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;; example with smoothing
;; Newcomb's speed of light data
(use '(incanter core stats charts))
;; A numeric vector giving the Third Series of measurements of the
;; passage time of light recorded by Newcomb in 1882. The given
;; values divided by 1000 plus 24 give the time in millionths of a
;; second for light to traverse a known distance. The 'true' value is
;; now considered to be 33.02.
(def x [28 -44 29 30 24 28 37 32 36
27 26 28 29 26 27 22 23 20
25 25 36 23 31 32 24 27 33
16 24 29 36 21 28 26 27 27
32 25 28 24 40 21 31 32 28
26 30 27 26 24 32 29 34 -2
25 19 36 29 30 22 28 33 39
25 16 23])
;; view histogram of data to see outlier observations
(view (histogram x
:nbins 20
:title "Newcomb's speed of light data"))
;; calculate the median of the data
(median x) ;; 27
;; define a function that converts Newcomb's data into speeds
(defn to-speed
"Converts Newcomb's data into speed (meters/sec)"
([x] (div 7400 (div (plus 24800 x) 1E9))))
;; convert the data to speeds and calculate the median
(median (to-speed x)) ;; 2.981E8
;; Draw 10000 bootstrap samples of the median
(def t* (bootstrap x median :size 10000))
;; view a histogram of the bootstrap medians
(view (histogram t*
:density true
:nbins 20
:title "Bootstrap sample of medians"
:x-label "median"))
;; Calculate the estimate of the median: 27.301
(mean t*)
;; Convert bootstrap median estimate to speed: 2.981E8
(to-speed (mean t*))
;; Calculate a 95% CI for the median: (26.0 28.5)
(quantile t* :probs [0.025 0.975])
;; Convert to speed and calculate 95% CI: (2.9804E8 2.9807E8)
(quantile (to-speed t*) :probs [0.025 0.975])
;; estimate the standard error of the median: 0.681
(sd t*)
;; estimate the bias of the sample median: -0.3
(- (mean t*) (median x))
;; draw 10000 smoothed bootstrap samples of the median
(def smooth-t* (bootstrap x median :size 10000 :smooth true))
(view (histogram smooth-t*
:density true
:nbins 20
:title "Smoothed bootstrap sample of medians"
:x-label "median"))(mean smooth-samp)
;; Calculate the estimate of the median
(mean smooth-t*)
;; Calculate a 95% CI: (25.905 28.446)
(quantile smooth-t* :probs [0.025 0.975])
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