Fourier Series Model Research Articles

Application Of Fourier Series Analysis To Temperature Data

This Paper seeks to model a periodic time series using Fourier Series Analysis Method and to use such model to forcast future values of such data. The mean monthly temperature of Uyo Metropolis consisting of 180 data points (1991 – 2006) are collected for the study. The parameter estimates of the Fourier series model are obtained by ordinary least squares method in multiple regression. The test of significance of the general model and parameters indicate that the model is statistically significant and the significant parameters provide a Fourier series model of the form: 26.82-1.163cosωt - 0.169 sinωt + 0.133cos2ωt +0.164sin2ωt - 0.116sin4ωt + 0.255et-1. The P – P plot is also used to test for the overall goodness-of-fit and it is found out that the model fits well to the data and can be used to forcast the future values of the data. Keywords: Fourier Series Analysis, Periodic Time Series, Forcasting.Global Journal of Mathematical Sciences Vol. 7 (1) 2008: pp. 5-14

Pseudo-Additive (Mixed) Fourier Series Model of Time Series

Pseudo-Additive (Mixed) Fourier Series Model of Time Series

Avaliação do perfil nictemeral de excreção de indicadores internos e de óxido crômico em ensaios de digestão com ruminantes

Objetivou-se com este trabalho avaliar a dinâmica de excreção fecal dos indicadores internos matéria seca indigestível (MSi), fibra em detergente neutro indigestível (FDNi), fibra em detergente ácido indigestível (FDAi) e do indicador externo óxido crômico (Cr2O3). Foram utilizados cinco bovinos mestiços Holandês × Zebu, com peso médio de 240 kg, em três períodos experimentais de 12 dias (os sete primeiros destinados à adaptação dos animais). Antes do início do período experimental, os animais passaram por um período de adaptação à dieta e ao manejo, com duração dez dias, no qual foram mantidos em baias individuais e alimentados à vontade com silagem de sorgo. O Cr2O3 foi fornecido diariamente em duas porções de 5 g, às 8h e 17h, durante todo o período experimental. Procedeu-se à caracterização do padrão diurno de excreção fecal dos indicadores, por meio do algoritmo de Gauss-Newton, em um modelo em série de Fourier. A avaliação descritiva das funções ajustadas evidenciou que a amplitude total dos ciclos foi menor para FDAi que para outros indicadores, com valores de 4,71; 1,83; 2,42 e 1,74% para Cr2O3, MSi, FDNi e FDAi, respectivamente, em relação à concentração média estimada para cada um desses indicadores. Por sua vez, o período fundamental de excreção dos indicadores em estudo foi significativo e evidenciou duração do ciclo de 3,25; 3,47; 3,46 e 3,74 horas para o Cr2O3, MSi, FDNi e FDAi, respectivamente. A concentração média fecal dos indicadores pode ser obtida por meio de duas coletas diárias de fezes, mantendo-se o espaço de tempo entre duas coletas consecutivas de 3, 9, 15 ou 21 horas.

Fourier Representation of Intermittent Flow in a Porous Medium

The field measurements and numerical results for intermittent flow regime in a sandy soil show that the time distributions of the soil water flux q(z, t), and the soil water content θ(z, t)at various depths are periodic in nature, where t is time and z is the depth (i.e., at the surface z = 0 and at depths z = − 5, − 10, − 15 cm, etc). The period of q(z, t) and θ(z, t) variations are generally determined by the sum of the duration of pulse and the duration between the initiation of two consecutive pulses of water at the soil surface. Fourier series models have been given for q(z, t) and θ(z, t) variations. The predicted Fourier results for these variations have been compared with the experimentally verified numerical results—designated as observed values. The results show that the amplitudes of these variations were damped exponentially with depth, and the phase shift increased linearly with depth.

Diurnal Variation of the AMSU-A Brightness Temperatures Over the Amazon Rainforest

Brightness temperatures over the Amazon rainforest are obtained from the Advanced Microwave Sounding Units (AMSU-A and AMSU-B) instruments onboard three NOAA satellites (NOAA-15, -16, and -17, respectively) for the months of July, August, and October, 2002. The three AMSU-A instruments provided six daily measurements, separated by 2.5-5.5 h of the diurnal time intervals, over the Amazon rainforest region, and these measurements offer a unique opportunity for investigation of the diurnal variation of the brightness temperatures over the Amazon rainforests. The angular distributions of brightness temperatures over the Amazon rainforest are very stable and can be simulated with a radiative transfer model, which consists of an atmospheric radiative component and a rainforest-canopy model that treats the rainforest as a uniform layer with an effective canopy temperature. The simulated results agree well with the observations. The diurnal variation of brightness temperatures over the Amazon rainforest is simulated with a Fourier-series model. It shows that a second order of Fourier series can reproduce the observed pattern of diurnal variation of the brightness temperatures at zenith angles of 0deg, 28.7deg, and 58.1deg, respectively. In a practical application, the coefficients of Fourier-series expansion can be used to generate the brightness temperatures as a function of diurnal hours. These results can be applied to postlaunch calibration of satellite-borne microwave radiometer with different equator crossing time. In addition, the results presented in this paper indicate that the Amazon rainforest can be used as a hot calibration reference target. The availability of a land calibration target is important for calibration and validation of spaceborne microwave radiometers

A step‐wise incremented Fourier series model for chemical reactivity prediction

AbstractIn this paper, chemical reactivity is modeled as a time series of events defined by a reactant's concentration decay measured at consecutive discrete time periods. Since traditional time series techniques such as ARIMA and current Artificial Neural Networks require large data sets that are typically not available for chemical reactions, we developed a Step Wise Incremented Fourier Series (SWIFS) algorithm to model and predict nonlinear short time series. The application of SWIFS to experimental data from first‐ and second‐order reactions produced a significant improvement in prediction accuracy over traditional integrated rate laws. In forward‐time prediction, SWIFS has achieved significantly higher prediction accuracy with first‐ and second‐order chemical reactions data. SWIFS also proved more robust in terms of error propagation caused by the effect of the size of the estimation set. The proposed SWIFS model also outperformed rate law models in backwards‐time prediction. The ability of SWIFS to provide high accuracy in predicting chemical reactions may have beneficial implications on the efficiency of industrial production of chemicals as well as on the effective control of hazardous materials degradation. Copyright © 2007 John Wiley & Sons, Ltd.

Three‐dimensional analysis of patterns of skin displacement over the equine radius

Surface markers are usually used to track bone movement. However, skin movement related to the bone has a large effect on the analysis of skeletal kinematics. A 2-dimensional (2D) skin displacement correction model has been successfully developed, but no 3D skin displacement model exists. To develop a 3-dimensional (3D) skin displacement model for the equine radial segment during trot. The 3D trajectories of 6 skin-based markers and a bone-fixed triad were captured at trot in 4 horses. Skin displacements in the bone-based coordinate system were calculated using a singular-value decomposition method. The truncated Fourier series models were developed for the skin displacements using a generalised cross-validatory spline. Mean + /- s.d. of peak skin displacement of the 3 markers on the proximal radius as percentage of radial length was 10.7 +/- 0.5, 4.6 +/- 1.5 and 14.5 +/- 2.9% in x, y and z direction, respectively. For the 3 markers on the distal radius, the equivalent displacements were 4.7 +/- 0.6, 1.7 +/- 0.8 and 7.3 +/- 18% in x, y and z direction, respectively. The 3D skin displacement model for correction of skin marker motion over the equine radius relative to the bone can be established using a truncated Fourier series, which has previously been used successfully to develop 2D models. This method of determining 3D skin displacement correction needs to be extended to the entire fore- and hindlimbs to provide a more sensitive measure of kinematic analysis. Accurate descriptions of the 3D motions of the limb segments and interactions between adjacent segments at the joints are necessary for understanding of the mechanics of different gaits and the gait aberrations that manifest as lameness.

Quantification of three-dimensional skin displacement artefacts on the equine tibia and third metatarsus

AbstractRoutine study of three-dimensional (3D) tarsal kinematics is hampered by errors due to the displacement of skin surface-tracking markers relative to the underlying bones. Reliable kinematics can be obtained with bone-fixed markers, but an accurate, non-invasive method would have more applications. Simultaneous kinematic data from skin-based and bone-fixed markers attached to the tibia and third metatarsus were collected from three trotting subjects. The motion of the skin-based markers was extracted relative to the underlying bone motion tracked using the bone-fixed markers. The 3D skin displacement patterns for the skin-based markers were parameterized using a truncated Fourier series model. These displacements were expressed in terms of the local coordinate system for each bone. Skin displacement artefacts were observed in all three axes of each bone segment, with the largest displacements occurring at the proximal tibia. The mean skin displacement amplitudes in the tibia were 6.7%, 3.2% and 10.5% of segment length, and for the third metatarsus were 2.6%, 1.4% and 3.8% of segment length, for the craniocaudal, mediolateral and longitudinal segment axes, respectively. Skin displacement patterns could be expressed concisely using the Fourier series model. Displacements were also consistent between subjects, which should allow them to be used as a basis for developing a correction procedure for 3D tarsal joint kinematics.

Statistical Data Validation Methods for Large Cheese Plant Database

Production data of the cheesemaking process are used to monitor milk fat and protein recoveries in cheese, cheese yield, and composition and eventually to predict these parameters. Due to the large impact of these factors on cheese quality and plant profitability, it is very important to use reliable data for analysis, modeling, and control of the process. This paper tested six methods for detecting erroneous data in industrial cheesemaking databases. The data analyzed came from 4 yr of stirred-curd Cheddar cheese production in an industrial cheesemaking facility, comprising over 10,000 vats. Single vat outliers were detected using a simple statistical criterion of x¯±3.6 SD on single variable distributions, Fourier series modeling of seasonal variables (fat, protein, lactose, and total solids in milk, and protein in whey), and the multivariate Mahalanobis outlier analysis. Detection of outlier productions (corresponding to several vats) was done by applying the x¯±3.6 SD criterion to variables obtained through calculating the fat mass balance, fat retention coefficient, and yield efficiency. Data treatment enabled the detection of outlier data, but also pinpointed variables with a low reliability (manually registered times). Single variable and multivariable methods proved complementary, and the use of both types of methods is recommended when validating an existing database.

To contemplate quantitative and qualitative water features by neural networks method

An application deals with calibration of neural model and Fourier series model for Ploučnice catchment. This approach has an advantage, that the network choice is independent of other example’s parameters. Each networks, and their variants (different units and hidden layer number) can be connected in as a black box and tested independently. A Stuttgart neural simulator SNNS and a multiagent hybrid system Bang2 developed in Institute of Computer Science, AS CR have been used for testing. A perceptron network has been constructed, which was trained by back propagation method improved with a momentum term. The network is capable of an accurate forecast of the next day runoff based on the runoff and rainfall values from previous day.

An Analysis of Cepheid Distances Using Bayesian Statistics

We have applied an approximately Bayesian and a fully Bayesian analysis to the calculation of Cepheid distances, radii and absolute magnitudes using the surface brightness (Baade–Wesselink) method. Both methods successfully account for errors in the data, provide unbiased distance estimates, and provide objective model selection for the radial velocity curve. In addition, the fully Bayesian analysis objectively selects a model for the magnitude curve; averages over models of various Fourier orders, properly weighted by the posterior probabilities of the individual models; and includes a Lutz–Kelker correction.The approximately Bayesian method is that described by Jefferys & Barnes (1999) and Barnes & Jefferys (1999). It is a maximum likelihood approach with objective selection of the order of the Fourier series model of the radial velocities.

A Fourier Series Model to Predict Hourly Heating and Cooling Energy Use in Commercial Buildings With Outdoor Temperature as the Only Weather Variable

Accurate modeling of hourly heating and cooling energy use in commercial buildings can be achieved by a Generalized Fourier Series (GFS) approach involving weather variables such as dry-bulb temperature, specific humidity and horizontal solar flux. However, there are situations when only temperature data is available. The objective of this paper is to (i) describe development of a variant of the GFS approach which allows modeling both heating and cooling hourly energy use in commercial buildings with outdoor temperature as the only weather variable and (ii) illustrate its application with monitored hourly data from several buildings in Texas. It is found that the new Temperature based Fourier Series (TFS) approach (i) provides better approximation to heating energy use than the existing GFS approach, ((ii) can indirectly account for humidity and solar effects in the cooling energy use, (iii) offers physical insight into the operating pattern of a building HVAC system and (iv) can be used for diagnostic purposes.

Blocking and desensitization in bipolar and MOSFET differential amplifiers

Blocking and desensitization are among the nonlinear effects that result when bipolar and MOSFET differential amplifiers are operating in a multisignal electromagnetic interference (EMI) environment. When a weak desired signal is processed along with a strong undesired interferer signal, the weak signal may experience a reduced gain; that is, desensitization. If the amplitude of the undesired signal is sufficiently large, the gain of the desired signal may drop to zero; that is, blocking. Using a Fourier-series model for the differential input-voltage/output-current characteristics, closed-form expressions for predicting the desensitization in bipolar and MOSFET differential amplifiers are presented. These expressions are valid over a wide range of desired and blocking interfering signals amplitudes.

Prediction of the Harmonic Distortion in Mosfet Gate Capacitors

A Fourier-series model is presented for the capacitance-voltage characteristic of a MOS gate capacitor using transistors biased in the strong inversion or the accumulation region. Using this model closed-form expressions are obtained for predicting the harmonic distortion of the MOS gate capacitor driven by a large-amplitude sinusoidal current. Results are compared with reported experimental observations.

An Analysis of Hourly Fluctuation of Supplied Water Consumption

In this paper, we propose a time series model for hourly fluctuation of supplied water consumption. This model gives us basic information about water distribution control. First, after analysis of the significant characteristics of the hourly fluctuation, we find that it has two component parts. One of them is the cyclic component, which regularly makes typical fluctuated patterns with a l-day cycle. The other is the deviational component, which deviates complexly and randomly from the typical patterns of the cyclic component. Second, we make an hourly fluctuation model, which is based on the two components mentioned above. We make the cyclic and deviational component models by applying the finite Fourier series and multiple ARIMA models respectively, and combine them to form an hourly fluctuation model. Third, we show a case study in Seoul City, and use this proposed model to estimate the supplied water consumption. From the results of this case study, we can obtain an effective model to describe the hourly fluctuation of supplied water consumption. Our proposed model is useful for water distribution control planning.

Comparison of B-mode, M-mode and echo-tracking methods for measurement of the arterial distension waveform

Measurements of arterial diameter throughout the cardiac cycle ( i.e., the arterial distension waveform) are conducted increasingly to study mechanical properties of the arterial wall and changes associated with disease. The distension waveform of peripheral arteries can be measured noninvasively via ultrasonic echo tracking, M-mode imaging, and B-mode imaging. Of these, echo tracking is the most popular method because of its single micrometer resolution during continuous measurements under ideal conditions However, high resolution within continuous measurements does not imply high reproducibility between measurements. Therefore, we compared repeated measurements of the amplitude of common carotid artery distension in 26 subjects, obtained sequentially in random order by: 1. Off-line echo tracking of digitized radiofrequency ultrasound; 2. M-mode imaging with automated edge detection; and 3. 30-Hz B-mode imaging with automated edge detection and model-based diameter estimation. In each case, the transducer was hand-held and was removed from the neck between repeated measurements. The amplitude of arterial distension was estimated from the serial diameter measurements by maximum likelihood (ML) estimation, by least-squares fit of a Fourier series model, and by application of a cubic smoothing spline. Within continuous measurements, the standard deviation of the ML distension amplitude for neighboring cardiac cycles was significantly smaller ( p < 0.05) with echo-tracking (0.023 mm) than with the B-mode (0.036 mm) or M-mode (0.074 mm) methods. However, between discontinuous measurements on the same subject, the standard deviation of the ML distension amplitude was similar for the echo-tracking (0.076 mm) and B-mode (0.073 mm) methods. The Fourier series model and the cubic smoothing spline slightly reduced the standard deviation of the B-mode and M-mode distension amplitudes, but also reduced the mean amplitude estimate. On the basis of this relative comparison of methods, we conclude that, although echo tracking offers high resolution for continuous measurements, the reproducibility of discontinuous measurements of carotid artery distension is no better with echo tracking than can be obtained from 30-Hz B-mode images.

Is driven transect sampling suitable for estimating red‐legged partridge Alectoris rufa densities?

An omnibus procedure for Alectoris partridges density estimation is still lacking. The suitability of driven line and strip transect sampling for estimating red‐legged partridge Alectoris rufa densities on mixed farmland, in southern Portugal, in March, May and July 1993 and 1994 was evaluated. Driven transect counts along a grid of dirt tracks were performed and compared to early spring territory mapping counts. Distance data sets exhibited prominent peaks near the origin, apparently caused by attraction of the birds to the tracks; nevertheless, line transect estimates computed using Fourier series models were similar to mapping counts in March of both years, as well as adult density estimates in July, but density was underestimated in May 1994. Line transect estimates were robust to the shift from individuals to clusters as sighting units, and to different grouping options, but not to data truncation. Driven strip transect estimates were computed using 10 m, 20 m, 40 m, 80 m, and 160 m strip widths. In March, estimates computed using the first three widths overestimated density and underestimated it in 1994 when the 160‐m strip was used. May and July adult density estimates were not significantly different from those in March. In this study, driven line transect sampling was an efficient and accurate method of estimating red‐legged partridge density on a farmland area in early spring and summer; however, further research is needed to evaluate its suitability in habitats of lower visibility, as detection models seem to require long‐distance sightings to compensate for the attractiveness of tracks. Strip transects are not recommended because of the difficulty of choosing an adequate strip width for density estimation. Both methods revealed similar utility as indices of relative population abundance.

Harmonic and intermodulation performance of spatial light modulators

A Fourier-series model describing the nonlinear characteristic of a spatial light modulator is presented. With the use of this model, closed-form expressions are obtained for the harmonic and intermodulation performance of a spatial light modulator excited by a multisinusoidal input-write light intensity.

An improved analysis for the bipolar linear transconductor

An improved analysis for the bipolar linear transconductor, assuming a finite value for the transistor current-gain β, is presented. A new expression for the normalized input-voltage as a function of the normalized output-current is obtained. A Fourier-series model is presented for this current-voltage relationship. Closed-form expressions are also derived for the harmonic and intermodulation components resulting from exciting the bipolar linear transconductor by a multisinusoidal input voltage. These expressions can be used to predict the limiting values of the input voltage amplitudes for a prescribed tolerable distortion percentage.

Constrained imaging: overcoming the limitations of the Fourier series

Magnetic resonance imaging (MRI) is usually implemented as a Fourier transform-based technique. During data acquisition, spatially resolved information relating to spin density, relaxation rates, chemical shifts, and other parameters is phase and frequency encoded in the measured data. Image reconstruction is accomplished through the use of the Fourier series model, which can be evaluated efficiently using a fast Fourier transform (FFT) algorithm. Theoretically, the Fourier series is capable of producing perfect images if the data space (often called k-space) is sufficiently covered. In practice, several problems arise with this model due to finite sampling. Specifically, finite sampling leads to a truncation or the Fourier series, which results in image blurring and ringing. Image blurring is attributed to a loss of spatial resolution. In fact, with the Fourier series model, the resulting image resolution is limited to roughly the reciprocal of the frequency interval over which the data are sampled. The ringing artifact is due to the well-known Gibbs phenomenon, which is more pronounced for images with sharp edges. In order to overcome these limitations associated with the direct application of the Fourier series model, many alternatives have been proposed in the past decade to incorporate a priori information into the imaging process. This article discusses the constrained imaging concept. Specifically, the authors review 3 model-based imaging techniques that the authors have developed in the past few years. An essential feature of these methods is that a parametric model in the form of a generalized series is superimposed on the underlying measured data or image.