9 · Evaluation, Metrics & Diagnostics · Regression Metrics & Residuals

Residual analysis

Plot residuals against predictions, key features and time. Structure, changing variance or systematic bias indicates missing relationships or data problems. The important practical question is not only how the technique is defined, but what assumptions it introduces, which data are allowed to influence it, and how its effect should be validated on unseen evidence.

Reference lessonPython exampleVisual explanation

Intuition first

What this concept means in practice

The practical value of Residual analysis comes from understanding both the transformation and the boundary around it: what information is allowed to enter, what assumption is being made, and how you know the result is still valid after the transformation.

A beginner-friendly way to reason about it is to start with a tiny case where the correct result can be checked independently. Once the mechanism is clear, scale the exact same reasoning to larger tables, pipelines or models.

PurposeUse for every regression model, even when headline metrics look good.

MechanismStudy residuals versus predictions, features, groups and time to detect bias, heteroscedasticity, nonlinear structure and drift.

EvidenceInspect intermediate and final output; compare with an independent expectation.

Main cautionA random-looking aggregate residual plot can still hide subgroup bias.

Mechanism

Trace the operation from input to decision

Study residuals versus predictions, features, groups and time to detect bias, heteroscedasticity, nonlinear structure and drift.

1Input→

2Apply rule→

3Inspect state→

4Validate→

5Use result

Key rule

residual e = y-ŷ

Visual explanation

Make the structure visible

The interactive view uses a concept-specific plot when the topic maps naturally to one; otherwise it uses a workflow view instead of leaving a broken placeholder.

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Practical example

Where you would use it

A funnel-shaped residual plot suggests error variance rises with predicted magnitude.

Use when

Use for every regression model, even when headline metrics look good.

Pitfall

What can make the result misleading

Watch out

A random-looking aggregate residual plot can still hide subgroup bias.

A useful diagnostic question is: Could the same code still run successfully if the analytical assumption were wrong? If yes, add an explicit validation check rather than relying on execution success.

Implementation

Miniature Python example

Keep the example small enough that you can inspect each stage manually.

Python

# Purpose: demonstrate Residual analysis with a small, inspectable example.
# Follow the comments and printed stages to connect each operation with its result.
# Import the library or helper used in this example.
# Step 1 — Import the module so its functions/classes are available to the rest of this example.
import numpy as np
# Import the library or helper used in this example.
# Step 2 — Import only the named objects needed by the following steps, keeping dependencies explicit.
from sklearn.metrics import mean_absolute_error, mean_squared_error, r2_score

# Create the numerical values used in the calculation.
# Step 3 — Construct `y` as an array so vectorised numerical operations can be applied consistently.
y = np.array([20,24,27,31,35,39,42,47,51,56,60,65],dtype=float)
# Step 4 — Construct `p` as an array so vectorised numerical operations can be applied consistently.
p = np.array([21,23,29,30,34,41,40,48,54,55,59,67],dtype=float)
# Store this intermediate value with a descriptive name for the next step.
# Step 5 — Compute the right-hand expression and store its result in `residual` for the next step.
residual = y-p
# Print this intermediate result so you can verify the workflow step by step.
# Step 6 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 1 · Mean residual:", round(residual.mean(),3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 7 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 2 · MAE:", round(mean_absolute_error(y,p),3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 8 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 2 · RMSE:", round(mean_squared_error(y,p)**0.5,3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 9 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 2 · R2:", round(r2_score(y,p),3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 10 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 3 · Largest absolute error:", round(np.abs(residual).max(),3))

Expected / illustrative output

STEP 1 · Mean residual: -0.333
STEP 2 · MAE: 1.5
STEP 2 · RMSE: 1.633
STEP 2 · R2: 0.986
STEP 3 · Largest absolute error: 3.0

Implementation checklist

Before you move on

Can you state what data or object enters the operation?
Can you explain what changes and what must remain invariant?
Have you checked the result on a tiny case you can verify independently?
Have you considered the main failure mode described above?
Can the operation be reproduced from code/formulas and documented assumptions?