Improving the methodology for spectral analysis of climatic time series

Abstract

The paper discusses a methodology able to estimate both the discrete and continuous spectra without any assumption on the shape of spectral densities. The approach to estimate the spectral density is based on a robust smoothing of the periodogram. Bandwidth, a quantity similar to the width of spectral windows traditionally used in spectral analysis, is estimated locally in contrast to intuitively chosen global window lengths. Detection and estimation of frequencies forming discrete spectra are also addressed. The procedure is applied to Central England temperature (CEt), North Atlantic Oscillation (NAO) index and Oxygen Isotope of North Greenland Ice Core Project (δ¹⁸O of NGRIP) data. Annual and half annual cycles were detected in CEt data, whilst 118.2- and 41.7-ky cycles were found in δ¹⁸O of NGRIP. This latter periodicity is almost as intense as the dominant longer cycle. Several local peaks of spectral densities were recognised in each time series that mostly cover earlier results. However, a few previous findings at low frequencies have not been reinforced by the present method. Identification of modest local peaks or discrete amplitudes at low frequencies is an extremely challenging task as climatic data generally have spectral densities rising to low frequencies.

Appendix

1.1 Local bandwidth selection

The mean squared error (MSE) to be minimised at every ω is ${\rm MSE} \left( \omega \right) = {u^2}\left( \omega \right) + {v^2}\left( \omega \right)$, where u and v ² denote the bias and variance of the estimate, respectively.

The asymptotic variance, utilising the sandwich formula of Huber (1967), is given by:

$$ {v^2}\left( \omega \right) = {{E\left[ {\sum\limits_{k = - L}^L {K_k^2\left( \omega \right){\Psi^2}\left( {I\left( {{\omega_k}} \right)/f\left( {{\omega_k}} \right) - 1} \right)} } \right]} \mathord{\left/{\vphantom {{E\left[ {\sum\limits_{k = - L}^L {K_k^2\left( \omega \right){\Psi^2}\left( {I\left( {{\omega_k}} \right)/f\left( {{\omega_k}} \right) - 1} \right)} } \right]} {{{\left( {E\left[ {\sum\limits_{k = - L}^L {{K_k}\left( \omega \right)\frac{d}{{ds}}\Psi \left( {I\left( {{\omega_k}} \right)/s - 1} \right)} \left| {_{s = f(\omega )}} \right.} \right]} \right)}^2}}}} \right.} {{{\left( {E\left[ {\sum\limits_{k = - L}^L {{K_k}\left( \omega \right)\frac{d}{{ds}}\Psi \left( {I\left( {{\omega_k}} \right)/s - 1} \right)} \left| {_{s = f(\omega )}} \right.} \right]} \right)}^2}}} = .729{f^2}\left( \omega \right)/\left( {Tb} \right), $$

where ${K_k}\left( \omega \right) = K\left( {\left( {\omega - {\omega_k}} \right)/b} \right)$. A simulation study was performed in order to compare the exact variance to asymptotic variance. It was found that the difference between the two variances is negligible even for b > 15/T. For smaller bandwidths, the exact variances are used.

When evaluating the bias u(ω), the relationship

$$ \int\limits_0^\infty {\sum\limits_{k = - L}^L {{K_k}\left( \omega \right)\Psi \left( {z - 1} \right){\lambda_k}\exp \left( { - {\lambda_k}z} \right)dz} } = 0 $$

should be satisfied due to Eq. 5, where ${\lambda_k} = S\left( \omega \right)/f\left( {{\omega_k}} \right)$ with $S\left( \omega \right) = E\left[ {s\left( \omega \right)} \right] = E\left[ {\hat f\left( \omega \right)} \right]$. The integration with some algebra results in the equation

$$ \sum\limits_{k = - L}^L {{K_k}\left( \omega \right)\frac{{\exp \left( { - \left( {{\lambda_k} + 1} \right)} \right) - \exp \left( { - 2{\lambda_k}} \right)}}{{{\lambda_k}}}} - (1 - {e^{ - 1}})\sum\limits_{k = - L}^L {{K_k}\left( \omega \right)} \frac{{{\lambda_k} - 1}}{{{\lambda_k}}} = 0, $$

and the solution $u\left( \omega \right) = S\left( \omega \right) - f\left( \omega \right)$ is obtained using a numerical procedure. The convergence is very fast when starting the procedure with

$$ 1/\left( {Tb} \right)\sum\limits_{k = - L}^L {{K_k}\left( \omega \right)} f\left( {{\omega_k}} \right) - f\left( \omega \right) $$

(8)

as the solution is close to Eq. 8.

Note that minimization of MSE requires the knowledge of f(ω). The unknown f(ω) can be substituted by its initial estimate obtained by Eq. 5 using a global bandwidth. This step is similar to the concept of Staniswalis (1989) introduced for local bandwidth selection for non-parametric curve fitting.