**4. The Lomb-Scargle-Tarantola (LST) periodogram**

In geophysical inverse theory, there are high mature methods to deal with large uncertainties in data and ill-conditioned models. It comes from the constant necessity; geophysicists have to build and evaluate physical models of the subsurface (or Earth's deep interior) based mostly on data acquired from the surface.

In Seismics, for example, geophysicists developed a procedure called *stacking* where a set of different signals, acquired by distinct geophones, are gathered under specific geometrical settings. The procedure aims to amplify the signal and to cancel noise, improving overall information about the same subsurface points. The stacking of seismic signals is one of the main reasons for the success of this technique on the oil industry nowadays, allowing the discovery of new oil fields and improving oil/gas recovering on reservoir structures.

Caminha-Maciel and Ernesto [16, 17] created a method to analyze spectral content (and its uncertainties) in irregularly sampled times series applying some principles from the geophysical inverse theory – the Popper-Bayes approach, as developed by Albert Tarantola [18–20]. This approach uses freely normalizable probability distributions to encapsulate the information and afterward operates with these distributions (the data information and the model/geometrical information on the problem). The results of these operations constitute a Bayesian physical inference method, and its *a posteriori* probability distributions are proper solutions to our inverse problem. In the following section, we will see how this applies to the Lomb-Scargle periodogram.

### **4.1 State of information periodogram-based functions**

There is a vast diversity of methods to analyze time series, both in time and in frequency domains. Fourier-derived methods still show continued interest since they are fast (due to FFT algorithm), intuitive, and have a straightforward extension to irregularly sampled time series.

However, in some applications, as the high-resolution deep-sea stratigraphic records, there are novel challenges for the extraction and interpretation of meaningful information. It is worth to mention the uncertainties in the measured times (plus the usual uncertainties on the other variables), non-stationarity of the dynamical systems observed, and shortness of the records compared to wavelengths of interest. In the stratigraphic time series, there is also a general dependence of the recording-sampling process on the unknown-climatic signal itself. These issues contribute to breaking the orthogonality among periodogram ordinates, which is necessary to perform appropriately statistical significance tests on supposedindependent ordinates.

Nevertheless, there are some less-known interesting properties of the Lomb-Scargle periodogram:

• Analytical independence among ordinates – non-statistical independence. Since we have a fixed set of points *X tj* , *tj* and a fixed frequency point *ω*0, *P*ð Þ *ω*<sup>0</sup> gives the same result regardless of how many different frequencies *ω<sup>i</sup>* we use in the analysis. The periodogram does not "see" other ordinates. The problem is when we try to compare different ordinates (statistical independence).

*Periodogram Analysis under the Popper-Bayes Approach DOI: http://dx.doi.org/10.5772/intechopen.93162*


In the LST periodogram, a freely normalized version of the Lomb-Scargle periodogram is initially defined over the broadest possible range of frequencies. The frequency set is chosen to include all wavelengths about which could exist information on the time series. The next step is to choose the minimum frequency as zero, the frequency grid spacing *δf*, or the total length of time series, or some fraction in between these values. We can choose the maximum frequency as the highest frequency about which we believe there is any information in the time series, as some *a priori* known Nyquist limit, up to the limit of the inverse of the gcd of time intervals. Above this limit, we probably have to deal we some folding in the periodogram. The frequency grid *δf* has to be chosen to allow the calculation to be computationally feasible.

### *4.1.1 Normalizing by the bandwidth total content*

We define

a peak is related to the peak width, usually in Fourier analysis defined as the peak half-width. For this reason, in periodogram analysis, Gaussian error bars should be

In geophysical inverse theory, there are high mature methods to deal with large

uncertainties in data and ill-conditioned models. It comes from the constant necessity; geophysicists have to build and evaluate physical models of the subsurface (or Earth's deep interior) based mostly on data acquired from the surface. In Seismics, for example, geophysicists developed a procedure called *stacking* where a set of different signals, acquired by distinct geophones, are gathered under specific geometrical settings. The procedure aims to amplify the signal and to cancel

noise, improving overall information about the same subsurface points. The stacking of seismic signals is one of the main reasons for the success of this technique on the oil industry nowadays, allowing the discovery of new oil fields and

Caminha-Maciel and Ernesto [16, 17] created a method to analyze spectral content (and its uncertainties) in irregularly sampled times series applying some principles from the geophysical inverse theory – the Popper-Bayes approach, as developed by Albert Tarantola [18–20]. This approach uses freely normalizable probability distributions to encapsulate the information and afterward operates with these distributions (the data information and the model/geometrical

information on the problem). The results of these operations constitute a Bayesian physical inference method, and its *a posteriori* probability distributions are proper solutions to our inverse problem. In the following section, we will see how this

There is a vast diversity of methods to analyze time series, both in time and in frequency domains. Fourier-derived methods still show continued interest since they are fast (due to FFT algorithm), intuitive, and have a straightforward

However, in some applications, as the high-resolution deep-sea stratigraphic records, there are novel challenges for the extraction and interpretation of meaningful information. It is worth to mention the uncertainties in the measured times (plus the usual uncertainties on the other variables), non-stationarity of the

dynamical systems observed, and shortness of the records compared to wavelengths of interest. In the stratigraphic time series, there is also a general dependence of the recording-sampling process on the unknown-climatic signal itself. These issues contribute to breaking the orthogonality among periodogram ordinates, which is necessary to perform appropriately statistical significance tests on supposed-

Nevertheless, there are some less-known interesting properties of the Lomb-

• Analytical independence among ordinates – non-statistical independence.

problem is when we try to compare different ordinates (statistical

*P*ð Þ *ω*<sup>0</sup> gives the same result regardless of how many different frequencies *ω<sup>i</sup>* we use in the analysis. The periodogram does not "see" other ordinates. The

, *tj*

and a fixed frequency point *ω*0,

avoided as a way to report uncertainties in frequency determinations.

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

**4. The Lomb-Scargle-Tarantola (LST) periodogram**

improving oil/gas recovering on reservoir structures.

applies to the Lomb-Scargle periodogram.

extension to irregularly sampled time series.

Since we have a fixed set of points *X tj*

independent ordinates.

Scargle periodogram:

independence).

**86**

**4.1 State of information periodogram-based functions**

$$P\_{LST}(a) = K \cdot P\_{LS}(a) \tag{11}$$

where the normalizing constant *<sup>K</sup>* is set as *<sup>K</sup>* <sup>¼</sup> <sup>P</sup> *<sup>ω</sup>P*ð Þ *<sup>ω</sup>* <sup>Δ</sup>*<sup>ω</sup>* � ��<sup>1</sup> . This values of the constant *K* is such that the function *PLST* normalizes to total area under the curve equals to 1 over the whole set of frequencies *ωi*. This area represents the total power in the bandwidth and reflect the times series total variance.

Note that the S/N ratio, as well as the ratio between any two distinct frequencies *P*ð Þ *ω<sup>a</sup> =P*ð Þ *ω<sup>b</sup>* , does not depend on the times series total power.

This procedure is equivalent to a stretching of the data series variable *X t*ð Þ in the time domain and also has the property of making comparable the total power of the various periodograms.

### **4.2 Periodogram analysis by combination of information**

The two main ideas of the Lomb-Scargle-Tarantola periodogram are

1.Smoothing the periodogram.

2. Stacking independent periodogram estimates.

Since its proposition, the periodogram is recognized as high noisy statistics, even for less noisy data. Smoothing the periodogram is not a new idea. There are several

attempts in this direction by averaging adjacent estimates (as in Daniel's averaged periodogram, for example) and in many Bayesian formulations.

A new idea on working with periodograms is stacking in the frequency domain: we can consider two or more distinct time series with information about the same dynamical variable as independent observations of the same phenomenon. For example, several stratigraphic sections covering the same time interval; different variables related to the same dynamics – sediment accumulation and *δ*18*O* time series, both related to seawater surface temperature; independent observations of the same variable – as astronomical observations from distinct geographic locations; and biological circadian rhythms from different organisms.

After that, we operate these periodogram distributions with two logical operators "OR" and "AND." The OR operator can be described as a generalization of doing histograms and is mathematically defined as the arithmetic average of the individual LST periodograms (very similar to usual stacking of seismic signals). The AND operator is a non-linear operator that represents the generalization of conditional probability and is mathematically defined as <sup>Π</sup>*iPi*ð Þ *<sup>ω</sup> <sup>μ</sup>* , where *μ* is the null information function – characterizing the geometry of the physical problem.

In some physical problems involving the dynamical variable *frequency*, *f*, itself, the null information function can be better written as 1*=f*. However, in spectral analysis, the variable frequency only means labels for some general class of eigenfunctions – as the Fourier basis sin and cos . Then we can fairly consider the null information function, *μ*, as a *constant function* over the entire domain.

### *4.2.1 Using LSTperiod software*

We have published [17] a proof-of-concept software to implement the LST periodogram. This software, the *LSTperiod* (Download it at http://www.iag.usp.br/ paleo/sites/default/files/LSTperiod-files.zip), exemplify one possible implementation of the LST periodogram. We have made a set of choices for the frequency grid, normalizations of the state of information functions, evaluation of the resulting models (candidates periodicities), and visualizations of the results. Those choices are for no means unique [27].

Drilling Project (ODP) [28]. These time series were subjected to spectral analysis and other statistical methods and show Milankovitch climatic cycles around 19, 23,

*Rose diagram showing the phases and amplitudes of the calculated period T* ¼ 40*:*92*kyr for each analyzed*

*Periodogram Analysis under the Popper-Bayes Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.93162*

combined (OR/AND) periodograms. **Figure 2** shows the amplitude and phase analysis for the � 41*kyr* Milankovitch period found, for each analyzed series.

constitutes the main body of physical evidence to understand and solve fundamental today's problems – as the origin and development of the *climatic*

alies ultimately prevent the use of standard Fourier techniques, as the FFT

In **Figure 1**, we can see the periodograms for these time series – individual and

With the recent advances in experimental sciences, there is an increased need to analyze and statistically evaluate the information contained in time series. In some areas, as paleoclimatic studies, the study of the information on this kind of data

Fourier methods still constitute an updated tool since they are simple and offer easy-to-understand results. However, in several of these applications, as in stratigraphic data, these time series come with severe sampling anomalies. These anom-

algorithm and periodograms. The Lomb-Scargle periodogram has been very useful to attack this sort of problem. However, its use, as seen in the literature, lacks a proper analysis of the uncertainties associated and, worse, is unfit to be applied in

and 41 kyr [28–30].

**Figure 2.**

*series.*

**89**

**5. Conclusions**

*change* observed in recent years.

the most poorly sampled time series.

To illustrate the use of the LSTperiod software, we show a set of five stratigraphic series of benthic *δ*<sup>18</sup>*O* from the sedimentary core drilled by the Ocean

### **Figure 1.**

*LST periodograms for the benthic δ*<sup>18</sup>*O series from ocean drilling project (ODP) cores. The windows show the periodogram calculated for each data file separately (bottom) and the combined results for the OR (middle) and AND (top) spectra.*

*Periodogram Analysis under the Popper-Bayes Approach DOI: http://dx.doi.org/10.5772/intechopen.93162*

attempts in this direction by averaging adjacent estimates (as in Daniel's averaged

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

A new idea on working with periodograms is stacking in the frequency domain: we can consider two or more distinct time series with information about the same dynamical variable as independent observations of the same phenomenon. For example, several stratigraphic sections covering the same time interval; different variables related to the same dynamics – sediment accumulation and *δ*18*O* time series, both related to seawater surface temperature; independent observations of the same variable – as astronomical observations from distinct geographic locations;

After that, we operate these periodogram distributions with two logical operators "OR" and "AND." The OR operator can be described as a generalization of doing histograms and is mathematically defined as the arithmetic average of the individual LST periodograms (very similar to usual stacking of seismic signals). The AND operator is a non-linear operator that represents the generalization of condi-

In some physical problems involving the dynamical variable *frequency*, *f*, itself, the null information function can be better written as 1*=f*. However, in spectral analysis, the variable frequency only means labels for some general class of eigenfunctions – as the Fourier basis sin and cos . Then we can fairly consider the

information function – characterizing the geometry of the physical problem.

null information function, *μ*, as a *constant function* over the entire domain.

We have published [17] a proof-of-concept software to implement the LST periodogram. This software, the *LSTperiod* (Download it at http://www.iag.usp.br/ paleo/sites/default/files/LSTperiod-files.zip), exemplify one possible implementation of the LST periodogram. We have made a set of choices for the frequency grid, normalizations of the state of information functions, evaluation of the resulting models (candidates periodicities), and visualizations of the results. Those choices

To illustrate the use of the LSTperiod software, we show a set of five stratigraphic series of benthic *δ*<sup>18</sup>*O* from the sedimentary core drilled by the Ocean

*LST periodograms for the benthic δ*<sup>18</sup>*O series from ocean drilling project (ODP) cores. The windows show the periodogram calculated for each data file separately (bottom) and the combined results for the OR (middle)*

*<sup>μ</sup>* , where *μ* is the null

periodogram, for example) and in many Bayesian formulations.

and biological circadian rhythms from different organisms.

tional probability and is mathematically defined as <sup>Π</sup>*iPi*ð Þ *<sup>ω</sup>*

*4.2.1 Using LSTperiod software*

are for no means unique [27].

**Figure 1.**

**88**

*and AND (top) spectra.*

**Figure 2.** *Rose diagram showing the phases and amplitudes of the calculated period T* ¼ 40*:*92*kyr for each analyzed series.*

Drilling Project (ODP) [28]. These time series were subjected to spectral analysis and other statistical methods and show Milankovitch climatic cycles around 19, 23, and 41 kyr [28–30].

In **Figure 1**, we can see the periodograms for these time series – individual and combined (OR/AND) periodograms. **Figure 2** shows the amplitude and phase analysis for the � 41*kyr* Milankovitch period found, for each analyzed series.

### **5. Conclusions**

With the recent advances in experimental sciences, there is an increased need to analyze and statistically evaluate the information contained in time series. In some areas, as paleoclimatic studies, the study of the information on this kind of data constitutes the main body of physical evidence to understand and solve fundamental today's problems – as the origin and development of the *climatic change* observed in recent years.

Fourier methods still constitute an updated tool since they are simple and offer easy-to-understand results. However, in several of these applications, as in stratigraphic data, these time series come with severe sampling anomalies. These anomalies ultimately prevent the use of standard Fourier techniques, as the FFT algorithm and periodograms. The Lomb-Scargle periodogram has been very useful to attack this sort of problem. However, its use, as seen in the literature, lacks a proper analysis of the uncertainties associated and, worse, is unfit to be applied in the most poorly sampled time series.

The spectral analysis of irregularly sampled times series represents a problem of studying a physical system from incomplete information. As we know, from geophysical inverse theory, this kind of problem cannot be solved without some input of *a priori* information – explicitly or not.

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*DOI: http://dx.doi.org/10.5772/intechopen.93162*

*Periodogram Analysis under the Popper-Bayes Approach*

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The Popper-Bayes approach for physical inference proposes to solve inference problems by a combination of information: theoretical usually expressed through the functional form of statistical distributions and by combining independent experimental data. The LST periodogram is a development of the Lomb-Scargle periodogram that brings these inference principles to the periodogram analysis of irregularly sampled time series. The main idea of the LST periodogram is to smooth the periodogram of a dynamical variable through the stacking of spectral information from multiple irregularly sampled times series.

The periodogram of an irregularly sampled times series cannot, by any means, become a set of independent ordinates for being submitted to a proper statistical test. With the LST periodogram, we propose to change the use of the periodogram: from an auxiliary tool to statistical decision theory (define a periodicity) to a dimension reduction problem – from a broad set of possible frequencies to very narrow set of periodogram local maxima (peaks).

## **Acknowledgements**

We are thankful to Dr. M. Ernesto for critically reading the early version of this manuscript. Thanks are also due to the editor Dr. Y.K.N. Truong whose suggestions greatly improved the manuscript.
