Question Type 1: Determining best fit model for a given set of data Exercises

A set of data $(x,y)$ is given by:

$(1, 2), (2, 4), (3, 8), (4, 16)$

Investigate which model form—a power model $y = a x^b$ or an exponential model $y = A B^x$—fits the data better.

Calculate the sum of squared residuals (SSR) for both models to support your conclusion.

A data set $(x, y)$ consists of the following points:

$(10, 20), (20, 80), (30, 180), (40, 320)$

Assuming a power model $y = ax^b$, use linearization to find estimates of $a$ and $b$.

The data below are recorded for $x$ and $y$:

(1, 2), (2, 5), (3, 10), (4, 17).

Compare the models

(a) quadratic $y=ax^2+bx+c$, (b) exponential $y=A B^x$,

by computing and comparing their sum of squared residuals (SSR) values. Which model fits better?

A scatter plot of data $(x, y)$ shows that as $x$ doubles, $y$ roughly quadruples. Propose a model of the form $y = ax^b$, explain your reasoning, and describe briefly how you would estimate $a$ and $b$.

Given that the sum of squared residuals ($\text{SSR}$) for three candidate models fitted to a data set are $\text{SSR}_{\text{linear}} = 150$, $\text{SSR}_{\text{quadratic}} = 120$, and $\text{SSR}_{\text{exponential}} = 130$.

State which model provides the best fit and justify your answer.

A bacterial culture grows according to an exponential law. The measured population at times $t$ (hours) is given:

$(0, 100), (1, 180), (2, 324), (3, 583)$

Find the exponential model $y = A B^t$ that best fits the data.

The following $\text{SSR}$ (sum of squared residuals) values are obtained for four candidate models fitted to the same data:

$\text{SSR}_{\text{linear}} = 500, \text{SSR}_{\text{quadratic}} = 200, \text{SSR}_{\text{cubic}} = 180, \text{SSR}_{\text{exponential}} = 190$

Select the best model and justify your choice, mentioning any trade-offs.

The data set below is collected for variables $x$ and $y$:

$(1, 3), (2, 5), (3, 7), (4, 9)$.

(a) Fit a linear model $y = mx + b$ to these points and compute the sum of squared residuals (SSR).

(b) Fit an exponential model $y = AB^x$ by linearizing $\ln y$ vs $x$ and compute the SSR.

(c) Determine which model is more appropriate.

After fitting a linear model and a cubic model to a given data set, you obtain $SSR_{\text{linear}} = 50$ and $SSR_{\text{cubic}} = 0$. Should you choose the cubic model? Explain any additional considerations beyond $SSR$.

You fit both an exponential model and a cubic model to a data set. The sum of squared residuals ($SSR$) values are $SSR_{\text{exp}} = 120$ (with 2 parameters) and $SSR_{\text{cubic}} = 110$ (with 4 parameters).

Discuss which model you would choose, referring to $SSR$ and model complexity.