.66444 83.59028 2.474 

35.46778 89.15861 1.811 33.95778 102.77611 0.376 

28.46667 97.70694 0.886 29.66139 96.25944 0.934 

30.26139 100.55500 0.938 32.37861 94.71167 2.229 

31.56056 94.86083 1.472 33.27333 99.21528 0.890 

33.39167 97.63972 1.585 37.61861 112.17278 0.237 

41.65833 111.89694 0.271 38.99833 110.16528 0.143 

41.35750 111.04861 0.172 42.87611 73.16333 2.412 

44.52833 72.86889 2.549 38.04056 78.54306 2.478 

37.33139 80.55750 1.650 38.52250 78.43583 2.360 

47.86000 123.93194 1.144 48.54056 121.44528 0.837 

46.83528 122.28667 0.635 46.76056 117.18472 0.255 

37.98000 80.95000 2.396 39.08972 79.66222 3.291 

45.79639 88.39944 1.054 45.05333 88.37278 1.457 

44.66444 89.65222 1.044 43.70194 90.56861 1.309 

46.05278 89.65306 1.132 42.57917 88.50056 1.809 

45.82278 91.87444 0.984 41.34028 106.19083 0.335 

42.73389 108.85000 0.236 42.49472 108.82917 0.313 

42.92889 109.78667 0.182 43.22278 109.99111 0.161 

43.87333 104.19222 0.306 44.91722 110.42028 0.210 

45.07611 72.67556 2.646 

run;

Fitting Curves and Surfaces Using SAS Software - ISS - University of ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?