Department of Electrical- and Information Technology. Dealing with stochastic processes

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1 Dealing with stochastic processes

2 The stochastic process (SP) Definition (in the following material): A stochastic process is random process that happens over time, i.e. the process is dynamic and changes over time. An SP can be continuous- or discrete-time If discrete-time, the events in the process are countable

3 Sampling refresher Let X1..Xn be independent random variables with same distribution function. Let Θ denote the mean and σ 2 denote the variance. Θ = E [Xi] and σ 2 = Var(Xi) Let X be the arithmetic average of X i,then: X = E[X] =E = nx i=1 X i n " n X i=1 nx i=1 X i n # E[X i ] n = n n = this shows that X is an unbiased estimator of

4 Sampling refresher How good an estimator is X of? Var X = E[(X! ) 2 ] 1 nx = Var X i n 1 = 1 X n n 2 Var(X i ) = 2 n 1 Therefore, X is a good estimator of when p n is small

5 Confidence Intervals We first need to revisit the Normal distribution. Consider N(0,1) P (Z apple w) = 1 p 2 Z w 1 e ( x µ ) 2 dx For example: P (Z apple 1) = 1 p 2 Z w 1 e ( x µ ) 2 dx =0.8413

6 Confidence Intervals We use this to calculate w for different values of P such as: P (Z apple w) =0.95 ) w =1.96 P (Z apple w) =0.99 ) w =2.58 P (Z apple w) =0.999 ) w =3.29

7 Confidence Intervals If X is the mean of a random sample of size n from N (µ, ) then the sampling distribution of X is N (µ, p n ) If X 1,X 2,...X n constitute random samples from an infinite population with mean µ and standard deviation then the sampling distribution of Z = X µ / p as n!1is N (0, 1) n This result is known as the Central Limit Theorem

8 Confidence Intervals What does this mean? If we sample a random process with finite variance, our samples tend to follow a gaussian distribution and we can estimate the probability that our true value is within a range from the sample with a certain confidence We often want to find some mean value from a process e.g. the mean number of people in a queue with arrival rate λ and mean service time μ Working with means we need one more piece of information:

9 Confidence Intervals The standard deviation of the mean is calculated as follows: Var(X) = Var(cX) =c 2 Var(X) 2 x Var(mean) =Var = 1 N 2 1 N! NX X i i=1 = 1 N 2 Var N X i=1 X i! NX Var(X i )= N N 2 Var(X) = 1 N Var(X) i=1 mean = p N

10 Confidence Intervals We can now set the intervals that give us the desired confidence in our results for mean values. For example, if we want to find the mean average queue length with a confidence of 95% we set: ConfidenceInterval = z ± 1.96 mean

11 Confidence Intervals Process for the example with the queue 1. Run the simulation to get mean q length 2. Calculate mean of the means sampled 3. Calculate stddev 4. Calculate confidence (in this case based on σmean) 5. Change random seed 6. Run again 7. until confidence interval small enough Can also run as a single longer experiment

12 Confidence Intervals Plotting confidence intervals in graphs make interpretation easier Think about overlapping confidence intervals

13 Stopping condition Question: How long should the simulation run for? I simulated for 10 minutes, not a good answer. Better idea is to use variance measurements Stopping condition: Variance < x

14 Warmup time Take as example a simulation of a M/M/1 queue Say λ = 9, μ = 10 This system has a queue that will grow to large sizes over time There is a lag before the system reaches steady state Consider carefully when to start collecting statistics

15 Warmup time 100 X = Z f(x)dx = X = 1 40 Z f(x)dx =

16 <latexit sha1_base64="g0wbpmy28dna82l2a7tfinzjtnw=">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</latexit> <latexit sha1_base64="g0wbpmy28dna82l2a7tfinzjtnw=">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</latexit> <latexit sha1_base64="g0wbpmy28dna82l2a7tfinzjtnw=">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</latexit> <latexit sha1_base64="g0wbpmy28dna82l2a7tfinzjtnw=">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</latexit> Variance reduction techniques There are ways of reducing variance to speed up convergence of confidence intervals from several runs One common approach is the use of Antithetic variates Consider the following. We have generated two identically distributed variables X1 and X2 with mean θ. Then: Var( X 1 + X 2 2 )= 1 4 [Var(X 1)+Var(X 2 )+2Cov(X 1,X 2 )]

17 Antithetic variables Variance will be reduced if X1,X2 negatively correlated (covariance negative) Consider X1 = f(u1,u2,u3 Um) where U are m independent random numbers. Then so are 1-Um Therefore, X2 = f(1-u1,1-u2,1-u3 1-Um) has the same distribution as X1 Good chance that X1 and X2 are negatively correlated when using Um and 1-Um for generation

18 Discussion For each seed, the two antithetic runs combined produce results closer to real mean with high probability. Not to be confused with correlation between samples in a series

19 Sample Correlation Calculating confidence intervals assumes uncorrelated samples If this is not the case, there is a problem Covariance again Let the sampled mean from a simulation run be â Then, cov(â, â) =cov 1 n nx x i, 1 n nx x j 1 A should be close to 0 i=1 j=1

20 Sample Correlation If the samples are not uncorrelated, the variance will be underestimated since: V (â) =cov 1 n nx x i, 1 n i=1 nx j=1 x j 1 A = 1 X X n 2 cov(x i,x j ) i j cov(â i, â j ) should be 0 when i 6= j When i = j, this reduces to the variance

21 Reducing covariance (1) Reducing covariance simplest way: sample less frequently Instead of sampling all events, generate a random variable for inter-sample times i.e. next sample = getsampletime( 1 e t ) This will also lower the data volume for a simulation run. could be useful Reduces risk of correlation because of cyclic phenomena

22 Reducing covariance (2) Batch means ˆbi ˆbi ˆbi ˆbi ˆbi ˆbi ˆbi Another fast way of lowering correlation between successive samples, use method of batch means. Take average from a sequence of samples and use as a sample Selecting current batch size a problem. Too large, confidence intervals become too large; too small, variance underestimated. From literature, 10 < size < 20 common

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sha1_base64="wild/kefyipygzdfigzipqwzsa4=">aaab8hicbvdlsgnbeoynrxhfuy9ebopgkeykomegf48rzeosjcxoosmqmdllzlyis77ciwdfvpo53vwbj8kenlggoajqprsrsgq31ve/vcla+sbmvng7tlo7t39qpjxqmjjvdbssfrfur9sg4aobllub7uqjlzhavjs+nfmtj9sgx+rbthimjr0qpucmwic9dioqs6jhp71yxa/6c5bveuskajnqvfjxtx+zvkkytfbjoogf2dcj2nimcfrqpgytysz0ib1hfzvowmx+8jscoavpbrf2psyzq78nmiqnmcjidupqr2bzm4n/ez3udq7djksktajyytegfctgzpy96xonziqji5rp7m4lbeq1zdzlvhihbmsvr5lmrtxwq8h9zav2k8drhbm4hxmi4apqcad1aaadcc/wcm+e9l68d+9j0vrw8plj+apv8wcbrzch</latexit> <latexit sha1_base64="wild/kefyipygzdfigzipqwzsa4=">aaab8hicbvdlsgnbeoynrxhfuy9ebopgkeykomegf48rzeosjcxoosmqmdllzlyis77ciwdfvpo53vwbj8kenlggoajqprsrsgq31ve/vcla+sbmvng7tlo7t39qpjxqmjjvdbssfrfur9sg4aobllub7uqjlzhavjs+nfmtj9sgx+rbthimjr0qpucmwic9dioqs6jhp71yxa/6c5bveuskajnqvfjxtx+zvkkytfbjoogf2dcj2nimcfrqpgytysz0ib1hfzvowmx+8jscoavpbrf2psyzq78nmiqnmcjidupqr2bzm4n/ez3udq7djksktajyytegfctgzpy96xonziqji5rp7m4lbeq1zdzlvhihbmsvr5lmrtxwq8h9zav2k8drhbm4hxmi4apqcad1aaadcc/wcm+e9l68d+9j0vrw8plj+apv8wcbrzch</latexit> <latexit sha1_base64="wild/kefyipygzdfigzipqwzsa4=">aaab8hicbvdlsgnbeoynrxhfuy9ebopgkeykomegf48rzeosjcxoosmqmdllzlyis77ciwdfvpo53vwbj8kenlggoajqprsrsgq31ve/vcla+sbmvng7tlo7t39qpjxqmjjvdbssfrfur9sg4aobllub7uqjlzhavjs+nfmtj9sgx+rbthimjr0qpucmwic9dioqs6jhp71yxa/6c5bveuskajnqvfjxtx+zvkkytfbjoogf2dcj2nimcfrqpgytysz0ib1hfzvowmx+8jscoavpbrf2psyzq78nmiqnmcjidupqr2bzm4n/ez3udq7djksktajyytegfctgzpy96xonziqji5rp7m4lbeq1zdzlvhihbmsvr5lmrtxwq8h9zav2k8drhbm4hxmi4apqcad1aaadcc/wcm+e9l68d+9j0vrw8plj+apv8wcbrzch</latexit> <latexit sha1_base64="wild/kefyipygzdfigzipqwzsa4=">aaab8hicbvdlsgnbeoynrxhfuy9ebopgkeykomegf48rzeosjcxoosmqmdllzlyis77ciwdfvpo53vwbj8kenlggoajqprsrsgq31ve/vcla+sbmvng7tlo7t39qpjxqmjjvdbssfrfur9sg4aobllub7uqjlzhavjs+nfmtj9sgx+rbthimjr0qpucmwic9dioqs6jhp71yxa/6c5bveuskajnqvfjxtx+zvkkytfbjoogf2dcj2nimcfrqpgytysz0ib1hfzvowmx+8jscoavpbrf2psyzq78nmiqnmcjidupqr2bzm4n/ez3udq7djksktajyytegfctgzpy96xonziqji5rp7m4lbeq1zdzlvhihbmsvr5lmrtxwq8h9zav2k8drhbm4hxmi4apqcad1aaadcc/wcm+e9l68d+9j0vrw8plj+apv8wcbrzch</latexit> <latexit sha1_base64="wild/kefyipygzdfigzipqwzsa4=">aaab8hicbvdlsgnbeoynrxhfuy9ebopgkeykomegf48rzeosjcxoosmqmdllzlyis77ciwdfvpo53vwbj8kenlggoajqprsrsgq31ve/vcla+sbmvng7tlo7t39qpjxqmjjvdbssfrfur9sg4aobllub7uqjlzhavjs+nfmtj9sgx+rbthimjr0qpucmwic9dioqs6jhp71yxa/6c5bveuskajnqvfjxtx+zvkkytfbjoogf2dcj2nimcfrqpgytysz0ib1hfzvowmx+8jscoavpbrf2psyzq78nmiqnmcjidupqr2bzm4n/ez3udq7djksktajyytegfctgzpy96xonziqji5rp7m4lbeq1zdzlvhihbmsvr5lmrtxwq8h9zav2k8drhbm4hxmi4apqcad1aaadcc/wcm+e9l68d+9j0vrw8plj+apv8wcbrzch</latexit> <latexit sha1_base64="diefvaexeqmv+9i5o9irqxxbt6s=">aaab9hicbvbns8naej34wetx1aoxxsj4kokieix68vjbfkabyma7azdunnf3uighv8olb0w8+mo8+w/ctjlo64mzhu/nslmvskqw6lrfztr6xubwdmmnvlu3f3byotpumtjvjddzlgpdcajhuijerigsdxlnarri3g7gdzo/pehaifg94jthfkshsoscubss3wuozuytypn+perw3dnikvekuoucjx7lqzeiwrpxhuxsy7qem6cfuy2csz6xe6nhcwvjourdsxwnupgz+de5obfkgisxtqwqznxfgxmnjjlggz2mki7msjct//o6kyy3fizukijxbpfqmeqcmzklqazcc4zyagllwthbcrtrtrnanmo2bg/5y6ukdvnz3jr3cfwt3xzxloauzuacpligotxda5ra4ame4rxeninz4rw7h4vrnafyoye/cd5/ae9mkna=</latexit> <latexit sha1_base64="diefvaexeqmv+9i5o9irqxxbt6s=">aaab9hicbvbns8naej34wetx1aoxxsj4kokieix68vjbfkabyma7azdunnf3uighv8olb0w8+mo8+w/ctjlo64mzhu/nslmvskqw6lrfztr6xubwdmmnvlu3f3byotpumtjvjddzlgpdcajhuijerigsdxlnarri3g7gdzo/pehaifg94jthfkshsoscubss3wuozuytypn+perw3dnikvekuoucjx7lqzeiwrpxhuxsy7qem6cfuy2csz6xe6nhcwvjourdsxwnupgz+de5obfkgisxtqwqznxfgxmnjjlggz2mki7msjct//o6kyy3fizukijxbpfqmeqcmzklqazcc4zyagllwthbcrtrtrnanmo2bg/5y6ukdvnz3jr3cfwt3xzxloauzuacpligotxda5ra4ame4rxeninz4rw7h4vrnafyoye/cd5/ae9mkna=</latexit> <latexit sha1_base64="diefvaexeqmv+9i5o9irqxxbt6s=">aaab9hicbvbns8naej34wetx1aoxxsj4kokieix68vjbfkabyma7azdunnf3uighv8olb0w8+mo8+w/ctjlo64mzhu/nslmvskqw6lrfztr6xubwdmmnvlu3f3byotpumtjvjddzlgpdcajhuijerigsdxlnarri3g7gdzo/pehaifg94jthfkshsoscubss3wuozuytypn+perw3dnikvekuoucjx7lqzeiwrpxhuxsy7qem6cfuy2csz6xe6nhcwvjourdsxwnupgz+de5obfkgisxtqwqznxfgxmnjjlggz2mki7msjct//o6kyy3fizukijxbpfqmeqcmzklqazcc4zyagllwthbcrtrtrnanmo2bg/5y6ukdvnz3jr3cfwt3xzxloauzuacpligotxda5ra4ame4rxeninz4rw7h4vrnafyoye/cd5/ae9mkna=</latexit> Testing correlation, deciding on batch size Mean of a batch = mean of all batches = batch size = N Test for correlation magnitude as follows: b i b<latexit sha1_base64="diefvaexeqmv+9i5o9irqxxbt6s=">aaab9hicbvbns8naej34wetx1aoxxsj4kokieix68vjbfkabyma7azdunnf3uighv8olb0w8+mo8+w/ctjlo64mzhu/nslmvskqw6lrfztr6xubwdmmnvlu3f3byotpumtjvjddzlgpdcajhuijerigsdxlnarri3g7gdzo/pehaifg94jthfkshsoscubss3wuozuytypn+perw3dnikvekuoucjx7lqzeiwrpxhuxsy7qem6cfuy2csz6xe6nhcwvjourdsxwnupgz+de5obfkgisxtqwqznxfgxmnjjlggz2mki7msjct//o6kyy3fizukijxbpfqmeqcmzklqazcc4zyagllwthbcrtrtrnanmo2bg/5y6ukdvnz3jr3cfwt3xzxloauzuacpligotxda5ra4ame4rxeninz4rw7h4vrnafyoye/cd5/ae9mkna=</latexit> Then, if: P N 1 i=1 C =1 ( b i bi+1 ) 2 P N 1 i=1 ( b i b) 2 C (N 2)(N 2 N (0, 1) 1) then batch size well chosen

24 <latexit sha1_base64="wu1+jfnzicoaq80d+wypnpzl8ek=">aaacfxicbvdlssnafj34rpuvdelmsagupgre0e2h6mznoyj9qbpdzdpph84kywzikse/4czfcencebeco//g6wohrqcuhm65l3vvcrlolhacb2tpewv1bb2wudzc2t7ztff2mypojaenevnytgoskgcrbwimow0nkmircnokbtdjv/vapwjxdkdhcfue7kuszarri/n2qrtgcyewat1qypkhpkvl0nvmuavdlqo/yxwu39fg0ge+xxlkzgrwkaazkyez6r795xzjkgoaackxuh3kjnrlsnsmcjox3vtrbjmb7tgoore2w71s8luoj43shwestuuatttfexkwso1eydof1n01743f/7xoqsnll2nrkmoakemimovqx3aceewysynmi0mwkczcckkfm3s0cbjoqkdzly+s5lkzowv0e16qxs3ikibdcaroaaixoapuqb00aagp4bm8gjfryxqx3q2paeusnzs5ah9gff4a/daefq==</latexit> <latexit sha1_base64="wu1+jfnzicoaq80d+wypnpzl8ek=">aaacfxicbvdlssnafj34rpuvdelmsagupgre0e2h6mznoyj9qbpdzdpph84kywzikse/4czfcencebeco//g6wohrqcuhm65l3vvcrlolhacb2tpewv1bb2wudzc2t7ztff2mypojaenevnytgoskgcrbwimow0nkmircnokbtdjv/vapwjxdkdhcfue7kuszarri/n2qrtgcyewat1qypkhpkvl0nvmuavdlqo/yxwu39fg0ge+xxlkzgrwkaazkyez6r795xzjkgoaackxuh3kjnrlsnsmcjox3vtrbjmb7tgoore2w71s8luoj43shwestuuatttfexkwso1eydof1n01743f/7xoqsnll2nrkmoakemimovqx3aceewysynmi0mwkczcckkfm3s0cbjoqkdzly+s5lkzowv0e16qxs3ikibdcaroaaixoapuqb00aagp4bm8gjfryxqx3q2paeusnzs5ah9gff4a/daefq==</latexit> <latexit sha1_base64="wu1+jfnzicoaq80d+wypnpzl8ek=">aaacfxicbvdlssnafj34rpuvdelmsagupgre0e2h6mznoyj9qbpdzdpph84kywzikse/4czfcencebeco//g6wohrqcuhm65l3vvcrlolhacb2tpewv1bb2wudzc2t7ztff2mypojaenevnytgoskgcrbwimow0nkmircnokbtdjv/vapwjxdkdhcfue7kuszarri/n2qrtgcyewat1qypkhpkvl0nvmuavdlqo/yxwu39fg0ge+xxlkzgrwkaazkyez6r795xzjkgoaackxuh3kjnrlsnsmcjox3vtrbjmb7tgoore2w71s8luoj43shwestuuatttfexkwso1eydof1n01743f/7xoqsnll2nrkmoakemimovqx3aceewysynmi0mwkczcckkfm3s0cbjoqkdzly+s5lkzowv0e16qxs3ikibdcaroaaixoapuqb00aagp4bm8gjfryxqx3q2paeusnzs5ah9gff4a/daefq==</latexit> <latexit sha1_base64="wu1+jfnzicoaq80d+wypnpzl8ek=">aaacfxicbvdlssnafj34rpuvdelmsagupgre0e2h6mznoyj9qbpdzdpph84kywzikse/4czfcencebeco//g6wohrqcuhm65l3vvcrlolhacb2tpewv1bb2wudzc2t7ztff2mypojaenevnytgoskgcrbwimow0nkmircnokbtdjv/vapwjxdkdhcfue7kuszarri/n2qrtgcyewat1qypkhpkvl0nvmuavdlqo/yxwu39fg0ge+xxlkzgrwkaazkyez6r795xzjkgoaackxuh3kjnrlsnsmcjox3vtrbjmb7tgoore2w71s8luoj43shwestuuatttfexkwso1eydof1n01743f/7xoqsnll2nrkmoakemimovqx3aceewysynmi0mwkczcckkfm3s0cbjoqkdzly+s5lkzowv0e16qxs3ikibdcaroaaixoapuqb00aagp4bm8gjfryxqx3q2paeusnzs5ah9gff4a/daefq==</latexit> Compensate for covariance contribution The below is typically more difficult but a possibility! There are statistical methods in literature e.g. 2 x = nx i=1 2 1 var(x i )+2 n nx j=i cov(x i,x j ) n If the process is stationary in regards to covariance and k-dependent, loosely, if a sample is dependent on k previous samples this becomes: 2 x = 1 n ( var(x i )+2 ) kx cov(x i,x i+1 ) i=1 We use the following estimate of the covariance from the literature: cov(w i,w i+s )= 1 n s P n 1 i=1 (w i w)(w i+s w) w = 1 M M X i=1 w i

25 <latexit sha1_base64="wu1+jfnzicoaq80d+wypnpzl8ek=">aaacfxicbvdlssnafj34rpuvdelmsagupgre0e2h6mznoyj9qbpdzdpph84kywzikse/4czfcencebeco//g6wohrqcuhm65l3vvcrlolhacb2tpewv1bb2wudzc2t7ztff2mypojaenevnytgoskgcrbwimow0nkmircnokbtdjv/vapwjxdkdhcfue7kuszarri/n2qrtgcyewat1qypkhpkvl0nvmuavdlqo/yxwu39fg0ge+xxlkzgrwkaazkyez6r795xzjkgoaackxuh3kjnrlsnsmcjox3vtrbjmb7tgoore2w71s8luoj43shwestuuatttfexkwso1eydof1n01743f/7xoqsnll2nrkmoakemimovqx3aceewysynmi0mwkczcckkfm3s0cbjoqkdzly+s5lkzowv0e16qxs3ikibdcaroaaixoapuqb00aagp4bm8gjfryxqx3q2paeusnzs5ah9gff4a/daefq==</latexit> <latexit sha1_base64="wu1+jfnzicoaq80d+wypnpzl8ek=">aaacfxicbvdlssnafj34rpuvdelmsagupgre0e2h6mznoyj9qbpdzdpph84kywzikse/4czfcencebeco//g6wohrqcuhm65l3vvcrlolhacb2tpewv1bb2wudzc2t7ztff2mypojaenevnytgoskgcrbwimow0nkmircnokbtdjv/vapwjxdkdhcfue7kuszarri/n2qrtgcyewat1qypkhpkvl0nvmuavdlqo/yxwu39fg0ge+xxlkzgrwkaazkyez6r795xzjkgoaackxuh3kjnrlsnsmcjox3vtrbjmb7tgoore2w71s8luoj43shwestuuatttfexkwso1eydof1n01743f/7xoqsnll2nrkmoakemimovqx3aceewysynmi0mwkczcckkfm3s0cbjoqkdzly+s5lkzowv0e16qxs3ikibdcaroaaixoapuqb00aagp4bm8gjfryxqx3q2paeusnzs5ah9gff4a/daefq==</latexit> <latexit sha1_base64="wu1+jfnzicoaq80d+wypnpzl8ek=">aaacfxicbvdlssnafj34rpuvdelmsagupgre0e2h6mznoyj9qbpdzdpph84kywzikse/4czfcencebeco//g6wohrqcuhm65l3vvcrlolhacb2tpewv1bb2wudzc2t7ztff2mypojaenevnytgoskgcrbwimow0nkmircnokbtdjv/vapwjxdkdhcfue7kuszarri/n2qrtgcyewat1qypkhpkvl0nvmuavdlqo/yxwu39fg0ge+xxlkzgrwkaazkyez6r795xzjkgoaackxuh3kjnrlsnsmcjox3vtrbjmb7tgoore2w71s8luoj43shwestuuatttfexkwso1eydof1n01743f/7xoqsnll2nrkmoakemimovqx3aceewysynmi0mwkczcckkfm3s0cbjoqkdzly+s5lkzowv0e16qxs3ikibdcaroaaixoapuqb00aagp4bm8gjfryxqx3q2paeusnzs5ah9gff4a/daefq==</latexit> <latexit sha1_base64="wu1+jfnzicoaq80d+wypnpzl8ek=">aaacfxicbvdlssnafj34rpuvdelmsagupgre0e2h6mznoyj9qbpdzdpph84kywzikse/4czfcencebeco//g6wohrqcuhm65l3vvcrlolhacb2tpewv1bb2wudzc2t7ztff2mypojaenevnytgoskgcrbwimow0nkmircnokbtdjv/vapwjxdkdhcfue7kuszarri/n2qrtgcyewat1qypkhpkvl0nvmuavdlqo/yxwu39fg0ge+xxlkzgrwkaazkyez6r795xzjkgoaackxuh3kjnrlsnsmcjox3vtrbjmb7tgoore2w71s8luoj43shwestuuatttfexkwso1eydof1n01743f/7xoqsnll2nrkmoakemimovqx3aceewysynmi0mwkczcckkfm3s0cbjoqkdzly+s5lkzowv0e16qxs3ikibdcaroaaixoapuqb00aagp4bm8gjfryxqx3q2paeusnzs5ah9gff4a/daefq==</latexit> <latexit sha1_base64="p2uvtdwtyfoqeesyusxy8hsur40=">aaachxicbvdlssnafj34rpvvdelmsaiualikuiy6cvnbpqaj4wyybyfojgfmoptqh3hjr7hxoygln+lfoe2z0nyda4dzzuxopuhcmdk2/w2trk6tb2ywtsrbo7t7+5wdw46ku0lom8q8lr0afousom3nnke9rfiqaafdyhw987v3vcowr3d6klbpwdbia0zag8mvnnwajh7abikwy81cch7mak9ggm/ru+ysmnbyvwwozkyenvqvql2zc+bl4hskigq0/mqng8ykfttshinsfcdotjeb1ixwoi27qaijkdemad/qcarvxpzfn8wnrgnxijbmrrrn6u+jdirsexgypaa9uovetpzp66d6colllepstsmyxzriodyxnlwfqyyp0xxicbdjzf8xgyeeok2hzvocs3jymunua45dc24b1ezvuucjhamtdiycdiga6aa1ubsr9iie0st6s56sf+vd+phhv6xi5gj9gfx1a2jjox0=</latexit> <latexit sha1_base64="p2uvtdwtyfoqeesyusxy8hsur40=">aaachxicbvdlssnafj34rpvvdelmsaiualikuiy6cvnbpqaj4wyybyfojgfmoptqh3hjr7hxoygln+lfoe2z0nyda4dzzuxopuhcmdk2/w2trk6tb2ywtsrbo7t7+5wdw46ku0lom8q8lr0afousom3nnke9rfiqaafdyhw987v3vcowr3d6klbpwdbia0zag8mvnnwajh7abikwy81cch7mak9ggm/ru+ysmnbyvwwozkyenvqvql2zc+bl4hskigq0/mqng8ykfttshinsfcdotjeb1ixwoi27qaijkdemad/qcarvxpzfn8wnrgnxijbmrrrn6u+jdirsexgypaa9uovetpzp66d6colllepstsmyxzriodyxnlwfqyyp0xxicbdjzf8xgyeeok2hzvocs3jymunua45dc24b1ezvuucjhamtdiycdiga6aa1ubsr9iie0st6s56sf+vd+phhv6xi5gj9gfx1a2jjox0=</latexit> <latexit sha1_base64="p2uvtdwtyfoqeesyusxy8hsur40=">aaachxicbvdlssnafj34rpvvdelmsaiualikuiy6cvnbpqaj4wyybyfojgfmoptqh3hjr7hxoygln+lfoe2z0nyda4dzzuxopuhcmdk2/w2trk6tb2ywtsrbo7t7+5wdw46ku0lom8q8lr0afousom3nnke9rfiqaafdyhw987v3vcowr3d6klbpwdbia0zag8mvnnwajh7abikwy81cch7mak9ggm/ru+ysmnbyvwwozkyenvqvql2zc+bl4hskigq0/mqng8ykfttshinsfcdotjeb1ixwoi27qaijkdemad/qcarvxpzfn8wnrgnxijbmrrrn6u+jdirsexgypaa9uovetpzp66d6colllepstsmyxzriodyxnlwfqyyp0xxicbdjzf8xgyeeok2hzvocs3jymunua45dc24b1ezvuucjhamtdiycdiga6aa1ubsr9iie0st6s56sf+vd+phhv6xi5gj9gfx1a2jjox0=</latexit> <latexit sha1_base64="p2uvtdwtyfoqeesyusxy8hsur40=">aaachxicbvdlssnafj34rpvvdelmsaiualikuiy6cvnbpqaj4wyybyfojgfmoptqh3hjr7hxoygln+lfoe2z0nyda4dzzuxopuhcmdk2/w2trk6tb2ywtsrbo7t7+5wdw46ku0lom8q8lr0afousom3nnke9rfiqaafdyhw987v3vcowr3d6klbpwdbia0zag8mvnnwajh7abikwy81cch7mak9ggm/ru+ysmnbyvwwozkyenvqvql2zc+bl4hskigq0/mqng8ykfttshinsfcdotjeb1ixwoi27qaijkdemad/qcarvxpzfn8wnrgnxijbmrrrn6u+jdirsexgypaa9uovetpzp66d6colllepstsmyxzriodyxnlwfqyyp0xxicbdjzf8xgyeeok2hzvocs3jymunua45dc24b1ezvuucjhamtdiycdiga6aa1ubsr9iie0st6s56sf+vd+phhv6xi5gj9gfx1a2jjox0=</latexit> In plain text, estimate mean from long run Sample mean values wi and calculate sampled average taking into account dependency Using adjusted σ 2. w = 1 M M X i=1 w i, estimate covariance and calculate Use this to estimate confidence interval with significance 1-α as: w ± /2 w

26 <latexit sha1_base64="ne/pbzx9gskdgu3axpfnoctgurs=">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</latexit> <latexit sha1_base64="ne/pbzx9gskdgu3axpfnoctgurs=">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</latexit> <latexit sha1_base64="ne/pbzx9gskdgu3axpfnoctgurs=">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</latexit> <latexit sha1_base64="ne/pbzx9gskdgu3axpfnoctgurs=">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</latexit> Compensate for covariance contribution Problem, estimate k. Can be done using the autoregressive approach. Translate {w} to a set {u} of independent variables u i = kx b s (w i s w), i = k +1,...,n i=0 We estimate ˆb s from a set of linear equations: kx ˆbs ˆRr s = ˆRr r =1,...,k s=1 Where ˆR s is the estimated covariance cov(w t,w t+s ) and ˆb s are estimated coefficients of b s

27 Compensate for covariance contribution The variance at the k th order is estimated to be: ˆ2u,k = kx ˆbs ˆRs ˆb0 1 s=0 pick k max = K and calculate ˆ2u,k for k =1,...,K formulate a 2 test where the density Q is Q = n 1 ˆu,K ˆu,k The hypothesis sequence w 1,...,w n is of order k is discarded with significance if Q> 2 (f) wheref = K k signifies degree of freedom After k is found: 2 w = 2 kx u,k nb 2 where b = s=0 ˆbs We have to use a t-distribution instead of the normal distribution for confidence limits where: w ± t a/2 (f) w and f = nb 2 Pk s=0 (k 2s)ˆb s

28 Sample Correlation Important! your aim is to capture the correlation between different tests on the process you do not want to capture correlation from dependencies between successive samples often, it is therefore useful to run many simulation runs with varying seeds rather than few long simulation runs not always though, depends on the nature of the simulation

29 Tradeoff Each simulation run brings warmup time (wasted time) Many shorter simulations are more costly in time Estimating and compensating for covariance possible but takes time can be tedious work to do the stats. However, if statistical analysis not carried out, results don t have meaning.

30 More information Very good resource is Sheldon Ross, Simulation 4th ed. academic press 2006 Books on time series analysis, especially sections on auto regression for methods on estimating and compensating for covariance