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bootstrap_regcoef

Bootstrap estimates of linear regression coefficient.

Available in version 6.4.0 and later.

Prototype

	function bootstrap_regcoef (
		x     [*] : numeric,  
		y         : numeric,  
		nBoot [1] : integer,  
		nDim  [*] : integer,  
		opt   [1] : logical   
	)

	return_val [ variable of type 'list' containing multiple estimates]

Arguments

x

A one-dimensional numeric array: x(N) where 'N' represents the sample size. This variable is referred as the 'explanatory' or 'independent' variable.

y

A numeric array of up to four dimensions: y(N), y(N,:), y(N,:,:), y(N,:,:,:) where 'N' represents the sample size.

nBoot

An integer specifying the number of bootstrap data samples to be generated.

nDim

The dimension(s) of y on which to calculate the statistic. Must be consecutive and monotonically increasing.

opt

A logical scalar to which optional attributes may be attached. If opt=False, default values are used. If opt=True and no optional attributes are present, default values will be used. If opt=True then:

  • opt@sample_size: specifies the size of the resampled array to be used for the bootstrapped statistics.
  • opt@sample_size=N is the default.
  • opt@sample_size=n where (n.le.N). When this option is used, n=toint(f*N) where 'f' represents (say) 0.10 to 0.20.

  • opt@rseed1=rseed1: allows user to set the first random seed integer value. Default is to use the system initial random seed. (See: random_setallseed)
  • opt@rseed2=rseed2: allows user to set the second random seed integer value. Default is to use the system initial random seed. (See: random_setallseed)
  • optrseed3="clock": tells NCL to use the 'date' clock to set the two random seeds. (See: random_setallseed)

Return value

A variable of type 'list'. Members of a list can be accessed directly. However, it is clearer if the members are explicity extracted and given meaningful names. 

                                    ; typeof(Bootstrap) is 'list'
   BootStrap  = bootstrap_regcoef(x, y, nBoot, 0, opt)
   rcBoot      = BootStrap[0]       ; Bootstrapped regression coefficients is ascending order
   rcBootAvg   = BootStrap[1]       ; Average of bootstrapped regression coefficients
   rcBootStd   = BootStrap[2]       ; Std. Deviation of bootstrapped regression coefficients
   delete(BootStrap)

Description

Bootstrapping is a statistical method that uses data resampling with replacement (see: generate_sample_indices) to estimate the properties of nearly any statistic. It is particularly useful when dealing with small sample sizes. A key feature is that bootstrapping makes no apriori assumption about the distribution of the sample data.

The current version resamples x and y pairs.

References: 

Computer Intensive Methods in Statistics 
   P. Diaconis and B. Efron 
   Scientific American (1983), 248:116-130  
   doi:10.1038/scientificamerican0583-116
   http://www.nature.com/scientificamerican/journal/v248/n5/pdf/scientificamerican0583-116.pdf
   
An Introduction to the Bootstrap 
   B. Efron and R.J. Tibshirani, Chapman and Hall (1993) 
   
Bootstrap Methods and Permutation Tests: Companion Chapter 18 to the Practice of Business Statistics
   Hesterberg, T. et al (2003)
   http://statweb.stanford.edu/~tibs/stat315a/Supplements/bootstrap.pdf

Climate Time Series Analysis: Classical Statistical and Bootstrap Methods
   M. Mudelsee (2014) Second edition. Springer, Cham Heidelberg New York Dordrecht London
   ISBN: 978-3-319-04449-1, e-ISBN: 978-3-319-04450-7
   doi: 10.1007/978-3-319-04450-7
   xxxii + 454 pp; Atmospheric and Oceanographic Sciences Library, Vol. 51

See Also

bootstrap_stat, bootstrap_diff, bootstrap_correl, bootstrap_estimate, generate_sample_indices, ListIndexFromName

Examples

Please see the Bootstrap and Resampling application page.

Example 1: Let x(100); y(100), N=100 


   nBoot        = 1000                ; user set
   nDim         = 0
   opt          = False               ; use defaults

   BootStrap    = bootstrap_regcoef(x, y, nBoot, nDim, opt)
   rcBoot       = BootStrap[0] ; Bootstrapped regression coefficients in ascending order
   rcBootAvg    = BootStrap[1] ; Average of the boot strapped regression coefficients
   rcBootStd    = BootStrap[2] ; Standard deviation of bootstrapped regression coefficients
   delete(BootStrap) ; no longer needed

   rcBootLow    = bootstrap_estimate(rcBoot, 0.025, False)   ;  2.5% lower confidence bound
   rcBootMed    = bootstrap_estimate(rcBoot, 0.500, False)   ; 50.0% median of bootstrapped estimates
   rcBootHi     = bootstrap_estimate(rcBoot, 0.975, False)   ; 97.5% upper confidence bound

   printVarSummary(rcBoot)        ; information only
   printVarSummary(rcBootMed)     ; examine meta data