Trigonometric Regression Research Articles

A note on trigonometric regression in the presence of Berkson‐type measurement error

In this note, we study how parameter vector estimation for a trigonometric regression model and the expected squared residual error computed from an estimated model are affected by Berkson‐type measurement error. Closed‐form expressions for the parameter vector and the expected squared residual error are obtained by assuming that the observed covariate data are sampled from an equispaced design and that measurement error is generated from a symmetric probability distribution with a mean of zero. Notably, these results indicate that estimates of the amplitude parameters for a trigonometric regression model suffer from attenuation bias when covariate data are mis‐measured, and that estimates of the phase‐shift parameters are unbiased.

Consistency of the LSE for Chirp Signal Parameters in the Models with Strongly and Weakly Dependent Noise

A time continuous statistical model of multiple chirp signal observed against the background of strongly-dependent stationary Gaussian noise is considered in the paper. Strong consistency of the least squares estimates for such a trigonometric regression model parameters is proved.

Виявлення прихованих перiодичностейв моделях з дискретним часом та сильно залежним випадковим шумом

Trigonometric regression models take a special place among various models of nonlinear regression analysis and signal processing theory. The problem of estimating the parameters of such models is called the problem of detecting hidden periodicities, and it has many applications in natural and technical sciences. The paper is devoted to the study of the problem of detecting hidden periodicities in the case when we observe only one harmonic oscillation with discrete time, where random noise is a local functional of Gaussian random sequence with singular spectrum. In particular, the random sequence in the model can be strongly dependent. For estimation of unknown parameters the periodogram estimator is chosen. Sufficient conditions of the consistency of the amplitude and angular frequency periodogram estimator of the model described above are obtained in the paper. The proof of Lemmas 1 and 2 gave an important asymptotic properties of the random noise functional related to the periodogram estimator which necessary for the proof of the main results. Series expansion of random noise in terms of Hermite polynomials and the Diagram formula are main tools that were used to obtain this lemmas.

Consistency of LSE for the many-dimensional symmetric textured surface parameters

A multivariate trigonometric regression model is considered. In the paper strong consistency of the least squares estimator for amplitudes and angular frequencies is obtained for such a multivariate model on the assumption that the random noise is a homogeneous or homogeneous and isotropic Gaussian, specifically, strongly dependent random field on ${\mathbb{R}^{M}},M\ge 3$.

A statistical learning framework for spatial-temporal feature selection and application to air quality index forecasting

Accurate air quality index (AQI) forecasting makes a difference to public health, local economic development, and ecological environment. As a typical geographical datum, the spatial autocorrelation (SAC) of the AQI is often ignored, which may violate the assumptions of some models, such as machine learning which requires variables to be independent and identically distributed. Considering the strong SAC of the AQI, this study proposes a novel statistical learning framework integrating SAC variables, feature selection, and support vector regression (SVR) for AQI prediction in which correlation analysis and time series analysis are used to extract the spatial-temporal features. In addition, the historical AQI series of the target site is adjusted by using trigonometric regression to eliminate the non-stationarity. To further improve prediction accuracy, a feature selection method combining reinforcement learning with a heuristic algorithm is adopted. To demonstrate the effectiveness of our proposed framework, we select the AQI data of 34 cities from the Yangtze River Delta, which is one of the most polluted areas in eastern China, and focus on the three largest cities, Nanjing, Hangzhou, and Shanghai. We compared the proposed framework with several baselines, and the experiment illustrates that the forecasting accuracy of the proposed framework is significantly better than the baselines at all selected key sites that can provide accurate predictions for air quality.

Unstable Circadian Rhythm of Heart Rate of Alzheimer Dementia Based on Biological Data of Mattress Sensor.

It is important to detect daily Alzheimer dementia (AD) possibility using unconstrained mattress sensors because dementia takes time before subjective symptoms appear and the main treatment is to slow the rate of progression. Forcusing on circadian rhythm disorder which tend to occur with AD, this paper analyzes the features of unstable circadian rhythms of heart rate associated with its weakening and misalignment in AD. It is found that the external feature of heart rate of AD that of up and down changes seen frequently compared to that of healthy subjects makes the multiple types of effects on the estimation process of the AD detection method based on the circadian instability represented by the trigonometric regression equation estimated from the heart rate. So, we designed two feature values from the effects and analyzed them in an analysis experiment. An analysis experiment was conducted on the heart rate of total 72 days data of one AD patient and total 30 days data of 21 healthy people, and the result confirmed significant differences between the AD and the healthy people at from 0.01% to 0.1 % level for the first feature and at from 0.1% to 5% level for the second feature. And the result shows the possibility that differences in the heart rate features is found between AD and healthy people.

Wave-shape function model order estimation by trigonometric regression

The adaptive non-harmonic (ANH) model is a powerful tool to compactly represent oscillating signals with time-varying amplitude and phase, and non-sinusoidal oscillating morphology. Given good estimators of instantaneous amplitude and phase we can construct an adaptive model, where the morphology of the oscillation is described by the wave-shape function (WSF), a 2π-periodic more general periodic function. In this paper, we address the problem of estimating the number of harmonic components of the WSF, a problem that remains underresearched, by adapting trigonometric regression model selection criteria into this context. We study the application of these criteria, originally developed in the context of stationary signals, to the case of signals with time-varying amplitudes and phases. We then incorporate the order estimation to the ANH model reconstruction procedure and analyze its performance for noisy AM-FM signals. Experimental results on synthetic signals indicate that these criteria enable the adaptive estimation of the waveform of non-stationary signals with non-sinusoidal oscillatory patterns, even in the presence of considerable amount of noise. We also apply our reconstruction procedure to the task of denoising simulated pulse wave signals and determine that the proposed technique performs competitively to other denoising schemes. We conclude this work by showing that our adaptive order estimation algorithm takes into account the interpatient waveform variability of the electrocardiogram (ECG) and respiratory signals by analyzing recordings from the Fantasia Database.

A study of univariate forecasting methods for crude oil price

PurposeThis paper aims to compare different univariate forecasting methods to provide a more accurate short-term forecasting model on the crude oil price for rendering a reference to manages.Design/methodology/approachSix different univariate methods, namely the classical decomposition model, the trigonometric regression model, the regression model with seasonal dummy variables, the grey forecast, the hybrid grey model and the seasonal autoregressive integrated moving average (SARIMA), have been used.FindingsThe authors found that the grey forecast is a reliable forecasting method for crude oil prices.Originality/valueThe contribution of this research study is using a small size of data and comparing the forecasting results of the six univariate methods. Three commonly used evaluation criteria, mean absolute error (MAE), root mean squared error (RMSE) and mean absolute percent error (MAPE), were adopted to evaluate the model performance. The outcome of this work can help predict the crude oil price.

On the least squares estimator asymptotic normality of the multivariate symmetric textured surface parameters

A multivariate trigonometric regression model is considered. Various discrete modifications of the similar bivariate model received serious attention in the literature on signal and image processing due to multiple applications in the analysis of symmetric textured surfaces. In the paper asymptotic normality of the least squares estimator for amplitudes and angular frequencies is obtained in multivariate trigonometric model assuming that the random noise is a homogeneous or homogeneous and isotropic Gaussian, in particular, strongly dependent random field on R M , M > 2. \mathbb {R}^M,\,\, M>2.

A comparative study of univariate models for container throughput forecasting of major ports in Asia

This study aims to make comparisons on different univariate forecasting methods and provides a more accurate short-term forecasting model on the container throughput for rendering a reference to relevant authorities. We collected monthly data regarding container throughput volumes for three major ports in Asia, Shanghai, Singapore, and Busan Ports. Six different univariate methods, including the grey forecasting model, the hybrid grey forecasting model, the multiplicative decomposition model, the trigonometric regression model, the regression model with seasonal dummy variables, and the seasonal autoregressive integrated moving average (SARIMA) model, were used. We found that the hybrid grey forecasting model outperforms the other univariate models. This study’s findings can provide a more accurate short-term forecasting model for container throughput to create a reference for port authorities.

COVID-19 mortality analysis from soft-data multivariate curve regression and machine learning.

A multiple objective space-time forecasting approach is presented involving cyclical curve log-regression, and multivariate time series spatial residual correlation analysis. Specifically, the mean quadratic loss function is minimized in the framework of trigonometric regression. While, in our subsequent spatial residual correlation analysis, maximization of the likelihood allows us to compute the posterior mode in a Bayesian multivariate time series soft-data framework. The presented approach is applied to the analysis of COVID-19 mortality in the first wave affecting the Spanish Communities, since March 8, 2020 until May 13, 2020. An empirical comparative study with Machine Learning (ML) regression, based on random k-fold cross-validation, and bootstrapping confidence interval and probability density estimation, is carried out. This empirical analysis also investigates the performance of ML regression models in a hard- and soft-data frameworks. The results could be extrapolated to other counts, countries, and posterior COVID-19 waves.Supplementary InformationThe online version contains supplementary material available at 10.1007/s00477-021-02021-0.

D-optimal designs of experiments for trigonometric regression on interval with heteroscedastic observations

In article the problem of construction continuous (number of observations is not fixed) D-optimal designs of experiments for trigonometric regression in a case when variance of errors of observations depend on a point in which is made is investigated. Class of functions which describe change variance of heteroscedastic observations is defined for which it is possible construct continuous D-optimal designs of experiments. For trigonometric regression with three factors it is constructed continuous D-optimal designs of experiments with different types heteroscedastic observations. For each of these types the own class of functions describing change variance of observations is defined.

Консистентнiсть оцiнки найменших квадратiв параметрiв тригонометричної моделi регресiї у присутностi лiнiйного випадкового шуму

Регресiйний аналiз є iстотною частиною математичної та прикладної статистики. Нелiнiйний регресiйний аналiз є значним розширенням та ускладненням класичного лiнiйного регресiйного аналiзу, завдяки використанню нелiнiйних або частково нелiнiйних за параметрами моделей, якi адекватнiше описують, нiж лiнiйнi моделi, явища, що потребують статистичного аналiзу. Велика кiлькiсть прикладних проблем у численних наукових, технiчних та гуманiтарних галузях знань дають поштовх розвитку нелiнiйного регресiйного аналiзу. Задача оцiнювання векторного параметра сигналу в моделях спостереження «сигнал + шум» є добре вiдомою проблемою статистики випадкових процесiв, та у випадку нелiнiйного параметра сигналу – задачею нелiнiйного регресiйного аналiзу. Серед рiзноманiтностi задач нелiнiйного регресiйного аналiзу оцiнювання амплiтуд та кутових частот суми гармонiчних коливань, що спостерiгається на фонi випадкового шуму, займає значне мiсце, завдяки її численним застосуванням. Статистичнi моделi такого типу називаються тригонометричними моделями регресiї, а проблема статистичного оцiнювання її параметрiв називається задачею виявлення прихованих перiодичностей. Статтю присвячено вивченню тригонометричної моделi регресiї, в якiй випадковий шум є лiнiйним Левi-керованим стацiонарним четвертого порядку випадковим процесом iз нульовим середнiм, iнтегровную та iнтегровную з квадратом iмпульсною перехiдною функцiєю. Це припущення призводить до iнтегровностi коварiацiйної функцiї та кумулянтної функцiї 4-го порядку. Для оцiнювання амплiтуд та кутових частот такої тригонометричної моделi ми використовуємо оцiнки найменших квадратiв у сенсi Уолкера, тобто розглянуто спецiальну множину параметрiв, щоб розрiзнити належним чином рiзнi кутовi частоти в сумi гармонiчних коливань. У статтi доведено теорему про сильну консистентнiсть оцiнки найменших квадратiв за описаними вище припущеннями щодо випадкового шуму. Для отримання такого результату було доведено дуже важливу лему про рiвномiрну збiжнiсть майже напевно середнього значення перетворення Фурьє лiнiйного Левi-керованого випадкового процеса. Ця лема є головним iнструментом доведення теореми про сильну консистентнiсть. Для доведення теореми, по-перше, знаходимо деякi представлення оцiнок найменших квадратiв амплiтуд через вiдповiднi оцiнки кутових частот. По-друге, ми пiдставляємо цi формули у функцiонал методу найменших квадратiв. Останнiй крок доведення полягає у перетвореннi L2-норми рiзницi мiж емпiричною тригонометричною функцiєю регресiї та iстиною функцiєю регресiї таким чином, що ця норма прямує до нуля майже напевно тодi i тiльки тодi, коли оцiнки є сильно консистентними.

CosinorPy: a python package for cosinor-based rhythmometry

BackgroundEven though several computational methods for rhythmicity detection and analysis of biological data have been proposed in recent years, classical trigonometric regression based on cosinor still has several advantages over these methods and is still widely used. Different software packages for cosinor-based rhythmometry exist, but lack certain functionalities and require data in different, non-unified input formats.ResultsWe present CosinorPy, a Python implementation of cosinor-based methods for rhythmicity detection and analysis. CosinorPy merges and extends the functionalities of existing cosinor packages. It supports the analysis of rhythmic data using single- or multi-component cosinor models, automatic selection of the best model, population-mean cosinor regression, and differential rhythmicity assessment. Moreover, it implements functions that can be used in a design of experiments, a synthetic data generator, and import and export of data in different formats.ConclusionCosinorPy is an easy-to-use Python package for straightforward detection and analysis of rhythmicity requiring minimal statistical knowledge, and produces publication-ready figures. Its code, examples, and documentation are available to download from https://github.com/mmoskon/CosinorPy. CosinorPy can be installed manually or by using pip, the package manager for Python packages. The implementation reported in this paper corresponds to the software release v1.1.

Estimation and forecasting of PM10 air pollution in Ankara via time series and harmonic regressions

In this study, monthly particulate matter (PM10) values in Ankara (39.9334° N, 32.8597° E) from January 1993 to December 2017 are examined. The PM10 are those thoracic particles whose aerodynamic diameter is less than 10 μm (micrometers), and it is of critical health importance due to the penetrability to the lower airways. As an alternative to classical unit root tests, a unit root test primarily based on periodograms is introduced owing to its advantages over alternatives. After examining the stationarity of the series through periodogram-based test as well as its standard rivals, periodic components in the series are examined and it is observed that the series has both periodic and seasonal components. These components are modeled, using the inherent dynamics of a time series alone, within a trigonometric harmonic regression setup, eventually yielding the forecast values for 2018 that turns out to be superior to those obtained by means of ARIMA (autoregressive integrated moving average). This is a striking result since the modeling framework requires no assumptions, no parameter estimations except for the variance of the white noise series, no simulations of the power of tests, no adjustments of test statistics with respect to sample size and no preliminary work as to independent variable which is simply time, i.e., the period of forecast.

Seasonal patterns in fecundability in North America and Denmark: a preconception cohort study.

To what extent does fecundability vary across seasons? After accounting for seasonal patterns in pregnancy planning, we observed higher fecundability in the fall and lower fecundability in the spring, particularly at lower latitudes. In human populations, there are strong seasonal patterns of births that vary across geographic regions and time periods. However, previous studies of seasonality and fecundity are limited because they examine season of birth rather than season of conception and therefore neglect to account for seasonal variation in initiating attempts to conceive or pregnancy loss or differences in gestational length. We conducted a preconception cohort study of 14 331 women residing in North America (June 2013-May 2018: n = 5827) and Denmark (June 2007-May 2018: n = 8504). Participants were attempting to conceive without fertility treatment and had been attempting pregnancy for ≤6 menstrual cycles at enrolment. We collected information on season of each pregnancy attempt using last menstrual period dates over the study period. Pregnancy was reported on female bi-monthly follow-up questionnaires. We fit log-binomial models with trigonometric regression to examine periodic variation in fecundability. We accounted for seasonal variation in initiation of pregnancy attempts by including indicator variables for menstrual cycle of attempt in the regression models. Initiation of pregnancy attempts peaked in September, with stronger seasonality in North America than in Denmark (48 vs. 16% higher probability initiating attempts in September compared with March). After accounting for seasonal variation in initiation of pregnancy attempts, we observed modest seasonal variation in fecundability, with a peak in the late fall and early winter in both cohorts, but stronger peak/low ratios in North America (1.16; 95% confidence interval [CI]: 1.05, 1.28) than in Denmark (1.08; 95% CI: 1.00, 1.16). When we stratified the North American data by latitude, we observed the strongest seasonal variation in the southern USA (peak/low ratio of 1.45 [95% CI: 1.14, 1.84]), with peak fecundability in late November. We estimated menstrual cycle dates between follow-up questionnaires, which may have introduced exposure misclassification, particularly when women skipped follow-up questionnaires. We were unable to measure seasonally varying factors that may have influenced fecundability, including ambient temperature, vitamin D levels or infectious disease. An understanding of how fecundability varies across seasons could help identify factors that can impair reproductive function. Neglecting to account for seasonal variation in initiation of pregnancy attempts could bias estimates of seasonal patterns in fecundability. This is the first preconception cohort study to examine seasonal variation in fecundability after accounting for seasonality in initiation of pregnancy attempts. Fecundability was highest in the fall and lowest in the spring, with stronger effects in southern latitudes of North America, suggesting that seasonal exposures may affect fecundity. This research was funded by the Eunice K. Shriver National Institute of Child Health and Human Development (R21-050264, R01-HD060680, R21-HD072326 and R01-HD086742) and the Danish Medical Research Council (271-07-0338). The authors declare no conflicts of interest.

Evaluation of Short-Term Freeway Speed Prediction Based on Periodic Analysis Using Statistical Models and Machine Learning Models

Accurate prediction of traffic information (i.e., traffic flow, travel time, traffic speed, etc.) is a key component of Intelligent Transportation System (ITS). Traffic speed is an important indicator to evaluate traffic efficiency. Up to date, although a few studies have considered the periodic feature in traffic prediction, very few studies comprehensively evaluate the impact of periodic component on statistical and machine learning prediction models. This paper selects several representative statistical models and machine learning models to analyze the influence of periodic component on short-term speed prediction under different scenarios: (1) multi-horizon ahead prediction (5, 15, 30, 60 minutes ahead predictions), (2) with and without periodic component, (3) two data aggregation levels (5-minute and 15-minute), (4) peak hours and off-peak hours. Specifically, three statistical models (i.e., space time (ST) model, vector autoregressive (VAR) model, autoregressive integrated moving average (ARIMA) model) and three machine learning approaches (i.e., support vector machines (SVM) model, multi-layer perceptron (MLP) model, recurrent neural network (RNN) model) are developed and examined. Furthermore, the periodic features of the speed data are considered via a hybrid prediction method, which assumes that the data consist of two components: a periodic component and a residual component. The periodic component is described by a trigonometric regression function, and the residual component is modeled by the statistical models or the machine learning approaches. The important conclusions can be summarized as follows: (1) the multi-step ahead prediction accuracy improves when considering the periodic component of speed data for both three statistical models and three machine learning models, especially in the peak hours; (2) considering the impact of periodic component for all models, the prediction performance improvement gradually becomes larger as the time step increases; (3) under the same prediction horizon, the prediction performance of all models for 15-minute speed data is generally better than that for 5-minute speed data. Overall, the findings in this paper suggest that the proposed hybrid prediction approach is effective for both statistical and machine learning models in short-term speed prediction.

Minimax designs for estimating slopes in a trigonometric regression model

Designs for the first order trigonometric regression model over an interval on the real line are considered for the situation where estimation of the slope of the response surface at various points in the factor space is of primary interest. Minimization of the variance of the estimated slope at a point maximized over all points in the region of interest is taken as the design criterion. Optimal designs under the minimax criterion are derived for the situation where the design region and the region of interest are identical and a symmetric “partial cycle”. Some comparisons of the minimax designs with the traditional D- and A-optimal designs are provided. Efficiencies of some exact designs under the minimax criterion are also investigated.

Parameter Estimation of Fractional Trigonometric Polynomial Regression Model

Trigonometric Polynomial Regression is a form of non-linear regression in which the relationship between the outcome variable and risk variable is Fractional modeled as 1/nth degree polynomial regression by combining the function of cos(nx) and sin(nx) on the value of natural numbers. The model was used to analyze the relationship between three continuous and periodic variables. Coefficients of the model were estimated using the Maximum Likelihood Estimate (MLE) method. From the results, the model obtained indicated that an increased in body mass index will increase the level of blood pressure while age may or may not have an influence on the blood pressure level. The values of the Coefficient of variation showed the variation in the dependent variable was well explained by the independent variables and the value of the adjusted coefficient of determination showed the model had a good fit with a high level of predictive power.

Properties of optimal regression designs under the second-order least squares estimator

We investigate properties of optimal designs under the second-order least squares estimator (SLSE) for linear and nonlinear regression models. First we derive equivalence theorems for optimal designs under the SLSE. We then obtain the number of support points in A-, c- and D-optimal designs analytically for several models. Using a generalized scale invariance concept we also study the scale invariance property of D-optimal designs. In addition, numerical algorithms are discussed for finding optimal designs. The results are quite general and can be applied for various linear and nonlinear models. Several applications are presented, including results for fractional polynomial, spline regression and trigonometric regression models.