predict

Class: NonLinearModel

Predict response of nonlinear regression model

Syntax

ypred = predict(mdl,Xnew)
[ypred,yci] = predict(mdl,Xnew)
[ypred,yci] = predict(mdl,Xnew,Name,Value)


Description

ypred = predict(mdl,Xnew) returns the predicted response of the mdl nonlinear regression model to the points in Xnew.

[ypred,yci] = predict(mdl,Xnew) returns confidence intervals for the true mean responses.

[ypred,yci] = predict(mdl,Xnew,Name,Value) predicts responses with additional options specified by one or more Name,Value pair arguments.

Input Arguments

mdl	Nonlinear regression model, constructed by fitnlm.
Xnew	Points at which mdl predicts responses. If Xnew is a table or dataset array, it must contain the predictor names in mdl. If Xnew is a numeric matrix, it must have the same number of variables (columns) as was used to create mdl. Furthermore, all variables used in creating mdl must be numeric.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'Alpha'	Positive scalar from 0 to 1. Confidence level of yci is 100(1 – alpha)%. Default: 0.05, meaning a 95% confidence interval.
'Prediction'	Type of prediction: 'curve' — predict predicts confidence bounds for the fitted mean values. 'observation' — predict predicts confidence bounds for the new observations. This results in wider bounds because the error in a new observation is equal to the error in the estimated mean value, plus the variability in the observation from the true mean. For details, see polyconf. Default: 'curve'
'Simultaneous'	Logical value specifying whether the confidence bounds are for all predictor values simultaneously (true), or hold for each individual predictor value (false). Simultaneous bounds are wider than separate bounds, because it is more stringent to require that the entire curve be within the bounds than to require that the curve at a single predictor value be within the bounds. For details, see polyconf. Default: false
'Weights'	Vector of real, positive value weights or a function handle. If you specify a vector, then it must have the same number of elements as the number of observations (or rows) in Xnew. If you specify a function handle, then the function must accept a vector of predicted response values as input, and return a vector of real positive weights as output. Given weights, W, predict estimates the error variance at observation i by MSE*(1/W(i)), where MSE is the mean squared error. Default: No weights

Output Arguments

ypred	Predicted mean values at Xnew. ypred is the same size as each component of Xnew.
yci	Confidence intervals, a two-column matrix with each row providing one interval. The meaning of the confidence interval depends on the settings of the name-value pairs.

Examples

Predict Responses

Create a nonlinear model of car mileage as a function of weight, and predict the response.

Create an exponential model of car mileage as a function of weight from the carsmall data. Scale the weight by a factor of 1000 so all the variables are roughly equal in size.

load carsmall
X = Weight;
y = MPG;
modelfun = 'y ~ b1 + b2*exp(-b3*x/1000)';
beta0 = [1 1 1];
mdl = fitnlm(X,y,modelfun,beta0);

Create predicted responses to the data.

Xnew = X;
ypred = predict(mdl,Xnew);

Plot the original responses and the predicted responses to see how they differ.

plot(X,y,'o',X,ypred,'x')
legend('Data','Predicted')

Confidence Intervals for Predictions

Create a nonlinear model of car mileage as a function of weight, and examine confidence intervals of some responses.

Create an exponential model of car mileage as a function of weight from the carsmall data. Scale the weight by a factor of 1000 so all the variables are roughly equal in size.

load carsmall
X = Weight;
y = MPG;
modelfun = 'y ~ b1 + b2*exp(-b3*x/1000)';
beta0 = [1 1 1];
mdl = fitnlm(X,y,modelfun,beta0);

Create predicted responses to the smallest, mean, and largest data points.

Xnew = [min(X);mean(X);max(X)];
[ypred,yci] = predict(mdl,Xnew)

ypred = 3×1

   34.9469
   22.6868
   10.0617

yci = 3×2

   32.5212   37.3726
   21.4061   23.9674
    7.0148   13.1086

Simultaneous Confidence Intervals for Robust Fit Curve

Generate sample data from the nonlinear regression model

$$y=b_1+b_2\exp (-b_{3}x)+ϵ$$

where $$b_1$$, $$b_2$$, and $$b_3$$ are coefficients, and the error term $$ϵ$$ is normally distributed with mean 0 and standard deviation 0.5.

modelfun = @(b,x)(b(1)+b(2)*exp(-b(3)*x));

rng('default') % For reproducibility
b = [1;3;2];
x = exprnd(2,100,1);
y = modelfun(b,x) + normrnd(0,0.5,100,1);

Fit the nonlinear model using robust fitting options.

opts = statset('nlinfit');
opts.RobustWgtFun = 'bisquare';
b0 = [2;2;2];
mdl = fitnlm(x,y,modelfun,b0,'Options',opts);

Plot the fitted regression model and simultaneous 95% confidence bounds.

xrange = [min(x):.01:max(x)]';
[ypred,yci] = predict(mdl,xrange,'Simultaneous',true);

figure()
plot(x,y,'ko') % observed data
hold on
plot(xrange,ypred,'k','LineWidth',2)
plot(xrange,yci','r--','LineWidth',1.5)

Confidence Interval Using Observation Weights

Load sample data.

S = load('reaction');
X = S.reactants;
y = S.rate;
beta0 = S.beta;

Specify a function handle for observation weights, then fit the Hougen-Watson model to the rate data using the specified observation weights function.

a = 1; b = 1;
weights = @(yhat) 1./((a + b*abs(yhat)).^2);
mdl = fitnlm(X,y,@hougen,beta0,'Weights',weights);

Compute the 95% prediction interval for a new observation with reactant levels [100,100,100] using the observation weight function.

[ypred,yci] = predict(mdl,[100,100,100],'Prediction','observation', ...
    'Weights',weights)

ypred = 1.8149

yci = 1×2

    1.5264    2.1033

Tips

• For predictions with added noise, use random.

• For a syntax that can be easier to use with models created from tables or dataset arrays, try feval.

References

[1] Lane, T. P. and W. H. DuMouchel. “Simultaneous Confidence Intervals in Multiple Regression.” The American Statistician. Vol. 48, No. 4, 1994, pp. 315–321.

[2] Seber, G. A. F., and C. J. Wild. Nonlinear Regression. Hoboken, NJ: Wiley-Interscience, 2003.