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The Leaning Tower of Pisa: Construction, Stability, and Predictive Analysis of Its Tilt

The Leaning Tower of Pisa, begun in 1173, began sinking soon after construction of its second floor in 1178. To address the tilt, engineers built taller levels on one side, completing the seventh floor in 1319 and adding a bell tower in 1372. Despite ongoing tilt monitoring, the tower was closed in 1990 due to structural concerns. Stabilization efforts completed in 2008 involved removing soil from the taller side. This analysis uses historical data to explore the relationship between the tower's angle of tilt and the years, offering insights into predictive modeling.

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The Leaning Tower of Pisa: Construction, Stability, and Predictive Analysis of Its Tilt

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  1. Topic 2: An Example

  2. Leaning Tower of Pisa • Construction began in 1173 and by 1178 (2nd floor), it began to sink • Construction resumed in 1272. To compensate for tilt, engineers built upper levels with one side taller • Seventh floor completed in 1319 with bell tower added in 1372 • Tilt continued to grow over time and was monitored. Closed in 1990. • Stabilization completed in 2008 by removing ground from taller side

  3. Leaning Tower of Pisa • Response variable the lean (Y) • Lean in tenths of mm past 2.9 meters • Explanatory variable year (X) • Construct scatterplot • Can we use a line to describe relationship? • Want to predict the future lean

  4. SAS Data Step data a1; input year lean @@; cards; 75 642 76 644 77 656 78 667 79 673 80 688 81 696 82 698 83 713 84 717 85 725 86 742 87 757 112 . ; data a1p; set a1; if lean ne .; run;

  5. SAS Output Settings • Version 9.3: all output is by default in HTML • May prefer output in RTF or listing format to cut and paste in editor (e.g., Microsoft Word) ods html close; ods rtf file="H:\pisa.rtf"; ….SAS commands…. ods rtf close;

  6. Proc Print Specify the data set to use proc print data=a1; run; Will print all variables if none are specified using var statement

  7. The data set arranged in columns. First row provides names for variables

  8. Proc Gplot symbol1 v=circle i=sm70; proc gplot data=a1p; plot lean*year; run; symbol1 v=circle i=rl; proc gplot data=a1p; plot lean*year; run; Requests a smoothed curve be added to the plot Requests the least-squares regression line be added to the plot

  9. Smoothed curve

  10. Regression line

  11. Proc Reg proc reg data=a1; model lean=year / clb p r; output out=a2 p=pred r=resid; id year; run;

  12. Background Reading • Appendix A. • A.3 : random variables • A.4 : probability distributions • Chapter 1 • 1.3 : simple linear regression • 1.6 : estimation of regression function • 1.7 : estimation of error variance • 1.8 : normal error regression model

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