ARX Model Research Articles

A regime-based approach for integrating wind information in spatio-temporal solar forecasting models

Spatio-temporal solar forecasting based on statistical models seldom integrates wind information. An AutoRegressive with eXogenous input (ARX) model was tested using global horizontal irradiation records from a set of pyranometers deployed in Oahu, Hawaii, USA, where northeasterly winds are predominant. When irradiance is forecasted 10-s ahead, interesting forecast skills (up to 30.8%) can be achieved when a site has upwind neighbors available. However, when forecast skill is mapped as a function of wind direction at 850 hPa (from an ERA 5 reanalysis), negative skill values can be observed when nondominant winds occur. A wind regime-based approach is proposed, where different ARX models are built for different wind direction intervals, which substantially improves the forecasting accuracy for the underperforming wind directions. When the regime definition also takes into account wind speed, the ARX model detects spatial patterns for faster winds, with several nondominant directions achieving skill scores higher than 20%. Replacing the wind reanalysis by historical forecasts from ERA 5 reduced the overall skill by less than 0.1%.

Robust Identification under Correlated and Non-Gaussian Noises: WMLLM Procedure

The paper represents the main ideas of Robust Identification Theory, which was initiated by Ya.Z. Tzypkin in 80s of the past century. Here we demonstrate that the parallel application of both the “Whitening Procedure” and the recurrent min-max version of the “Maximum Likelihood Method” (MLLM), guarantees the property of Asymptotic Consistency of this procedure. The information Cramer–Rao inequality is also obtained and it is shown that this combined procedure attains this information bound. This means that for wide class of regular observations of ARX (auto regression with exogenous input) models, there does not exist any other identification algorithm, estimating asymptotically unknown parameters “quicker” then the procedure discussed here, if the distribution of the white noise at the input of the forming filter is known exactly. The main specific feature of this method consists in the possible consideration of external non-Gaussian white noise (defined on the given class of admissible distributions) in the input of the forming filter, making the noise sequence, affecting the input of ARX-dynamics, non-Gaussian, and correlated. The almost sure convergence as well as the Asymptotic Normality of the estimation error are formulated. If the information on the input of the forming filter is uncertain, that is, when the distribution belongs to a given class, then the Huber’s approach is applied using the robust version of MLLM.

H2, H∞, H2/H∞, and μ‐synthesis controllers for the speed and temperature control of a real gas turbine unit in a combined cycle power plant

AbstractThis paper designs, implements, and compares the performance of a H2, H∞, H2/H∞, and μ‐synthesis approach for a V94.2 gas turbine mounted in Damavand combined cycle power plant. The controllers are designed to maintain the speed and exhaust temperature within their desired intervals and to ensure the robust performance of the gas turbine power plant (GTPP) in the presence of uncertainties and load demand variations. A linear model of the GTPP is first estimated using V94.2 gas turbine real‐time data and an autoregressive with exogenous input (ARX) identification approach, and then verified by residual analysis tests and steady‐state performance. The H2, H∞, H2/H∞, and μ‐synthesis controllers are then designed and implemented to the ARX model of the GTPP. The performance of the approaches is assessed and compared in terms of tracking capability, robustness, and transient performance. Additionally, the controllers' performance is compared to that of a conventional PID approach. Despite the slight variations in the performance, all the controllers exhibited robust stability and good overall performance in the presence of model uncertainties and load variations.

Choosing between linear and nonlinear models and avoiding overfitting for short and long term groundwater level forecasting in a linear system

Groundwater level forecasting is a useful tool for a more efficient and sustainable groundwater resource management. Developing models that can accurately reproduce groundwater level response to meteorological conditions can lead to a better understanding of the groundwater resource availability. Here an autoregressive neural network (NNARx) approach is proposed and compared with autoregressive linear models with exogenous input (ARx) in order to forecast groundwater level in an aquifer system where a linear groundwater level response to recharge by rainfall is observed. A well known problem regarding neural networks consists in the high risk of overfitting. Here, three NNARx model were trained using different methods to avoid overfitting: Early stopping, Bayesian regularization and a combination of both. The results show that on the short term forecasting (up to 15 days) the performance of NNARx and ARx are comparable but the ARx model generalizes better, while the NNARx trained with Bayesian regularization outperforms the linear models and the other NNARx models on longer scenarios on the test set. As linear models are less time demanding and do not require high computational power, they can be considered as suitable tools for short term groundwater level forecasting in linear systems while when longer scenarios are needed neural networks can be considered more reliable, and training them with Bayesian regularization allows to minimize the risk of overfitting.

ARX model for interstitial glucose prediction during and after physical activities

This paper presents the first autoregressive with exogenous input (ARX) model using energy expenditure, carbohydrates on board, and insulin on board as input to predict interstitial glucose (IG). The proposed model may be used for predicting IG even during physical activity (PA). A population-based model, obtained from a first database composed of 14 type 1 diabetes (T1D) patients, achieved a root-mean-square error (RMSE) of 16.7±15.6 mg/dL, on IG prediction (30-min ahead) at the end of a PA, on a second database (15 T1D patients). Patient-specific ARX models, obtained on the second database, improved prediction accuracy (RMSE = 7.8±4.5 mg/dL), outperforming the results found in the literature.

Hybrid Wavelet-ARX approach for modeling association between rainfall and meteorological forcings at river basin scale

Interaction between meteorological and hydrologic processes is challenging to model owing to their high spatio-temporal variability. The understanding of their associations can help to ensure future fresh water security with changing climate. In this study, due to continuously evolving nature of these interactions, the hydrological and meteorological variables are studied on wavelet component level. Multi-Resolution Stationary Wavelet Transformation (MRSWT) is used to transform the independent (climatic variable) and dependent (hydrological variable) time series into their components. The components of the dependent time series are modeled using a kernel-based auto-regressive (AR) model for separating their memory part. The residuals are hypothesized to be the effect of interaction of predictor variables and thus, are modeled using the MRSWT components of meteorological variables in an auto-regressive model with exogenous inputs (ARX). Finally, the predicted residuals (effect of climatic variables) are added to the component estimated by kernel-based AR estimator (memory of dependent series components) to obtain the predicted components of the dependent hydrologic variable, which are then inverse-transformed to obtain the predicted dependent hydrologic variable. The developed hybrid Wavelet-ARX is found to capture the information about relationship between synthetically generated data better than a simple ARX model. The model is then applied to predict total monthly rainfall over Upper Mahanadi Basin and is found to effectively extract the information from the poorly associated hydro meteorological variables. While the potential of Wavelet-ARX is found to be impressive for hydro meteorological applications, additionally, discarding some climatic inputs on the basis of their relative importance may lead to better prediction by the developed model. The developed model is suitable for extracting climatic forcings and is desirable in a changing climate.

EMG-Based Control in a Test Platform for Exoskeleton with One Degree of Freedom.

There are several efforts to use the electrical signals generated by the human muscles to control virtual or even physical devices. It is expected that, the development of this method will provide a natural way to control these devices, requiring little user training, depending on the task complexity. With respect to the control of exoskeletons from the electric signals generated by the muscles, it is desirable that the exoskeleton acts in synergy with the user using surface electromyography (sEMG) signals to detect user intentions. One of the challenges of this approach is the variability of the sEMG signals due to factors such as electrode positioning and conditions of the volunteer at the time of acquisition. In previous work, a procedure based on an Autoregressive with Exogenous Input (ARX) linear model was developed to translate sEMG from biceps, triceps and brachioradialis muscles to elbow joint angle. In this work, we developed a method based on a Genetic Algorithm (GA) to update the ARX model coefficients online to minimize the training periods and we have used the EMG signals to control a one-degree of freedom exoskeleton.vThe GA was able to obtain ARX model coefficients that generate the joint angle reference from the EMG signals. In addition, the joint angle references generated from the offline sEMG from three muscles via an ARX model were used to control a device. At this point we are carrying out tests with the exoskeleton using real-time signals from sEMG.

ARX model decomposed on Meixner-Like orthonormal bases

The present study provides a new modeling of linear slowly starting systems. More precisely, this new approach extends the technique of filtering only the input using Meixner-Like (M-L) filters to filter both the output and input of the system outlined by an ARX model. Therefore, the idea is to develop the input and output parameters of ARX modeling over 2 M-L bases. So as to ensure an optimal representation, the two M-L poles are optimized using Newton–Raphson (N-R) and Genetic Algorithms (GA) methods. A new method is proposed for Model Predictive Control (MPC) using the obtained optimal model that is called ARXMeixner-Like (ARXM-L). A numerical example of system having delay and three examples of experimental research: A supersonic jet engine inlet, a Process Trainer PT326 and a Quanser aero experiment with one degree of freedom attitude control are made.

Mapping the pitfalls in the characterisation of the heat loss coefficient from on-board monitoring data using ARX models

Several studies have demonstrated the capability of data-driven modelling based on on-site measurements to characterise the thermal performance of building envelopes.Currently, such methods include steady-state and dynamic heating experiments and have mainly been applied to scale models and unoccupied test buildings. Nonetheless, it is proposed to upscale these concepts to characterise the thermal performance of in-use buildings based on on-board monitoring (OBM) devices which gather long-term operational data (e.g., room temperatures, gas and electricity consumption…). It remains, however, to be proven whether in-use data could be a cost-effective, practical and reliable alternative for the dedicated tests whose more intrusive measurements require on-site inspections. Furthermore, it is presently unclear what the optimal experimental design of the OBM would be and which data analysis methods would be adequate.This paper presents a first step in bridging this knowledge gap, by using on-board monitoring data to characterise the overall heat loss coefficient (HLC) [W/K] of an occupied, well-insulated single-family house in the UK. With the aid of a detailed building physical framework and specifically selected data subsets a sensitivity analysis is carried out to analyse the impact of the measurement set-up, the duration of the measurement campaign and the applied data analysis method.Although the exact HLC of the building is unknown and no absolute errors could hence be calculated, this paper provides a new understanding of the decisions that have to be made during the process from design of experiment to data analysis. It is demonstrated that such judgements can lead to differences in the mean HLC estimate of up to 89.5%.

Towards the characterization of the heat loss coefficient via on-board monitoring: Physical interpretation of ARX model coefficients

This paper explores the concept of characterizing the as-built Heat Loss Coefficient (HLC) of buildings based on-board monitoring (OBM), via energy consumption and temperature sensors, and time series analysis. It is examined (1) how the coefficients of different Auto-Regressive with eXogenous inputs (ARX) models can be interpreted and (2) whether these conclusions are sensitive to the building envelope assembly or the applied indoor temperature profile.The paper presents a theoretical case study whereby detailed building energy simulations are used to accurately map the impact of physical phenomena on the characterization process. The simulation models and boundary conditions are composed to focus on the link between the estimated ARX-coefficients and the physical driving forces for transmission heat loss to the ground and the exterior environment.The results show how the various ARX model coefficients are linked to specific components of the HLC (e.g. heat transfer through the walls and roof or through the slab-on-ground floor) and to what extent they are affected by the selection of input variables. By monitoring the ground temperature, the transmission heat losses can rather accurately be assigned to either the slab-on-ground or the walls and roof. Without this measurement data, the uncertainty on the estimates increases (ranges of the 95% confidence interval of up to 35% of the mean estimate). Modeling the ground heat losses by a constant intercept term leads to underestimations of the reference HLC of up to 59%, whereas adding heat flux sensors to monitor the transmission heat losses to the ground to the measurement set-up allows to assess the transmission heat transfer coefficient to the exterior environment HLCe within 2%.

Neural network, ARX, and extreme learning machine models for the short-term prediction of temperature in buildings

In this paper, the possibilities of developing machine learning based data-driven models for the short-term prediction of indoor temperature within prediction horizons ranging from 1 hour up to 12 hours are systematically investigated. The study was based on a TRNSYS emulation of a residential building heated by a heat pump, combined with measured weather data for a typical winter season in Ljubljana, Slovenia. Autoregressive models with exogenous inputs (ARX), neural network models (NN), and extreme learning machine models (ELM) are considered. The results confirm the finding that nonlinear models, particularly the NN model trained by regularization, consistently outperform linear models in both fitting and generalization performance, so they are the recommended choice as predictive models. The availability of future weather data considerably improved the predictive performance of all the tested models. Besides data about the future outdoor temperature, also data about future expected solar radiation significantly improve predictions of temperature in buildings. The linear models required embedding dimensions of 24 hours for accurate predictions, whereas the nonlinear models were not very sensitive to the use of past data. Nonlinear models required about three months of training data to reach good predictive performance, whereas the linear models converged to accurate predictions within six weeks. The RMSE prediction errors, averaged over all the data sets and all the prediction horizons, are within the range between 0.155 °C for the linear ARX model (in the case of no future available weather data), and 0.065 °C for the neural network model (in the case of available future weather data).

Flexible dispatch of a building energy system using building thermal storage and battery energy storage

The increasing development of renewable energy sources requires more flexible technologies to be applied in building energy systems and a flexible controlled resource for the power grid. This work focuses on investigating the flexibility potential of building thermal storage and battery energy storage. Firstly, an autoregressive model with exogenous inputs is proposed to forecast the dynamic cooling demand and, based on that, a mixed integer linear model is formulated to optimize the dispatch of building energy systems with minimal operating costs. A factory building located in Huizhou, China is used as a case study. The results show that the ARX model can accurately predict the thermal load hourly. The model’s level of fit is above 81%. The optimization objectives influence the development of the flexibility potential when building thermal storage and battery energy storage are considered. In this work, the economic objective is applied first to discuss the flexibility potential. The results show that, in this studied case, the operational cost decreased by 5.3% by using battery energy storage and further decreased by 4.0% by using building thermal storage. However, the results also reveal that unilaterally pursuing minimal operational costs results in larger peak valley difference of feeder power. The peak-valley difference of the feeder power increased from 714 kW to 1245 kW and 1689 kW respectively when the battery energy storage and building thermal storage were employed for the economic dispatch of the building energy system. Therefore, this work retests the flexibility potential of battery energy storage and building thermal storage by adding a constraint for feeder power difference. The results show that the operational costs can be still reduced greatly without damaging the stability of the power feeder.

Can semi-parametric additive models outperform linear models, when forecasting indoor temperatures in free-running buildings?

A novel application combining semi-parametric Generalized Additive Models (GAMs) with logistic GAMs was developed to forecast indoor temperatures and window opening states during prolonged heatwaves. GAM models were compared to AutoRegressive models with eXogenous inputs (ARX) and validated against monitored data from two case study dwellings, located near to Loughborough in the UK, during the 2013 heatwave. Input variables were selected using backward stepwise regressions based on minimisation of the Akaike Information Criterion (AIC) and Mean Absolute Error (MAE), for the ARX and GAM models respectively. Comparison of the models showed that whilst GAMs are capable of improving the forecasting accuracy, the improvements are significant only up to 3–6 h ahead. During heatwaves and over longer forecasting horizons, GAMs were found to be less reliable and accurate than ARX models. The marginal improvement in forecasting accuracy at shorter horizons did not justify the additional computational time and risk of instability associated with more complex GAMs, at longer forecasting horizons. Whilst, logistic GAMs were shown to adequately predict the window opening state, incorporating knowledge of the window state did not significantly improve the accuracy of the indoor temperature predictions.

Groundwater level forecasting using linear time series modeling: the case study of the thermal aquifer system of Monsummano Terme (central Italy)

Groundwater level forecasting can support a more efficient and sustainable groundwater management. In the present study, linear models are implemented on hydrogeological and meteorological time series related to the thermal aquifer system of Monsummano Terme (Pistoia province, Tuscany region in central Italy), in order to develop a tool for short-term groundwater level forecasting.Groundwater level data were analyzed along with daily precipitation and used for training Autoregressive (AR) models and Autoregressive models with Exogeous Input (ARx). The best performing AR model consisted in a linear combination of thepast 9 groundwater level measurements whereas the ARx model concerned, in addition, the precipitation values of the past 10 days. As a result, these models show an optimal performance in terms of Normalized Root Mean Square, and their residuals (i.e. modeled – measured values) can be associated to a white noise containing no relevant information. This study shows how linear models can beuseful and easy-applicable tools for groundwater level forecasting in systems characterized by a linear relationship between recharge and groundwater level increase.

Control-oriented dynamic model of an inductively coupled plasma torch by artificial intelligence methodology

Inductively coupled plasma (ICP) torches have been widely used in various materials processing. To improve the control of their processes, there has been a growing demand for simplified numerical models which can rapidly predict the dynamics of plasma jets subject to time-varying inputs or external disturbances. In this paper, control-oriented dynamic models of an ICP torch are developed as an alternative to the complex, high-level 2D time-dependent numerical model (i.e. a model based on magneto-hydrodynamic equations). Prior to model development, the detailed dynamic and nonlinear nature of the ICP torch to its time-varying operation conditions was numerically investigated using a 2D numerical model to gain insight into choosing appropriate dynamic model structures. Linear ARX (AutoRegressive with eXogenous input) and ARX-type neural network models were selected in the model identification, and their parameters or weightings were determined using the input/output data obtained from the 2D time-dependent numerical model. The dynamic behaviours of the ICP torch predicted from the developed models were in good agreement with the data from the 2D time-dependent numerical model. Using the developed models, a simple control system for the plasma temperature and axial velocity regulations was also designed and tested. The feedback control simulations demonstrated good set point tracking and disturbance rejection performances, indicating that the developed approach can be directly applied to the model-based control system design of an ICP torch as well as other thermal plasma torches.

Monitoring Drought Effects on Vegetation Productivity Using Satellite Solar-Induced Chlorophyll Fluorescence

Around the world, the increasing drought, which is exacerbated by climate change, has significant impacts on vegetation carbon assimilation. Identifying how short-term climate anomalies influence vegetation productivity in a timely and accurate manner at the satellite scale is crucial to monitoring drought. Satellite solar-induced chlorophyll fluorescence (SIF) has recently been reported as a direct proxy of actual vegetation photosynthesis and has more advantages than traditional vegetation indices (e.g., the Normalized Difference Vegetation Index, NDVI and the Enhanced Vegetation Index, EVI) in monitoring vegetation vitality. This study aims to evaluate the feasibility of SIF in interpreting drought effects on vegetation productivity in Victoria, Australia, where heat stress and drought are often reported. Drought-induced variations in SIF and absorbed photosynthetically active radiation (APAR) estimations based on NDVI and EVI were investigated and validated against results indicated by gross primary production (GPP). We first compared drought responses of GPP and vegetation proxies (SIF and APAR) during the 2009 drought event, considering potential biome-dependency. Results showed that SIF exhibited more consistent declines with GPP losses induced by drought than did APAR estimations during the 2009 drought period in space and time, where APAR had obvious lagged responses compared with SIF, especially in evergreen broadleaf forest land. We then estimated the sensitivities of the aforementioned variables to meteorology anomalies using the ARx model, where memory effects were considered, and compared the correlations of GPP anomaly with the anomalies of vegetation proxies during a relatively long period (2007–2013). Compared with APAR, GPP and SIF are more sensitive to temperature anomalies for the general Victoria region. For crop land, GPP and vegetation proxies showed similar sensitivities to temperature and water availability. For evergreen broadleaf forest land, SIF anomaly was explained better by meteorology anomalies than APAR anomalies. GPP anomaly showed a stronger linear relationship with SIF anomaly than with APAR anomalies, especially for evergreen broadleaf forest land. We showed that SIF might be a promising tool for effectively evaluating short-term drought impacts on vegetation productivity, especially in drought-vulnerable areas, such as Victoria.

Estimating models with high-order noise dynamics using semi-parametric weighted null-space fitting

Standard system identification methods often provide inconsistent estimates with closed-loop data. With the prediction error method (PEM), this issue is solved by using a noise model that is flexible enough to capture the noise spectrum. However, a too flexible noise model (i.e., too many parameters) increases the model complexity, which can cause additional numerical problems for PEM. In this paper, we consider the weighted null-space fitting (WNSF) method. With this method, the system is first modeled using a non-parametric ARX model, which is then reduced to a parametric model of interest using weighted least squares. In the reduction step, a parametric noise model does not need to be estimated if it is not of interest. Because the flexibility of the noise model is increased with the sample size, this will still provide consistent estimates in closed loop and asymptotically efficient estimates in open loop. In this paper, we prove these results, and we derive the asymptotic covariance for the estimation error obtained in closed loop, which is optimal for an infinite-order noise model. For this purpose, we also derive a new technical result for geometric variance analysis, instrumental to our end. Finally, we perform a simulation study to illustrate the benefits of the method when the noise model cannot be parametrized by a low-order model.

Comparison of model identification techniques for MPC in all-air HVAC systems in an educational building

In school and office buildings the ventilation system has a large contribution to the total energy use. A control strategy that adjusts the operation to the actual demand can significantly reduce the energy use. This is important in rooms with a highly fluctuating occupancy profile. However, a standard rule-based control is reactive, making the installation ‘lag behind’ in relation to the demand. A model predictive control (MPC) might be a solution. To implement an MPC control first a suitable model must be identified for reliable predictions of room temperature and CO2 concentration. For CO2 predictions three scenarios are proposed respectively counting camera, lecture schedule and motion sensor. Two model identification techniques are evaluated: ARX and RC models. For identifying the heating dynamics of the case study building the 3 state RC model showed a good performance, a 5 step ahead prediction on a 15 minute time interval indicated a RMSE of approximately 0.60 °C. The 3rd order ARX model indicated similar results, however the cross validation demonstrated that the RC model outperforms the ARX model. For CO2 predictions the counting scenario resulted in the most accurate n-step ahead predictions. The RMSE found for the RC model is at maximum 90 ppm while 140 ppm for the ARX model. RC models are recommended for modelling all-air HVAC systems attributed by the higher prediction accuracy over ARX models. In addition, these models still contain physical parameters compared to ARX models.

Empirical Analysis of VDAX and VSTOXX as Major Volatility Indices in the EU Including Forecasting Tools

This study reviews various time series forecasting models in order to find the best fit for the VDAX and VSTOXX for one month and one year. Additionally, the influence of the trading volume of the DAX is examined. Both durations are found to be stationary by the Phillips-Perron test, that is why non-integrated models are used. For a duration of one month, a GARCHX(1,1) model is the best fit in-sample as well as out-of-sample, while the best fit for a duration of one year is found to be a ARX(1) model. Based on the forecasts, two trading strategies are tested for each duration, which is a long only strategy and a combination of long and short trades. The performance of both strategies is compared with a simple buy and hold strategy on each VDAX and VSTOXX. It is found that an excess return over the buy and hold strategy can be generated for both durations even with transaction costs.

Parametric Identification Using Weighted Null-Space Fitting

In identification of dynamical systems, the prediction error method with a quadratic cost function provides asymptotically efficient estimates under Gaussian noise, but in general it requires solving a nonconvex optimization problem, which may imply convergence to nonglobal minima. An alternative class of methods uses a nonparametric model as intermediate step to obtain the model of interest. Weighted null-space fitting (WNSF) belongs to this class, starting with the estimate of a nonparametric ARX model with least squares. Then, the reduction to a parametric model is a multistep procedure where each step consists of the solution of a quadratic optimization problem, which can be obtained with weighted least squares. The method is suitable for both open- and closed-loop data, and can be applied to many common parametric model structures, including output-error, ARMAX, and Box–Jenkins. The price to pay is the increase of dimensionality in the nonparametric model, which needs to tend to infinity as function of the sample size for certain asymptotic statistical properties to hold. In this paper, we conduct a rigorous analysis of these properties: namely, consistency, and asymptotic efficiency. Also, we perform a simulation study illustrating the performance of WNSF and identify scenarios where it can be particularly advantageous compared with state-of-the-art methods.