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R package with a set of functions to select the optimal block-length for a dependent bootstrap (block-bootstrap). Includes the Hall, Horowitz, and Jing (1995) cross-validation method and the Politis and White (2004) Spectral Density Plug-in method.
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Alec-Stashevsky/blocklength
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blocklength is an R package used to automatically select theblock-length parameter for a block-bootstrap. It is meant for use withdependent data such as stationary time series.
Regular bootstrap methods rely on assumptions that observations areindependent and identically distributed (i.i.d.), but this assumptionfails for many types of time series because we would expect theobservation in the previous period to have some explanatory power overthe current observation. This could occur in any time series fromunemployment rates, stock prices, biological data, etc. A time seriesthat isi.i.d. would look like white noise, since the followingobservation would be totally independent of the previous one (random).
To get around this problem, we can retain some of thistime-dependenceby breaking-up a time series into a number of blocks with lengthl.Instead of sampling each observation randomly (with replacement) like aregular bootstrap, we can resample theseblocks at random. This waywithin each block the time-dependence is preserved.
The problem with the block bootstrap is the high sensitivity to thechoice of block-length, or the number of blocks to break the time seriesinto.
The goal ofblocklength is to simplify and automate the process ofselecting a block-length to perform a bootstrap on dependent data.blocklength has several functions that take their name from theauthors who have proposed them. Currently, there are three methodsavailable:
hhj()takes its name from theHall, Horowitz, and Jing(1995) “HHJ” method toselect the optimal block-length using a cross-validation algorithmwhich minimizes the mean squared error(MSE) incurred by thebootstrap at various block-lengths.pwsd()takes its name from thePolitis and White(2004) Spectral Density“PWSD” Plug-in method to automatically select the optimalblock-length using spectral density estimation via “flat-top” lagwindows ofPolitis and Romano(1995).nppi()takes its name from theLahiri, Furukawa, and Lee(2007) NonparametricPlug-In “NPPI” method to select the optimal block-length for blockbootstrap procedures. The NPPI method estimates the leading term inthe first-order expansion of the theoretically optimal block lengthby using resampling methods to construct consistent bias andvariance estimators for the block-bootstrap. Specifically, thispackage implements the Moving Block Bootstrap (MBB) method ofKünsch (1989) and theMoving Blocks Jackknife (MBJ) ofLiu and Singh(1992) as the bias andvariance estimators, respectively.
Under the hood,hhj() uses the moving block bootstrap (MBB) procedureaccording toKünsch(1989) which resamplesblocks from a set of overlapping sub-samples with a fixed block-length.However, the results ofhhj() may be generalized to other blockbootstrap procedures such as thestationary bootstrap ofPolitis andRomano (1994).
Compared topwsd(),hhj() is more computationally intensive as itrelies on iterative sub-sampling processes that optimize the MSEfunction over each possible block-length (or a select grid ofblock-lengths), whilepwsd() is a simpler “plug-in” rule that usesauto-correlations, auto-covariance, and the spectral density of theseries to optimize the choice of block-length. Similarly,nppi() isanother “plug-in” rule, however, due to its heavy reliance onresampling, it can also be computationally intensive compared topwsd().
For a detailed comparison, see the table below:
| NPPI (Lahiri et al., 2007) | PWSD (Politis & White, 2004) | HHJ (Hall, Horowitz & Jing, 1995) | |
|---|---|---|---|
| Method Type | Nonparametric resampling | Spectral density estimation | Subsampling-based cross-validation |
| Computational Cost | Medium (bootstrap resampling & jackknife) | Low (direct ACF computation) | High (subsampling & cross-validation) |
| Primary Goal | Minimize MSE of bootstrap estimator | Estimate block length using spectral density | Minimize MSE via cross-validation |
| Variance Estimation | Moving Blocks Jackknife-After-Bootstrap (JAB) | Implicitly estimated via spectral density | Uses subsample-based variance estimation |
| Bias Estimation | Directly estimates bias from bootstrap | Indirectly accounts for bias via ACF decay | Uses subsample-based bias estimation |
| Best for | General-purpose estimators, small sample sizes, and quantile estimation | Block-length selection for circular and stationary bootstrap, time series with strong autocorrelation | Estimating functionals with strong dependencies |
| Estimation Capacity | Bootstrap bias, variance, distribution function, and quantile estimation | Bootstrap sample mean only | Bootstrap variance and distribution function estimation |
| Dependency* | User-defined parameters for initial block-lengthl and number of deletion blocksm | User-defined parameters for autocorrelation lag and implied hypothesis tests (4 total) | Requires user-defined parameters forpilot_block_length (l*) andsub_sample size (m) |
* All algorithms have default user-defined parameters recomended by therespective authors.
You can install the released version fromCRAN with:
install.packages("blocklength")You can install the development version fromGitHub with:
# install.packages("devtools")devtools::install_github("Alec-Stashevsky/blocklength")
We want to select the optimal block-length to perform a block bootstrapon a simulated autoregressiveAR(1) time series.
First we will generate the time series:
library(blocklength)# Simulate AR(1) time seriesseries<-stats::arima.sim(model=list(order= c(1,0,0),ar=0.5),n=500,rand.gen=rnorm)# Coerce time series to data.frame (not necessary)data<-data.frame("AR1"=series)
Now, we can find the optimal block-length to perform a block-bootstrap.We do this using the three available methods.
## Using the HHJ Algorithm with overlapping subsamples of width 10hhj(series,sub_sample=10,k="bias/variance")#> Pilot block length is: 3#> Registered S3 method overwritten by 'quantmod':#> method from#> as.zoo.data.frame zoo#> Performing minimization may take some time#> Calculating MSE for each level in subsample: 10 function evaluations required.
#> Chosen block length: 11 After iteration: 1
#> Converged at block length (l): 11#> $`Optimal Block Length`#> [1] 11#> #> $`Subsample block size (m)`#> [1] 10#> #> $`MSE Data`#> Iteration BlockLength MSE#> 1 1 4 0.5428141#> 2 1 7 0.5159969#> 3 1 11 0.5058036#> 4 1 15 0.5101443#> 5 1 18 0.5173569#> 6 1 22 0.5285706#> 7 1 26 0.5307042#> 8 1 29 0.5459579#> 9 1 33 0.5596887#> 10 1 37 0.5536393#> 11 2 4 0.5404021#> 12 2 7 0.5165052#> 13 2 11 0.5083257#> 14 2 15 0.5096941#> 15 2 18 0.5099022#> 16 2 22 0.5251345#> 17 2 26 0.5297922#> 18 2 29 0.5411978#> 19 2 33 0.5595094#> 20 2 37 0.5541540#> #> $Iterations#> [1] 2#> #> $Series#> [1] "series"#> #> $Call#> hhj(series = series, sub_sample = 10, k = "bias/variance")#> #> attr(,"class")#> [1] "hhj"# Using Politis and White (2004) Spectral Density Estimationpwsd(data)
#> $BlockLength#> b_Stationary b_Circular#> AR1 9.327923 10.67781#> #> $Acf#> $Acf$AR1#> #> Autocorrelations of series 'data[, i]', by lag#> #> 0 1 2 3 4 5 6 7 8 9 10 #> 1.000 0.550 0.291 0.175 0.107 0.066 0.049 0.010 -0.078 -0.057 -0.071 #> 11 12 13 14 15 16 17 18 19 20 21 #> -0.081 -0.071 -0.097 -0.073 -0.054 -0.010 0.014 -0.030 -0.042 -0.020 -0.073 #> 22 23 24 25 26 27 28 #> -0.088 -0.050 -0.058 -0.097 -0.049 0.021 0.078 #> #> #> $parameters#> n k c K_N M_max b_max m_hat M rho_k_critical#> [1,] 500 1 1.959964 5 28 68 3 6 0.1439999#> #> $Call#> pwsd(data = data)#> #> attr(,"class")#> [1] "pwsd"We can see that both methods produce similar results for a block-lengthof 9 or 11 depending on the type of bootstrap method used.
# Using Lahiri, Furukawa, and Lee (2007) Nonparametric Plug-Innppi(data,m=8)#> Setting l to recomended value: 3
#> $optimal_block_length#> [1] 0.01073086#> #> $bias#> [1] 0.01152002#> #> $variance#> [1] 0.03309448#> #> $jab_point_values#> [1] -0.0327374132 0.0086065594 0.0025366424 0.0406465091 0.0406465091#> [6] 0.0027838449 -0.0030100569 0.0406465091 -0.0971565707 -0.0371154550#> [11] -0.0755599178 -0.0755599178 0.0339749181 -0.0153968136 0.0145322529#> [16] 0.0169365682 -0.0333963967 -0.0333963967 -0.0333963967 0.0005655976#> [21] 0.0005655976 0.0005655976 -0.0708860990 -0.0270703095 -0.0270703095#> [26] -0.0659660916 0.0105882914 -0.0534955263 0.0206597579 0.0206597579#> [31] -0.0011544990 -0.0011544990 0.0183578992 -0.0301712541 0.0054632495#> [36] -0.0266459338 -0.0054879294 0.0134722464 0.0134722464 0.0449865026#> [41] 0.0449865026 0.0449865026 0.0449865026 0.0449865026 0.0100703950#> [46] -0.0889593545 0.0310801474 -0.0243945020 0.0152544405 -0.0278311082#> [51] -0.0116288527 -0.0320142219 -0.0320142219 -0.0142486081 -0.0052232227#> [56] -0.0052232227 -0.0052232227 -0.0276679506 0.0423320598 -0.0917146684#> [61] -0.0672953105 0.0678550387 -0.0378347310 -0.0378347310 -0.0818476398#> [66] -0.0427210926 -0.0250356679 -0.0250356679 0.0150383573 -0.0180655321#> [71] -0.0011544990 -0.0446551570 -0.0994751979 -0.0994751979 -0.0576207962#> [76] -0.0396419555 -0.0396419555 -0.0396419555 -0.0003119844 -0.0540061018#> [81] 0.0803656842 -0.0528692544 -0.0646321946 -0.0646321946 -0.0402261089#> [86] -0.0208329391 -0.0389210110 0.0489710594 -0.0099764121 -0.0355391996#> [91] -0.0536628938 -0.0378180773 0.0124084253 0.0124084253 -0.0644349773#> [96] -0.0644349773 -0.0644349773 -0.0369211728 -0.0238227629 -0.0238227629#> [101] 0.0333384171 0.0333384171 0.0140114628 0.0447980697 -0.1019902258#> [106] -0.0428492854 -0.0142220238 -0.1019902258 -0.1019902258 -0.0835156159#> [111] -0.0819563010 -0.0819563010 -0.0185758839 -0.0266705165 -0.0144722001#> [116] -0.0107257918 0.0487134593 0.0487134593 0.0002546279 -0.0931438608#> [121] -0.0011345209 -0.0011345209 0.0375753481 0.0062359912 0.0866643452#> [126] 0.0866643452 0.0866643452 0.0246924924 -0.0187366056 -0.0934362236#> [131] -0.0603000732 -0.0603000732 -0.0417117852 -0.0074732676 -0.0074732676#> #> $jab_pseudo_values#> [1] 0.64920441 -1.88311391 -1.51133149 -3.84556083 -3.84556083 -1.52647265#> [7] -1.17159616 -3.84556083 4.59487780 0.91735947 3.27208282 3.27208282#> [13] -3.43692588 -0.41290732 -2.24606264 -2.39332695 0.68956715 0.68956715#> [19] 0.68956715 -1.39060500 -1.39060500 -1.39060500 2.98581141 0.30209431#> [25] 0.30209431 2.68446096 -2.00449499 1.92063884 -2.62137232 -2.62137232#> [31] -1.28524909 -1.28524909 -2.48038348 0.49202717 -1.69058618 0.27610129#> [37] -1.01982647 -2.18113724 -2.18113724 -4.11138543 -4.11138543 -4.11138543#> [43] -4.11138543 -4.11138543 -1.97277384 4.09279831 -3.25962118 0.13820110#> [49] -2.29029663 0.34869323 -0.64369492 0.60490894 0.60490894 -0.48323491#> [55] -1.03603976 -1.03603976 -1.03603976 0.33869982 -3.94880081 4.26156129#> [61] 2.76587562 -5.51208327 0.96141512 0.96141512 3.65720579 1.26070477#> [67] 0.17747251 0.17747251 -2.27706154 -0.24944831 -1.28524909 1.37916621#> [73] 4.73689372 4.73689372 2.17331162 1.07210762 1.07210762 1.07210762#> [79] -1.33685310 1.95191159 -6.27836030 1.88227969 2.60275977 2.60275977#> [85] 1.10788702 -0.07994463 1.02794977 -4.35543954 -0.74490691 0.82081383#> [91] 1.93089010 0.96039509 -2.11597820 -2.11597820 2.59068021 2.59068021#> [97] 2.59068021 0.90545968 0.10318208 0.10318208 -3.39794020 -3.39794020#> [103] -2.21416425 -4.09984392 4.89093918 1.26855658 -0.48486319 4.89093918#> [109] 4.89093918 3.75936933 3.66386129 3.66386129 -0.21818926 0.27760699#> [115] -0.46953990 -0.69900740 -4.33966153 -4.33966153 -1.37155811 4.34909933#> [121] -1.28647275 -1.28647275 -3.65745222 -1.73791661 -6.66415329 -6.66415329#> [127] -6.66415329 -2.86837731 -0.20834505 4.36700654 2.33741733 2.33741733#> [133] 1.19888469 -0.89822451 -0.89822451#> #> $l#> [1] 3#> #> $m#> [1] 8#> #> attr(,"class")#> [1] "nppi"A big shoutout to Malina Cheeneebash for designing theblocklength hexsticker! Also to Sergio Armella andSimon P.Couch for their help and feedback!
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R package with a set of functions to select the optimal block-length for a dependent bootstrap (block-bootstrap). Includes the Hall, Horowitz, and Jing (1995) cross-validation method and the Politis and White (2004) Spectral Density Plug-in method.
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