scholarly journals Chaos in rainfall: variability, temporal scale and zeros

2005 ◽  
Vol 7 (3) ◽  
pp. 175-184 ◽  
Author(s):  
B. Sivakumar
Keyword(s):  
Correlation Dimension ◽  
Rainfall Variability ◽  
Prediction Method ◽  
Rainfall Series ◽  
Rainfall Data ◽  
Nonlinear Prediction ◽  
Number Of Zeros ◽  
Dimension Analysis ◽  
Dimensional Phase Space ◽  
Zero Values

A recent study on rainfall observed at the Leaf River basin reports that the presence of a large number of zeros in the data significantly underestimates the correlation dimension. The present study attempts to verify such a claim, by making predictions and comparing the results with the correlation dimensions. A nonlinear prediction method, which uses the concept of reconstruction of a single-variable series in a multi-dimensional phase space to represent the underlying dynamics, is employed. Correlation dimension analysis of only the non-zero rainfall series is also carried out for further verification. Rainfall data of four different temporal resolutions (or scales), i.e. daily, 2-day, 4-day and 8-day, are analyzed. The predictions for the finer-resolution (i.e. higher-resolution) rainfall are found to be better than those obtained for the coarser-resolution (i.e. lower-resolution) rainfall and seem to be consistent with the variability vs. predictability logic in a deterministic sense, i.e. higher prediction accuracy for data with lower correlation dimension and vice versa. An important implication of this result is that the presence of (a large number of) zeros in the rainfall data may not always result in an underestimation of the correlation dimension. The correlation dimensions estimated for the non-zero rainfall series are not significantly different when compared to those obtained for series including zero values, supporting the above. These results suggest that the low correlation dimensions for rainfall (in particular finer-resolution ones that commonly have a large number of zeros), as reported by past studies, could well be, or at least closer to, the actual dimensions of the rainfall processes studied.

Degrees of Freedom

2006 ◽  
Author(s):  
Huug van den Dool
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Prediction Method ◽  
Data Sets ◽  
Nonlinear Prediction ◽  
Orthogonal Direction ◽  
Empirical Prediction ◽  
Long Run ◽  
Dimensional Phase Space ◽  
Natural Analogues

How many degrees of freedom are evident in a physical process represented by f(s, t)? In some form questions about “degrees of freedom” (d.o.f.) are common in mathematics, physics, statistics, and geophysics. This would mean, for instance, in how many independent directions a weight suspended from the ceiling could move. Dofs are important for three reasons that will become apparent in the remaining chapters. First, dofs are critically important in understanding why natural analogues can (or cannot) be applied as a forecast method in a particular problem (Chapter 7). Secondly, understanding dofs leads to ideas about truncating data sets efficiently, which is very important for just about any empirical prediction method (Chapters 7 and 8). Lastly, the number of dofs retained is one aspect that has a bearing on how nonlinear prediction methods can be (Chapter 10). In view of Chapter 5 one might think that the total number of orthogonal directions required to reproduce a data set is the dof. However, this is impractical as the dimension would increase (to infinity) with ever denser and slightly imperfect observations. Rather we need a measure that takes into account the amount of variance represented by each orthogonal direction, because some directions are more important than others. This allows truncation in EOF space without lowering the “effective” dof very much. We here think schematically of the total atmospheric or oceanic variance about the mean state as being made up by N equal additive variance processes. N can be thought of as the dimension of a phase space in which the atmospheric state at one moment in time is a point. This point moves around over time in the N-dimensional phase space. The climatology is the origin of the phase space. The trajectory of a sequence of atmospheric states is thus a complicated Lissajous figure in N dimensions, where, importantly, the range of the excursions in each of the N dimensions is the same in the long run. The phase space is a hypersphere with an equal probability radius in all N directions.

scholarly journals Rainfall dynamics at different temporal scales: A chaotic perspective

2001 ◽  
Vol 5 (4) ◽  
pp. 645-652 ◽  
Author(s):  
B. Sivakumar
Keyword(s):  
Time Series ◽  
Coefficient Of Variation ◽  
Correlation Dimension ◽  
Rainfall Data ◽  
Resolution Time ◽  
Scale Invariant ◽  
Temporal Scales ◽  
Number Of Zeros ◽  
Correlation Dimension Method ◽  
Dimension Number

Abstract. This study of the behaviour of rainfall dynamics at different temporal scales identifies the type of approach most suitable for transformation of rainfall data from one scale to another. Rainfall data of four different temporal scales, i.e. daily, 2-day, 4-day and 8-day, observed over a period of about 25 years at the Leaf River basin, Mississippi, USA, are analysed. The correlation dimension method is employed to identify the behaviour of rainfall dynamics. The finite correlation dimensions obtained for the four rainfall series (4.82, 5.26, 6.42 and 8.87, respectively) indicate the possible existence of chaotic behaviour in the rainfall observed at the four scales. A possible implication of this might be that the rainfall processes at these scales are related through a chaotic (scale-invariant) behaviour. However, a comparison of the correlation dimension and coefficient of variation of each of the time series reveals an inverse relationship between the two (higher dimension for lower coefficient of variation and vice versa). The presence of a large number of zeros in the higher resolution time series (that could result in an underestimation of the dimension) and the possible presence of a higher level of noise in the lower resolution time series (that could result in an overestimation of the dimension) might account for such results. In view of these problems, it is concluded that the results must be verified using other chaos identification methods and the existence of chaos must be substantiated with additional evidence. Keywords: rainfall, chaos, scaling, correlation dimension, number of variables, coefficient of variation, data size, noise, zeros

scholarly journals Nonlinear time series analysis of the fluctuations of the geomagnetic horizontal field

2002 ◽  
Vol 20 (2) ◽  
pp. 175-183 ◽  
Author(s):  
B. George ◽  
G. Renuka ◽  
K. Satheesh Kumar ◽  
C. P. Anil Kumar ◽  
C. Venugopal
Keyword(s):  
Time Series ◽  
Time Series Analysis ◽  
Correlation Dimension ◽  
Prediction Method ◽  
Nonlinear Time Series ◽  
Nonlinear Time Series Analysis ◽  
Suitable Model ◽  
Nonlinear Prediction ◽  
Series Analysis ◽  
Hourly Data

Abstract. A detailed nonlinear time series analysis of the hourly data of the geomagnetic horizontal intensity H measured at Kodaikanal (10.2° N; 77.5° E; mag: dip 3.5° N) has been carried out to investigate the dynamical behaviour of the fluctuations of H. The recurrence plots, spatiotemporal entropy and the result of the surrogate data test show the deterministic nature of the fluctuations, rejecting the hypothesis that H belong to the family of linear stochastic signals. The low dimensional character of the dynamics is evident from the estimated value of the correlation dimension and the fraction of false neighbours calculated for various embedding dimensions. The exponential decay of the power spectrum and the positive Lyapunov exponent indicate chaotic behaviour of the underlying dynamics of H. This is also supported by the results of the comparison of the chaotic characteristics of the time series of H with the pseudo-chaotic characteristics of coloured noise time series. We have also shown that the error involved in the short-term prediction of successive values of H, using a simple but robust, zero-order nonlinear prediction method, increases exponentially. It has also been suggested that there exists the possibility of characterizing the geomagnetic fluctuations in terms of the invariants in chaos theory, such as Lyapunov exponents and correlation dimension. The results of the analysis could also have implications in the development of a suitable model for the daily fluctuations of geomagnetic horizontal intensity.Key words. Geomagnetism and paleomagnetism (time variations, diurnal to secular) – History of geophysics (solar-planetary relationships) Magnetospheric physics (storms and substorms)

scholarly journals Dynamics of monthly rainfall-runoff process at the Gota basin: A search for chaos

2000 ◽  
Vol 4 (3) ◽  
pp. 407-417 ◽  
Author(s):  
B. Sivakumar ◽  
R. Berndtsson ◽  
J. Olsson ◽  
K. Jinno ◽  
A. Kawamura
Keyword(s):  
Prediction Accuracy ◽  
Correlation Dimension ◽  
Prediction Method ◽  
Monthly Rainfall ◽  
Runoff Coefficient ◽  
Embedding Dimension ◽  
Rainfall Runoff ◽  
Nonlinear Prediction ◽  
Runoff Series ◽  
Correlation Dimension Method

Abstract. Sivakumar et al. (2000a), by employing the correlation dimension method, provided preliminary evidence of the existence of chaos in the monthly rainfall-runoff process at the Gota basin in Sweden. The present study verifies and supports the earlier results and strengthens such evidence. The study analyses the monthly rainfall, runoff and runoff coefficient series using the nonlinear prediction method, and the presence of chaos is investigated through an inverse approach, i.e. identifying chaos from the results of the prediction. The presence of an optimal embedding dimension (the embedding dimension with the best prediction accuracy) for each of the three series indicates the existence of chaos in the rainfall-runoff process, providing additional support to the results obtained using the correlation dimension method. The reasonably good predictions achieved, particularly for the runoff series, suggest that the dynamics of the rainfall-runoff process could be understood from a chaotic perspective. The predictions are also consistent with the correlation dimension results obtained in the earlier study, i.e. higher prediction accuracy for series with a lower dimension and vice-versa, so that the correlation dimension method can indeed be used as a preliminary indicator of chaos. However, the optimal embedding dimensions obtained from the prediction method are considerably less than the minimum dimensions essential to embed the attractor, as obtained by the correlation dimension method. A possible explanation for this could be the presence of noise in the series, since the effects of noise at higher embedding dimensions could be significantly greater than that at lower embedding dimensions. Keywords: Rainfall-runoff; runoff coefficient; chaos; phase-space; correlation dimension; nonlinear prediction; noise

Stormwater inflow prediction using radar rainfall data compressed by principal component analysis

2006 ◽  
Vol 1 (1) ◽  
Author(s):  
K. Katayama ◽  
K. Kimijima ◽  
O. Yamanaka ◽  
A. Nagaiwa ◽  
Y. Ono
Keyword(s):  
Principal Component Analysis ◽  
Prediction Model ◽  
Principal Components ◽  
Prediction Method ◽  
Principal Component ◽  
Component Analysis ◽  
Rainfall Data ◽  
Radar Rainfall ◽  
Input Variables ◽  
Inflow Prediction

This paper proposes a method of stormwater inflow prediction using radar rainfall data as the input of the prediction model constructed by system identification. The aim of the proposal is to construct a compact system by reducing the dimension of the input data. In this paper, Principal Component Analysis (PCA), which is widely used as a statistical method for data analysis and compression, is applied to pre-processing radar rainfall data. Then we evaluate the proposed method using the radar rainfall data and the inflow data acquired in a certain combined sewer system. This study reveals that a few principal components of radar rainfall data can be appropriate as the input variables to storm water inflow prediction model. Consequently, we have established a procedure for the stormwater prediction method using a few principal components of radar rainfall data.

Effect of Ethanol on Human Sleep EEG Using Correlation Dimension Analysis

2002 ◽  
Vol 46 (2) ◽  
pp. 104-110 ◽  
Author(s):  
Toshio Kobayashi ◽  
Shigeki Madokoro ◽  
Yuji Wada ◽  
Kiwamu Misaki ◽  
Hiroki Nakagawa
Keyword(s):  
Correlation Dimension ◽  
Sleep Eeg ◽  
Human Sleep ◽  
Dimension Analysis

scholarly journals Inhomogeneity detection in the rainfall series for the Mae Klong River Basin, Thailand

2021 ◽  
Vol 11 (9) ◽  
Author(s):  
Alamgir Khalil
Keyword(s):  
River Basin ◽  
Curve Analysis ◽  
Rainfall Series ◽  
Rainfall Data ◽  
Homogeneity Test ◽  
Significance Level ◽  
Von Neumann ◽  
Standard Normal Homogeneity Test ◽  
Double Mass ◽  
Climate Studies

AbstractAn accurate and complete rainfall record is prerequisite for climate studies. The purpose of this research study was to evaluate the homogeneity of the rainfall series for the Mae Klong River Basin in Thailand. Monthly rainfall data of eight stations in the Mae Klong River Basin for the period 1971–2015 were used. The double mass curve analysis was used to check the consistency of rainfall data, whereas the absolute homogeneity was assessed using the Pettitt test, standard normal homogeneity test, Buishand test, and von Neumann test at a 5% significance level. The results of these tests were qualitatively classified as ‘useful’, ‘doubtful’, and ‘suspect’ according to the null hypothesis. Results of the monthly time series indicated the rainfall data as ‘useful’ for 75% of the stations, while two stations’ data were classified as ‘doubtful’ (Stn130221) and ‘suspect’ (Stn376401). On an annual scale, seven out of eight stations data were classified as ‘useful,’ while one station (Stn376401) data were classified as ‘suspect’. Double mass curve analysis technique was used for the adjustment of inhomogeneous data. The results of this study can help provide reliable rainfall data for climate studies in the basin.

scholarly journals Modeling Rainfall Variability over Urban Areas: A Case Study for Kuwait

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Jaber Almedeij
Keyword(s):  
Urban Areas ◽  
Rainfall Variability ◽  
Rainfall Data ◽  
Point Estimate ◽  
Spatial And Temporal Variability ◽  
Time Duration ◽  
Reliable Model ◽  
Monthly Total ◽  
Seasonal Basis

This study examines the spatial and temporal variability of monthly total rainfall data obtained from weather stations located in the urban areas of Kuwait. The rainfall data are analyzed by considering statistics on a seasonal basis and by means of periodogram technique to reveal the periods responsible for the variable pattern. The results demonstrate similarity implying that a point estimate of rainfall data can be considered spatially representative over the urban areas of Kuwait. A sinusoidal model triggering the influence of the detected periods is developed accordingly for the time duration from January 1965 to December 2009. The model is capable of describing the rainfall data with some discrepancies between the actual and calculated values resulting from hidden periods that have not been taken into account. This finding suggests that the ability to construct a more reliable model would require a wider range of historical data to detect the other periods affecting the rainfall pattern.

scholarly journals Climatologia da pluviometria do município de Teresina, Piauí, Brasil

2016 ◽  
Vol 11 (4) ◽  
pp. 135
Author(s):  
Hudson Ellen Alencar Menezes ◽  
Raimundo Mainar de Medeiros ◽  
José Lucas Guilherme Santos
Keyword(s):  
Water Resources ◽  
Rainfall Variability ◽  
Intertropical Convergence Zone ◽  
Atmospheric Dynamics ◽  
Rainfall Data ◽  
Annual Rainfall ◽  
Convergence Zone ◽  
Northeast Brazil ◽  
Urban Structures ◽  
Urbanized Areas

<p>As variações nas precipitações refletem claramente a dinâmica atmosférica da região, marcada pela intensa variabilidade, onde se observa a atuação da Zona de Convergência Intertropical (ZCIT) com sua atuação entre os meses de janeiro a março, sendo esse período mais chuvoso. As variabilidades espaço temporal no comportamento das chuvas tem sido analisadas e diagnosticadas por vários autores no Nordeste do Brasil (NEB), portanto objetivou-se diagnosticar a variabilidade dos índices pluviométricos em Teresina no Estado do Piauí no período de 1913 a 2010. A análise do comportamento da precipitação nas cidades de grande e médio porte é de extrema importância para o gerenciamento dos recursos hídricos, uma vez que se trata de áreas densamente urbanizadas. Muitas vezes, sem uma estruturação urbana adequada, estas cidades se encaixam perfeitamente nesse contexto. Foram utilizados dados mensais observados e anuais de precipitação pluviométrica no período de 1913 a 2010, com 97 anos de observações. Os resultados mostraram a recorrência de valores máximos de precipitação anual dentro de um intervalo de 18, 11 e 8 anos. Na análise dos desvios-padrões, os resultados mostraram predominância dos desvios negativos em relação aos desvios positivos.</p><p align="center"><strong><em>Climatology of rainfall in the Teresina city, Piauí state, Brazil</em></strong></p><p>Variations in precipitation clearly reflect the atmospheric dynamics of the region, marked by intense variability, where we observe the performance of the Intertropical Convergence Zone (ITCZ) with his performance in the months of January-March, this being more rain tem period. The timeline of rainfall variability in behavior has been analyzed and diagnosed by several authors in Northeast Brazil (NEB), so let's study this variability between the periods 1913 to 2010 of Teresina city.  The behavior of rainfall in cities large and medium sized is of utmost importance to the managerial of water resources, since it is densely urbanized areas. Often without adequate urban structures these cities fit perfectly in this context. We used observed monthly and annual rainfall data for the period 1913-2010, 97 years of observations. The results showed recurrence of maximum values of annual precipitation an interval of 18, 11 and 8 years. In the analysis of standard deviations, the results showed a predominance of negative deviations from the positive deviations.<strong></strong></p><p align="center"><strong><em><br /></em></strong></p>

Wavelet decomposition and nonlinear prediction of nonstationary vibration signals

2020 ◽  
Vol 51 (3) ◽  
pp. 52-59 ◽  
Author(s):  
Xiao-bin Fan ◽  
Bin Zhao ◽  
Bing-xu Fan
Keyword(s):  
Wavelet Decomposition ◽  
Wavelet Packet ◽  
Prediction Method ◽  
Vibration Signal ◽  
Signal Denoising ◽  
Maximum Lyapunov Exponent ◽  
Nonlinear Prediction ◽  
Local Method ◽  
Time Frequency ◽  
Nonstationary Vibration

In order to overcome the shortcomings (such as the time–frequency localization and the nonstationary signal analysis ability) of the Fourier transform, time–frequency analysis has been carried out by wavelet packet decomposition and reconstruction according to the actual nonstationary vibration signal from a large equipment located in a large Steel Corporation in this article. The effect of wavelet decomposition on signal denoising and the selection of high-frequency weight coefficients for each layer on signal denoising were analyzed. The nonlinear prediction of the chaotic time series was made by global method, local method, weighted first-order local method, and maximum Lyapunov exponent prediction method correspondingly. It was found the multi-step prediction method is better than other prediction methods.

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