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6(3) + 1 = 18 + 1 = 19 Therefore, 19 is a prime number. Example 3: Find if 53 is a prime number or not. Solution: The only factors of 53 are 1 and 53. So, 53 is a prime number. Example 4: Check if 64 is a prime number or not. Solution: The factors of 64 are 1, 2, 4, 8, 16, 32, 64. Hence, it is a composite number and not a prime number. So total number of ways = n-1 P r = 5-1 P 3 = 4 P 3 = 24. Number of permutations of n different things taking all at a time, in which m specified things always come together = m!(n-m+1). Example 11: In how many ways can we arrange the five vowels, a, e, i, o and u if.
- Authors: Luke Johnston
- Research field: Nutritional Sciences
- Lesson content URL: https://github.com/UofTCoders/studyGroup/tree/gh-pages/lessons/r/resampling
In statistics, we use samples to infer about the population given some set ofconditions. We generally do not know or can ever know (without substantial costand time) the entire characteristics of the population, so we use the sample tomake guesses. Because the sample is not the entire population, the sample doesnot completely represent the population. That’s why there are standarddeviations, standard errors, and confidence intervals with all statistics. If wetook another sample, it would be slightly different from the original sample andwould have slightly different statistics. Since standard errors of thestatistics are calculated based on the sample, these estimates can be biased tothe sample and have certain mathematical assumptions about the distribution.What resampling does is to take randomly drawn (sub)samples of the sample andcalculate the statistic from that (sub)sample. Do this enough times and you canget a distribution of statistic values that can provide an empirical measure ofthe accuracy/precision of the test statistic, with less rigid assumptions. Inthis lesson, I’ll cover bootstrapping and permutation testing. The differencescome down to essentially:
- Bootstrapping is best used to estimate confidence intervals of test statistics.
- Permutation testing is best used for testing hypotheses.
Permute 3 1000 Mg
Bootstrapping
Bootstrapping is the process of resampling with replacement (all values inthe sample have an equal probability of being selected, including multipletimes, so a value could have a duplicate). Resample, calculate a statistic (e.g.the mean), repeat this hundreds or thousands of times and you are able toestimate a precise/accurate uncertainty of the mean (confidence interval) of thedata’s distribution. There are less assumptions about the underlyingdistribution using bootstrap compared to calculating the standard errordirectly.
There is a R package that does boostrapping, called
boot
. The boot
functionneeds a function that calculates the mean based on the resample of the data. Ittakes two arguments, the values (x
) and the resample vector of the values(i
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- Following this algorithm, the next lexicographic permutation will be 1,3,2,4, and the 24th permutation will be 4,3,2,1 at which point ak 3 comparisons and 1.5 swaps per permutation, amortized over the whole sequence, not counting the initial sort.
If you bootstrap for a statistic with two values, the individual rows areresampled. However, the rows are still paired together (no reshuffling of theorder occurs).
Permutation testing
Similar to bootstrapping, except permutation testing resamples withoutreplacement (meaning when a value is selected, it can not be selected again, sono value can be a duplicate). This simply shuffles the values. In the case of aunivariate statistic (e.g. mean), this will not change anything. However, whenthere are two or more variables, reshuffling one variable will change the teststatistic (e.g. correlation or regression). Usually this is done on theresponse/outcome/y variable, and usually for tests that use discrete orcategorical (“yes”, “no”) variables.
Taskpaper 3 7 5 – simple to do listening. There is a
coin
package that does permutation testing for several tests.Permute 3 1000 Watt
There are many more tests found within the
coin
package. Astropad 1 0. Check them out in the vignette: