The First Law of Thermodynamics is the Principle of Conservation of Energy applied to the interaction between Systems. Such interaction is partially observed at a macroscopic scale, in the form of Work. The remaining interaction, taking... more
The First Law of Thermodynamics is the Principle of Conservation of Energy applied to the interaction between Systems. Such interaction is partially observed at a macroscopic scale, in the form of Work. The remaining interaction, taking place at the microscopic scale and not observed as macroscopic work, is denoted as Heat. Thus, the change in energy of a system can be interpreted as the sum of energies transferred in the form of (macroscopic) Work and (microscopic) Heat. However, there are different types of heat. The most common type of heat is proportional to the temperature difference between the systems, but there are other types which are independent of the systems temperatures. To avoid the incorrect use of the First Law, it is important to clearly understand the concepts of Heat and Work. In the first part of these series, these fundamental concepts are discussed in detail, and a general formulation of the First Law is presented. In the second part of the series, this general formulation is applied to a wide variety of representative interacting systems.
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The error involved in the estimation of the mean value of a population depends on both the sample size and the population size. Conventional expressions for determining the standard error in the estimation of the mean have been obtained... more
The error involved in the estimation of the mean value of a population depends on both the sample size and the population size. Conventional expressions for determining the standard error in the estimation of the mean have been obtained under the assumption of independence between the elements in the sample. Unfortunately, for finite populations, the elements are not independent from each other, but they are correlated since the distribution of remaining elements in the population changes after an element is sampled. In this report, a general expression for the estimation error of the mean of finite populations is derived. As the population size increases, the estimation error approaches the conventional expression for infinite populations. An illustrative example is used to show the validity of the general expression obtained.
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The properties of molecular systems are typically fluctuating due to the permanent motion and interaction (including collisions) of their molecules. Due to our inability to track the position and determine the energy of all molecules in... more
The properties of molecular systems are typically fluctuating due to the permanent motion and interaction (including collisions) of their molecules. Due to our inability to track the position and determine the energy of all molecules in the system at all times, those fluctuations seem to be random. Thus, randomistic models (combining deterministic and random terms) can be used to describe the behavior of local properties in a molecular system. In particular, a microcanonical (NVE) system is considered for the present analysis. As an illustrative example, the randomistic models for describing the fluctuations expected in monoatomic ideal gas systems are reported.
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The binomial distribution is a well-known example of discrete probability distribution. Only two outcomes are possible for each independent trial in a binomial experiment. In this report, a continuous approximation is proposed for... more
The binomial distribution is a well-known example of discrete probability distribution. Only two outcomes are possible for each independent trial in a binomial experiment. In this report, a continuous approximation is proposed for describing the discrete binomial probability function, which can then be used to represent an analogous binomial continuous variable. The proposed approximation consists of a correction to the combinatorial number approximated by using Stirling's equation, followed by a Taylor series approximation truncated after the second power. As a result, a normal or Gaussian distribution function is obtained. The error of the proposed approximation decays with the number of trials considered. However, even for small numbers of trials (e.g. less than 10), the approximation can be considered satisfactory.
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Local indistinguishability of the values of a randomistic variable (due to resolution limitations, measurement uncertainty or any other cause), have a discretization effect on the probability distribution function of the variable. In this... more
Local indistinguishability of the values of a randomistic variable (due to resolution limitations, measurement uncertainty or any other cause), have a discretization effect on the probability distribution function of the variable. In this report, analytical expressions for determining the probability distributions after locally averaging variable values are presented. As a particular case, local conditional averaging is observed when the discretization of a variable affects the probability distribution function of a dependent variable. These expressions are then applied to some representative examples in order to illustrate the procedure. In the case of continuous variables, after local averaging a variable, the original probability density function transforms into a series of step-like, local uniform functions, resembling a histogram. As the size of the local region considered decreases, the resulting probability distribution function coincides with the original, exact distribution function. On the other hand, as the local region size increases, the distribution function resembles a histogram with fewer bins, until a single uniform distribution is finally obtained.
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Randomistic variables integrate the realms of deterministic and random variables. Randomistic variables are represented by probability distribution functions, and in the case of continuous variables, also by probability density functions... more
Randomistic variables integrate the realms of deterministic and random variables. Randomistic variables are represented by probability distribution functions, and in the case of continuous variables, also by probability density functions (just like random variables). Any randomistic variable can be subject to external constraints on its possible values. Thus, the resulting probability distribution of the constrained variable may be different from the probability distribution of the original variable. In this report, general expressions for analytically determining the probability distribution functions (or probability density functions) of constrained randomistic variables are presented. These expressions are extended to constraints involving multiple, independent randomistic variables. Several illustrative examples, with different degrees of difficulty, are included. These examples show that constrained randomistic variables represent the solution to a wide variety of problems, including algebraic systems of equations, inequalities, magic squares, etc. Further improvements in analytical and numerical methods for finding constrained probability functions would be highly desirable.
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Any rounding operation of a value causes loss of information, and thus, introduces error. Two types of error are involved: Systematic error (bias) and random error (uncertainty). Uncertainty is always introduced for any type of rounding... more
Any rounding operation of a value causes loss of information, and thus, introduces error. Two types of error are involved: Systematic error (bias) and random error (uncertainty). Uncertainty is always introduced for any type of rounding employed. Bias is directly introduced only when lower ("floor") and upper ("ceiling") types of rounding are used. Central rounding is in principle unbiased, but bias may emerge in the case of nonlinear operations. The purpose of this report is discussing the propagation of both types of rounding error when rounded values are used in common mathematical operations. The basic mathematical operations considered are addition/subtraction, product, and natural powers. These operations can be used to evaluate the propagation of error in power series, which then are used to describe error propagation for any arbitrary nonlinear function. Even when power series approximations can be obtained for any arbitrary reference value, it is highly recommended using the corresponding rounded value as reference. The error propagation expressions obtained are implemented in R language to facilitate the calculations. A couple of examples are included to illustrate the evaluation of error propagation. These examples also show that truncating the power series after the linear term already provides a good estimation of error propagation (using the rounded value as reference point for the power series expansion).
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This report summarizes the principles of the calculus of probabilities applied to real, quantitative randomistic variables. These principles are consistent with the conventional theories of probability, sets and logic. In addition, this... more
This report summarizes the principles of the calculus of probabilities applied to real, quantitative randomistic variables. These principles are consistent with the conventional theories of probability, sets and logic. In addition, this calculus of probabilities applies to both random and deterministic variables, as well as their linear combinations (randomistic variables). Most equations involved in the calculus of probabilities are expressed in terms of set membership functions, which can be either Boolean (binary values of 0 and 1) or Fuzzy (real values between 0 and 1). A direct extension of the calculus of probabilities to multivariate situations is also included.
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Nonlinear regression consists in finding the best possible model parameter values of a given homoscedastic mathematical structure with nonlinear functions of the model parameters. In this report, the second part of the series, the... more
Nonlinear regression consists in finding the best possible model parameter values of a given homoscedastic mathematical structure with nonlinear functions of the model parameters. In this report, the second part of the series, the mathematical structure of models with nonlinear functions of their parameters is optimized, resulting in the minimum estimation of model error variance. The uncertainty in the estimation of model parameters is evaluated using a linear approximation of the model about the optimal model parameter values found. The homoscedasticity of model residuals must be evaluated to validate this important assumption. The model structure identification procedure is implemented in R language and shown in the Appendix. Several examples are considered for illustrating the optimization procedure. In many practical situations, the optimal model obtained has heteroscedastic residuals. If the purpose of the model is only describing the experimental observations, the violation of the homoscedastic assumption may not be critical. However, for explanatory or extrapolating models, the presence of heteroscedastic residuals may lead to flawed conclusions.
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A statistical test of scedasticity indicates, with a given confidence, whether a set of observations has a constant (homoscedastic) or a variable (heteroscedastic) standard deviation with respect to any associated reference variable. Many... more
A statistical test of scedasticity indicates, with a given confidence, whether a set of observations has a constant (homoscedastic) or a variable (heteroscedastic) standard deviation with respect to any associated reference variable. Many different tests of scedasticity are available, in part due to the difficulty for unequivocally determining the scedasticity of a data set, particularly for non-normal and for small samples. In addition, the lack of an objective criterion for decision (significance level) increases the uncertainty involved in the evaluation. In this report, a new test of scedasticity is proposed based on the statistical distribution of the R 2 coefficient describing the behavior of the standard deviation of the data, and considering an optimal significance level that minimizes the total test error. The decision of the test is determined by a proposed H-value, resulting from the logarithm of the ratio between the Pvalue of the test and the optimal significance level. If H>0 then the data is homoscedastic. If H<0 then the data is heteroscedastic. The performance of the proposed test was found satisfactory and competitive compared to established tests of scedasticity.
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Material collisions (and interaction processes in general) play an important role in most, if not all, physicochemical phenomena observed in Nature including (but not limited to): Chemical reactions, diffusion, viscosity, adhesion,... more
Material collisions (and interaction processes in general) play an important role in most, if not all, physicochemical phenomena observed in Nature including (but not limited to): Chemical reactions, diffusion, viscosity, adhesion, pressure, transmission of forces, sound, and momentum and heat transfer, just to mention a few. It is quite surprising that a unique, clear, objective definition of "collision" is missing in most scientific textbooks and encyclopedias. In this report, some missing definitions in collision theory are proposed aiming at providing a more clear language, and at avoiding the confusion emerging from the lack of objective definitions. In addition, the illusion of elasticity of collisions is discussed. While elastic collisions are clearly defined as collisions with no change in the macroscopic translational kinetic energy of the bodies, the subjective definition of the bodies, and the inevitable simultaneous occurrence of multiple additional collisions involving internal components and/or external bodies may lead to different conclusions about the elastic character of a collision. Interaction processes involving composite bodies (having multiple components and an internal structure, like all bodies known to us so far) are typically inelastic or superelastic, but the overall result of many consecutive interactions, may resemble an elastic behavior. True perfectly elastic interactions can only be observed between isolated pairs of rigid, indivisible, structureless bodies, like the hypothetical "true atoms" proposed by the ancient Greeks.
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Beginning the 19 th century, Gay-Lussac proposed a free expansion experiment where gas is allowed to flow from one flask into another identical but empty flask, to show that thermal effects (cooling of the first vessel and warming of the... more
Beginning the 19 th century, Gay-Lussac proposed a free expansion experiment where gas is allowed to flow from one flask into another identical but empty flask, to show that thermal effects (cooling of the first vessel and warming of the second) were not caused by residual air present in the empty flask. While he successfully rejected such hypothesis, no alternative explanation was proposed for these effects. Classical and statistical thermodynamics have been used to explain the experimental results, but unfortunately, they are not entirely satisfactory. In this report, a different hypothesis is proposed where temperature changes in the flasks are caused by an unbalanced distribution of molecules, since the empty vessel is initially filled by the fastest molecules. Due to the low molecular density initially observed in the empty flask, temperature measurements are strongly influenced by the thermal behavior of the thermometer. A theoretical model and a simplified numerical simulation of the system are found to qualitatively support the proposed hypothesis as a potential explanation of the experimental results obtained by Gay-Lussac and other researchers.
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This is the first part of a series of reports discussing different strategies for optimizing the structure of mathematical models fitted from experimental data. In this report, the concept of randomistic models is introduced along with... more
This is the first part of a series of reports discussing different strategies for optimizing the structure of mathematical models fitted from experimental data. In this report, the concept of randomistic models is introduced along with the general formulation of the multi-objective optimization problem of model structure identification. Different approaches can be used to solve this problem, depending on the set of possible models considered. In the case of mathematical models with linear parameters, a stepwise multiple linear regression procedure can be used. In particular, a stepwise strategy in both directions (backward elimination and forward selection) is suggested based on the selection of relevant terms for the model prioritized on their absolute linear correlation coefficients with respect to the response variable, followed by the identification of statistically significant or explanatory terms based on optimal significant levels. Two additional constraints can be included, considering a lower limit in the normality value of the residuals (normality assumption check), as well as a lower limit in standard residual error (avoiding model overfitting). This stepwise strategy, which successfully overcomes several limitations of conventional stepwise regression, is implemented as a function (steplm) in R language, and different examples are presented to illustrate its use.
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In this report, a standard Maxwell-Boltzmann distribution (B) is defined by analogy to the concept of the standard Gaussian distribution. The most important statistical properties of B, as well as a simple method for generating random... more
In this report, a standard Maxwell-Boltzmann distribution (B) is defined by analogy to the concept of the standard Gaussian distribution. The most important statistical properties of B, as well as a simple method for generating random numbers from the standard Maxwell-Boltzmann distribution are presented. Given that the properties of B are already known, it is advantageous to describe any arbitrary Maxwell-Boltzmann distribution as a function of the standard Maxwell-Boltzmann distribution B. By using this approach, it is possible to demonstrate that the temperature of a material is a function only of the fluctuating component of the average molecular kinetic energy, and that it is independent of its macroscopic kinetic energy.
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The purpose of this paper is to demonstrate, from a mathematical point of view, the universal validity of the laws of conservation of energy and momentum. It will also be shown that these conservation laws are a natural consequence of the... more
The purpose of this paper is to demonstrate, from a mathematical point of view, the universal validity of the laws of conservation of energy and momentum. It will also be shown that these conservation laws are a natural consequence of the motion of matter. Finally, the implications of energy and momentum conservation to the collision between two particles are considered, and the validity of the Born-Mayer interaction potential as the reason for collision is discussed.
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In this paper, the basic concept of variance algebra is used for describing fluctuation in dynamical systems. By expressing any random variable as a function of standard random variables (e.g. standard white noise, standard Markovian),... more
In this paper, the basic concept of variance algebra is used for describing fluctuation in dynamical systems. By expressing any random variable as a function of standard random variables (e.g. standard white noise, standard Markovian), the algebra of the expected value and the variance is greatly simplified and easier to apply. Through three different case studies, the validity and usefulness of variance algebra for modeling fluctuation in dynamical systems is demonstrated. The stochastic model can be simulated using Monte Carlo simulation by generating the corresponding random numbers for each type of random variable.
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Styrene is the classical monomer obeying zero-one kinetics in radical emulsion polymerization. Accordingly, particles that are less than 100 nm in diameter contain either one or no growing radical(s). We describe a unique photoinitiated... more
Styrene is the classical monomer obeying zero-one kinetics in radical emulsion polymerization. Accordingly, particles that are less than 100 nm in diameter contain either one or no growing radical(s). We describe a unique photoinitiated polymerization reaction accelerated by snowballing radical generation in a continuous flow reactor. Even in comparison to classical emulsion polymerization, these unprecedented snowballing reactions are rapid and high-yielding, with each particle simultaneously containing more than one growing radical. This is a consequence of photoinitiator incorporation into the nascent polymer backbone and repeated radical generation upon photo-irradiation.
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The No Free Lunch (NFL) Theorem states that the average success rate of all optimization algorithms is basically the same, considering that certain algorithms work well for some types of problems, but fail for other types of problems.... more
The No Free Lunch (NFL) Theorem states that the average success rate of all optimization algorithms is basically the same, considering that certain algorithms work well for some types of problems, but fail for other types of problems. Another interpretation of the NFL Theorem is that "there is no universal optimizer", capable of successfully and efficiently solving any type of optimization problem. In this report, a Multi-Algorithm Optimization strategy is presented which allows increasing the average success rate at a reasonable cost, by running a sequence of different optimization algorithms, starting from multiple random points. Optimization of different benchmark problems performed with this algorithm illustrated that the particular sequence employed and the number of starting points greatly influence the success rate and cost of the optimization. A suggested sequence consisting on using the Broyden-Fletcher-Goldfarb-Shanno, Nelder-Mead, and adaptive step-size One-at-a-time optimization algorithms and using random starting points, achieved overall success rate, with average optimization time for a benchmark set of different global optimization problems. The proposed method (implemented in R language) is included in the Appendix.
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This is the third part of a series of reports presenting a molecular model of phase change, and in this case, heat and rates of evaporation are discussed. The heat of evaporation represents the difference in average intermolecular... more
This is the third part of a series of reports presenting a molecular model of phase change, and in this case, heat and rates of evaporation are discussed. The heat of evaporation represents the difference in average intermolecular potential energy between the liquid and gas phases, which is difficult to determine theoretically due to the mathematical complexity involved. Alternatively, the heat of evaporation is determined as the difference between the average kinetic energy of molecules overcoming an energy barrier for evaporation at the vapor/liquid interface, and the average kinetic energy of all molecules in the liquid phase. According to this model, the heat of evaporation at the normal boiling temperature is found to be correlated with the cohesion temperature, which is associated to the overall energy barrier for evaporation of a pure compound. Such correlation has been found to satisfy both the Clausius-Clapeyron relation and Trouton's rule. In fact, a molecular explanation of Trouton's rule is obtained. In addition, a molecular model for the rate of evaporation of liquids is derived and presented.
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The R² coefficient (a generalization of the determination coefficient defined in linear regression) has been widely used as a criterion for assessing and comparing the performance of mathematical models with respect to a given set of... more
The R² coefficient (a generalization of the determination coefficient defined in linear regression) has been widely used as a criterion for assessing and comparing the performance of mathematical models with respect to a given set of experimental data. Unfortunately, the R² coefficient can only be used to confidently compare linear models with different terms, fitted by ordinary least-squares (OLS) regression, satisfying all assumptions of OLS regression, and using the same experimental data set. In addition, the R² coefficient actually represents the relative performance of a model compared to the best constant model for the data. A new fitness coefficient (C F) is proposed as an alternative to R², where the performance of the model is now relative to the corresponding measurement error in the data. A modeling selection procedure is suggested where the best model maximizes the fitness coefficient and the normality of the residuals, while minimizing the number of fitted parameters (parsimony principle).
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The permanent motion and interaction of microscopic entities result in the transport of those entities from one region to another, and the subsequent change in overall properties of the regions. The purpose of this report is presenting a... more
The permanent motion and interaction of microscopic entities result in the transport of those entities from one region to another, and the subsequent change in overall properties of the regions. The purpose of this report is presenting a general framework for describing the change in overall properties of a system as a result of the transport of microscopic entities. This approach is valid for any property directly associated to the microscopic entities, such as number, mass, electric charge, density, concentration, velocity, linear momentum, mechanical energy, temperature, pressure, and many others. Different scenarios of microscopic transport are considered (non-interacting or ideal, Brownian, Uniform, and constant external field) which allows deriving a general expression for the flux of entities across a boundary. Also, different types of properties are considered (additive, average, reciprocal average, and weighted average) covering most common properties of practical interest. In addition, different types of transport coefficients were presented depending on the particular "driving force" considered (global, local, directional, and positional) representing the most common coefficients found in the literature. Finally, some relevant considerations are discussed, providing additional clarity to the concepts introduced here.
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One of the most important tools for data analysis is statistical regression. This technique consists on identifying the best parameters of a given mathematical model describing a particular set of experimental observations. This method... more
One of the most important tools for data analysis is statistical regression. This technique consists on identifying the best parameters of a given mathematical model describing a particular set of experimental observations. This method implicitly assumes that the model error has a constant variance (homoscedasticity) over the whole range of observations. However, this is not always the case, leading to inadequate or incomplete models as the changing variance (heteroscedasticity) is neglected. In this report, a method is proposed for describing the heteroscedastic behavior of the regression model residuals. The method uses weighted least squares minimization to fit the confidence intervals of the regression from a model of the standard error. The weights used are related to the confidence level considered. In addition, a test of heteroscedasticity is proposed based on the coefficient of variation of the model of standard error obtained by optimization. Various practical examples are presented for illustrating the proposed method.
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This is a continuation of the fictional dialogue taking place by Descartes' hypothetical characters (Eudoxius, Epistemon and Polyander). In this opportunity, the discussion revolves around the common confusion between models and reality.... more
This is a continuation of the fictional dialogue taking place by Descartes' hypothetical characters (Eudoxius, Epistemon and Polyander). In this opportunity, the discussion revolves around the common confusion between models and reality. Knowledge is built upon models, which are approximate representations of reality. However, it is practically impossible for us to determine the correctness of those models, and therefore, we will never know for sure which model is an accurate description of reality. Unfortunately, scientific models taught in schools and universities are tacitly assumed by most students (and educators) to be our reality, and this misconception limits scientific progress. For this reason, students, educators and scientists are invited to continuously and openly question all of our current scientific paradigms, even when those paradigms can be regarded as "universally accepted truths".
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Thermodynamics embraces both the concepts of entropy and enthalpy. Strangely, both concepts express work done by an expanding system in terms of W=PdV, without providing the necessary clarity concerning what PdV actually represents. In... more
Thermodynamics embraces both the concepts of entropy and enthalpy. Strangely, both concepts express work done by an expanding system in terms of W=PdV, without providing the necessary clarity concerning what PdV actually represents. In this report, we shall discuss that such work is external to the expanding system. This means that one should question the validity of accepted notions in thermodynamics including those concerning both enthalpy and enthalpy.
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Thermodynamics embraces both the concepts of entropy and enthalpy. Strangely, both concepts express work done by an expanding system in terms of W=PdV, without providing the necessary clarity concerning what PdV actually represents. In... more
Thermodynamics embraces both the concepts of entropy and enthalpy. Strangely, both concepts express work done by an expanding system in terms of W=PdV, without providing the necessary clarity concerning what PdV actually represents. In this report, we shall discuss that such work is external to the expanding system. This means that one should question the validity of accepted notions in thermodynamics including those concerning both enthalpy and enthalpy.
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Automatic control is an important tool in process systems engineering for achieving high productivity and quality with reduced variability and improved safety during operation. Nowadays, the most common type of automatic controller used... more
Automatic control is an important tool in process systems engineering for achieving high productivity and quality with reduced variability and improved safety during operation. Nowadays, the most common type of automatic controller used in industry is the PID controller. The PID controller is a general type of linear controller incorporating three effects: Proportional to feedback error, to the derivative of error and to the integral of error over time. While the PID is highly versatile and can provide a good balance between performance and implementation costs in most processes (including nonlinear processes), the main disadvantage of this controller is that an adequate set of tuning parameters is required for achieving such performance. PID tuning is best done if an approximate linear model of the process is known. Reaction curve methods can be used for identifying the parameters of the linear model. Almost all PID tuning methods have been obtained considering the model in the Laplace-transform domain. The purpose of this report is illustrating the fact that PID parameters ensuring stability can be obtained from a linear model solely expressed in the time domain. A set of tuning equations for PID and PI controllers for different types of systems (selfregulating, runaway and capacitive) is presented. The performance of this tuning method is illustrated considering different examples of control in chemical processes.
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This brief report introduces the periodic equality operator (≗) for representing the equivalence between periodic arguments of functions. Such equivalence is valid for integer multiples of the period, while for non-integer multiples the... more
This brief report introduces the periodic equality operator (≗) for representing the equivalence between periodic arguments of functions. Such equivalence is valid for integer multiples of the period, while for non-integer multiples the equivalence is not necessarily valid. The periodic equality can be easily mistaken with conventional equality relations (unconditionally valid), leading to erroneous or paradoxical results. Two representative examples of the necessity of this periodic equality operator are presented: 1) The periodic behavior of angles in a Cartesian plane, and 2) the periodic behavior of the logarithm of negative numbers.
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There are different methods for scale estimation, that is, the estimation of standard deviation from a data sample. However, all of them present different degrees of bias from the true population value depending on sample size, on the... more
There are different methods for scale estimation, that is, the estimation of standard deviation from a data sample. However, all of them present different degrees of bias from the true population value depending on sample size, on the nature of the population distribution, and on the presence of contaminants (elements not belonging to the target population). Particularly the sample standard deviation is slightly biased for clean samples (without contaminants), and strongly biased in the presence of contaminants. More robust alternatives for estimating standard deviation in contaminated samples are available, which unfortunately do not work well for clean samples, or only work for normal or for symmetrical distributions. In this report, three strategies for determining an unbiased sample standard deviation after outlier removal are proposed. The strategies are: 1) Decreasing the risk of false positive outlier detection for a normal distribution, 2) Evaluating the change in non-normality as a criterion for outlier removal, and 3) Adapting the outlier removal method to the particular skewness and Normality value observed in the sample. Even when no universal robust scale estimator providing the best performance (low bias and high efficiency) for all possible populations is available, the adaptive strategy proposed provides a more reliable approach, with the best overall rank among the different estimators considered.
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Since the 19 th century, it has been known that the estimation of the population variance from a sample needs to be corrected to remove bias (Bessel's correction), and that even when the estimation of the variance can be unbiased, the... more
Since the 19 th century, it has been known that the estimation of the population variance from a sample needs to be corrected to remove bias (Bessel's correction), and that even when the estimation of the variance can be unbiased, the corresponding estimation of the standard deviation is still biased. Unfortunately, the probability distribution of the standard deviation of a sample strongly depends on the particular distribution of the elements in the population, and only for very well-known distributions, such as the normal distribution, a distribution model of the sample standard deviation is possible (i.e. Helmert or Chi distribution). In this report, the general formulation of the probability density of the sample standard deviation is presented, along with different approximations for reducing its mathematical complexity. These approximations are also particularly considered for normal populations. In addition, a Monte Carlo simulation was performed to obtain a general empirical approximation of estimation bias for arbitrary distributions, based on the kurtosis of the population (if known) or alternatively, on the kurtosis of the sample. While normal populations can be satisfactorily estimated using Helmert's distribution, non-normal distributions require the proposed empirical correction.
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Any mathematical operation or transformation of a random variable results in a new random variable with different properties, such as expected value, variance, and probability density function. The new probability density function can be... more
Any mathematical operation or transformation of a random variable results in a new random variable with different properties, such as expected value, variance, and probability density function. The new probability density function can be obtained by employing the change of variable theorem. For multivariate functions, the change of variable theorem introduces one or more definite integrals. In some cases, a simple analytical expression of those integrals is not possible, and numerical methods must be employed (such as numerical integration or numerical change of variable). The expected value and variance of the mathematical function can be obtained by integration using the corresponding probability density function, or in some cases, they can be more easily obtained by using the algebra of the expected value and variance. The current report summarizes the behavior of different representative mathematical transformations involving one or more standard normal random variables, including powers, absolute value, logarithm, exponential, sums, products, and some of their combinations.
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After the original definition of Entropy and the Second Law of Thermodynamics given by Clausius in the 1860's, many different additional definitions and interpretations have emerged. Perhaps one of the most representative contributions... more
After the original definition of Entropy and the Second Law of Thermodynamics given by Clausius in the 1860's, many different additional definitions and interpretations have emerged. Perhaps one of the most representative contributions was Boltzmann's Entropy, which connected the microscopic behavior of the system with the macroscopic observations. While both notions can be interpreted as measures of the deviation with respect to equilibrium, and are similar under certain conditions, it can also be shown that they are, in general, different mathematical constructs which cannot be considered equivalent.
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The random deviation operator represents the difference in the expected value of a function of one or more randomistic variables with respect to the corresponding deterministic behavior. Thus, it can be used to measure the effect of... more
The random deviation operator represents the difference in the expected value of a function of one or more randomistic variables with respect to the corresponding deterministic behavior. Thus, it can be used to measure the effect of randomness on a particular function. A general distribution deviation operator is introduced in order to compare the effect of randomness with respect to any arbitrary reference probability distribution. This general distribution deviation operator is closely related to the similitude between probability density functions. It can also be further generalized by means of the incomplete distribution deviation. Different examples are included for illustrating the use of the random and general distribution deviation operators.