# Category Archives: Mathematics

# Example of Bootstrapping

In a statistical analysis, bootstrapping is a popular technique, especially useful when the sample size is small. It is involved with resampling and the technique is assume nothing about the distribution of our data. It is noted that a small sample (<40), we can assuming a normal or a t-distributions.

Bootstrapping can be run in many statistical software in the market. For me, I like to use SPSS, since the software has integrating bootstrapping in each analysis. Many codes that implementing bootstrapping can be found in Matlab and R-language.

The following example is how this technique works that been obtained from this source:

Example SampleWe begin with a statistical sample from a population that we know nothing about. Our goal will be a 90% confidence interval about the mean of the sample. Although other statistical techniques used to determine confidence intervals assume that we know the mean or standard deviation of our population, bootstrapping does not require anything other than the sample. For purposes of our example, we will assume that the sample is 1, 2, 4, 4, 10.Example – Bootstrap SampleWe now resample with replacement from our sample to form what are known as bootstrap samples. Each bootstrap sample will have a size of five, just like our original sample. Since we randomly selecting and then are replacing each value, the bootstrap samples may be different from the original sample and from each other. For examples that we would run into in the real world we would do this resampling hundreds if not thousands of times. In what follows below, we will see an example of 20 bootstrap samples: 2, 1, 10, 4, 2 4, 10, 10, 2, 4 1, 4, 1, 4, 4 4, 1, 1, 4, 10 4, 4, 1, 4, 2 4, 10, 10, 10, 4 2, 4, 4, 2, 1 2, 4, 1, 10, 4 1, 10, 2, 10, 10 4, 1, 10, 1, 10 4, 4, 4, 4, 1 1, 2, 4, 4, 2 4, 4, 10, 10, 2 4, 2, 1, 4, 4 4, 4, 4, 4, 4 4, 2, 4, 1, 1 4, 4, 4, 2, 4 10, 4, 1, 4, 4 4, 2, 1, 1, 2 10, 2, 2, 1, 1Example – MeanSince we are using bootstrapping to calculate a confidence interval about the population mean, we now calculate the means of each of our bootstrap samples. These means, arranged in ascending order are: 2, 2.4, 2.6, 2.6, 2.8, 3, 3, 3.2, 3.4, 3.6, 3.8, 4, 4, 4.2, 4.6, 5.2, 6, 6, 6.6, 7.6.Example – Confidence IntervalWe now obtain from our list of bootstrap sample means a confidence interval. Since we want a 90% confidence interval, we use the 95th and 5th percentiles as the endpoints of the intervals. The reason for this is that we split 100% - 90% = 10% in half so that we will have the middle 90% of all of the bootstrap sample means. For our example above we have a confidence interval of 2.4 to 6.6.

# Monte Carlo Simulation 1 (Bookmark & Planning)

Monte-Carlo, is very famous nowadays. If you work with uncertainty analysis, you will become familiar with this term. From my knowledge (maybe I wrong), Monte-Carlo can be related as sampling method: same like bootstrapping or random generation. Since my study related with climate change, uncertainty became a dominant topic. I very confuse, with this topic and currently, I try to understand and applied it to my study. In this post, I find and collect all possible links and references in which related to Monte-Carlo simulation. In the end, I will summaries those collections into a new post, entitled Monte Carlo Simulation 2. Please wait for the future post.

Bookmarks:

- http://blog.minitab.com/blog/understanding-statistics/monte-carlo-is-not-as-difficult-as-you-think
- http://www.minitab.com/en-us/Published-Articles/Doing-Monte-Carlo-Simulation-in-Minitab-Statistical-Software/

# Resampling Data