ts_normstrap documentation

ts_normstrap performs a bootstrap uncertainty analysis on a time series given an uncertainty value at each step assuming a normal probability distribution. Bootstrapping means estimating a value at each time point within particular uncertainty bounds with a given probability distribution. Ultimately, the goal is to generate several realizations of the time series and provide confidence intervals at each time step.

Back to Climate Data Tools Contents

Syntax

tsb = ts_normstrap(ts)
tsb = ts_normstrap(ts,e)
tsb = ts_normstrap(ts,E)
tsb = ts_normstrap(ts,'nboot',nboot)
[tsb,Nts] = ts_normstrap(...)

Description

tsb = ts_normstrap(ts) calculates confidence intervals for a given timeseries after randomly subsampling vector ts 1000 times at each point with a normal probability, assuming an uncertainty of 1 standard deviation of the overall time series ts. The output tbs is a length(ts) x2 matrix containg the +/- 1 standard deviation bounds of time series ts. Note that ts is a vector without time dimensions as the bounds are returned at the points of query.

tsb = ts_normstrap(ts,e) specifies the uncertainty value e from which the uncertainty distribution at each step in the vector ts is calculated and thereby overrides the default value of 1 standard deviation of ts.

tsb = ts_normstrap(ts,E) specifies a vector E containing uncertainty values at each step from which the uncertainty distribution in vector ts is calculated.

tsb = ts_normstrap(...,'nboot',nboot) specifies the number of bootstrap samples. Default is 1000, meaning 1000 random time series are calculated.

[tsb,Nts] = ts_normstrap(...) also returns the 1000 (or the specified number of) randomly generated time series with given uncertainty.

[tsb,Nts] = ts_normstrap(...) also returns the 1000 (or the specified number of) randomly generated time series subsampled with the specified uncertainty.

Example

This example performs a bootstrap analysis of a randomly generated time series of 50 points, which we shall assume to be 50 measurements of oxygen isotopes in water with a mean value around -5 permil VSMOW.

iso_ts = -5 + randn(50,1);

% Let's say they were continuously sampled over 50 days in 2018
t1 = datetime(2018,1,1,8,0,0);
t = t1:t1+49;

Let's plot the data

figure
plot(t,iso_ts)
box off
axis tight
ylabel 'Oxygen Isotope Composition (permil VSMOW)'
set(gca,'ydir','reverse') % flips the direction of the Y-Axis Now we want uncertainty bounds for the time series

tsb = ts_normstrap(iso_ts);

% This will, by default, give uncertainty bounds close the the standard
% deviation of the overall |iso_ts| timeseries
overall_sd = std(iso_ts)
default_bootstrap_uncertainty = mean(tsb)

% Let's plot it up on the original plot as a 2-sigma bound (multiply by 2)
hold on;
plot(t,iso_ts+2.*tsb,':r')
plot(t,iso_ts-2.*tsb,':r')
overall_sd =
1.0830
default_bootstrap_uncertainty =
1.0817 Now let's be a little more specific and specify the analytical uncertainty of the oxygen isotopic measurement as 0.1 permil

tsb = ts_normstrap(iso_ts,0.1);

specified_bootstrap_uncertainty = mean(tsb)
specified_bootstrap_uncertainty =
0.1002

Let's plot the new, better uncertainty, also as a 2-sigma bound

hold on;
plot(t,iso_ts+2.*tsb,':k')
plot(t,iso_ts-2.*tsb,':k') Now let's compare a low number of bootstrap samples to a high number of samples

tsb_low = ts_normstrap(iso_ts,0.1,'nboot',3);
tsb_high = ts_normstrap(iso_ts,0.1,'nboot',500);

low_bootstrap_uncertainty = mean(tsb_low)
high_bootstrap_uncertainty = mean(tsb_high)
low_bootstrap_uncertainty =
0.0956
high_bootstrap_uncertainty =
0.0997

Let's plot them both now:

figure; hold on;
plot(t,iso_ts)
box off
axis tight
ylabel 'Oxygen Isotope Composition (permil VSMOW)'
set(gca,'ydir','reverse') % flips the direction of the Y-Axis

plot(t,iso_ts+2.*tsb_low,'--r')
plot(t,iso_ts-2.*tsb_low,'--r')

plot(t,iso_ts+2.*tsb_high,'--g')
plot(t,iso_ts-2.*tsb_high,'--g') Notice that sometimes the low number indicates higher uncertainty and sometimes it shows a lower uncertainty. Keep in mind that larger number of bootstrap samples will generally converge to some value.

For fun, let's plot all the time series that we have generated!

[tsb,Nts] = ts_normstrap(iso_ts,0.08,'nboot',500);

figure;hold on;
plot(t,Nts,'color',[0.7 0.7 0.7]) % In gray
plot(t,iso_ts,'ko-','linewidth',1.5); % In black References

The original bootstrap was introduced formally to the literature by Eforon in 1979: Efron, B., 1979: Bootstrap methods: another look at the jackknife. Ann. Stat. 7, 1-26. doi:10.1007/978-1-4612-4380-9_41.

Although this is a rather dense read! Here are some (paleoclimate) applications:

Thirumalai, K., T. M. Quinn, and G. Marino, 2016: Constraining past seawater delta-18-O and temperature records developed from foraminiferal geochemistry, Paleoceanography doi:10.1002/2016PA002970.

Carré, M., J. P. Sachs, J. M. Wallace, and C. Favier, 2012: Exploring errors in paleoclimate proxy reconstructions using Monte Carlo simulations: paleotemperature from mollusk and coral geochemistry, Clim. Past, 8(2), 433-450. doi:10.5194/cp-8-433-2012.

The first figure in both of these papers provide a flowchart to understand basic schematics of the bootstrap in age uncertainty.

Author Info

The ts_normstrap function and supporting information were written for The Climate Data Toolbox for Matlab by Kaustubh Thirumalai of the University of Arizona, January 2019.