IB AI: Mathematical Models Overview

Mathematical models are the engines of prediction and understanding in our data-driven world. In the IB AI course, you move beyond simply recognizing equations to mastering the process of selecting, building, and critiquing models that translate real-world phenomena into mathematical language. This skill is fundamental, whether you're forecasting population growth, analyzing business trends, or modeling the swing of a pendulum.

Core Model Types and Their Real-World Signatures

The first step in modeling is knowing your toolkit. Each type of model represents a distinct pattern of change, and identifying this pattern in your data is crucial.

Linear models, of the form $y=mx+c$, describe a constant rate of change. They are appropriate when a quantity increases or decreases by a fixed amount for each unit increase in another. For example, the total cost of producing items might have a fixed startup cost $c$ and a constant production cost per item $m$.

Quadratic models, with the general form $y=a{x}^2+bx+c$, describe parabolic relationships where the rate of change itself is changing at a constant rate. They are ideal for objects in free fall (height over time) or scenarios involving area optimization.

Exponential models, expressed as $y=a\cdot b^x$ or $y=a\cdot e^{kx}$, model growth or decay that is proportional to the current value. This leads to rapid increase or decrease. Real-world applications include compound interest, population growth under ideal conditions, and radioactive decay.

Sinusoidal models, typically $y=a\sin (b(x-c))+d$ or using cosine, capture periodic, wave-like behavior. The parameters control amplitude ($a$), period ($\frac{2\pi }{b}$), phase shift ($c$), and vertical shift ($d$). These are indispensable for modeling tides, seasonal temperature variations, or alternating current.

Logistic models represent growth in a constrained environment. Starting with exponential growth, it slows as it approaches a maximum capacity or carrying capacity, $L$. A common form is $y=\frac{L}{1+C{e}^{-kx}}$. This S-shaped curve is the definitive model for the spread of a virus within a finite population or the adoption of a new technology in a saturated market.

Fitting Models and Analyzing Goodness-of-Fit

Once you suspect a model type, you use technology—like your graphing calculator or software such as Desmos—to perform fitting models to data, often via regression. This process calculates the optimal parameters (like $m$, $c$, $a$, $L$) that minimize the overall difference between the model's predictions and the actual data points.

To evaluate how well a model fits, you must go beyond just looking at the curve on a graph. Residual analysis is your primary diagnostic tool. A residual is the vertical difference between an observed data point and the value predicted by your model: $\text{residual}=y_{\text{observed}}-y_{\text{predicted}}$.

By plotting residuals against the independent variable (or the predicted values), you can assess model appropriateness. A good fit will show residuals randomly scattered around zero with no discernible pattern. A clear pattern—like a U-shape in the residuals of a linear fit—is a strong signal that a different model type (perhaps quadratic) is needed. Systematic patterns in residuals reveal the model's limitations in capturing the underlying trend.

Advanced Modeling Techniques and Selection

Real-world data is often messy and may not conform perfectly to a single, simple function. This is where more flexible techniques come into play.

Piecewise models are built by defining different functions over different intervals of the domain. For instance, a taxi fare might be modeled with one function for the initial flag drop and a different linear function for each subsequent kilometer. Piecewise modeling is powerful for capturing abrupt changes in behavior that a single global model cannot.

Your final responsibility is evaluating model limitations. Every model is a simplification. An exponential population model ignores resource limits; a linear revenue forecast ignores market saturation. You must explicitly state the domain over which your model is reasonable and the assumptions (e.g., "assuming constant growth rate, no policy changes") that underpin it. The process of using technology for model selection involves comparing the fit of different candidate models—often by comparing the correlation coefficient ($r$ or $R^2$) and, more importantly, by critically examining their residual plots and real-world plausibility.

Common Pitfalls

1. Forcing a Linear Fit onto Non-Linear Data: This is the most common error. Just because you can fit a line doesn't mean you should. Always check the residual plot. A curved pattern in the data demands a curved model.
2. Extrapolating Far Beyond the Data: Models are built to fit a specific dataset within a specific range. Using an exponential growth model to predict population 500 years into the future will almost certainly produce a nonsensical result, as it ignores the inevitable carrying capacity of the environment. Always qualify your predictions and recognize the domain of validity.
3. Ignoring Context When Selecting a Model: Choosing a model based solely on a high $R^2$ value is a trap. A sinusoidal model might fit 12 monthly data points well, but if you're modeling a company's steadily growing profits, it is contextually inappropriate. The mathematical fit must be interpreted through the lens of the real-world phenomenon.
4. Misinterpreting the Parameters of a Transformed Model: When fitting an exponential model $y=a\cdot b^x$, the parameter $b$ is the growth/decay factor per unit x, not the percentage rate. The percentage growth rate is $(b-1)\times 100\%$. Confusing these leads to incorrect interpretations.

Summary

• Mathematical models (linear, quadratic, exponential, sinusoidal, logistic) are defined by their characteristic equations and the patterns of change they represent.
• Technology is essential for fitting these models to data via regression and for generating residual plots, which are the key tool for diagnosing the appropriateness of a fit.
• Piecewise models provide the flexibility to model complex, non-uniform behaviors by combining different functions over specific intervals.
• Model selection is a critical process that balances statistical measures (like $R^2$) with residual analysis and, most importantly, the contextual plausibility of the model for the real-world scenario.
• Every model has limitations. A proficient modeller clearly states the assumptions and domain of their model and avoids the dangerous practice of unjustified extrapolation.