Glossary
Bad Model (in context of residuals)
A model is considered a bad fit if its residual plot displays a clear pattern (e.g., a curve or a line), indicating that the model systematically misrepresents the data.
Example:
If a weather prediction model consistently overestimates temperature on sunny days, its residual plot would show a pattern, indicating it's a bad model.
Changing rate of change
A characteristic of non-linear functions (like exponential and quadratic) where the rate at which the output changes is not constant but varies depending on the input value.
Example:
The speed of a roller coaster as it goes down a hill demonstrates a changing rate of change, accelerating as it descends.
Constant rate of change
A characteristic of linear functions where the output changes by the same amount for each unit increase in the input, represented by the slope.
Example:
A car traveling at a steady 60 mph has a constant rate of change in distance over time.
Error
The general term for the difference between a predicted value from a model and the actual observed value, indicating how much the model 'got wrong.'
Example:
When a GPS predicts your arrival time, the difference between that prediction and your actual arrival is the error.
Exponential Functions
Functions of the form $f(x) = ab^x$ that describe data exhibiting growth or decay where the rate of change is proportional to the current value.
Example:
The spread of a rumor often follows an exponential function, rapidly increasing as more people hear it.
Good Model (in context of residuals)
A model is considered a good fit if its residual plot shows a random scattering of points around zero, indicating no discernible pattern.
Example:
A scientist analyzing experimental data would look for a good model where the errors are just random noise, not a systematic trend.
Linear Functions
Functions of the form $f(x) = b + mx$ that describe data exhibiting a constant rate of change, resulting in a straight-line graph.
Example:
If a plant grows 2 cm every week, its height over time can be described by a linear function.
Model Validation
The process of evaluating how well a mathematical model fits a given set of data and its appropriateness for making predictions.
Example:
Before launching a new app, a company uses model validation to ensure their user growth prediction model accurately reflects real-world trends.
Overestimate
Occurs when a model's predicted value is higher than the actual observed value.
Example:
A sales forecast that predicts 1000 units sold when only 800 are actually sold is an overestimate.
Quadratic Functions
Functions of the form $f(x) = ax^2 + bx + c$ that describe data exhibiting a parabolic (U-shaped) pattern, where the rate of change itself changes linearly.
Example:
The trajectory of a basketball shot follows a quadratic function before it reaches the hoop.
Residuals
The differences between the actual observed data values and the corresponding predicted values from a statistical model, calculated as Actual - Predicted.
Example:
If your model predicts a student will score 85 on a test, but they actually score 90, the residual is 5.
Underestimate
Occurs when a model's predicted value is lower than the actual observed value.
Example:
If a budget model predicts expenses of 600, the model underestimate the cost.