Linear vs GLM Models
User: Thava Alagu (thavamuni@gmail.com)
Created: 3/25/2025 20:13
Updated: 3/25/2025 21:45
Exported: 3/25/2025 22:58
In statistics, we have linear models, Generalized Linear Models, Non-Linear Models; What makes a model linear or non-linear ? What is the difference between linear model and generalized linear model ? Use examples if necessary.
A model is considered linear if it is linear in parameters (coefficients), regardless of whether it is linear in features (predictors). This means the dependent variable (response) is a linear combination of the model’s parameters. If the model does not satisfy this condition, it is non-linear.
Linear Model: The model equation has the form:
where:
Non-Linear Model: At least one parameter appears in a non-linear way (e.g., squared, exponentiated, inside a sine function). Examples:
A Generalized Linear Model (GLM) extends the concept of linear models by allowing:
Assumes normal distribution of residuals (
Uses identity link function:
Example: Simple linear regression predicting house prices based on square footage.
Relaxed distribution assumption: Response variable can be Poisson (count data), Binomial (probabilities), Gamma, etc.
Uses a link function
Common GLMs:
Logistic Regression (Binary Outcomes, Binomial Distribution)
(Logit link function)
Poisson Regression (Count Data, Poisson Distribution)
(Log link function)
| Feature | Linear Model | Generalized Linear Model (GLM) |
|---|---|---|
| Response Variable | Continuous, Normal | Can be Normal, Poisson, Binomial, Gamma, etc. |
| Link Function | Identity ( |
Can be Log, Logit, Identity, etc. |
| Error Distribution | Normal | Flexible (Poisson, Binomial, etc.) |
| Example | Predicting house price from square footage | Predicting probability of customer churn |
Would you like an example implementation in Python? 🚀
I know Normal distribution. What is Poisson, Binomial and Gamma ?
Great! Since you already know the Normal distribution, let’s go over Poisson, Binomial, and Gamma distributions, which are commonly used in Generalized Linear Models (GLMs).
The Poisson distribution models the number of times an event occurs in a fixed interval of time or space, assuming the events happen independently at a constant average rate.
📌 Key Properties:
📊 Formula:
where:
🔹 Example:
If a website gets 5 visitors per minute on average, the probability of exactly 3 visitors in a minute is:
🛠 Used in:
The Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes: success (1) or failure (0).
📌 Key Properties:
Used when outcomes are binary (yes/no, pass/fail, click/no-click).
Defined by two parameters:
Values range from 0 to n (number of successes).
📊 Formula:
where:
🔹 Example:
If a basketball player scores a basket 70% of the time (p = 0.7) and takes 10 shots (n = 10), the probability of scoring exactly 7 times is:
🛠 Used in:
The Gamma distribution models the time until a certain number of events happen, assuming they occur continuously and independently.
📌 Key Properties:
Used for continuous, positive values (time, wait times, rainfall amounts).
Defined by two parameters:
A generalization of the Exponential distribution, which models only one event.
📊 Formula:
where:
🔹 Example:
If the average time between customer arrivals is 5 minutes (
(where
🛠 Used in:
| Distribution | Type of Data | Parameters | Used For |
|---|---|---|---|
| Normal | Continuous (real numbers) | Mean |
Heights, test scores, measurement errors |
| Poisson | Counts (0, 1, 2, …) | Rate |
Number of calls per hour, website visits |
| Binomial | Integer (0 to |
Number of heads in coin flips, customer conversions | |
| Gamma | Continuous, positive | Shape |
Time until failure, waiting times |
Would you like a visual representation of these distributions or a Python example? 😊
Explain GLM with concrete simple example with concrete link function and how it works.
To understand Generalized Linear Models (GLM), let’s walk through an example using Poisson Regression, a common GLM.
📌 Problem:
A call center records the number of customer calls received per hour. We want to predict the expected number of calls based on the number of agents available.
Since the response variable (number of calls) is a count, we use Poisson Regression, which assumes:
where
💡 Why Poisson? The number of calls per hour is a non-negative integer (0, 1, 2, …) and follows a rate.
GLMs require a link function to relate the predictor
For Poisson regression, the log-link function is used:
🔹 This ensures that
The GLM equation in our case is:
Rearrange to express
🔹 Unlike a simple linear model, where
Assume we estimated our model and got:
If
If
🛠 Key Takeaway: The number of expected calls grows exponentially as the number of agents increases.
A standard linear regression would assume:
By using a GLM with a log-link, we ensure predictions remain positive and reflect real-world exponential growth in counts.
| Step | Explanation |
|---|---|
| 1. Select Distribution | Poisson (for count data). |
| 2. Choose Link Function | Log-link: |
| 3. Build Model | Fit parameters |
| 4. Make Predictions | Transform using |
| 5. Interpret Results | Each agent increases call volume by 22%. |
A linear model would fail for count data because it could predict negative values.
A GLM with a log-link function ensures the predicted call counts remain positive and capture real-world behavior.
Other GLMs work similarly:
Would you like a Python implementation of this example? 🚀
Explain another GLM Scenario, this time with logistic regression example.
Let’s go through Logistic Regression, which is another type of Generalized Linear Model (GLM).
📌 Problem:
A bank wants to predict whether a customer will default on a loan (Yes/No) based on their monthly income.
Since the response variable is binary (0 or 1), we use Logistic Regression, which assumes:
where
💡 Why Bernoulli? The outcome is binary (Default = 1, No Default = 0).
GLMs require a link function to relate the predictor
For Logistic Regression, we use the logit function (log-odds):
🔹 This ensures that
Rearrange to express
This is the logistic function, which produces an S-shaped curve.
The GLM equation in our case is:
🔹 Unlike a linear model (
Assume we estimated the model and got:
If
✅ Very high probability of NOT defaulting.
If
✅ Still a high probability of NOT defaulting, but lower than before.
If
❌ Now, there is a 38% chance of default.
A standard linear regression would assume:
By using a GLM with a logit-link, we ensure predictions remain valid probabilities and capture the S-shaped relationship between income and default risk.
| Step | Explanation |
|---|---|
| 1. Select Distribution | Bernoulli (binary data). |
| 2. Choose Link Function | Logit-link: |
| 3. Build Model | Fit parameters |
| 4. Make Predictions | Transform using |
| 5. Interpret Results | Higher income → Lower chance of default. |
A linear model would fail because it could predict invalid probabilities.
A GLM with a logit-link function ensures the predicted default probability is between 0 and 1.
Logistic Regression is widely used for classification tasks like:
Would you like a Python implementation of this example? 🚀
Just give me a python program which just plots Poisson, Binomial, Gamma probability density functions for visual illustration purpose.
Here’s a simple Python program that plots the Probability Density Function (PDF) of Poisson, Binomial, and Gamma distributions using matplotlib and scipy.stats. 🚀
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
# Set figure size
plt.figure(figsize=(12, 4))
# 1. Poisson Distribution (λ = 4)
plt.subplot(1, 3, 1)
x_pois = np.arange(0, 15) # Poisson takes discrete values
pois_pmf = stats.poisson.pmf(x_pois, mu=4)
plt.stem(x_pois, pois_pmf, basefmt=" ", use_line_collection=True)
plt.title("Poisson Distribution (λ=4)")
plt.xlabel("k")
plt.ylabel("P(X=k)")
# 2. Binomial Distribution (n=10, p=0.5)
plt.subplot(1, 3, 2)
x_binom = np.arange(0, 11)
binom_pmf = stats.binom.pmf(x_binom, n=10, p=0.5)
plt.stem(x_binom, binom_pmf, basefmt=" ", use_line_collection=True)
plt.title("Binomial Distribution (n=10, p=0.5)")
plt.xlabel("k")
plt.ylabel("P(X=k)")
# 3. Gamma Distribution (α=2, β=1)
plt.subplot(1, 3, 3)
x_gamma = np.linspace(0, 10, 100)
gamma_pdf = stats.gamma.pdf(x_gamma, a=2, scale=1)
plt.plot(x_gamma, gamma_pdf, color="r")
plt.title("Gamma Distribution (α=2, β=1)")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.tight_layout()
plt.show()
Poisson Distribution (λ = 4)
Binomial Distribution (n = 10, p = 0.5)
n trials.Gamma Distribution (α = 2, β = 1)
🔹 Poisson and Binomial use PMF (probability mass function), while Gamma uses PDF (probability density function).
Would you like modifications, like changing parameters dynamically? 🚀
Now I understood LM and GLM; Now give concrete example of truly non-linear model (not a GLM) with explanation.
A non-linear model is one where the relationship between the predictors
Let’s consider a logistic growth model, often used in population dynamics or disease spread.
A linear or GLM approach would fail because bacterial growth is not linear—it follows an S-shaped (sigmoid) curve.
A commonly used non-linear model for population growth is the logistic growth model:
where:
Unlike GLMs, parameters appear inside an exponent and division, making it impossible to rewrite in a linear form:
This cannot be transformed into a linear form, so it’s truly non-linear.
Here’s how to simulate bacterial growth data and fit a non-linear logistic model:
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
# Simulated data: Time (hours) and observed bacterial population
time = np.linspace(0, 10, 20) # 20 time points from 0 to 10 hours
true_K = 1000 # Carrying capacity
true_beta0 = -2
true_beta1 = 1.2
# Logistic Growth Function (used for generating synthetic data)
def logistic_growth(t, K, beta0, beta1):
return K / (1 + np.exp(-(beta0 + beta1 * t)))
# Simulated bacterial growth data with some noise
population = logistic_growth(time, true_K, true_beta0, true_beta1) + np.random.normal(0, 30, len(time))
# Fit the non-linear model using curve_fit
popt, _ = curve_fit(logistic_growth, time, population, p0=[1200, -1, 1])
# Plotting
plt.figure(figsize=(6, 4))
plt.scatter(time, population, label="Observed Data", color="red")
plt.plot(time, logistic_growth(time, *popt), label="Fitted Logistic Model", color="blue")
plt.xlabel("Time (hours)")
plt.ylabel("Population Size")
plt.title("Non-Linear Logistic Growth Model")
plt.legend()
plt.show()
The logistic function starts slow, grows rapidly, and then levels off.
The fitted model gives estimates for:
Unlike linear regression or GLMs, this model cannot be rewritten as a simple linear equation.
| Feature | GLM (e.g., Logistic Regression) | Non-Linear Model (Logistic Growth) |
|---|---|---|
| Equation | Logit link function |
Growth curve |
| Linear in Parameters? | Yes | No (parameters inside exponent and fraction) |
| Transformable to Linear? | Yes | No |
| Example Use Case | Binary classification (Spam detection, loan default) | Population growth, epidemiology |
🚀 This logistic growth model is a truly non-linear model because:
✔️ Parameters appear in a non-linear way (exponents, fractions).
✔️ It cannot be transformed into a linear or GLM form.
✔️ Requires non-linear optimization techniques (like curve_fit) to estimate parameters.
Would you like another non-linear example, like neural networks or exponential decay? 🤖📈
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