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Least Squares Demonstration

Least Squares Demonstration
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2015-03-25 23:01
VectorFunctionCoordinate GeometryOptimization
This example demonstrates how the distance between a line and a set of points is calculated (SSE) for linear regression. Key math concepts include linear regression, least squares method, residuals, and fitted line. GeoGebra commands used are Point, Line, Intersect, Segment, Residual, FitLineY, and If. Suitable for high school and above students to understand the geometric meaning and calculation of linear regression.

GeoGebra Commands & Parameters

  1. Point({1.95, 0.35}) → A
  2. Point({1.01, 1.4}) → B
  3. Point({1.74, 1.27}) → C
  4. Point({2.17, 1.71}) → D
  5. Point({2.33, 0.61}) → E
  6. Point({3.75, 3.18}) → F
  7. Point({5.33, 3.84}) → G
  8. Point({4.17, 2.14}) → H
  9. Point({4.6, 2.66}) → I
  10. Line(B, yAxis) → b
  11. Line(J, K) → a
  12. Line(B, yAxis) → c
  13. Intersect(a, c) → L
  14. Segment(L, B) → d
  15. Residual(a, I) → e
  16. Residual(a, F) → f
  17. Residual(a, G) → g
  18. Residual(a, D) → h
  19. Residual(a, E) → i
  20. Residual(a, C) → j
  21. Residual(a, A) → k
  22. Residual(a, H) → l
  23. FitLineY({d, e, f, g, h, i, j, k, l}) →
  24. FitLineY({A, B, C, D, E, F, G, H, I}) → m
  25. Residual(m, B) → n
  26. Residual(m, C) → p
  27. Residual(m, A) → q
  28. Residual(m, D) → r
  29. Residual(m, E) → s
  30. Residual(m, F) → t
  31. Residual(m, H) → a_1
  32. Residual(m, I) → b_1
  33. Residual(m, G) → c_1
  34. If(showLine, false, true) → u
  35. If(showLine, false, true) → v

AI Drawing Prompt Reference

Please draw a graph demonstrating linear regression. The graph should include 9 data points (A, B, C, D, E, F, G, H, I) located in a coordinate system. Then draw a fitted line (m) that passes through all data points via the least squares fit. Calculate the vertical distances between each data point and the fitted line (residuals), including e, f, g, h, i, j, k, l, a_1, b_1, c_1. Show the relative positions of the fitted line and data points, and explain that residuals represent the distance between points and the fitted line. The construction order should be: first draw the data points, then draw the fitted line, then calculate the residuals, and finally display the graph.
The prompt above was compiled from the original drawing commands and is for reference only.