Straight line fit
A well-defined collection of distinct objects called elements or members.
Learning Outcomes
● Associate the study on lines with real-world problems. ● Predict the suitable line among the given set of lines for a given set of points. ● Find the best fit line among the given set of lines using the Sum Squared Error Method for a given data.
Exercise Questions 🤯
1) If a line fit $ y = x + 1 $ is given for the data as shown in Table AQ-3.1, then compute the Sum Squares Error (SSE).
| x | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| y | 1 | 1 | 2 | 5 | 7 |
Concept:
- The Sum of Squares Error (SSE) is a measure of how well a line fits the data.
- It is computed as:
where $y_i$ are the actual values and $\hat{y}_i$ are the predicted values from the line.
Step-by-Step Solution:
- For each $x$, compute predicted $y = x + 1$, then square the difference from actual $y$.
- $x=1$: predicted $y=2$, actual $y=1$; error = $(1-2)^2 = 1$
- $x=2$: predicted $y=3$, actual $y=1$; error = $(1-3)^2 = 4$
- $x=3$: predicted $y=4$, actual $y=2$; error = $(2-4)^2 = 4$
- $x=4$: predicted $y=5$, actual $y=5$; error = $(5-5)^2 = 0$
- $x=6$: predicted $y=7$, actual $y=7$; error = $(7-7)^2 = 0$
Add up errors: $1 + 4 + 4 + 0 + 0 = 9$
Final Answer: $ \boxed{9} $
Concept Taught:
- SSE quantifies the total squared “distance” between data points and the regression line; the smaller, the better the fit.
2) If the relation between $x$ and $y$ is as shown in Table AQ-3.2, then which among the following lines is the best fit?
Given options:
- $y = 2x$ with SSE=143
- $y = x$ with SSE=40
- $y = 3x$ with SSE=428
- $y = 4x$ with SSE=895
Concept:
- The best fit line is the one with the smallest SSE (Sum of Squares Error).
Answer:
- SSEs: 143, 40, 428, 895. The smallest is 40 (for $y = x$).
Final Answer: $y = x$ with SSE=40 is the best fit.
Concept Taught:
- The SSE gives an objective criterion to select the best model among given regression lines; always prefer the model/line with the lowest SSE for best fit.
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