Mitigating parameter uncertainty in business forecasting

Organisations make multiple decisions, and each layer requires different types of information. The main task of forecasting in these different situations is to support each decision with relevant future information in point and interval forecasts. The consequence is that we have multiple and correlated time series. To produce point and interval forecasts for multiple decisions, we have three modeling options, namely: • modeling each time series with a set of univariate Exponential Smoothing methods, • modeling all time series with a Vector Exponential Smoothing model, and • utilising state-of-the-art forecast reconciliation. Each option has the same idea: to approximate the ‘true’ data-generating process. Consequently, we have uncertainties around each modeling option, namely (a) model structure, (b) parameter, and (c) sampling uncertainty. The literature mainly focuses on mitigating the model structure uncertainty, which is believed to harm forecast accuracy significantly. On the other hand, this thesis mitigates the parameter uncertainty in each modeling case (Exponential Smoothing, Vector Exponential Smoothing, and Forecast Reconciliation). We propose parameter shrinkage in each modeling option. Specifically, we propose a shrinkage estimator for the univariate and the multivariate exponential smoothing. We also suggest forcing some covariances to zero to mitigate the covariance matrix estimation uncertainty in the forecast reconciliation. Our study relies on theoretical investigations, simulation, and empirical studies. The theoretical analysis provides solid and rational arguments to mitigate the parameter uncertainty. We complement it with empirical findings, where the difference between the simulation and the empirical study is how much we can control the experimental designs. We also ensure that each design follows sound principles of forecasting evaluation. Our findings show that the shrinkage estimator improves forecast accuracy. However, the results are mixed for the Vector Exponential Smoothing. We also find that forcing some covariances in the covariance matrix approximation improves both the forecast accuracy and the variability of the forecasting performance. By understanding the parameter uncertainty, we find important correlations between parameters that may affect forecast accuracy. We also propose the concept of stochastic coherency to encapsulate the overlooked uncertainties in forecast reconciliation. Our thesis emphasises the importance of revisiting uncertainty in business forecasting. We decipher it via the bias-variance decomposition and understand how the interdependence between parameters affects our understanding of the uncertainty. It is not only essential to address each uncertainty individually but also to address all uncertainties comprehensively. In particular, we propose different types of parameter shrinkage. The implementation depends on whether we have sufficient information to estimate parameters in the model. In the univariate case, the parameters’ estimates tend to be inefficient when the sample size is limited. In the multivariate case, either the shrinkage estimator or forcing some parameters to zero by design is also a potential solution to the problem. These forms of shrinkage avoid overfitting and potentially improve foecast accuracy. Concerning decision makers, our understanding of uncertainty highlights the importance of reliability in forecasting, i.e., unmitigated parameter uncertainty results in unreliable forecasting performance. This reliability is essential to gain the decision-maker’s trust in our forecasts. It is a new business forecasting concept and is open to investigation

Shrinkage Estimator for Exponential Smoothing Models

Exponential smoothing is widely used in practice and has shown its efficacy and reliability in many business applications. Yet there are cases, for example when the estimation sample is limited, where the estimated smoothing parameters can be erroneous, often unnecessarily large. This can lead to over-reactive forecasts and high forecast errors. Motivated by these challenges, we investigate the use of shrinkage estimators for exponential smoothing. This can help with parameter estimation and mitigating parameter uncertainty. Building on the shrinkage literature, we explore ℓ 1 and ℓ 2 shrinkage for different time series and exponential smoothing model specifications. From a simulation and an empirical study, we find that using shrinkage in exponential smoothing results in forecast accuracy improvements and better prediction intervals. In addition, using bias–variance decomposition, we show the interdependence between smoothing parameters and initial values, and the importance of the initial value estimation on point forecasts and prediction intervals

Stochastic coherency in forecast reconciliation

Hierarchical forecasting has been receiving increasing attention in the literature. The notion of coherency is central to this, which implies that the hierarchical time series follows some linear aggregation constraints. This notion, however, does not take several modelling uncertainties into account. We propose to redefine coherency as stochastic. This allows to accommodate overlooked uncertainties in forecast reconciliation. We show analytically that there are two potential sources of uncertainty in forecast reconciliation. We use simulated data to demonstrate how these uncertainties propagate to the covariance matrix estimation, introducing uncertainty in the reconciliation weights matrix. This then increases the uncertainty of the reconciled forecasts. We apply our understanding to modelling accident and emergency admissions in a UK hospital. Our analysis confirms the insights from stochastic coherency in forecast reconciliation. It shows that we gain accuracy improvement from forecast reconciliation, on average, at the cost of the variability of the forecast error distribution. Users may opt to prefer less volatile error distributions to assist decision making

Quantitative Evaluation of the Spatial Variation of Surface Soil Properties in Continuous Paddy Growing Fields

Soil degradation caused by poor land management practices is a major impediment to optimal land productivity. Soil spatial variability is required for agricultural productivity, food safety and environmental modeling. Rice is one of the important food resources for most of the world’s population, especially in India and feeds more than 60 per cent population of the country. Telangana is on track to become India's rice bowl as rice production is expected to reach 1.3 crore tons in 2019–20.The present study was conducted in continuous paddy cultivated field of Machapur village of Siddipet district, Telangana, India to know the spatial variability of soil properties with a help of geostatistical model. For this, a total of 100 composite samples at 20*20 m grids in an area of 4 ha were collected. The pH of the soil, electrical conductivity (EC), organic carbon (OC), available nitrogen (N), phosphorus (P) and potassium (K) were all determined. The semivariogram model was used to create surface maps of soil properties using the ordinary kriging technique. The skewness values showed a normal distribution for all analyzed parameters except for Available K. Coefficient of variation ranged from 1.92% for pH to 34.08% for EC in topsoil indicating the heterogeneity of soil properties. Spherical model fits well with experimental semivariogram of pH, EC and AK. Exponential model better described the variation of soil OC and AN while the variation of AP was best described by Gaussian model. The soil pH, OC and available P were moderately spatially dependent whereas EC, available N and K were strongly spatially dependent. The cross validation results demonstrated the spatial prediction's smoothing effect. According to the findings of this study, a geostatistical model can directly reveal the spatial variability of lateritic soils and will assist farmers and decision makers in improving soil-water management

Crop Classification by Forward Neural Network with Adaptive Chaotic Particle Swarm Optimization

This paper proposes a hybrid crop classifier for polarimetric synthetic aperture radar (SAR) images. The feature sets consisted of span image, the H/A/α decomposition, and the gray-level co-occurrence matrix (GLCM) based texture features. Then, the features were reduced by principle component analysis (PCA). Finally, a two-hidden-layer forward neural network (NN) was constructed and trained by adaptive chaotic particle swarm optimization (ACPSO). K-fold cross validation was employed to enhance generation. The experimental results on Flevoland sites demonstrate the superiority of ACPSO to back-propagation (BP), adaptive BP (ABP), momentum BP (MBP), Particle Swarm Optimization (PSO), and Resilient back-propagation (RPROP) methods. Moreover, the computation time for each pixel is only 1.08 × 10−7 s