How to Find Good Start Values for nls Function in R

The nls (nonlinear least squares) function in R is used for fitting nonlinear models to data. Unlike linear models, nonlinear models can be more challenging to fit because they often require careful selection of starting values for the parameters. Poor starting values can lead to convergence issues or suboptimal fits. This article provides a comprehensive guide on finding good starting values for the nls function in R.

Understanding Nonlinear Models

Nonlinear models are used when the relationship between the independent and dependent variables is not linear. They are represented by equations that involve nonlinear functions of the parameters. For example, a common nonlinear model is the exponential growth model:

y = a \cdot e^{b \cdot x}

where a and b are parameters to be estimated.

The Importance of Good Starting Values

In nonlinear regression, starting values are initial guesses for the parameters in the model. The nls function uses iterative algorithms to minimize the sum of squared residuals, and the quality of the fit depends on these initial values. Good starting values can:

• Improve convergence.
• Ensure the algorithm finds the global minimum rather than a local minimum.
• Speed up the fitting process.

Strategies for Finding Good Starting Values

Now we will discuss different Strategies for Finding Good Starting Values using R Programming Language.

1: Use Domain Knowledge

Understanding the subject matter and the expected range of parameters can provide valuable insights. For example, if you know that a growth rate is usually between 0.1 and 1, you can use this information to set reasonable starting values for the growth rate parameter.

2: Plot the Data

Visualizing the data can help you estimate reasonable starting values. For example, if you are fitting an exponential model, plot the data and look for the initial value of aaa (the intercept) and bbb (the growth rate) visually. You might need to transform the data or use linear approximations to get initial estimates.

# Example of plotting data
plot(x, y, main = "Data Plot", xlab = "x", ylab = "y")

3: Linearize the Model

Some nonlinear models can be linearized by transforming them into linear form. This transformed model can be fit using linear regression to get initial estimates for log⁡(a) and b. From these estimates, you can derive the starting values for the original nonlinear model.

# Fit a linear model to the transformed data
fit_linear <- lm(log(y) ~ x)
summary(fit_linear)

# Extract starting values
start_a <- exp(coef(fit_linear)[1])
start_b <- coef(fit_linear)[2]

Output:

Call:
lm(formula = log(y) ~ x)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.107195 -0.029341 -0.008143  0.005164  0.145996 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.689013   0.048369   14.24 5.75e-07 ***
x           0.301735   0.007795   38.71 2.18e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0708 on 8 degrees of freedom
Multiple R-squared:  0.9947,	Adjusted R-squared:  0.994 
F-statistic:  1498 on 1 and 8 DF,  p-value: 2.181e-10


(Intercept) 
   1.991749 

        x 
0.3017347 

4: Use Diagnostic Plots

Diagnostic plots from linear models or initial fits can give clues about the parameters. For example, residual plots or leverage plots might help in refining starting values. In practice, fitting a simpler model first and examining the results can provide hints for more complex models.

5: Try a Grid Search

A grid search involves trying different combinations of starting values systematically. This approach can be time-consuming but ensures you explore a range of potential starting values. This is particularly useful if you have no prior information about the expected values.

# Define a range of starting values
start_values <- expand.grid(a = seq(0.5, 1.5, by = 0.1), b = seq(0.1, 1, by = 0.1))

# Try each combination
results <- lapply(1:nrow(start_values), function(i) {
  tryCatch({
    nls(y ~ a * exp(b * x), data = data.frame(x, y), start = start_values[i, ])
  }, error = function(e) NULL)
})
results 

Output:

Nonlinear regression model
  model: y ~ a * exp(b * x)
   data: data.frame(x, y)
     a      b 
2.1030 0.2938 
 residual sum-of-squares: 1.656

Number of iterations to convergence: 4 
Achieved convergence tolerance: 3.365e-06......................................................

6: Use nlsLM from the minpack.lm Package

The nlsLM function from the minpack.lm package can be more robust than nls and might handle poor starting values better. It uses the Levenberg-Marquardt algorithm, which is often more reliable for finding optimal parameters.

library(minpack.lm)

# Fit the model using nlsLM
fit <- nlsLM(y ~ a * exp(b * x), data = data.frame(x, y), 
                           start = list(a = 1, b = 0.5))
summary(fit)

Output:

Formula: y ~ a * exp(b * x)

Parameters:
  Estimate Std. Error t value Pr(>|t|)    
a 2.102991   0.090195   23.32 1.22e-08 ***
b 0.293800   0.004808   61.11 5.72e-12 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.455 on 8 degrees of freedom

Number of iterations to convergence: 6 
Achieved convergence tolerance: 1.49e-08

Conclusion

Finding good starting values for the nls function in R is crucial for obtaining accurate and reliable results in nonlinear regression. Key strategies include leveraging domain knowledge, visualizing the data, linearizing the model, using diagnostic plots, trying a grid search, and utilizing robust algorithms like nlsLM. By applying these techniques, you can improve the fit of your nonlinear models and ensure better convergence and performance.