Climacogram versus autocovariance and power spectrum in stochastic modelling for Markovian and Hurst–Kolmogorov processes

Abstract

Three common stochastic tools, the climacogram i.e. variance of the time averaged process over averaging time scale, the autocovariance function and the power spectrum are compared to each other to assess each one’s advantages and disadvantages in stochastic modelling and statistical inference. Although in theory, all three are equivalent to each other (transformations one another expressing second order stochastic properties), in practical application their ability to characterize a geophysical process and their utility as statistical estimators may vary. In the analysis both Markovian and non Markovian stochastic processes, which have exponential and power-type autocovariances, respectively, are used. It is shown that, due to high bias in autocovariance estimation, as well as effects of process discretization and finite sample size, the power spectrum is also prone to bias and discretization errors as well as high uncertainty, which may misrepresent the process behaviour (e.g. Hurst phenomenon) if not taken into account. Moreover, it is shown that the classical climacogram estimator has small error as well as an expected value always positive, well-behaved and close to its mode (most probable value), all of which are important advantages in stochastic model building. In contrast, the power spectrum and the autocovariance do not have some of these properties. Therefore, when building a stochastic model, it seems beneficial to start from the climacogram, rather than the power spectrum or the autocovariance. The results are illustrated by a real world application based on the analysis of a long time series of high-frequency turbulent flow measurements.

Appendix

Here, we express the expected value of the discrete time autocovariance in terms only of its true continuous time value using the corresponding true climacogram. This is very useful in stochastic modelling as it saves computational time (compared to a direct calculation where a sum throughout all the discrete time autocovariances is needed) and also because it gives a physical interpretation of the expected discrete time autocovariance.

Equation 2 can be expressed in terms of the true discrete autocovariance:

$$ \gamma (k\Delta ) = \frac{1}{{k^{2} }}\mathop \sum \limits_{i = 1}^{k} \mathop \sum \limits_{j = 1}^{k} c_{d}^{(\Delta )} (i - j) = \frac{2}{{k^{2} }}\mathop \sum \limits_{i = 1}^{k - 1} \left( {k - i} \right)c_{d}^{(\Delta )} (i) + \frac{\gamma (\Delta )}{k} $$

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The estimation of autocovariance in Eq. 9 can be analysed to:

$$ \begin{aligned}{\text{E}}\left[ {\widehat{{\underline{c}}}_{d}^{\Delta } \left( j \right)} \right] &= {\text{E}}\left[{\frac{1}{\zeta \left( j \right)}\sum\limits_{l = 1}^{n} {\left( {\widehat{{\underline{x} }}_{l}^{\left( \Delta \right)} - \frac{1}{n}\left( {\sum\limits_{l = 1}^{n} {\widehat{{\underline{x}}}_{l}^{\left( \Delta \right)} } } \right)} \right)\;\left( {\widehat{{\underline{x} }}_{i + j}^{\left( \Delta \right)} - \frac{1}{n}\left( {\sum\limits_{l = 1}^{n} {\widehat{{\underline{x} }}_{l}^{\left( \Delta \right)} } } \right)} \right)} } \right]\\& = \frac{1}{\zeta (j)}\sum\limits_{i = 1}^{n - j} {{\text{E}}\left[ {\left( {\left( {\widehat{{\underline{x}}}_{i}^{\left( \Delta \right)} - \mu } \right) - \left({\frac{1}{n}\left( {\sum\limits_{l = 1}^{n} {\widehat{{\underline{x}}}_{l}^{\left( \Delta \right)} } } \right) - \mu } \right)}\right)\;\left( {\left( {\widehat{{\underline{x} }}_{i+j}^{\left(\Delta \right)} - \mu } \right) - \left( {\frac{1}{n}\left({\sum\limits_{l = 1}^{n} {\widehat{{\underline{x} }}_{l}^{\left(\Delta \right)} } } \right) - \mu } \right)} \right)} \right]}\\& = \frac{1}{\zeta \left( j \right)}\sum\limits_{i = 1}^{n - j} \left\{\overbrace {{{\text{E}}\left[ {\left( {\widehat{{\underline{x} }}_{i}^{\left( \Delta \right)} - \mu } \right)\left( {\widehat{{\underline{x} }}_{i + j}^{\left( \Delta \right)} - \mu } \right)} \right]}}^{{{\text{E}}1}} - \overbrace{{{\text{E}}\left[ {\left( {\widehat{{\underline{x} }}_{i}^{\left(\Delta \right)} - \mu } \right)\left( {\frac{1}{n}\left({\sum\limits_{l = 1}^{n} {\widehat{{\underline{x} }}_{l}^{\left(\Delta \right)} } } \right) - \mu } \right)} \right]}}^{{{\text{E}}2}} - \overbrace {{{\text{E}}\left[ {\left( {\widehat{{\underline{x} }}_{i}^{\left( \Delta \right)} - \mu } \right)\left( {\frac{1}{n}\left( {\sum\limits_{l = 1}^{n} {\widehat{{\underline{x} }}_{l}^{\left( \Delta \right)} } } \right) - \mu } \right)} \right]}}^{{{\text{E}}3}}\right.\\ &+\left. \overbrace{{\text{E}}\left[\left(\frac{1}{n}\left( {\sum\limits_{l = 1}^{n} {\widehat{{\underline{x} }}_{l}^{\left( \Delta \right)} } }\right)- \mu \right)^{2}\right]}^{\text{E4}} \right\}\end{aligned}$$

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where $\mu = {\text{E}}\left[ {\widehat{{\underline{x} }}_{i}^{(\Delta )} } \right]$.

Below we will express the above sums of expressions E1, E2, E3 and E4 in terms of the true climacogram $\gamma (\Delta k)$ and true autocovariance in discrete time $c_{d}^{(\Delta )} (j)$ for j ≥ 1. Firstly, the sum of E1 is:

$$ \mathop \sum \limits_{i = 1}^{n - j} {\text{E}}1 = \left( {n - j} \right)c_{d}^{(\Delta )} (j) $$

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We observe that $\mathop \sum \nolimits_{i = 1}^{n - j} {\text{E}}2 = \mathop \sum \nolimits_{i = 1}^{n - j} {\text{E}}3$ and thus, we only calculate the sum of E3:

$$ \begin{aligned} & \mathop \sum \limits_{i = 1}^{n - j} {\text{E}}3 = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n - j} \mathop \sum \limits_{l = 1}^{n} {\rm E}\left[ {\left( {\widehat{{\underline{x} }}_{i + j}^{\left( \Delta \right)} - \mu } \right)\left( {\widehat{{\underline{x} }}_{l}^{\left( \Delta \right)} - \mu } \right)} \right] = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n - j} \mathop \sum \limits_{l = 1}^{n} c_{d}^{\left( \Delta \right)} \left( {l - i - j} \right) \\ & = \frac{{\left( {n - j} \right)^{2} \gamma (\Delta \left( {n - j} \right))}}{n} + \frac{1}{n}\overbrace {{\sum\limits_{i = 1}^{n - j} {\sum\limits_{l = 1}^{j} {c_{d}^{\left( \Delta \right)} \left( {l - i - j} \right)} } }}^{{{\text{E}}5}} \\ \end{aligned} $$

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The sum of E4 can be expressed in terms of the true climacogram:

$$ \mathop \sum \limits_{i = 1}^{n - j} {\rm E4} = \left( {n - j} \right) {\text{Var}}\left[ {\frac{1}{n}\left( {\mathop \sum \limits_{l = 1}^{n} \widehat{{\underline{x} }}_{l}^{\left( \Delta \right)} } \right)} \right] = \left( {n - j} \right) \gamma \left( {n\Delta } \right) $$

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For the estimation of E5, we distinguish two cases, j ≤ n/2 and j > n/2. For the first case, we have:

$$ {\text{E}}5\left( {j \le \frac{n}{2}} \right) = \frac{j}{n}\mathop \sum \limits_{i = j}^{n - j} c_{d}^{\left( \Delta \right)} \left( i \right) + \frac{1}{n}\mathop \sum \limits_{i = 1}^{j - 1} ic_{d}^{\left( \Delta \right)} \left( i \right) + \frac{1}{n}\mathop \sum \limits_{i = n - j + 1}^{n - 1} c_{d}^{\left( \Delta \right)} \left( i \right)\left( {n - i} \right) = \frac{{j\gamma \left( \Delta \right) - j^{2} \gamma (j\Delta )}}{2n} + \overbrace {{\frac{1}{n}\mathop \sum \limits_{i = n - j + 1}^{n - 1} c_{d}^{\left( \Delta \right)} \left( i \right)\left( {n - i} \right) + j\mathop \sum \limits_{i = 1}^{n - j} c_{d}^{\left( \Delta \right)} \left( i \right)}}^{{{\text{E}}6}} $$

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For the estimation of E6, we have:

$$ {\text{E}}6 = n\gamma \left( {n\Delta } \right)/2 - \gamma \left( \Delta \right)/2 + \overbrace {{\frac{1}{n}\mathop \sum \limits_{i = 1}^{n - j} c_{d}^{\left( \Delta \right)} \left( {j - n + i} \right)}}^{{{\text{E}}7}} $$

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and E7 can be expressed as:

$$ {\text{E}}7 = \left( {n - j} \right)\gamma \left( \Delta \right)/\left( {2n} \right) - \left( {n - j} \right)^{2} \gamma \left( {\left( {n - j} \right)\Delta } \right)/\left( {2{\text{n}}} \right) $$

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For j > n/2, E5 is the same as for j ≤ n/2 but with replacing j with n−j and thus, in the general case of E5:

$$ {\text{E}}5 = n\gamma \left( {n\Delta } \right)/2 - j^{2} \gamma \left( {j\Delta } \right)/\left( {2n} \right) - \left( {n - j} \right)^{2} \gamma \left( {\left( {n - j} \right)\Delta } \right)/\left( {2n} \right) $$

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Thus, Eq. A.2 results in:

$$ {\text{E}}\left[ {\widehat{{\underline{c} }}_{d}^{\left( \Delta \right)} \left( j \right)} \right] = \frac{1}{\zeta \left( j \right)}\left( {\left( {n - j} \right)c_{d}^{\left( \Delta \right)} \left( j \right) + \frac{{j^{2} }}{n}\gamma \left( {j\Delta } \right) - j\gamma \left( {n\Delta } \right) - \frac{{\left( {n - j} \right)^{2} }}{n}\gamma \left( {\left( {n - j} \right)\Delta } \right)} \right) $$

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where $\zeta \left( j \right)$ is usually taken as: n or n−1 or n−j.

It is interesting to notice that using Eq. 7 we can express the expected discrete time autocovariance of the above using only the true climacogram.