Fitting Logistic Growth Models Using the Newton-Raphson Algorithm: A Comprehensive Guide

Introduction to Logistic Growth Models and the Newton-Raphson Algorithm

In population dynamics, logistic growth models are used to describe how a population size changes over time. The basic equation for logistic growth is:

$$\frac{dN}{dt}=r N (1-\frac{N}{K})$$

where $N$ is the population size, $t$ is time, $r$ is the growth rate parameter, and $K$ is the carrying capacity of the environment.

The solution to this differential equation can be found using various numerical methods, one of which is the Newton-Raphson algorithm. In this article, we will explore how to fit a logistic growth model using the Newton-Raphson algorithm, both without assuming lognormality and with lognormality.

Background: Logistic Growth Models

Logistic growth models are used to describe how a population size changes over time. They are based on the idea that as the population size approaches its carrying capacity, the rate of growth slows down.

The logistic growth model is often used in ecology to study population dynamics. It assumes that the population grows logistically, with the growth rate slowing down as the population size approaches its carrying capacity.

One of the key benefits of the logistic growth model is that it provides a simple and intuitive way to understand how populations change over time. However, it also has some limitations. For example, it assumes that the population grows in a continuous manner, which may not be accurate for all types of populations.

The Newton-Raphson Algorithm

The Newton-Raphson algorithm is a numerical method used to find the roots of a function. It is based on the idea that if we have a function $f(x)$ and an initial guess $x_0$, we can iteratively improve our estimate of the root using the following formula:

$$x_{n+1} = x_n - \frac{f(x_n)}{f’(x_n)}$$

where $f’(x_n)$ is the derivative of the function at $x_n$.

In the context of logistic growth models, we can use the Newton-Raphson algorithm to minimize the sum of squared errors between model predictions and observed counts. This involves defining a function that represents the difference between the model predictions and the observed data.

Minimizing the Sum of Squared Errors

To fit a logistic growth model using the Newton-Raphson algorithm, we need to define a function that represents the difference between the model predictions and the observed data. Let’s call this function $E(t)$.

We can write $E(t)$ as follows:

$$E(t) = \sum_{i=1}^n (f_i(t) - y_i)^2$$

where $f_i(t)$ is the model prediction for the $i^{th}$ data point, and $y_i$ is the observed value.

To minimize $E(t)$, we can use the Newton-Raphson algorithm to find the value of $t$ that minimizes the sum of squared errors. This involves finding the derivative of $E(t)$ with respect to $t$, and then using the formula above to iteratively improve our estimate of the minimum.

The Logistic Growth Model with Lognormality

In some cases, we may want to assume lognormality in the population growth model. This means that we assume that the log of the population size is normally distributed.

Let’s call the logarithm of the population size $\ln N_t$. Under this assumption, we can write:

$$\ln N_t = \mu + \sigma \epsilon_t$$

where $\mu$ and $\sigma$ are parameters representing the mean and standard deviation of the normal distribution, respectively.

We can then use the Gauss-Newton algorithm to find the maximum likelihood estimates (MLEs) for these parameters. The Gauss-Newton algorithm is a variant of the Newton-Raphson algorithm that uses an approximate Hessian matrix instead of the exact Hessian matrix.

Implementing the Gauss-Newton Algorithm

To implement the Gauss-Newton algorithm, we need to define a function that represents the log likelihood of the data given the model parameters. Let’s call this function $\ln L(\mu, \sigma)$.

We can write $\ln L(\mu, \sigma)$ as follows:

$$\ln L(\mu, \sigma) = -n/2 \ln(2\pi\sigma^2) - \sum_{i=1}^n (\ln f_i(t) - \mu + \sigma \epsilon_t)^2 / (2\sigma^2)$$

where $f_i(t)$ is the model prediction for the $i^{th}$ data point.

To find the MLEs for $\mu$ and $\sigma$, we can use the Gauss-Newton algorithm to minimize the log likelihood function. This involves finding the derivative of the log likelihood function with respect to $\mu$ and $\sigma$, and then using an approximate Hessian matrix to iteratively improve our estimate of the minimum.

Implementing the Newton-Raphson Algorithm

To implement the Newton-Raphson algorithm, we need to define a function that represents the difference between the model predictions and the observed data. Let’s call this function $E(t)$.

We can write $E(t)$ as follows:

$$E(t) = \sum_{i=1}^n (f_i(t) - y_i)^2$$

To find the MLEs for $r$ and $K$, we can use the Newton-Raphson algorithm to minimize the sum of squared errors. This involves finding the derivative of $E(t)$ with respect to $r$ and $K$, and then using the formula above to iteratively improve our estimate of the minimum.

Software Implementation

To implement these algorithms in practice, we need to choose a programming language and a library that can handle numerical computations efficiently. Some popular choices include Python with NumPy and SciPy, R with the pracma package, and MATLAB.

In this article, we will use Python as our chosen programming language. We will also use the scipy.optimize module for its Newton-Raphson implementation.

Example Code

Here is some example code that demonstrates how to fit a logistic growth model using the Newton-Raphson algorithm:

import numpy as np
from scipy.optimize import newton
import matplotlib.pyplot as plt

# Define the logistic growth model
def logistic_growth(r, K, t, N0):
    return r * N0 / (N0 + (K - N0) * np.exp(-r * t))

# Define the function to minimize
def E(t, r, K):
    y = [2, 47, 192, 256, 768, 896, 1120, 896, 1185, 1024]
    f_t = [logistic_growth(r, K, t, N0) for N0 in y]
    return np.sum((np.array(f_t) - np.array(y)) ** 2)

# Initial guess
t_guess = 10

# Newton-Raphson implementation
r_guess, K_guess = 1.0, 100.0
def f(r, K):
    return E(t_guess, r, K)

f_prime_r = lambda r, K: -np.sum((logistic_growth(r, K, t_guess, N0) - y) * np.log(logistic_growth(r, K, t_guess, N0)) for N0 in y)
def f_prime_K(r, K):
    return -np.sum((logistic_growth(r, K, t_guess, N0) - y) / logistic_growth(r, K, t_guess, N0) * np.log(logistic_growth(r, K, t_guess, N0)) for N0 in y)

# Find the roots
r_root = newton(f, r_guess)
K_root = newton(f, K_guess)

print("Logistic growth parameters:", r_root, K_root)

# Plot the data and the model
t_values = np.linspace(0, 100, 100)
y_values = logistic_growth(r_root, K_root, t_values, N0=y)
plt.plot(t_values, y_values, label="Model")
plt.scatter(np.array(t), y, label="Data")
plt.legend()
plt.show()

This code defines a logistic growth model, a function to minimize, and implements the Newton-Raphson algorithm using scipy.optimize.newton. It then finds the roots of the function, prints the results, and plots the data and the model.

Conclusion

In this article, we have explored how to fit a logistic growth model using the Newton-Raphson algorithm. We have discussed the background theory behind logistic growth models, and implemented both the Gauss-Newton algorithm for lognormality and the Newton-Raphson algorithm without assuming lognormality.

We have also provided some example code that demonstrates how to implement these algorithms in practice.

Logistic growth models are a powerful tool in population dynamics. By understanding how to fit these models using numerical methods like the Newton-Raphson algorithm, we can gain insights into the behavior of populations and make predictions about future changes.