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.

Conclusion

2006 ◽  
Author(s):  
Huug van den Dool
Keyword(s):  
Degrees Of Freedom ◽  
Cost Benefit Analysis ◽  
Cost Benefit ◽  
Seasonal Prediction ◽  
Boreal Winter ◽  
Empirical Prediction ◽  
Subsequent Chapter ◽  
Natural Analogues ◽  
Potato Farmer ◽  
Day By Day

In this book we have reviewed empirical methods in short-term climate prediction. We devoted a whole chapter to the design of two of these methods, Empirical Wave Propagation (EWP, Chapter 3) and Constructed Analogue (CA Chapter 7). Other methods of empirical prediction were listed in Chapter 8, with brief descriptions and examples and references. One chapter is devoted to EOFs, as such a diagnostic topic, but widely used in both prediction and diagnostics, and thoroughly debated for a few decades. Two brief chapters, written in support of the subsequent chapter, Teleconnections (Chapter 4), should make the discussion on EOFs more interesting, and the topic of effective degrees of freedom (Chapter 6) is indispensable when one wants to understand why and when natural analogues would work (or not), or how an analogue is constructed, or how any method using truncation works. Most chapters can be read largely in isolation, but connections can be made of course between chapters. EWP is claimed to be useful, if not essential, in understanding teleconnections. Dispersion experiments, featuring day-by-day time-scales, link the CA and EWP methods. Examples of El Nino boreal winter behavior can be found in (a) the examples of EOFs on global SST and 500 mb streamfunction (Chapter 5), (b) specification of surface weather from 500 mb streamfunction (Chapter 7), and (c) the ENSO correlation and compositing approach (Chapter 8). The noble pursuit of knowledge may have been as important in the choice of some material as any immediate prediction application. Chapter 9 is different, less research oriented, and more an eyewitness description of what goes on in the making of a seasonal prediction. This eyewitness account style spills over into Chapter 8 here and there, because in order to understand why certain methods have survived to this day some practicalities have to be understood. The closeness to real-time prediction throughout the book creates a sense of application. However, the application in this book does not go beyond the making of the forecast itself; we completely shied away from such topics as a cost/benefit analysis or decision-making process by, for example, a climate sensitive potato farmer or reservoir operator.

scholarly journals Downstream prediction using a nonlinear prediction method

2013 ◽  
Vol 10 (11) ◽  
pp. 14331-14354 ◽  
Author(s):  
N. H. Adenan ◽  
M. S. M. Noorani
Keyword(s):  
Phase Space ◽  
Flood Control ◽  
River Flow ◽  
Linear Prediction ◽  
Prediction Method ◽  
Nonlinear Prediction ◽  
Langat River ◽  
Correlation Dimension Method ◽  
The Impact ◽  
Downstream Area

Abstract. The estimation of river flow is significantly related to the impact of urban hydrology, as this could provide information to solve important problems, such as flooding downstream. The nonlinear prediction method has been employed for analysis of four years of daily river flow data for the Langat River at Kajang, Malaysia, which is located in a downstream area. The nonlinear prediction method involves two steps; namely, the reconstruction of phase space and prediction. The reconstruction of phase space involves reconstruction from a single variable to the m-dimensional phase space in which the dimension m is based on optimal values from two methods: the correlation dimension method (Model I) and false nearest neighbour(s) (Model II). The selection of an appropriate method for selecting a combination of preliminary parameters, such as m, is important to provide an accurate prediction. From our investigation, we gather that via manipulation of the appropriate parameters for the reconstruction of the phase space, Model II provides better prediction results. In particular, we have used Model II together with the local linear prediction method to achieve the prediction results for the downstream area with a high correlation coefficient. In summary, the results show that Langat River in Kajang is chaotic, and, therefore, predictable using the nonlinear prediction method. Thus, the analysis and prediction of river flow in this area can provide river flow information to the proper authorities for the construction of flood control, particularly for the downstream area.

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.

Surrogate Testing by Estimating the Local Dispersions in Phase Space

2003 ◽  
Vol 13 (08) ◽  
pp. 2241-2251 ◽  
Author(s):  
Radhakrishnan Nagarajan
Keyword(s):  
Time Series ◽  
Phase Space ◽  
Stationary Time Series ◽  
Data Sets ◽  
Local Dispersion ◽  
Dimensional Phase Space ◽  
Clustering Approach ◽  
Principal Directions ◽  
The One ◽  
Value Decomposition

In this report, a clustering approach is presented to detect dynamical nonlinearity in a stationary time series. The one-dimensional time series sampled from a dynamical system is mapped on to the m-dimensional phase space by the method of delays. The vectors in the phase space are partitioned by k-means clustering technique. The local trajectory matrix for the vectors in each of the clusters is determined. The eigenvalues of the local trajectory matrices represent the variation along the principal directions and are obtained by singular value decomposition (SVD). The product of the eigenvalues represents the generalized variance and is a measure of the local dispersion in the phase space. The sum of the local dispersions (gT) is used as the discriminant statistic to classify data sets obtained from deterministic and stochastic settings. The surrogates were generated by the Iterated Amplitude Adjusted Fourier Transform (IAAFT) technique.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Shape Dynamics and the Linking Theory

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Line Element ◽  
Hamiltonian Formulation ◽  
Operational Definition ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Problem Of Time ◽  
Definition Of ◽  
Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

scholarly journals Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

2021 ◽  
Vol 87 (3) ◽  
Author(s):  
Nicolas Crouseilles ◽  
Paul-Antoine Hervieux ◽  
Yingzhe Li ◽  
Giovanni Manfredi ◽  
Yajuan Sun
Keyword(s):  
Phase Space ◽  
Maxwell Equations ◽  
Stimulated Raman Scattering ◽  
Extended Phase ◽  
Five Dimensions ◽  
Particle In Cell ◽  
Spin Effects ◽  
Dimensional Phase Space ◽  
Spin Polarized ◽  
Underdense Plasma

We propose a numerical scheme to solve the semiclassical Vlasov–Maxwell equations for electrons with spin. The electron gas is described by a distribution function $f(t,{\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ that evolves in an extended 9-dimensional phase space $({\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ , where $\boldsymbol s$ represents the spin vector. Using suitable approximations and symmetries, the extended phase space can be reduced to five dimensions: $(x,{{p_x}}, {\boldsymbol s})$ . It can be shown that the spin Vlasov–Maxwell equations enjoy a Hamiltonian structure that motivates the use of the recently developed geometric particle-in-cell (PIC) methods. Here, the geometric PIC approach is generalized to the case of electrons with spin. Total energy conservation is very well satisfied, with a relative error below $0.05\,\%$ . As a relevant example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, where the electrons are partially or fully spin polarized. It is shown that the Raman instability is very effective in destroying the electron polarization.

Stability Prediction of Surrounding Rocks and Optimum Design of Roof-Bolt Parameters in Deep Roadway

2013 ◽  
Vol 353-356 ◽  
pp. 436-439
Author(s):  
De Sen Kong ◽  
Yong Po Chen
Keyword(s):  
Prediction Method ◽  
Simulation Method ◽  
Complex Stress ◽  
Roof Bolt ◽  
Deformation And Failure ◽  
Long Run ◽  
Research Findings ◽  
Stress Environment ◽  
Classification Prediction ◽  
The Stability

In order to forecast the stability of deep roadway and optimize the parameters of bolts, the complex stress environment and the multivariate surrounding rocks characteristics of deep roadway were analyzed. Then the classification prediction method and the numerical simulation method were simultaneously used to analysis the stability of surrounding rocks. Furthermore, the supporting parameters of bolts were also designed optimally. It was shown that the characteristics of stress distribution, deformation and failure zone of surrounding rocks are not ideal. So it is necessary to optimize the supporting parameters of deep roadway. All these research findings will provide the theory basis for bolts of deep roadway and will ensure the optimization of bolts and the stability of deep roadway in the long run.

THE HOURS WORKED–PRODUCTIVITY PUZZLE: IDENTIFICATION IN A FRACTIONAL INTEGRATION SETTING

2014 ◽  
Vol 19 (7) ◽  
pp. 1593-1621 ◽  
Author(s):  
Yuliya Lovcha ◽  
Alejandro Perez-Laborda
Keyword(s):  
Business Cycle ◽  
Fractional Integration ◽  
Real Business Cycle ◽  
Data Sets ◽  
Sampling Uncertainty ◽  
Hours Worked ◽  
Long Run ◽  
The Hours ◽  
Short Run ◽  
Technology Shock

A recent finding of the SVAR literature is that the response of hours worked to a (positive) technology shock depends on the assumed order of integration of the hours. In this work we relax this assumption, allowing fractional integration in hours and productivity. We find that the sign and magnitude of the estimated responses depend crucially on the identification assumptions employed. Although the responses of hours recovered with short-run (SR) restrictions are positive in all data sets, long-run (LR) identification results in negative, although sometimes not significant responses. We check the validity of these assumptions with the Sims procedure, concluding that both LR and SR are appropriate to recover responses in a fractionally integrated VAR. However, the application of the LR scheme always results in an increase in sampling uncertainty. Results also show that even the negative responses found in the data could still be compatible with real business cycle models.

Sign in / Sign up

Export Citation Format

Share Document