MATLAB Examples

Predict or Simulate Responses Using a Nonlinear Model

This example shows how to use the methods predict , feval , and random to predict and simulate responses to new data.

Contents

Randomly generate a sample from a Cauchy distribution.

rng('default')
X = rand(100,1);
X = tan(pi*X - pi/2);

Generate the response according to the model y = b1*(pi /2 + atan((x - b2) / b3)) and add noise to the response.

modelfun = @(b,x) b(1) * ...
    (pi/2 + atan((x - b(2))/b(3)));
y = modelfun([12 5 10],X) + randn(100,1);

Fit a model starting from the arbitrary parameters b = [1,1,1].

beta0 = [1 1 1]; % An arbitrary guess
mdl = fitnlm(X,y,modelfun,beta0)
mdl = 


Nonlinear regression model:
    y ~ b1*(pi/2 + atan((x - b2)/b3))

Estimated Coefficients:
          Estimate      SE       tStat       pValue  
          ________    _______    ______    __________

    b1    12.082      0.80028    15.097    3.3151e-27
    b2    5.0603       1.0825    4.6747    9.5063e-06
    b3      9.64      0.46499    20.732    2.0382e-37


Number of observations: 100, Error degrees of freedom: 97
Root Mean Squared Error: 1.02
R-Squared: 0.92,  Adjusted R-Squared 0.918
F-statistic vs. zero model: 6.45e+03, p-value = 1.72e-111

The fitted values are within a few percent of the parameters [12,5,10].

Examine the fit.

plotSlice(mdl)

predict

The predict method predicts the mean responses and, if requested, gives confidence bounds. Find the predicted response values and predicted confidence intervals about the response at X values [-15;5;12].

Xnew = [-15;5;12];
[ynew,ynewci] = predict(mdl,Xnew)
ynew =

    5.4122
   18.9022
   26.5161


ynewci =

    4.8233    6.0010
   18.4555   19.3490
   25.0170   28.0151

The confidence intervals are reflected in the slice plot.

feval

The feval method predicts the mean responses. feval is often more convenient to use than predict when you construct a model from a dataset array.

Create the nonlinear model from a dataset array.

ds = dataset({X,'X'},{y,'y'});
mdl2 = fitnlm(ds,modelfun,beta0);

Find the predicted model responses (CDF) at X values [-15;5;12].

Xnew = [-15;5;12];
ynew = feval(mdl2,Xnew)
ynew =

    5.4122
   18.9022
   26.5161

random

The random method simulates new random response values, equal to the mean prediction plus a random disturbance with the same variance as the training data.

Xnew = [-15;5;12];
ysim = random(mdl,Xnew)
ysim =

    6.0505
   19.0893
   25.4647

Rerun the random method. The results change.

ysim = random(mdl,Xnew)
ysim =

    6.3813
   19.2157
   26.6541