Abstract
In Chap. 2 we introduced random variables, their corresponding densities, and quantities characterizing these densities. We have also seen how to compute with random variables. Up to now, we have avoided a discussion about the interpretation of the probability density. Of course, what we always had in mind is the so-called relative frequency interpretation of the probability density: One assumes that many realizations of the random variable exist. Then a histogram where the relative frequencies of the measured values in so-called bins, i.e., small intervals, are plotted against the intervals themselves, should converge with increasing number of realizations to the probability density.
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References
Abragam, A., Proctor, W.: Experiments on Spin Temperature, Phys. Rev. 106, 160 (1957)
Abragam, A., Proctor, W.: Spin Temperature, Phys. Rev. 109, 1441 (1958)
Ahlers, G.: Thermodynamics and experimental tests of static scaling and universality near the superfluid transition in He4 under pressure. Phys. Rev. A 8(1), 530–568 (1973)
Anderson, M., Ensher, J., Matthews, M., Wieman, C., Cornell, E.: Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269, 198–201 (1995)
Ashcroft, N.W., Mermin, N.: Solid State Physics. Holt–Saunders, Fort Worth (1976)
Baccala, L.A., Sameshima, K.: Partial directed coherence: a new concept in neural structure determination. Biol. Cybern. 84, 463–474 (2001)
Baehr, H.: Thermodynamik, 9th edn. Springer, Berlin/Heidelberg (1996)
Balescu, R.: Equilibrium and Nonequilibrium Statistical Mechanics. Wiley, New York (1975)
Barker, J., Henderson, D.: What is a liquid? Understanding the states of matter. Rev. Mod. Phys. 48, 487 (1976)
Baxter, R.: Exactly Soluble Models in Statistical Mechanics. Academic, New York (1984)
Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York (1978)
Berne, B., Pecora, R.: Dynamic Light Scattering. Wiley, New York (1976)
Besag, J.: Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B 36, 192–236 (1974)
Binder, K.: Theory and technical aspects of Monte Carlo simulation. In: Binder, K. (ed.) Monte Carlo Methods in Statistical Physics. Springer, Berlin/Heidelberg (1979)
Binder, K.: Applications of the Monte-Carlo Method in Statistical Physics. Springer, Berlin/Heidelberg (1987)
Binder, K.: The Monte Carlo Method in Condensed Matter Physics. Springer, Berlin/Heidelberg (1995)
Binder, K., Heermann, D.: Monte Carlo Simulation in Statistical Physics. Springer, Berlin/ Heidelberg (1997)
Bloomfield, P.: Fourier Analysis of Time Series: An Introduction. Wiley, New York (1976)
Brenig, W.: Statistische Theorie der Wärme. Springer, Berlin/Heidelberg (1992)
Breuer, H.-P., Petruccione, F.: Stochastic dynamics of quantum jumps. Phys. Rev. E 52(1), 428–441 (1995)
Breuer, H., Huber, W., Petruccione, F.: The macroscopic limit in a Stochastic Reaction–Diffusion process. Europhys. Lett. 30(2), 69–74 (1995)
Brockwell, P.J., Davies, R.A.: Time Series: Theory and Methods. Springer, New York (1987)
Collins, G.: Gaseous BoseEinstein Condensate Finally Observed, Phys. Today 48, 17-20 (1995)
Dahlhaus, R.: Graphical interaction models for multivariate time series. Metrika 51, 157–172 (2000)
Davis, P., Rabinowitz, P.: Methods of Numerical Integration. Academic, New York (1975)
Dawydow, A.S.: Quantum Mechanics. Pergamon Press, Oxford (1965)
Delves, L., Mohamed, J.: Computational Methods for Integral Equations. Cambridge University Press, Cambridge (1985)
Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum Likelihood from Incomplete Data via the EM-Algorithm. J. R. Stat. Soc. B 39, 1 (1977)
Diu, B., Guthmann, C., Lederer, D., Roulet, B.: Grundlagen der Statistischen Physik. de Gruyter, Berlin (1994)
Domb, C., Green, M.: Phase Transitions and Critical Phenomena, vol. 3. Academic, New York (1974)
Dudley, R.: Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove (1989)
Efron, B., Tibshirani, R.: An Introduction to Bootstrap. Chapman & Hall, New York (1993)
Eichenauer, J., Grothe, H., Lehn, J.: Marsaglia’s lattice test and non-linear congruential pseudo random number generators. Metrika 35, 241 (1988)
Eichler, M.: Graphical modeling of dynamic relationships in multivariate time series. In: Schelter, B., Winterhalder, M., Timmer, J. (eds.) Handbook of Time Series Analysis. Wiley, New York (2006)
Ellis, R.S.: Entropy, Large Deviations, and Statistcal Mechanics. Springer, Berlin/Heidelberg (1985)
Feller, W.: An Introduction to Probability Theory, vol. 2. Wiley, New York (1957)
Fitzgerald, W.J., Ó Ruanaidh, J.J.K.: Numerical Bayesian Methods Applied to Signal Processing. Springer, New York/Berlin/Heidelberg (1996)
Frodesen, A.G., Skjeggestad, O., Tofte, H.: Probability and Statistics in Particle Physics. Universitätsforlaget, Bergen/Oslo/Tromso (1979)
Gardiner, C.: Handbook of Stochastic Methods. Springer, Berlin/Heidelberg (1985)
Gelb, A. (ed.): Applied Optimal Estimation. MIT-Press, Cambridge (1974)
Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6(6), 721–741 (1984)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison, Reading (1994)
Granger, C.J.W.: Investigating causal relations by econometric models and cross-spectral methods. Econometrics 37, 424–438 (1969)
Griffin, A., Snoke, D., Stringari, S. (eds.): Bose–Einstein Condensation. Cambridge University Press, Cambridge (1995)
Groetsch, C.W.: The Theory of Tikhonovs Regularisation for Fredholm Equations of the First Kind. Pitman, Boston (1984)
Hammersley, J.M., Clifford, P.: Markov Field on Finite Graphs and Lattices. unpublished (1971)
Hartung, J., Elpelt, B., Klösener, K.: Statistik. Oldenbourg, Munich/Vienna (1986)
Heller, P.: Experimental investigations of critical phenomena, Rep. Progr. Phys. 30(II), 731 (1967)
Hengartner, W., Theodorescu, R.: Einführung in die Monte-Carlo Methode. Carl Hanser, Munich/ Vienna (1978)
Herring, C.: Direct exchange. In: Rado, G., Suhl, H. (eds.) Magnetism, vol. 2B. Academic, New York (1966)
Hirschfelder, J., Curtiss, C., Bird, R.: Molecular Theory of Gases and Liquids. Wiley, New York (1954)
Ho, J., Litster, J.: Magnetic equation of state of CrBr3 near the critical point. Phys. Rev. Lett. 22(12), 603–606 (1969)
Honerkamp, J.: Stochastic Dynamical Systems. VCH, Weinheim (1994)
Honerkamp, J., Römer, H.: Theoretical Physics: A Classical Approach. Springer, Berlin/ Heidelberg (1993)
Honerkamp, J., Weese, J.: Determination of the Relaxation Spectrum by a Regularization Method. Macromolecules 22, 4372–4377 (1989)
Honerkamp, J., Weese, J.: Tikhonovs regularization method for Ill-posed problems: a comparision of different methods. Contin. Mech. Thermodyn. 2, 17–30 (1990)
Honerkamp, J., Weese, J.: A nonlinear regularization method for the calculation of the relaxation spectrum. Rheol. Acta 32, 32–65 (1993)
Honerkamp, J., Maier, D., Weese, J.: A nonlinear regularization method for the analysis of photon correlation spectroscopy data. J. Chem. Phys. 98(2), 865–872 (1993)
Huang, K.: Statistical Mechanics, 2nd edn. Wiley, New York (1987)
Ising, E.: Beitrag zur theorie des ferromagnetismus. Z. Phys. 31, 253–258 (1925)
Jachan, M., Henschel, K., Nawrath, J., Schad, A., Timmer, J., Schelter, B.: Inferring direct directed-information flow from multivariate nonlinear time series. Phys. Rev. E 80, 011138 (2009)
Jaynes, E.: Papers on Probability, Statistics and Statistical Physics. Reidel, Dordrecht (1982)
Jiu-li, L., Van den Broeck, C., Nicolis, G.: Stability criteria and fluctuations around nonequilibrium states. Z. Phys. B Condens. Matter 56, 165–170 (1984)
Julier, S., Uhlmann, J.: A general method for approximating nonlinear transformations of probability distributions. Technical Report, Department of Engineering Science, University of Oxford (1996)
Julier, S.J., Uhlmann, J.K., Durrant-Whyte, H.F.: A new approach for filtering nonlinear systems. In: Proceedings of the American Control Conference, Seattle, vol. 3, pp. 1628–1632 (1995)
Julier, S.J., Uhlmann, J.K., Durrant-Whyte, H.F.: A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Trans. Autom. Control 45, 477–482 (2000)
Kadanoff, L.: Scaling Laws for Ising Models Near Tc, Physics 2, 263 (1966)
Kitagawa, G., Gersch, W.: Smoothness Priors Analysis of Time Series. Lecture Notes in Statistics, vol. 116. Springer, New York (1996)
Kittel, C.: Thermal Physics. Freeman, San Francisco (1980)
Klein, M.J.: Phys. Rev. 104, 589 (1956)
Klitzing, K.V., Dorda, G., Pepper, M.: New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys. Rev. Lett. 45, 494–497 (1980)
Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin/Heidelberg (1995)
Knuth, D.E.: Seminumerical Algorithms, vol. 2: The Art of Computer Programming. Addison, Reading (1969)
Kolmogoroff, A. Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin/Heidelberg (1933)
Kose, V., Melchert, F.: Quantenmaße in der Elektrischen Meßtechnik. VCH, Weinheim (1991)
Kouvel, J., Comly, J.: Magnetic equation of state for nickel near its curie point. Phys. Rev. Lett. 20(22), 1237–1239 (1968)
Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)
Landau, L.D., Lifschitz, E.M.: Theory of Elasticity. Pergamon, London (1959)
Lehmann, E.: Theory of Point Estimation. Wadsworth & Brooks/Cole, New York (1991)
Lehmer, D.H.: Mathematical methods in large-scale computing units. In: Nr. 141. Proceedings of the 2nd Symposium on Large-Scale Digital Calculating Machinery. Harvard University Press, Cambridge (1951)
Levine, R.D., Tribus, M. (eds.): The Maximum Entropy Formalism. MIT Press, Cambridge (1979)
Lien, W.H., Phillips, N.E.: Low-temperature heat capacities of potassium, rubidium and cesium. Phys. Rev. A 133(5A), 1370 (1964)
Lütkepohl, H., Krätzig, M.: Applied Time Series Econometrics. Cambridge University Press, Cambridge (2004)
Malek Mansour, M., Van den Broeck, C., Nicolis, G., Turner, J.: Asymptotic properties of Markovian master equations. Ann. Phys. 131, 283–313 (1981)
Mandelbrot, B.: The Fractal Geometry of Nature. Freeman, San Francisco (1982)
Mandl, F.: Statistical Physics. Wiley, New York (1971)
Mayer, J.E.: The Statistical Mechanics of Condensing Systems. I, J. Chem. Phys. 5, 67 (1937)
McCoy, B., Wu, T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973)
McQuarrie, D.A.: Statistical Mechanics. Harper & Row, New York (1976)
McQuarrie, D.A.: Quantum Chemistry. Oxford University Press, Oxford (1983)
Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087 (1953)
Michels, A., Blaisse, B., Michels, C.: Proc. R. Soc. A 160, 358 (1937)
Miller, G.F.: In: Delves, L.M., Walsh, J. (eds.) Numerical Solution of Integral Equations. Clarendon, Oxford (1974)
Misiti, M., Misiti, Y., Oppenheim, G., Poggi, J.-M.: Wavelet Toolbox, for Use with MatLab. The MathWorks, Natick (1996)
Montroll, E.: Some remarks on the integral equations of statistical mechanics In: Cohen, E. (ed.) Fundamental Problems in Statistical Mechanics, p. 230. North-Holland, Amsterdam (1962)
Montroll, E., West, B.: On an enriched collection of Stochastic processes. In: Montroll, E., Lebowitz, J. (eds.) Fluctuation Phenomena, Chap. 2, p. 62–206. North-Holland, Amsterdam (1987)
Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)
Nicolis, G., Prigogine, I.: Self-Organization in Non-equilibrium Systems. Wiley, New York (1977)
Niederreiter, H.: Recent trends in random number and random vector generation. Ann. Oper. Res., Proc. 5th Int. Conf. Stoch. Programm. Ann. Arbor. 35, 241 (1989)
Nolte, G., Bai, O., Wheaton, L., Mari, Z., Vorbach, S., Hallett, M.: Identifying true brain interaction from EEG data using the imaginary part of coherency. Clin. Neurophysiol. 115, 2292–2307 (2004)
Oppenheimer, A., Schafer, R.: Discrete-Time Signal Processing. Prentice-Hall, Englewood Cliffs (1989)
Papoulis, A.: Probability, Random Variables, and Stochastic Processes. McGraw-Hill, New York (1984)
Penzias, A.A.,Wilson, R.: A Measurement of Excess Antenna Temperature at 4080 mc/s., Astrophys. J. 142, 419-421 (1965)
Prange, R., Girvin, S. (eds.): The Quantum Hall Effect. Springer, Berlin/Heidelberg (1987)
Press, W., Teukolsky, S., Vetterling, W., Flannery, B.: Numerical Recipes. Cambridge University Press, Cambridge (2007)
Priestley, M.: Spectral Analysis and Time Series. Academic, New York (1981)
Prigogine, I., Resibois, P.: On the kinetics of the approach to equilibrium, Physica 27(629) (1961)
Purcell, E., Pound, R.: A Nuclear Spin System at Negative Temperature,, Phys. Rev. 81, 279 (1951)
Rabiner, L., Gold, B.: Theory and Application of Digital Signal Processing. Prentice-Hall, Englewood Cliffs (1975)
Rahman, A: Correlations in the Motion of Atoms in Liquid Argon,.: Phys. Rev. A 136, A405 (1964)
Ramsey, N.: Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures, Phys. Rev. 103, 20 (1956)
Ree, F.H., Hoover, W.G.: Fifth and Sixth Virial Coefficients for Hard Spheres and Hard Disks, J. Chem. Phys. 40, 939 (1964)
Reichl, L.E.: A Modern Course in Statistical Physics. Arnold, London (1980)
Römer, H., Filk, T.: Statistische Mechanik. VCH, Weinheim (1994)
Roths, T., Maier, D., Friedrich, C., Marth, M., Honerkamp, J.: Determination of the relaxation time spectrum from dynamic moduli using an edge preserving regularization method. Rheol. Acta 39, 163–173 (2000)
Rubinstein, R.Y.: Simulation and the Monte-Carlo Method. Wiley, New York (1981)
Samorodnitzky, G., Taqqu, M.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York/London (1994)
Saupe, D.: Algorithms for random fractals. In: Peitgen, H., Saupe, D. (eds.) The Science of Fractal Images. Springer, Berlin/Heidelberg (1988)
Schelter, B., Winterhalder, M., Eichler, M., Peifer, M., Hellwig, B., Guschlbauer, B., Lücking, C.H., Dahlhaus, R., Timmer, J.: Testing for directed influences among neural signals using partial directed coherence. J. Neurosci. Method 152, 210–219 (2006a)
Schelter, B., Winterhalder, M., Timmer, J. (eds.): Handbook of Time Series Analysis. Wiley-VCH, Berlin (2006b)
Schlittgen, R., Streitberg, H.: Zeitreihenanalyse. R. Oldenbourg, Munich (1987)
Schneider, I. (ed.): Die Entwicklung der Wahrscheinlichkeitsrechnung von den Anfängen bis 1933. Wissenschaftliche Buchgesellschaft, Darmstadt (1986)
Shannon, C.: A Mathematical Theory of Communication. Bell Syst. Technol. J. 27, 379–423 (1948)
Shwartz, A., Weiss, A.: Large Deviations for Performance Analysis. Chapman & Hall, New York/London (1995)
Sommerlade, L., Thiel, M., Platt, B., Plano, A., Riedel, G., Grebogi, C., Timmer, J., Schelter, B. Inference of Granger causal time-dependent influences in noisy multivariate time series. J. Neurosci. Method 203, 173–185 (2012)
Stanley, H.E.: Phase Transition and Critical Phenomena. Clarendon Press, Oxford (1971)
Strang, G., Nguyen, T.: Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley (1996)
Straumann, N.: Thermodynamik. Springer, Berlin/Heidelberg (1986)
Stroud, A.: Approximate Calculation of Multiple Integrals. Prentice Hall, Englewood Cliffs (1971)
Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Wiley, New York (1977)
Ursell, H.D.: The evaluation of Gibb’s phase-integral for imperfect gases, Proc. Cambridge Phil. Soc. 23, 685 (1927)
Verlet, L.: Computer “Experiments” on Classical Fluids. II. Equilibrium Correlation Functions, Phys. Rev. 165, 201 (1968)
van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1985)
von Randow, G.: Das Ziegenproblem. Rowohlt, Reinbek (1992)
Voss, H., Timmer, J., Kurths, J.: Nonlinear dynamical system identification from uncertain and indirect measurements. Int. J. Bifurc. Chaos 6, 1905–1933 (2004).
Wan, E.A., Nelson, A.T.: Neural dual extended Kalman filtering: applications in speechenhancement and monaural blind signal separation. In: IEEE Proceedings in Neural Networks for Signal Processing, Munich, pp. 466–475.
Wegner, F.: Corrections to Scaling Laws, Phys. Rev. B 5, 4529 (1972)
Wilson, K.G.: Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture, Phys. Rev. B 4, 31743184 (1971)
Wilson, K.G., Kogut, J.: The renormalization group and the ε expansion. Phys. Rep. 12 C, 75–200 (1974)
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Honerkamp, J. (2012). Estimation of Parameters. In: Statistical Physics. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28684-1_8
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