Wavelet Analysis of Red Noise and Its Application in Climate Diagnosis

Signals are often destroyed by various kinds of noises. A common way to statistically assess the significance of a broad spectral peak in signals and the synchronization between signals is to compare with simple noise processes. At present, wavelet analysis of red noise is studied limitedly and there is no general formula on the distribution of the wavelet power spectrum of red noise. Moreover, the distribution of the wavelet phase of red noise is also unknown. In this paper, for any given real/analytic wavelet, we will use a rigorous statistical framework to obtain the distribution of the wavelet power spectrum and wavelet phase of red noise and apply these formulas in climate diagnosis.

Time series analysis of observed maximum and minimum air temperature at four urban cities of India during 1951-2015

A sixty - five year (1951-2015) long data for monthly minimum temperature (TMIN) and maximum temperature (TMAX), observed by the India Meteorological Department (IMD), is statistically analyzed at four urban stations namely Bhubaneswar, Delhi, Mumbai and Chennai of India. A bimodal nature in seasonality is noticed for TMAX and TMIN at all locations. Two peaks for TMAX and TMIN are observed in May and September. Exceptionally, Mumbai shows TMAX peaks during May and November and Delhi shows TMIN peaks during June and September. Higher standard deviations (SD) for TMAX is noted at Delhi with a maximum in March (1.78 °C), while for Chennai, the SD for TMIN is lesser compared to other cities. Two different periods 1951-1980 (P1, the first half of the study period) and 1981-2015 (P2, the second half of the study period) were identified from the time series of both TMAX and TMIN. A higher increasing trend is observed during P2 than P1 in all the cities except in TMIN at Mumbai. The highest increasing trend (0.040 °C/year) is observed for TMIN in Mumbai during P1 time, but the trend is almost constant (0.001 °C/year) during P2 time. The highest increasing trend for TMIN at Mumbai is mainly contributed by the increasing trend in post-monsoon and winter months in P1. Surprisingly, in both P1 and P2, the trends are less during monsoon months for all the cities. A consistent 5-year (3-year) band is observed throughout the wavelet power spectrum at the coastal cities Bhubaneswar, Mumbai (Chennai). However, the 5-year signal is not consistent at Delhi and it is observed only during the year 1975-1980. The global wavelet power spectrum showed that TMIN at Chennai has less power (0.6 °C2) corresponding to 3-year signal and Mumbai has highest power (12 °C2) corresponding to the 5-year signal in comparison to other cities.

Artificial Detection of Lower-Frequency Periodicity in Climatic Studies by Wavelet Analysis Demonstrated on Synthetic Time Series

The continuous wavelet transform (CWT) is a frequently used tool to study periodicity in climate and other time series. Periodicity plays a significant role in climate reconstruction and prediction. In numerous studies, the use of CWT revealed dominant periodicity (DP) in climatic time series. Several studies suggested that these “natural oscillations” would even reverse global warming. It is shown here that the results of wavelet analysis for detecting DPs can be misinterpreted in the presence of local singularities that are manifested in lower frequencies. This may lead to false DP detection. CWT analysis of synthetic and real-data climatic time series, with local singularities, indicates a low-frequency DP even if there is no true periodicity in the time series. Therefore, it is argued that this is an inherent general property of CWT. Hence, applying CWT to climatic time series should be reevaluated, and more careful analysis of the entire wavelet power spectrum is required, with a focus on high frequencies as well. A conelike shape in the wavelet power spectrum most likely indicates the presence of a local singularity in the time series rather than a DP, even if the local singularity has an observational or a physical basis. It is shown that analyzing the derivatives of the time series may be helpful in interpreting the wavelet power spectrum. Nevertheless, these tests are only a partial remedy that does not completely neutralize the effects caused by the presence of local singularities.

Changes in interictal pretreatment and posttreatment EEG in childhood absence epilepsy

Spike and wave discharges (SWDs) are the characteristic manifestation of childhood absence epilepsy (CAE). It has long been believed that they unpredictably emerge from otherwise almost normal interictal EEG. Herein, we demonstrate that pretreatment closed-eyes theta and beta EEG wavelet powers of CAE patients (20 girls and 10 boys, mean age 7.4 ± 1.9 years) are much higher than those of age-matched controls at multiple sites of 10-20 system. For example, at C4 site, we observed a 91% and 62% increase in power of theta and beta rhythms, respectively. We were able to compare the baseline and posttreatment wavelet power in 16 patients. The pharmacotherapy brought about a statistically significant decrease in delta and theta wavelet power in all the channels, e.g. for C4 the reduction was equal to 45% (delta) and 65% (theta). We also observed a less pronounced attenuation of posttreatment beta rhythm in several channels. We hypothesize that the increased theta and beta powers result from cortical hyperexcitability and propensity for epileptic spikes generation, respectively. We argue that the distinct features of CAE wavelet power spectrum may be used to define an EEG biomarker which could be used for diagnosis and monitoring of patients.

Analisis Data Curah Hujan Kabupaten Ketapang dan Kota Pontianak Menggunakan Transformasi Wavelet

Data curah hujan di Kabupaten Ketapang dan Kota Pontianak dianalisis untuk mengetahui pola curah hujan di Kabupaten Ketapang dan Kota Pontianak. Data yang digunakan adalah data curah hujan bulanan selama 30 tahun (1986–2016), dengan menggunakan transformasi wavelet. Hasil penelitian ini menunjukkan bahwa metode tersebut dapat memperlihatkan pola curah hujan melalui Wavelet Power Spectrum (WPS) dan Global Wavelet Spectrum (GWS) disetiap stasiun pengamatan. Hasil dari WPS menunjukkan bahwa intensitas curah hujan yang tinggi di Kabupaten Ketapang adalah pada periode 1 tahun dan periode 3 tahun dengan pola hujan tahunan. Tingginya intensitas curah hujan dipengaruhi oleh fenomena El Niño Southern Oscillation (ENSO), yang merupakan hasil dari pengolahan data indeks ENSO dengan data curah hujan menggunakan cross wavelet. Di Kota Pontianak intensitas curah hujanyang tinggi terjadi pada periode 0,5 tahun dan periode 1 tahun dengan pola hujan musiman dan tahunan.

Quasi ~500-year Cycle Signals in Solar Activity

Direct observations of solar activity are available for the past four century, so some proxies reflecting solar activity such as 14C, 10Be and geomagnetic variations are used to reconstruct solar activity in the past. In this present paper, the authors use rectified wavelet power transform and time-averaged wavelet power spectrum to investigate long-term fluctuations of the reconstructed solar activity series. Results show obvious a quasi ~500-year cycle exists in the past solar activity. Three reconstructed solar activity series from 14C variations confirm the periodic signals.