Regression and modelling with functions - Mathematics: Applications & Interpretation IB Study Notes

Regression is the statistical process of finding the mathematical function that best fits a set of data points, enabling prediction and analysis of relationships between variables. In Cambridge IB Mathematics: AI, you'll work with several regression models:

Linear Regression (y = ax + b): Models data with constant rate of change. The correlation coefficient (r) measures strength of linear relationship, where |r| close to 1 indicates strong correlation, while r ≈ 0 suggests weak correlation. The coefficient of determination (r²) represents the proportion of variance explained by the model.

Quadratic Regression (y = ax² + bx + c): Appropriate for data showing one turning point (maximum or minimum). Useful for projectile motion, profit optimization, and biological growth patterns.

Exponential Regression (y = a·b^x or y = a·e^(kx)): Models situations with constant percentage growth/decay. The parameter b or k determines growth (b > 1 or k > 0) versus decay (0 < b < 1 or k < 0).

Power Regression (y = ax^b): Models relationships where rate of change varies proportionally with the variable itself.

Sinusoidal Regression (y = a·sin(b(x - c)) + d): Models periodic phenomena with parameters: a (amplitude), b (frequency), c (phase shift), d (vertical shift).

Key Concept: Residuals are the vertical distances between actual data points and predicted values. A good model minimizes residual sum of squares.

Model Selection involves examining scatter plots, residual plots, and comparing r² values. The model with highest r² and randomly distributed residuals is typically best.

Understanding when to apply each regression type is crucial for real-world modelling:

Linear Regression in Action: Imagine analyzing the relationship between study hours and exam scores. If each additional hour consistently adds approximately 5 marks, a linear model is appropriate. Think of it as climbing stairs with equal steps – each step (hour) produces predictable progress (marks).

Exponential Growth – Viral Spread: During a pandemic's early stages, each infected person infects multiple others, creating exponential growth. If one person infects 2 people, who each infect 2 more, the pattern follows y = a·2^x. This is like a chain letter multiplying – not adding but multiplying at each stage.

Quadratic Models – Business Revenue: A company raising prices initially increases revenue, but eventually high prices reduce sales volume, decreasing total revenue. This creates an inverted parabola with a maximum point – the optimal pricing strategy. Imagine throwing a ball upward; it rises to a peak (maximum revenue) then falls (losses from overpricing).

Sinusoidal Patterns – Tidal Heights: Ocean tides oscillate predictably due to lunar gravitational effects. The height follows y = a·sin(b(x - c)) + d, where the average sea level is d, maximum deviation from average is a (amplitude), and the cycle repeats every 12.4 hours (determined by b). Picture a Ferris wheel rotating – riders repeatedly rise and fall in smooth, periodic motion.

Power Regression – Planetary Motion: Kepler's Third Law states T² ∝ R³ (orbital period squared proportional to orbital radius cubed), exemplifying power relationships in physics. The relationship isn't linear or exponential but follows a specific power pattern.