使用R语言nls拟合Sigmoid函数时出现singular gradient错误的解决建议咨询

Hey there! That singular gradient error is super common when using nls() for curve fitting—it usually boils down to poor initial parameter guesses or a model parameterization that causes collinearity between parameters. Let's break down how to fix this for your sigmoid fit.

First, Understand the Root Cause #

Your original sigmoid function phi1/(1 + exp(-phi3 * x - phi2)) is mathematically valid, but the way parameters are combined (-phi3*x - phi2) makes it hard to pick good starting values, and can lead the optimizer to hit a "singular gradient" where it can't adjust parameters effectively. Also, your initial guesses (especially phi3 = 0.0005) were way too small to match your data's growth rate.

Step 1: Re-Parameterize the Sigmoid for Better Interpretability #

Switch to a more intuitive sigmoid form where parameters map directly to meaningful features of your data:

• phi1: The upper asymptote (matches the maximum value in your cumulativo column)
• phi3: The x-value where the curve inflects (the point of fastest growth)
• phi2: The steepness of the curve (negative for an increasing sigmoid)

Here's the adjusted function:

f <- function(x, phi1, phi2, phi3) {
  phi1 / (1 + exp(phi2 * (x - phi3)))
}

Step 2: Set Realistic Initial Values #

Looking at your data:

• cumulativo tops out around 19639, so set phi1 = 19700 (a little higher than the max)
• The fastest growth happens between 2000-2005, so set phi3 = 2005 (the inflection point)
• For a moderate growth steepness, start with phi2 = -0.5 (negative because we want an increasing S-curve)

Your initial values list becomes:

st <- list(phi1 = 19700, phi2 = -0.5, phi3 = 2005)

Step 3: Use a More Robust Fitting Function (Optional but Recommended) #

The base nls() is sensitive to initial values. The nlsLM() function from the minpack.lm package uses a more robust algorithm that handles poor initial guesses better.

Full Working Code #

library(ggplot2)
library(minpack.lm) # Load the package for nlsLM

# Your original data
data <- structure(list(cumulativo = c(2, 3, 17, 191, 819, 1699, 2679, 3907, 5535, 7254, 9226, 11543, 13809, 15542, 16852, 17709, 18246, 18661, 18976, 19256, 19412, 19539, 19639), eixox = 1994:2016), class = "data.frame", row.names = c(NA, -23L))

# Plot raw data to visualize trend
plot(cumulativo ~ eixox, data = data)

# Re-parameterized sigmoid function
f <- function(x, phi1, phi2, phi3) {
  phi1 / (1 + exp(phi2 * (x - phi3)))
}

# Realistic initial values
st <- list(phi1 = 19700, phi2 = -0.5, phi3 = 2005)

# Fit the model with nlsLM (more robust than base nls)
curvaS <- nlsLM(cumulativo ~ f(eixox, phi1, phi2, phi3), start = st, data = data, trace = TRUE)

# Check the fit summary
summary(curvaS)

# Add predicted values to your data frame
data$predicted <- predict(curvaS, newdata = data)

# Plot the fit against raw data
ggplot(data, aes(x = eixox)) +
  geom_point(aes(y = cumulativo), color = "black") +
  geom_line(aes(y = predicted), color = "red", linewidth = 1) +
  labs(title = "Sigmoid Fit to Cumulative Data", x = "Year", y = "Cumulative Value")

What If You Want to Stick With Your Original Model? #

If you prefer your original sigmoid form, adjust the initial values to match the re-parameterized version. For example, since -phi3*x - phi2 = phi2_new*(x - phi3_new), you can calculate:

• phi3_original = -phi2_new = 0.5
• phi2_original = phi2_new * phi3_new = -0.5 * 2005 = -1002.5

Your initial values would be:

st <- list(phi1=20000, phi2=-1002.5, phi3=0.5)

Then you can use this with base nls()—though nlsLM() is still more reliable.

内容的提问来源于stack exchange，提问作者Antonio Abrantes