11 Mathematical Modeling Free Online FlashCards

What is a mathematical model?

A mathematical model is a simplified representation of a real situation, built to investigate relationships and make estimates or predictions.

What are the main stages of the modeling cycle?

The cycle is: define the question, identify variables, examine data, select and fit a model, interpret and validate it, then use it within appropriate limits.

How should a model’s domain and range be chosen?

The domain contains allowable inputs, while the range contains possible outputs. Both should reflect mathematical, physical, and practical constraints.

When is a linear model preferable to an exponential model?

Approximately constant additive changes support a linear model; approximately constant multiplicative or percentage changes support an exponential model.

What does least squares minimize?

Least squares chooses parameters to minimize $\sum_{i=1}^{n}(y_i-\hat y_i)^2$. Squaring prevents opposite errors from canceling and emphasizes larger errors.

What do the slope and intercept mean in a fitted line?

In $\hat y=mx+b$, the slope $m$ is the estimated output change per input unit, while $b$ is the predicted output at $x=0$, when meaningful.

How is a residual interpreted?

A residual is $e_i=y_i-\hat y_i$. A positive residual means the model underpredicted; a negative residual means it overpredicted.

What can patterns in a residual plot reveal?

A curved residual pattern suggests missing curvature, while a funnel shape suggests changing variability. Runs or cycles can indicate dependence or an omitted time effect.

How do MAE and RMSE differ?

MAE averages absolute errors in the original units. RMSE also uses the original units but penalizes large errors more strongly because it squares the errors.

Why is a high $R^2$ insufficient for model validation?

A high $R^2$ does not by itself show that a model is appropriate, causal, or reliable outside the observed data range.

What do $P_0$ and $b$ represent in an exponential model?

In $P(t)=P_0b^t$, $P_0$ is the value at $t=0$, and $b$ is the multiplicative factor per time unit. The percentage change is $(b-1)\times100\%$.

What is the difference between interpolation and extrapolation?

Interpolation estimates within the observed input range and is generally more reliable. Extrapolation predicts outside that range and is riskier because the relationship may change.