A Statistical Journey through Historic Haverford
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1 A Statistical Journey through Historic Haverford Arthur Berg University of California, San Diego
2 Arthur Berg A Statistical Journey through Historic Haverford 2/ 18
3 The Dataset Histograms Oldest? Arthur Berg KDE Simulations Youngest? Correlation Smallest? Poisson Conclusions Largest? A Statistical Journey through Historic Haverford 3/ 18
4 The Dataset Histograms Woodside Cottage Built: 1811 Size: 6,485 ft2 Arthur Berg KDE Simulations Athletic Center Built: 2005 Size: 100,000 ft2 Correlation Skating House Built: 1949 Size: 651 ft2 Poisson Conclusions Science Center Built: 2002 Size: 188,000 ft2 A Statistical Journey through Historic Haverford 3/ 18
5 Arthur Berg A Statistical Journey through Historic Haverford 4/ 18
6 Arthur Berg A Statistical Journey through Historic Haverford 4/ 18
7 Arthur Berg A Statistical Journey through Historic Haverford 4/ 18
8 Arthur Berg A Statistical Journey through Historic Haverford 4/ 18
9 Arthur Berg A Statistical Journey through Historic Haverford 5/ 18
10 The negative kernel function K(x) satisfies x j K(x) = 0 for all j = 1, 2, 3,.... Arthur Berg A Statistical Journey through Historic Haverford 6/ 18
11 Which one is better? Arthur Berg A Statistical Journey through Historic Haverford 7/ 18
12 Statistical Experiments Computer Simulations Arthur Berg A Statistical Journey through Historic Haverford 8/ 18
13 Statistical Experiments Computer Simulations Arthur Berg A Statistical Journey through Historic Haverford 8/ 18
14 Start with a known density like N (0, 1). Arthur Berg A Statistical Journey through Historic Haverford 9/ 18
15 1 2 3 N RMSE flat-top RMSE Gaussian Arthur Berg A Statistical Journey through Historic Haverford 10/ 18
16 Testing Correlation H 0 : There isn t a linear relationship between Year and Area H 1 : There is a linear relationship between Year and Area Arthur Berg A Statistical Journey through Historic Haverford 11/ 18
17 Testing Normality A Pretest H 0 : Data is normally distributed H 1 : Data is not normally distributed Normally Distributed Parametric Test (e.g. Pearson test) Not Normally Distributed Nonparametric Test (e.g. Spearman test) Arthur Berg A Statistical Journey through Historic Haverford 12/ 18
18 Poisson Processes Does construction at Haverford follow a Poisson process? Figure: Siméon-Denis Poisson ( ) Number of customers arriving in line Number of telephone calls to a switchboard Number of particle emissions during radioactive decay Number of web page requests to a server Arthur Berg A Statistical Journey through Historic Haverford 13/ 18
19 The interarrival times are independent and exponentially distributed. The number of arrivals in disjoint intervals are independently distributed Poisson random variables. Arthur Berg A Statistical Journey through Historic Haverford 14/ 18
20 years= years= years sorted= Interval Length (yrs) Observed (O i ) Expected (E i ) (O i E i ) 2 /E i > Σ Arthur Berg A Statistical Journey through Historic Haverford 15/ 18
21 Hypothesis Test H 0 : Interarrival times of Haverford s buildings are exponentially distributed. H 1 : Interarrival times of Haverford s buildings are not exponentially distributed, i.e. construction does not follow a Poisson process. Significance level: α =.05 Rejection region: reject H 0 if χ 2 3 > Test statistic: χ 2 3 =.795 p-value:.851 Conclusion: Fail to reject H 0. Arthur Berg A Statistical Journey through Historic Haverford 16/ 18
22 Binned Years Observed (O i ) Expected (E i ) (O i E i ) 2 /E i Σ Hypothesis Test H 0 : The number of buildings built every 49 years follows a Poisson distribution. H 1 : The number of buildings built every 49 years does not follow a Poisson distribution., i.e. construction does not follow a Poisson process. Significance level: α =.05 Rejection region: reject H 0 if χ 2 3 > Test statistic: χ 2 3 = p-value:.570 Conclusion: Fail to reject H 0. Arthur Berg A Statistical Journey through Historic Haverford 17/ 18
23 Conclusions & Extensions Conclusions: Statistically speaking, construction at Haverford does follow a Poisson process. Future construction is not influenced by past construction Time to the next building is independent of the elapsed time since the last building. There are lots of practical things Statistics enables you to do! Extensions: Compare construction with economic events. Incorporate the map use geographical information in the analysis Test for iidness and clumping possibly new research topics?! Arthur Berg A Statistical Journey through Historic Haverford 18/ 18
n =10,220 observations. Smaller samples analyzed here to illustrate sample size effect.
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