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Resampling techniques. Why resampling? Jacknife Cross-validation Bootstrap Examples of application of bootstrap. Why resampling?.
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Resampling techniques • Why resampling? • Jacknife • Cross-validation • Bootstrap • Examples of application of bootstrap
Why resampling? One of the purposes of statistics is to estimate some parameters and their reliability. Since estimators are functions of sample points they are random variables. If we could find distribution of this random variable (sample statistic) then we could estimate reliability of the estimators. Unfortunately apart from the simplest cases, sampling distribution is not easy to derive. There are several techniques to approximate them. These include: Edgeworth series, Laplace approximation, saddle-point approximations. They give analytical forms for the approximate distributions. With advent of computers computationally intensive methods are emerging. They work in many cases satisfactorily. Examples of simplest cases where sample distributions are known include: • Sample mean, when sample is from a population with normal distribution, has normal distribution with mean value equal to the population mean and variance equal to variance of the population divided by the sample size if population variance is known. If population variance is not known then variance of sample mean is the sample variance divided by n. • Sample variance has the distribution of multiple of 2 distribution. Again it is valid if population distribution is normal and sample points are independent. • Sample mean divided by square root of sample variance has the multiple of the t distribution – again normal and independent case • For independent samples and normal distribution: sample variance divided by sample variance has the multiple of F-distribution.
Resampling techniques Three of the popular computer intensive resampling techniques are: • Jacknife. It is a useful tool for bias removal. It may work fine for medium and large samples. • Cross-validation. Very useful technique for model selection. It may help to choose “best” model among those under consideration. • Bootstrap. Perhaps one of the most important resampling techniques. It can reduce bias as well as can give variance of the estimator. Moreover it can give the distribution of the statistic under consideration. This distribution can be used for such wide variety purposes as interval estimation, hypothesis testing.
Jacknife Jacknife is used for bias removal. As we know, mean-square error of an estimator is equal to the square of the bias plus the variance of the estimator. If the bias is much higher than variance then under some circumstances Jacknife could be used. Description of Jacknife: Let us assume that we have a sample of size n. We estimate some sample statistics using all the data – tn. Then by removing one point at a time we estimate tn-1,i, where subscript indicates the size of the sample and the index of the removed sample point. Then new estimator is derived as: If the order of the bias of the statistic tn is O(n-1) then after the jacknife the order of the bias becomes O(n-2). Variance is estimated using: This procedure can be applied iteratively. I.e. for the new estimator jacknife can be applied again. First application of Jacknife can reduce bias without changing variance of the estimator. But its second and higher order application can in general increases the variance of the estimator.
Jacknife: An example data mean 0) 368 390 379 260 404 318 352 359 216 222 283 332 323.5833 Jacknife samples 1) 390 379 260 404 318 352 359 216 222 283 332 319.5455 2) 368 379 260 404 318 352 359 216 222 283 332 317.5455 3) 368 390 260 404 318 352 359 216 222 283 332 318.5455 4) 368 390 379 404 318 352 359 216 222 283 332 329.3636 5) 368 390 379 260 318 352 359 216 222 283 332 316.2727 6) 368 390 379 260 404 352 359 216 222 283 332 324.0909 7) 368 390 379 260 404 318 359 216 222 283 332 321.0000 8) 368 390 379 260 404 318 352 216 222 283 332 320.3636 9) 368 390 379 260 404 318 352 359 222 283 332 333.3636 10) 368 390 379 260 404 318 352 359 216 283 332 332.8182 11) 368 390 379 260 404 318 352 359 216 222 332 327.2727 12) 368 390 379 260 404 318 352 359 216 222 283 322.8182 tjack = 12*323.5833-11*mean(t) =323.5833. It is an unbiased estimator var(t) = 345.7058 Let us take a data set of size 12 and perform jacknife for mean value.
Cross-validation Cross-validation is a resampling technique to overcome overfitting. Let us consider a least-squares technique. Let us assume that we have a sample of size ny=(y1,y2,,,yn). We want to estimate the parameters =(1, 2,,, m). Now let us further assume that mean value of the observations is a function of these parameters (we may not know form of this function). Then we can postulate that function has a form g. Then we can find values of the parameters using least-squares techniques. Where X is a fixed matrix or a matrix of random variables. After minimisation of h we will have values of the parameters, therefore complete definition of the function. Form of the function g defines model we want to use. We may have several forms of the function. Obviously if we have more parameters, the fit will be “better”. Question is what would happen if we would have new observations. Using estimated values of the parameters we could estimate the square of differences. Let us say we have new observations (yn+1,,,yn+l). Can our function predict these new observations? Which function predicts better? To answer to these questions we can calculate new differences: Where PE is the prediction error. Function g that gives smallest value for PE have higher predictive power. Model that gives smaller h but larger PE corresponds to overfitted model.
Cross-validation: Cont. If we have a sample of observations, can we use this sample and choose among given models. Cross validation attempts to reduce overfitting thus help model selection. Description of cross-validation: We have a sample of the size n. • Divide sample into K roughly equal size parts. • For the k-th part, estimate parameters using K-1 parts excluding k-th part. Calculate prediction error for k-th part. • Repeat it for all k=1,2,,,K and combine all prediction errors and get cross-validation prediction error. If K=n then we will have leave-one-out cross-validation technique. Let us denote an estimate at the k-th step by k (it is a vector of parameters). Let k-th subset of the sample be Akand number of points in this subset is Nk.. Then prediction error per observation is: Then we would choose the function that gives the smallest prediction error. We can expect that in future when we will have new observations this function will give smallest prediction error. This technique is widely used in modern statistical analysis. It is not restricted to least-squares technique. Instead of least-squares we could could use other techniques such as . maximum-likelihood, Bayesian estimation.
Bootstrap Bootstrap is one of the computationally expensive techniques. Its simplicity and increasing computational power makes this technique as a method of choice in many applications. In a very simple form it works as follows. We have a sample of size n. We want to estimate some parameter . The estimator for this parameter gives t. For each sample point we assign probability (usually equal to 1/n, i.e. all sample points have equal probability). Then from this sample with replacement we draw another random sample of size n and estimate . Let us denote an estimate of the parameter by ti*at the j-th resampling stage. Bootstrap estimator for and its variance is calculated as: It is a very simple form of application of the bootstrap resampling. For the parameter estimation, the number of the bootstrap samples is usually chosen to be around 200. When distribution is desired then the recommended number is around 1000-2000 Let us analyse the working of bootstrap in one simple case. Consider a random variable X with sample space x=(x1,,,,xM). Each point have the probability fj. I.e. f=(f1,,,fM) represents the distribution of the population. The sample of size n will have relative frequencies for each sample point as
Bootstrap: Cont. Then distribution of conditional on f will be multinomial distribution: Multinomial distribution is the extension of the binomial distribution and expressed as: Limiting distribution of: Is multinormal distribution. If we resample from the given sample then we should consider conditional distribution of the following (that is also multinomial distribution): Limiting distribution of is the same as the conditional distribution of the original sample. Since these two distribution converge to the same distribution then well behaved function of them also will have same limiting distributions. Thus if we use bootstrap to derive distribution of the sample statistic we can expect that in the limit it will converge to the distribution of sample statistic. I.e. following two function will have the same limiting distributions:
Bootstrap: Cont. If we could enumerate all possible resamples from our sample then we could build “ideal” bootstrap distribution. In practice even with modern computers it is impossible to achieve. Instead Monte Carlo simulation is used. Usually it works like: • Draw a random sample of size of n with replacement from the given sample of size n. • Estimate parameter and get the estimate tj. • Repeat step 1) and 2) B times and build frequency and cumulative distributions for t
Bootstrap: Cont. While resampling we did not use any assumption about the population distribution. Si this bootstrap is a non-parametric bootstrap. If we have some idea about the population distribution then we can use it in resampling. I.e. when we draw randomly from our sample we can use population distribution. For example if we know that population distribution is normal then we can estimate its parameters using our sample (sample mean and variance). Then we can approximate population distribution with this sample distribution and use it to draw new samples. As it can be expected if assumption about population distribution is correct then parametric bootstrap will perform better. If it is not correct then non-parametric bootstrap will overperform its parametric counterpart.
Balanced bootstrap One of the variation of bootstrap resampling is balanced bootstrap. In this case, when resampling one makes sure that number of occurrences of each sample point is the same. I.e. if we make B bootstrap we try to make the number of xiequal to B in all bootstrap samples. Of course, in each sample some of the observation will be present several times and other will be missing. But for all of them we want to make sure that all sample points are present and their number of occurrences is the same. It can be achieved as follows: Let us assume that the number of sample points is n. • Repeat numbers from 1 to n, B times • Find a random permutation of numbers from 1 to nB. Call it a vector N(nB) • Take the first n points from N and the corresponding sample points. Estimate parameter of interest. Then take the second n points (from n+1 to 2n) and corresponding sample points and do estimation. Repeat it B times and find bootstrap estimators, distributions and etc.
Balanced bootstrap: Example. Let us assume that we have 3 sample points and number of bootstraps we want is 3. Our observations are: (x1,x2,x3) Then we repeat numbers from 1 to 3 three times: 1 2 3 1 2 3 1 2 3 Then we take one of the random permutations of numbers from 1 to 3x3=9. E.g. 4 3 9 5 6 1 2 8 7 First we take observations x1,x3,x3 estimate the parameter Then we take x2,x3,x1 and estimate the parameter Then we take x2,x2,x1 and we estimate parameter. As it can be seen each observation is present 3 times. This technique meant to improve the results of bootstrap resampling.
Bootstrap: Example. Let us take the example we used for Jackknife. We generate 10000 (simple) bootstrap samples and estimate for each of them the mean value. Here is the bootstrap distribution of the estimated parameter. This distribution now can be used for various purposes (for variance estimation, for interval estimation, hypothesis testing and so on). For comparison the normal distribution with mean equal to the sample mean and variance equal to the sample variance divided by number of elements is also given (black line) . It seems that the approximation with the normal distribution was sufficiently good.
References • Efron, B (1979) Bootstrap methods: another look at the jacknife. Ann Statist. 7, 1-26 • Efron, B Tibshirani, RJ (1993) “An Introduction to the Bootstrap” • Chernick, MR. (1999) Bootstrap Methods: A practitioner’s Guide. • Berthold, M and Hand, DJ (2003) Intelligent Data Analysis • Kendall’s advanced statistics, Vol 1 and 2
Exercise 2 Differences between means and bootstrap confidence intervals Take data set from R - sleep. Find differences between means for treatments 1 and 2. Find confidence intervals using t.test and bootstrap technique. Necessary commands: data(sleep) a = sleep$extra[sleep$group==1] b = sleep$extra[sleep$group==2] t.test - standard t-test boot_mean - It is available from mres_course/2006 webpage. Write a report.