STATISTICAL THINKING IN PYTHON II. Generating bootstrap replicates
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1 STATISTICAL THINKING IN PYTHON II Generating bootstrap replicates
2 Michelson's speed of light measurements Data: Michelson, 1880
3 Resampling an array Data: [23.3, 27.1, 24.3, 25.3, 26.0] Mean = 25.2 Resampled data: [,,,, ]
4 Resampling an array Data: [23.3,, 24.3, 25.3, 26.0] Mean = 25.2 Resampled data: [27.1,,,, ]
5 Resampling an array Data: [23.3, 27.1, 24.3, 25.3, 26.0] Mean = 25.2 Resampled data: [27.1,,,, ]
6 Resampling an array Data: [23.3, 27.1, 24.3, 25.3, 26.0] Mean = 25.2 Resampled data: [27.1, 26.0,,, ]
7 Resampling an array Data: [23.3, 27.1, 24.3, 25.7, 26.0] Mean = 25.2 Resampled data: [27.1, 26.0, 23.3, 25.7, 23.3] Mean = 25.08
8 Mean of resampled Michelson measurements Statistical Thinking in Python II
9 Bootstrapping The use of resampled data to perform statistical inference
10 Bootstrap sample A resampled array of the data
11 Bootstrap replicate A statistic computed from a resampled array
12 Resampling engine: np.random.choice() In [1]: import numpy as np In [2]: np.random.choice([1,2,3,4,5], size=5) Out[2]: array([5, 3, 5, 5, 2])
13 Computing a bootstrap replicate In [1]: bs_sample = np.random.choice(michelson_speed_of_light,...: size=100) In [2]: np.mean(bs_sample) Out[2]: In [3]: np.median(bs_sample) Out[3]: In [4]: np.std(bs_sample) Out[4]:
14 STATISTICAL THINKING IN PYTHON II Let s practice!
15 STATISTICAL THINKING WITH PYTHON II Bootstrap confidence intervals
16 Bootstrap replicate function In [1]: def bootstrap_replicate_1d(data, func):...: """Generate bootstrap replicate of 1D data."""...: bs_sample = np.random.choice(data, len(data))...: return func(bs_sample)...: In [2]: bootstrap_replicate_1d(michelson_speed_of_light, np.mean) Out[2]: In [3]: bootstrap_replicate_1d(michelson_speed_of_light, np.mean) Out[3]: In [4]: bootstrap_replicate_1d(michelson_speed_of_light, np.mean) Out[4]:
17 Many bootstrap replicates In [1]: bs_replicates = np.empty(10000) In [2]: for i in range(10000):...: bs_replicates[i] = bootstrap_replicate_1d(...: michelson_speed_of_light, np.mean)...:
18 Plotting a histogram of bootstrap replicates In [1]: _ = plt.hist(bs_replicates, bins=30, normed=true) In [2]: _ = plt.xlabel('mean speed of light (km/s)') In [3]: _ = plt.ylabel('pdf') In [4]: plt.show()
19 Bootstrap estimate of the mean Statistical Thinking in Python II
20 Confidence interval of a statistic If we repeated measurements over and over again, p% of the observed values would lie within the p% confidence interval.
![Bootstrap confidence interval In [1]: conf_int = np.](/docs-images/81/84152817/images/21-1.jpg "percentile(bs_replicates, [2.5, 97.5]) Out[1]: array([ 299837.")
21 Bootstrap confidence interval In [1]: conf_int = np.percentile(bs_replicates, [2.5, 97.5]) Out[1]: array([ , ])
22 STATISTICAL THINKING WITH PYTHON II Let s practice!
23 STATISTICAL THINKING IN PYTHON II Pairs bootstrap
24 Nonparametric inference Make no assumptions about the model or probability distribution underlying the data
25 2008 US swing state election results Data retrieved from Data.gov (
26 Pairs bootstrap for linear regression Resample data in pairs Compute slope and intercept from resampled data Each slope and intercept is a bootstrap replicate Compute confidence intervals from percentiles of bootstrap replicates
27 Generating a pairs bootstrap sample In [1]: np.arange(7) Out[1]: array([0, 1, 2, 3, 4, 5, 6]) In [1]: inds = np.arange(len(total_votes)) In [2]: bs_inds = np.random.choice(inds, len(inds)) In [3]: bs_total_votes = total_votes[bs_inds] In [4]: bs_dem_share = dem_share[bs_inds]
28 Computing a pairs bootstrap replicate In [1]: bs_slope, bs_intercept = np.polyfit(bs_total_votes,...: bs_dem_share, 1) In [2]: bs_slope, bs_intercept Out[2]: ( e-05, ) In [3]: np.polyfit(total_votes, dem_share, 1) # fit of original Out[3]: array([ e-05, e+01])
29 2008 US swing state election results Data retrieved from Data.gov (
30 STATISTICAL THINKING IN PYTHON II Let s practice!
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