Exponential Smoothing State Space Model Research Articles

Ranking of univariate forecasting techniques for seasonal time series using analytical hierarchy process

The choice of a suitable forecasting method carries noteworthy significance for organisations in adequately accomplishing their business targets. The selection of forecasting method becomes more sophisticated when there is a significant impact of seasonality on the business of an organisation. To deal with the situation of selecting the most relevant forecasting method for seasonal data, this paper proposes a framework using analytical hierarchy process (AHP) to rank various forecasting techniques for long time series. Accuracy measures namely Theil's U, mean absolute error (MAE), root mean square error (RMSE) and mean absolute percentage error (MAPE) are used as AHP criteria for performance measurement of various univariate time series methods such as naïve + season level trend (SLT), error, trend, seasonal (ETS), seasonal autoregressive moving average (SARIMA), exponential smoothing state space model with Box-Cox transformation (BATS) and trigonometric exponential smoothing state space model with Box-Cox transformation (TBATS) for seasonal data. The proposed framework is validated through real-time data provided by a public sector company in India. Ranking obtained from the developed AHP framework suggests that SARIMA is ranked top amongst all the techniques for short-term forecasting of seasonal data.

Comparison of ARIMA, ETS, NNAR and Hybrid Models to Forecast the Second Wave of COVID-19 Hospitalizations in Italy

Coronavirus disease (COVID-19) is a severe ongoing novel pandemic that has emerged in Wuhan, China, in December 2019. As of October 13, the outbreak has spread rapidly across the world, affecting over 38 million people, and causing over 1 million deaths. In this article, I analysed several time series forecasting methods to predict the spread of COVID-19 second wave in Italy, over the period after October 13, 2020. I used an autoregressive model (ARIMA), an exponential smoothing state space model (ETS), a neural network autoregression model (NNAR), and the following hybrid combinations of them: ARIMA-ETS, ARIMA-NNAR, ETS-NNAR, and ARIMA-ETS-NNAR. About the data, I forecasted the number of patients hospitalized with mild symptoms, and in intensive care units (ICU). The data refer to the period February 21, 2020–October 13, 2020 and are extracted from the website of the Italian Ministry of Health (www.salute.gov.it). The results show that i) the hybrid models, except for ARIMA-ETS, are better at capturing the linear and non-linear epidemic patterns, by outperforming the respective single models; and ii) the number of COVID-19-related hospitalized with mild symptoms and in ICU will rapidly increase in the next weeks, by reaching the peak in about 50-60 days, i.e. in mid-December 2020, at least. To tackle the upcoming COVID-19 second wave, on one hand, it is necessary to hire healthcare workers and implement sufficient hospital facilities, protective equipment, and ordinary and intensive care beds; and on the other hand, it may be useful to enhance social distancing by improving public transport and adopting the double-shifts schooling system, for example.

Combined TBATS and SVM model of minimum and maximum air temperatures applied to wheat yield prediction at different locations in Europe

This paper explores the idea of combining Trigonometric Exponential Smoothing State Space model with Box-Cox transformation, ARMA errors, Trend and Seasonal Components (TBATS) with Support Vector Machine (SVM) model to estimate time series of the minimum and maximum daily air temperatures in a period of six years for various climatic localizations in Europe. It was found that a combined SVM/TBATS model can predict not only seasonality but also local temperature variation between subsequent days observed in daily data. Because the SVM sub-model uses not only results of TBATS prediction as an input data, but also several meteorological values, such modelling cannot be treated as a future time series estimation. Therefore, it has a potential to be used for filling gaps in the air temperature data. As is shown in our results, the precision of air temperature prediction improves when using the combined SVM/TBATS modelling, compared with pure TBATS or SVM modelling. For various locations, which can be related with different climatic conditions, this improvement ranged from 3% up to 14% for the maximum daily air temperature and from 5% to 25% for the minimum daily air temperature. The temperature sums calculated on the base of air temperatures predicted with SVM/TBATS models and from measured values did not differ more than 300 °C (less than 1 °C per day) in majority of cases. The average error in wheat yield prediction by WOFOST and DNDC models did not exceed 12.8% and 13.3%, respectively.

Application of Discrete-Interval Moving Seasonalities to Spanish Electricity Demand Forecasting during Easter

Forecasting electricity demand through time series is a tool used by transmission system operators to establish future operating conditions. The accuracy of these forecasts is essential for the precise development of activity. However, the accuracy of the forecasts is enormously subject to the calendar effect. The multiple seasonal Holt–Winters models are widely used due to the great precision and simplicity that they offer. Usually, these models relate this calendar effect to external variables that contribute to modification of their forecasts a posteriori. In this work, a new point of view is presented, where the calendar effect constitutes a built-in part of the Holt–Winters model. In particular, the proposed model incorporates discrete-interval moving seasonalities. Moreover, a clear example of the application of this methodology to situations that are difficult to treat, such as the days of Easter, is presented. The results show that the proposed model performs well, outperforming the regular Holt–Winters model and other methods such as artificial neural networks and Exponential Smoothing State Space Model with Box-Cox Transformation, ARMA Errors, Trend and Seasonal Components (TBATS) methods.

Time Series Analysis of Monthly Suicide Rates in West of Iran, 2006-2013.

Introduction:Iran's western provinces have higher suicide rate compared to the other provinces of the country. Although suicide rates fluctuate over time, suitable statistical models can describe their underlying stochastic dynamics.Methods:This study was conducted to explore the fluctuations of the monthly suicide rates in the most populated western province of Iran using exponential smoothing state space model to compute the forecasts. For this reason, the monthly frequencies of completed suicides were converted to rates per 100,000 and a state-space approach was identified and fitted to the monthly suicide rates. Diagnostic checks were performed to determine the adequacy of the fitted model.Results:A significant seasonal variation was detected in completed suicide with a peak in August. Diagnostic checks and the time series graph of the observed monthly suicide rates against predicted values from the fitted model showed that in the study period (from March 2006 to September 2013), the observed and predicted values were in agreement. Thus, the model was used to obtain the short-term forecasts of the monthly suicide rates.Conclusions:In this study, we had no significant trend but seasonal variations in the suicide rates that were identified. However, additional data from other parts of the country with longer duration are needed to visualize the reliable trend of suicide and identify seasonality of suicide across the country.

Forecasting Atmospheric Radiocarbon Decline to Pre-Bomb Values

AbstractIn this manuscript, I present an estimation of the rate of decline in atmospheric radiocarbon and the amplitude of its seasonal cycle for the past four decades for the northern and southern hemispheres, and forecast the time required to reach pre-1950 levels (i.e. Δ14C<0‰). Using a set of 30 different exponential smoothing state-space models, the time series were decomposed into their error, trend, and seasonal components, choosing the model that best represented the observed data. According to the best model, the rate of change in Δ14C has decreased considerably since the 1970s and reached values below −5‰ per year since 2005. Overall, the time-series showed larger rates of radiocarbon decline in the northern than in the southern hemisphere, and relatively stable seasonal cycles for both hemispheres. A forecast of the exponential smoothing models predicts that radiocarbon values will reach pre-1950 levels by 2021 in the northern hemisphere with 20% probability, and by around 2035 in the southern hemisphere. However, at regional levels radiocarbon concentrations have already reached pre-1950 levels in several industrialized regions and cities around the world as a consequence of fossil-fuel emissions.

Non-linear exponential smoothing and positive data

We consider the properties of nonlinear exponential smoothing state space models under various assumptions about the innovations, or error, process. Our interest is restricted to those models that are used to describe non-negative observations, because many series of practical interest are so constrained. We first demonstrate that when the innovations process is assumed to be Gaussian, the resulting prediction distribution may have an infinite variance beyond a certain forecasting horizon. Further, such processes may converge almost surely to zero; an examination of purely multiplicative models reveals the circumstances under which this condition arises. We then explore effects of using an (invalid) Gaussian distribution to describe the innovations process when the underlying distribution is lognormal. Our results suggest that this approximation causes no serious problems for parameter estimation or for forecasting one or two steps ahead. However, for longer-term forecasts the true prediction intervals become increasingly skewed, whereas those based on the Gaussian approximation may have a progressively larger negative component. In addition, the Gaussian approximation is clearly inappropriate for simulation purposes. The performance of the Gaussian approximation is compared with those of two lognormal models for short-term forecasting using data on the weekly sales of over three hundred items of costume jewelry.

Sustainability in Oman: Energy Consumption Forecasting using R

Objective: Smart city projects are still in their initial research stages in Oman. This paper aims to prove the effectiveness of smart cities by using Data Mining Techniques (DMT) to predict energy consumption in Oman. Methods: Data collected from thirteen residential and eight industrial meters are used for electricity consumption forecast. Detailed data analysis is carried out using K-means clustering and time-series forecasting in R. Energy consumption data is modeled using average, naive, seasonal naive, Seasonal decomposition of Time Series by Loess (STL) +Random Walk with Drift (RWD), Exponential smoothing state space model with Box-Cox transformation, ARMA errors, Trend and Seasonal component (TBATS) and Autoregressive Integrated Moving Average (ARIMA) models. Findings: Even though the dataset isn’t characterized by seasons or trends, Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) error measures suggest that electricity consumption for residential sector is more accurately forecasted using TBATS model. Energy consumption for small, medium and large scale industries, on the other hand are more accurately predicted by TBATS, Average and STL + RWD models respectively. Applications: The obtained results confirm the efficiency in forecasting energy consumption in Oman using time series models in order to initiate smart city implementation. Keywords: Data Mining, Energy Consumption, Smart City, Clustering, Time-Series Forecasting, R

Forecasting electricity load demand using hybrid exponential smoothing-artificial neural network model

Short-term electricity load demand forecast is a vital requirements for power systems. This research considers the combination of exponential smoothing for double seasonal patterns and neural network model. The linear version of Holt-Winter method is extended to accommodate a second seasonal component. In this work, the Fourier with time varying coefficient is presented as a means of seasonal extraction. The methodological contribution of this paper is to demonstrate how these methods can be adapted to model the time series data with multiple seasonal pattern, correlated non stationary error and nonlinearity components together. The proposed hybrid model is started by implementing exponential smoothing state space model to obtain the level, trend, seasonal and irregular components and then use them as inputs of neural network. Forecasts of future values are then can be obtained by using the hybrid model. The forecast performance was characterized by root mean square error and mean absolute percentage error. The proposed hybrid model is applied to two real load series that are energy consumption in Bawen substation and in Java-Bali area. Comparing with other existing models, results show that the proposed hybrid model generate the most accurate forecast

Forecasting Arabian Sea level rise using exponential smoothing state space models and ARIMA from TOPEX and Jason satellite radar altimeter data

Sea level rise is a threat to coastal habitation and is corroborating evidence for global warming. The present study investigated the combined use of quantitative forecasting methods for sea level rise using exponential smoothing state space models (ESMs) and an autoregressive integrated moving average (ARIMA) model fed with sea level data over 17 years (1994–2010). Two levels of ESMs were employed: double (model levels with trend) and triple (model levels, trend and seasonal decomposition). The overall data analysis revealed the better performance of ARIMA in terms of index of agreement (d = 0.79), root-mean-square error (RMSE = 32.8 mm) and mean absolute error (MAE = 25.55 mm) than the triple ESM (d = 0.76; RMSE = 39.86 mm; MAE = 35.02 mm) and double ESM (d = 0.14; RMSE = 52.71 mm; MAE = 45.99 mm) models. The present study results suggest that the rate of Arabian Sea level rise is high, and if this is not taken into consideration many coastal areas may become subject to climate-change-induced habitat loss in future.

Predicting the Evolution of CO2 Emissions in Bahrain with Automated Forecasting Methods

The 2012 Doha meeting established the continuation of the Kyoto protocol, the legally-binding global agreement under which signatory countries had agreed to reduce their carbon emissions. Contrary to this assumed obligation, all G20 countries with the exception of France and the UK saw significant increases in their CO2 emissions over the last 25 years, surpassing 300% in the case of China. This paper attempts to forecast the evolution of carbon dioxide emissions in Bahrain over the 2012–2021 decade by employing seven Automated Forecasting Methods, including the exponential smoothing state space model (ETS), the Holt–Winters Model, the BATS/TBATS model, ARIMA, the structural time series model (STS), the naive model, and the neural network time series forecasting method (NNAR). Results indicate a reversal of the current decreasing trend of pollution in the country, with a point estimate of 2309 metric tons per capita at the end of 2020 and 2317 at the end of 2021, as compared to the 1934 level achieved in 2010. The country’s baseline level corresponding to year 1990 (as specified by the Doha amendment of the Kyoto protocol) is approximately 25.54 metric tons per capita, which implies a maximum level of 20.96 metric tons per capita for the year 2020 (corresponding to a decrease of 18% relative to the baseline level) in order for Bahrain to comply with the protocol. Our results therefore suggest that Bahrain cannot meet its assumed target.

Past and projected trends of body mass index and weight status in South Australia: 2003 to 2019

Objective:Functional data analysis (FDA) is a forecasting approach that, to date, has not been applied to obesity, and that may provide more accurate forecasting analysis to manage uncertainty in public health. This paper uses FDA to provide projections of Body Mass Index (BMI), overweight and obesity in an Australian population through to 2019. Methods:Data from the South Australian Monitoring and Surveillance System (January 2003 to December 2012, n=51,618 adults) were collected via telephone interview survey. FDA was conducted in four steps: 1) age‐gender specific BMIs for each year were smoothed using a weighted regression; 2) the functional principal components decomposition was applied to estimate the basis functions; 3) an exponential smoothing state space model was used for forecasting the coefficient series; and 4) forecast coefficients were combined with the basis function. Results:The forecast models suggest that between 2012 and 2019 average BMI will increase from 27.2 kg/m2 to 28.0 kg/m2 in males and 26.4 kg/m2 to 27.6 kg/m2 in females. The prevalence of obesity is forecast to increase by 6–7 percentage points by 2019 (to 28.7% in males and 29.2% in females). Conclusions:Projections identify age–gender groups at greatest risk of obesity over time. The novel approach will be useful to facilitate more accurate planning and policy development.

Forecasting Age-Specific Brain Cancer Mortality Rates Using Functional Data Analysis Models

Incidence and mortality rates are considered as a guideline for planning public health strategies and allocating resources. We apply functional data analysis techniques to model age-specific brain cancer mortality trend and forecast entire age-specific functions using exponential smoothing state-space models. The age-specific mortality curves are decomposed using principal component analysis and fit functional time series model with basis functions. Nonparametric smoothing methods are used to mitigate the existing randomness in the observed data. We use functional time series model on age-specific brain cancer mortality rates and forecast mortality curves with prediction intervals using exponential smoothing state-space model. We also present a disparity of brain cancer mortality rates among the age groups together with the rate of change of mortality rates. The data were obtained from the Surveillance, Epidemiology and End Results (SEER) program of the United States. The brain cancer mortality rates, classified under International Classification Disease code ICD-O-3, were extracted from SEER*Stat software.

Satellite image analysis and a hybrid ESSS/ANN model to forecast solar irradiance in the tropics

We forecast hourly solar irradiance time series using satellite image analysis and a hybrid exponential smoothing state space (ESSS) model together with artificial neural networks (ANN). Since cloud cover is the major factor affecting solar irradiance, cloud detection and classification are crucial to forecast solar irradiance. Geostationary satellite images provide cloud information, allowing a cloud cover index to be derived and analysed using self-organizing maps (SOM). Owing to the stochastic nature of cloud generation in tropical regions, the ESSS model is used to forecast cloud cover index. Among different models applied in ANN, we favour the multi-layer perceptron (MLP) to derive solar irradiance based on the cloud cover index. This hybrid model has been used to forecast hourly solar irradiance in Singapore and the technique is found to outperform traditional forecasting models.

Short-term solar irradiance forecasting using exponential smoothing state space model

We forecast high-resolution solar irradiance time series using an exponential smoothing state space (ESSS) model. To stationarize the irradiance data before applying linear time series models, we propose a novel Fourier trend model and compare the performance with other popular trend models using residual analysis and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) stationarity test. Using the optimized Fourier trend, an ESSS model is implemented to forecast the stationary residual series of datasets from Singapore and Colorado, USA. To compare the performance with other time series models, autoregressive integrated moving average (ARIMA), linear exponential smoothing (LES), simple exponential smoothing (SES) and random walk (RW) models are tested using the same data. The simulation results show that the ESSS model has generally better performance than other time series forecasting models. To assess the reliability of the forecasting model in real-time applications, a complementary study of the forecasting 95% confidence interval and forecasting horizon of the ESSS model has been conducted.

Forecasting age-related changes in breast cancer mortality among white and black US women: A functional data approach

Background: The disparity in breast cancer mortality rates among white and black US women is widening, with higher mortality rates among black women. We apply functional time series models on age-specific breast cancer mortality rates for each group of women, and forecast their mortality curves using exponential smoothing state-space models with damping. Materials and Methods: The data were obtained from the Surveillance, Epidemiology and End Results (SEER) program of the US [1]. Mortality data were obtained from the National Centre for Health Statistics (NCHS) available on the SEER*Stat database. We use annual unadjusted breast cancer mortality rates from 1969 to 2004 in 5-year age groups (45–49, 50–54, 55–59, 60–64, 65–69, 70–74, 75–79, 80–84). Age-specific mortality curves were obtained using nonparametric smoothing methods. The curves are then decomposed using functional principal components and we fit functional time series models with four basis functions for each population separately. The curves from each population are forecast and prediction intervals are calculated. Results: Twenty-year forecasts indicate an overall decline in future breast cancer mortality rates for both groups of women. This decline appears to be steeper among white women aged 55–73 and black women aged 60–84. For black women under 55 years of age, the forecast rates are relatively stable indicating there is no significant change in future breast cancer mortality rates among young black women in the next 20 years. Conclusion: White women have smooth and consistent patterns in breast cancer mortality rates for all age-groups whereas the mortality rates for black women are much more variable. The projections suggest, for some age groups, black American women may not benefit equally from the overall decline in breast cancer mortality in the United States.

EXPONENTIAL SMOOTHING AND NON‐NEGATIVE DATA

SummaryThe most common forecasting methods in business are based on exponential smoothing, and the most common time series in business are inherently non‐negative. Therefore it is of interest to consider the properties of the potential stochastic models underlying exponential smoothing when applied to non‐negative data. We explore exponential smoothing state space models for non‐negative data under various assumptions about the innovations, or error, process. We first demonstrate that prediction distributions from some commonly used state space models may have an infinite variance beyond a certain forecasting horizon. For multiplicative error models that do not have this flaw, we show that sample paths will converge almost surely to zero even when the error distribution is non‐Gaussian. We propose a new model with similar properties to exponential smoothing, but which does not have these problems, and we develop some distributional properties for our new model. We then explore the implications of our results for inference, and compare the short‐term forecasting performance of the various models using data on the weekly sales of over 300 items of costume jewelry. The main findings of the research are that the Gaussian approximation is adequate for estimation and one‐step‐ahead forecasting. However, as the forecasting horizon increases, the approximate prediction intervals become increasingly problematic. When the model is to be used for simulation purposes, a suitably specified scheme must be employed.