A bootstrap test for the comparison of nonlinear time series

Hypothesis tests

Test procedures for assessing whether two stationary and independent time series with unequal lengths have the same spectral density (or same auto‐covariance function) are investigated. A new test statistic is proposed based on the wavelet transform. It relies on empirical wavelet coefficients of the logarithm of two spectral densities' ratio. Under the null hypothesis that two spectral densities are the same, the asymptotic normal distribution of the empirical wavelet coeffcients is derived. Furthermore, these empirical wavelet coefficients are asymptotically uncorrelated. A test statistic is proposed based on these results. The performance of the new test statistic is compared to several recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed test is very comparable to the current test statistics in most cases. The main advantage of our proposed test statistic is that it is constructed very simply and is easy to implement.

The prime goal of this paper is to establish sharp lower and upper bounds for useful functions such as the exponential functions, with a focus on exp(−x^2), the trigonometric functions (cosine and sine) and the hyperbolic functions (cosine and sine). The bounds obtained for hyperbolic cosine are very sharp. New proofs, refinements as well as new results are offered. Some graphical and numerical results illustrate the findings.

The student is introduced to numerical calculations at the initial stages of studying mathematics. The student masters the execution of four arithmetic operations (addition, subtraction, multiplication, division) manually. For calculating trigonometric and exponential functions, square and cubic roots, arithmetic calculators or tables are usually used; this also applies to logarithms. At the same time, confident proficiency in mathematics necessarily involves the ability to perform calculations manually, with at least some accuracy (2–3 significant figures), for any task in natural sciences and mathematics within the school curriculum. This competence enables the student not to perceive computer computations or calculator-based calculations as something incomprehensible — like a trick or a "black box," where input data is provided and output results are obtained. It is understood that the authors do not advocate abandoning calculators when solving school tasks. However, the student should not become entirely dependent on this tool. The article demonstrates methods for calculating the values of exponential, logarithmic, and trigonometric functions, extends the set of exact values of trigonometric functions beyond what is known from the school curriculum (for arguments , , , , ), and illustrates methods for finding square and cubic roots of a number.

The technic of the processing of the meteorological data for conclusion on the presence of the time trends in the quantitative characteristics of the weather on the example of the analysis of the average yearly atmospheric temperature change at the meteorological station Minsk from 1989 is presented. The average yearly atmospheric temperature received from the measurements is approximated by the least square error method in the linear time dependence regression function. The linear time dependence regression function received in such a way has some positive growing (positive trend). The aim of this paper is to clarify the significance of this growth. For this aim, the usage of the regression analysis with its procedures of hypotheses testing is proposed. First of all, the performing of the demands presented to the regression analysis is checked: normality of the distribution of the disturbance and the homogeneity of the variance (dispersion) of the disturbance. The normality of the distribution of the disturbance was checked and confirmed by the Kolmogorov test. The homogeneity of the dispersion of the disturbance was checked and confirmed both by checking the hypotheses on the equality of the dispersions of two normal distributions and by the Smirnov test for checking the hypotheses on the equality of two distributions. For checking the significance of the positive trend of the yearly mean temperature, the hypotheses on the significance of the coefficients of the linear regression function by the Student t-statistics and the hypothesis on the linear connection presence by the analysis of variance were checked. As the result, the insignificance of the positive linear trend from 1998 to 2016 and from 1998 to 2017 and its significance from 1998 to 2018 and from 1998 to 2019 on the level of significance 0.05 for mean average yearly atmospheric temperature at the meteorological station Minsk was stated.

CHAPTER 1. AN INTRODUCTION TO DATA AND FUNCTIONS. 1.1 Describing Single-Variable Data. 1.2 Describing Relationships between Two Variables. 1.3 An Introduction to Functions. 1.4 The Language of Functions. 1.5 Visualizing Functions. CHAPTER 2. RATES OF CHANGE AND LINEAR FUNCTIONS. 2.1 Average Rates of Change. 2.2 Change in the Average Rate of Change. 2.3 The Average Rate of Change Is a Slope. 2.4 Putting a Slant on Data. 2.5 Linear Functions: When Rates of Change Are Constant. 2.6 Visualizing Linear Functions. 2.7 Constructing Graphs and Equations of Linear Functions. 2.8 Special Cases. 2.9 Breaking the Line: Piecewise Linear Functions. 2.10 Constructing Linear Models of Data. 2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines. CHAPTER 3. WHEN LINES MEET: LINEAR SYSTEMS. 3.1 Interpreting Intersection Points: Linear and Nonlinear Systems. 3.2 Visualizing and Solving Linear Systems. 3.3 Reading between the Lines: Linear Inequalities. 3.4 Systems with Piecewise Linear Functions: Tax Plans. CHAPTER 4. THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE. 4.1 The Numbers of Science: Measuring Time and Space. 4.2 Positive Integer Exponents. 4.3 Zero, Negative, and Fractional Exponents. 4.4 Converting Units. 4.5 Orders of Magnitude. 4.6 Logarithms as Numbers. CHAPTER 5. GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS. 5.1 Exponential Growth. 5.2 Exponential Decay. 5.3 Comparing Linear and Exponential Functions. 5.4 Visualizing Exponential Functions. 5.5 Exponential Functions: A Constant Percent Change. 5.6 More Examples of Exponential Growth and Decay. 5.7 Compound Interest and the Number e. 5.8 Semi-Log Plots of Exponential Functions. CHAPTER 6. LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 6.1 Using Logarithms to Solve Exponential Equations. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e. 6.3 Visualizing and Applying Logarithmic Functions. 6.4 Using Semi-Log Plots to Construct Exponential Models for Data. C H A P T E R 7. POWER FUNCTIONS. 7.1 The Tension between Surface Area and Volume. 7.2 Direct Proportionality: Power Functions with Positive Powers. 7.3 Visualizing Positive Integer Power Functions. 7.4 Comparing Power and Exponential Functions. 7.5 Inverse Proportionality: Power Functions with Negative Powers. 7.6 Visualizing Negative Integer Power Functions. 7.7 Using Logarithmic Scales to Find the Best Functional Model. CHAPTER 8. QUADRATICS AND THE MATHEMATICS OF MOTION. 8.1 An Introduction to Quadratic Functions: The Standard Form. 8.2 Visualizing Quadratics: The Vertex Form. 8.3 The Standard Form vs. the Vertex Form. 8.4 Finding the Horizontal Intercepts: The Factored Form. 8.5 The Average Rate of Change of a Quadratic Function. 8.6 The Mathematics of Motion. CHAPTER 9. NEW FUNCTIONS FROM OLD. 9.1 Transformations. 9.2 The Algebra of Functions. 9.3 Polynomials: The Sum of Power Functions. 9.4 Rational Functions: The Quotient of Polynomials. 9.5 Composition and Inverse Functions. 9.6 Exploring, Extending & Expanding. APPENDIX Student Data Tables for Exploration 2.1. Data Dictionary for FAM1000 Data. SOLUTIONS For all Algebra Aerobics and Check Your Understanding problems for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1. INDEX.

Checking the adequacy of functional linear quantile regression model

Semiparametric Innovation-Based Tests of Orthogonality and Causality between Two Infinite-Order Cointegrated Ceries with Application to Canada/US Monetary Interactions

This paper exploits a known but insufficiently attended to idea for the autogressive modeling of multivariate stationary and nonstationary covariance time series. (The term multichannel is common in the engineering literature. Here, we use both terminologies, multivariate and muItich annel.) The idea is to model multichannel thing s one channel ar-a-time as an instantaneous response-orthogonal innovations autoregressive model. Such modeling can be realized using only scalar computations. That idea was implicit in Pagano (1978) and was explicit in a program MULMAR by Kitagawa (Akaike et al, 1979 ), in Newton (1982) and also Sakai (1982). Each of those papers only treated multichannel stationary time series AR modeling. Here , we exploit that idea to address time series analysis problems via the realization of autoregressive models of botti multi channel stationary and multi channel nonstationary covariance times series data.

Euler’s formula \(e^{it} = \cos (t) + i\sin (t)\) points to the intimate link between the exponential and the trigonometric functions. (Physicist Richard Feynman called the formula “the most remarkable formula in mathematics.” [32]) The generalized trigonometric and hyperbolic functions we have explored thus far lead to generalizations of the exponential and logarithmic functions as well. We engage here in a short exploration of these transcendental functions.

The ice core time series from Vostok Station in Antarctica and the North Greenland Ice Core Project have seasonal variation corresponding to the Milankovitch cycles. After removing these cycles, and interpolating to equal time intervals, stationary time series models are fitted. The series show clear directionality and this feature is modelled by either non-Gaussian errors or non-linear time series models. Threshold autoregressive models are fitted by penalized least squares and compared with non-threshold autoregressive models. Since both ice core time series are reasonably modelled as first order autoregressive series with parameters close to one, directionality will arise from non-symmetric error distributions. However, two regime threshold autoregressive models, of order one and two for Greenland and Vostok, respectively, give an improved match to the observed directionality and a reduced sum of squared residuals. Realizations from the threshold autoregressive models are noticeably different from the non-threshold models. Since the non-threshold models are a restricted case of the threshold models, and the threshold models are a better fit to the observed time series, threshold models should provide more realistic realizations. References C. Chatfield. The Analysis of Time Series: An Introduction . CRC Press, 2004. https://www.crcpress.com/The-Analysis-of-Time-Series-An-Introduction-Sixth-Edition/Chatfield/p/book/9781584883173 A. J. Lawrance. Directionality and reversibility in time series. Int. Stat. Rev. 59(1):67–79, 1991. doi:10.2307/1403575 M. M. Mansor, M. E. Glonek, D. A. Green and A. V. Metcalfe. Threshold autoregressive models for directional time series. In I. Rojas and H. Pomares (Eds.), Time Series Analysis and Forecasting Selected Contributions from the ITISE Conference (ITISE 2015) . pp. 13–25, 2016. doi:10.1007/978-3-319-28725-6 M. M. Mansor, M. E. Glonek, D. A. Green and A. V. Metcalfe. Modelling directionality in stationary geophysical time series. International work-conference on Time Series (ITISE 2015) . http://www.researchgate.net/publication/281835075 M. M. Mansor, D. A. Green and A. V. Metcalfe. Modelling and simulation of directional financial time series. Proceedings of the 21st International Congress on Modelling and Simulation (MODSIM 2015) , pp. 1022–1028, 2015. http://www.mssanz.org.au/modsim2015/E4/mansor.pdf M. M. Mansor, D. A. Green and A. V. Metcalfe. Directionality and volatility in electroencephalogram time series. Proceedings of the 2nd International Conference on Mathematical Sciences and Statistics (ICMSS 2016), AIP Conf. Proc. 1739:020080, 2016. doi:10.1063/1.4952560 North Greenland Ice Core Project members. High-resolution record of Northern Hemisphere climate extending into the last interglacial period. Nature , 431:147–151, 2004. doi:10.1038/nature02805 J. R. Petit, J. Jouzel, D. Raynaud, N. I. Barkov, J.-M. Barnola, I. Basile, M. Bender, J. Chappellaz, M. Davis, G. Delaygue, M. Delmotte, V. M. Kotlyakov, M. Legrand, V. Y. Lipenkov, C. Lorius, L. Pepin, C. Ritz, E. Saltzman and M. Stievenard. Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica. Nature , 399:429–436, 1999. doi:10.1038/20859 S. Soubeyrand, C. E. Morris and E. K. Bigg. Analysis of fragmented time directionality in time series to elucidate feedbacks in climate data. Environ. Modell. Softw. 61:78–86, 2014. doi:10.1016/j.envsoft.2014.07.003

Neural networks are universal approximators for nonlinear functions. This study aimed to develop an algorithm for functional link artificial neural network (FLANN), and to simulate insect’s food intake dynamics using the algorithm. Complete Matlab codes for FLANN algorithm were given in the paper. Conventional models and FLANN were used to modeling accumulated food intake of the larva of a holometabolous insect, Spodoptera litura. Simulation performance of FLANN was compared against conventional models and sensitivity analysis was conducted on FLANN. The results showed that the FLANN algorithm performed better than conventional models in the simulation of dynamics and temperature–time dependent relationship of larva’s food intake. The conventional models like fractional function, polynomial function, and exponential function were indicated to simulate the food intake dynamics at a higher accuracy but their performances were worse than FLANN. Both multivariate linear regression and trend surface model were used to describe temperature–time dependent relationship of food intake. The overall trend for this relationship could be simulated using these models, however, the simulation accuracy of these models was lower than FLANN. Sensitivity analysis showed that Legendre functions, Chebyshov functions, and trigonometric functions, used as the basis functions in FLANN, yielded better fitness than Laguerre functions and Hermite functions. The mean squared error of simulation using Legendre functions, Chebyshov functions, and trigonometric functions decreased as the increase of the number of these basis functions. Simulation performance also varied with the change of type of nonlinear functions and parameter values in the function. Linear function, negative exponential function and power function were the best nonlinear functions, which yielded more stable outputs as the change of parameter values.

Phi / k transforms are widely used to predict permeability. Some of the difficulties of this exercise are well identified, such as the homogeneity of the population (rock typing), the matching of cores and logs (especially depth matching), and the problem of permeability upscaling. Not so well known, however, are the pitfalls of a statistical and geostatistical nature that may create significant biases, always in the same direction, an underestimation of permeability. The passage from Phi to k is performed in three steps: (1) in cored wells, an exponential regression equation is established between core porosity and core permeability k; (2) in uncored wells, log porosity is used instead as input; (3) the same equation is sometimes used again to populate the cells of a dynamic reservoir model in 3D, where input porosity values are obtained by interpolation. The core-scale regression equation generally underestimates permeability by at least a factor of 2. The origin of the bias lies in the reverse transformation from logarithmic to arithmetic scale. To avoid this pitfall a new permeability estimator is proposed, based on the quantile curves of the Phi / k crossplot. This estimator is data-driven and does not a priori assume any particular functional relationship between Phi and k, such as an exponential regression function. One of the simplest diagnostic tools to check the agreement between log and core porosity is a crossplot of one against the other. In the absence of bias the points are expected to be distributed along the Y = X line. In reality they are or they are not, according to which variable is plotted along the X axis. This apparent paradox is elucidated by bivariate regression theory and related to the difference of investigated volume between core and log data. Direct input of upscaled cell porosity into an exponential core-scale permeability transform amounts to forcing geometric permeability averaging, which may again lead to serious underestimation of the true upscaled permeability when heterogeneity is significant.

We propose a test for the equality of the autocovariance functions of two independent and stationary time series. The test statistic is a quadratic form in the vector of differences of the first J + 1 autocovariances. Its asymptotic distribution is derived under the null hypothesis, and the finite‐sample properties of the test, namely the bias and the power, are investigated by Monte Carlo methods. A by‐product of this study is a new estimator of the covariance between two sample autocovariances which provides a positive definite covariance matrix. We establish the convergence of this estimator in the L1 norm.

The bispectrum and third-order moment can be viewed as equivalent tools for testing for the presence of nonlinearity in stationary time series. This is because the bispectrum is the Fourier transform of the third-order moment. An advantage of the bispectrum is that its estimator comprises terms that are asymptotically independent at distinct bifrequencies under the null hypothesis of linearity. An advantage of the third-order moment is that its values in any subset of joint lags can be used in the test, whereas when using the bispectrum the entire (or truncated) third-order moment is required to construct the Fourier transform. We propose a test for nonlinearity based upon the estimated third-order moment. We use the phase scrambling bootstrap method to give a nonparametric estimate of the variance of our test statistic under the null hypothesis. Using a simulation study, we demonstrate that the test obtains its target significance level, with large power, when compared to an existing standard parametric test that uses the bispectrum. Further we show how the proposed test can be used to identify the source of nonlinearity due to interactions at specific frequencies. We also investigate implications for heuristic diagnosis of nonstationarity.