Conditional Predictive Distribution Research Articles

A novel day-ahead regional and probabilistic wind power forecasting framework using deep CNNs and conformalized regression forests

Regional forecasting is crucial for a balanced energy delivery system and for achieving the global transition to clean energy. However, regional wind forecasting is challenging due to uncertain weather prediction and its high dimensional nature. Most solutions are limited to single-turbine or farm/park forecasting; therefore, this work proposes a day-ahead regional wind power forecasting framework using deep Convolutional Neural Networks (CNN) with context-aware turbine maps and Conformal Quantile Regression (CQR) to generate quantile forecasts with valid coverage.Additionally, this work introduces the use of the Split Conformal Predictive System (SCPS) to generate valid prediction distributions, which has not yet been proposed for wind power forecasting in general. As well as a new method to generate calibrated prediction distributions based on SCPS and Quantile Regression Forests (QRF). This new method, named Split Conformal Distribution Regression Forests (SCDRF), allows for conditional conformal predictive distribution that increases efficiency compared to SCPS while maintaining valid coverage. SCDRF, together with CNNs and context-aware turbine maps, outperforms the existing models on the evaluated dataset, reducing the pinball loss by 5.89% while having more flexibility due to the generation of prediction distributions that can be used to generate any quantile prediction without retraining the model.

Capturing Macro‐Economic Tail Risks with Bayesian Vector Autoregressions

AbstractMany studies using quantile regressions (QRs) have found that downside risk to output growth varies more than upside risk. We show that Bayesian vector autoregressions (BVARs) with stochastic volatility are able to capture tail risks in forecast distributions. Even though the one‐step‐ahead conditional predictive distributions from the conventional stochastic volatility specification are symmetric, forecasts of downside risks to output growth are more variable than upside risks, and the reverse applies in the case of inflation and unemployment. Overall, BVAR models perform comparably to QR for estimating and forecasting tail risks, complementing BVARs' established performance for forecasting and structural analysis.

Full Bayesian analysis of double seasonal autoregressive models with real applications

We present a full Bayesian analysis of multiplicative double seasonal autoregressive (DSAR) models in a unified way, considering identification (best subset selection), estimation, and prediction problems. We assume that the DSAR model errors are normally distributed and introduce latent variables for the model lags, and then we embed the DSAR model in a hierarchical Bayes normal mixture structure. By employing the Bernoulli prior for each latent variable and the mixture normal and inverse gamma priors for the DSAR model coefficients and variance, respectively, we derive the full conditional posterior and predictive distributions in closed form. Using these derived conditional posterior and predictive distributions, we present the full Bayesian analysis of DSAR models by proposing the Gibbs sampling algorithm to approximate the posterior and predictive distributions and provide multi-step-ahead predictions. We evaluate the efficiency of the proposed full Bayesian analysis of DSAR models using an extensive simulation study, and we then apply our work to several real-world hourly electricity load time series datasets in 16 European countries.

Incorporating historical weather information in crop insurance rating

AbstractCrop insurance programs rely on conditional predictive distributions of loss random variables (e.g., yield, revenue, loss costs, etc.) to determine probabilities and magnitudes of loss. The loss variables may be related to stochastic variables that are not known at the time the policy is priced. Such is the case for weather; weather is stochastic, realizations are not known when the crop insurance policy is sold, and there is often additional historical information on weather relative to the loss variable itself. We provide a Bayesian methodology for incorporating historical weather information in crop insurance rating. We apply the method in empirical applications to county‐level U.S. corn yields and loss cost ratios in the Midwest. The historical weather‐conditioned distributions differ from those based on shorter samples. In the yield distribution setting, additional temporal weather information leads to economic gains relative to other rating approaches; the magnitude of these gains increases with the amount of historical weather information included in the analysis.

Postprocessing continental-scale, medium-range ensemble streamflow forecasts in South America using Ensemble Model Output Statistics and Ensemble Copula Coupling

Probabilistic hydrological forecasting and ensemble techniques have leveraged streamflow prediction at regional to continental scales up to several weeks in advance. However, ensembles that only account for meteorological forecast uncertainty are typically biased and subject to dispersion errors, thus limiting their use for rational decision-making and optimization systems. Statistical postprocessing is therefore necessary to convert ensemble forecasts into calibrated and sharp predictive distributions, and it should also account for dependencies between lead times to enable realistic forecast trajectories. This work provides a continental-scale assessment of the use of statistical postprocessing on medium-range, ensemble streamflow forecasts over South America (SA). These forecasts were produced through a large-scale hydrologic–hydrodynamic model forced with a global precipitation dataset and ECMWF reforecast data. The Ensemble Model Output Statistics (EMOS) technique was used to generate conditional predictive distributions in 488 locations at each forecast lead time, while the Ensemble Copula Coupling method with the transformation scheme (ECC-T) was applied to derive ensemble traces from EMOS distributions. Postprocessed streamflow forecasts were cross-validated for the period from 1996 to 2014 using a range of verification metrics. Results showed that the skill and reliability of EMOS forecasts substantially improve over the raw ensembles, and that EMOS leads to skillful predictions relative to discharge climatology and persistence forecasts up to 15 days in advance in most locations. Furthermore, EMOS results in predictive distributions that are generally sharper than the climatology. Limitations in depicting autocorrelations of forecast trajectories were observed in rivers for which the raw ensemble spread is very low and EMOS has to largely increase dispersion, especially at short lead times. The study’s findings suggest that combining a continental-scale hydrological model with EMOS and ECC-T methods can lead to skillful predictions and realistic ensemble traces in several locations in SA, even if in situ hydrometeorological observations are not available in real time.

Noncausal counting processes: A queuing perspective

We introduce noncausal counting processes, defined by time-reversing an INAR(1) process, a non-INAR(1) Markov affine counting process, or a random coefficient INAR(1) [RCINAR(1)] process. The noncausal processes are shown to be generically time irreversible and their calendar time dynamic properties are unreplicable by existing causal models. In particular, they allow for locally bubble-like explosion, while at the same time preserving stationarity. Many of these processes have also closed form calendar time conditional predictive distribution, and allow for a simple queuing interpretation, similar as their causal counterparts.

Time Series Simulation by Conditional Generative Adversarial Net

Generative Adversarial Net (GAN) has proved to be a powerful machine learning tool in image data analysis and generation. In this paper, we propose to use Conditional Generative Adversarial Net (CGAN) to learn and simulate time series data. The conditions include both categorical and continuous variables with different auxiliary information. Our simulation studies show that CGAN has the capability to learn different types of normal and heavy-tailed distributions, as well as dependent structures of different time series. It also has the capability to generate conditional predictive distributions consistent with training data distributions. We also provide an in-depth discussion on the rationale behind GAN and the neural networks as hierarchical splines to establish a clear connection with existing statistical methods of distribution generation. In practice, CGAN has a wide range of applications in market risk and counterparty risk analysis: it can be applied to learn historical data and generate scenarios for the calculation of Value-at-Risk (VaR) and Expected Shortfall (ES), and it can also predict the movement of the market risk factors. We present a real data analysis including a backtesting to demonstrate that CGAN can outperform Historical Simulation (HS), a popular method in market risk analysis to calculate VaR. CGAN can also be applied in economic time series modeling and forecasting. In this regard, we have included an example of hypothetical shock analysis for economic models and the generation of potential CCAR scenarios by CGAN at the end of the paper

Inflation at Risk

We investigate how macroeconomic drivers affect the predictive inflation distribution as well as the probability that inflation will run above or below certain thresholds over the near term. This is what we refer to as Inflation-at-Risk–a measure of the tail risks to the inflation outlook. We find that the recent muted response of the conditional mean of inflation to economic conditions does not convey an adequate representation of the overall pattern of inflation dynamics. Analyzing data from the 1970s reveals ample variability in the conditional predictive distribution of inflation that remains even when focusing on the post-2000 period of stable and low mean inflation. We also document that in the United States and in the Euro Area tight financial conditions carry substantial downside inflation risks, a feature overlooked by much of the literature. Our paper offers a new empirical perspective to existing macroeconomic models, showing that changes in credit conditions are also key to understand the dynamics of the inflation tails.

Capturing Macroeconomic Tail Risks with Bayesian Vector Autoregressions

A rapidly growing body of research has examined tail risks in macroeconomic outcomes. Most of this work has focused on the risks of significant declines in GDP, and it has relied on quantile regression methods to estimate tail risks. Although much of this work discusses asymmetries in conditional predictive distributions, the analysis often focuses on evidence of downside risk varying more than upside risk. We note that this pattern in risk estimates over time could obtain with conditional distributions that are symmetric but subject to simultaneous shifts in conditional means (down) and variances (up). Building on that insight, we examine the ability of Bayesian VARs with stochastic volatility to capture tail risks in macroeconomic forecast distributions and outcomes. We consider both a conventional stochastic volatility specification and a specification with a common factor in volatility that enters the VAR’s conditional mean. Even though the one-step-ahead conditional predictive distributions from the conventional stochastic volatility specification are symmetric, the model estimates yield more time variation in downside risk as compared to upside risk. Results from the model that includes a volatility factor in the conditional mean and thereby allows for asymmetries in conditional distributions are very similar. Our paper also extends the recent literature by formally evaluating the accuracy of tail risk forecasts and assessing the performance of Bayesian quantile regression, as well as our Bayesian VARs, in this context. Overall, the BVAR models perform comparably to quantile regression for estimating and forecasting tail risks, complementing BVARs’ established performance for forecasting and structural analysis.

Gibbs Sampling for Bayesian Prediction of SARMA Processes

In this article we present a Bayesian prediction of multiplicative seasonal autoregressive moving average (SARMA) processes using the Gibbs sampling algorithm. First, we estimate the unobserved errors using the nonlinear least squares (NLS) method to approximate the likelihood function. Second, we employ conjugate priors on the model parameters and initial values and assume the model errors are normally distributed to derive the conditional posterior and predictive distributions. In particular, we show that the conditional posterior distribution of the model parameters and the variance are multivariate normal and inverse gamma respectively, and the conditional predictive distribution of the future observations is a multivariate normal. Finally, we use these closed-form conditional posterior and predictive distributions to apply the Gibbs sampling algorithm to approximate empirically the marginal posterior and predictive distributions, enabling us easily to carry out multiple-step ahead predictions. We evaluate our proposed Bayesian method using simulation study and real-world time series datasets.

Hydrological post-processing based on approximate Bayesian computation (ABC)

This study introduces a method to quantify the conditional predictive uncertainty in hydrological post-processing contexts when it is cumbersome to calculate the likelihood (intractable likelihood). Sometimes, it can be difficult to calculate the likelihood itself in hydrological modelling, specially working with complex models or with ungauged catchments. Therefore, we propose the ABC post-processor that exchanges the requirement of calculating the likelihood function by the use of some sufficient summary statistics and synthetic datasets. The aim is to show that the conditional predictive distribution is qualitatively similar produced by the exact predictive (MCMC post-processor) or the approximate predictive (ABC post-processor). We also use MCMC post-processor as a benchmark to make results more comparable with the proposed method. We test the ABC post-processor in two scenarios: (1) the Aipe catchment with tropical climate and a spatially-lumped hydrological model (Colombia) and (2) the Oria catchment with oceanic climate and a spatially-distributed hydrological model (Spain). The main finding of the study is that the approximate (ABC post-processor) conditional predictive uncertainty is almost equivalent to the exact predictive (MCMC post-processor) in both scenarios.

Bayesian Analysis of Double Seasonal Autoregressive Models

In this paper we use the Gibbs sampling algorithm to present a Bayesian analysis to multiplicative double seasonal autoregressive (DSAR) models, considering both estimation and prediction problems. Assuming the model errors are normally distributed and using natural conjugate and g priors on the initial values and model parameters, we show that the conditional posterior distributions of the model parameters and variance are multivariate normal and inverse gamma respectively, and the conditional predictive distribution of the future observations is a multivariate normal. Using these closed-form conditional posterior and predictive distributions, we apply the Gibbs sampling to approximate empirically the marginal posterior and predictive distributions, enabling us easily to carry out multiple-step ahead predictions. The proposed Bayesian method is evaluated using simulation study and real-world time series dataset.

Time Series Simulation by Conditional Generative Adversarial Net

Generative Adversarial Net (GAN) has been proven to be a powerful machine learning tool in image data analysis and generation. In this paper, we propose to use Conditional Generative Adversarial Net (CGAN) to learn and simulate time series data. The conditions can be both categorical and continuous variables containing different kinds of auxiliary information. Our simulation studies show that CGAN is able to learn different kinds of normal and heavy tail distributions, as well as dependent structures of different time series and it can further generate conditional predictive distributions consistent with the training data distributions. We also provide an in-depth discussion on the rationale of GAN and the neural network as hierarchical splines to draw a clear connection with the existing statistical method for distribution generation. In practice, CGAN has a wide range of applications in the market risk and counterparty risk analysis: it can be applied to learn the historical data and generate scenarios for the calculation of Value-at-Risk (VaR) and Expected Shortfall (ES) and predict the movement of the market risk factors. We present a real data analysis including a backtesting to demonstrate CGAN is able to outperform the Historic Simulation, a popular method in market risk analysis for the calculation of VaR. CGAN can also be applied in the economic time series modeling and forecasting, and an example of hypothetical shock analysis for economic models and the generation of potential CCAR scenarios by CGAN is given at the end of the paper.

Extending Extended Logistic Regression: Extended versus Separate versus Ordered versus Censored

Abstract Extended logistic regression is a recent ensemble calibration method that extends logistic regression to provide full continuous probability distribution forecasts. It assumes conditional logistic distributions for the (transformed) predictand and fits these using selected predictand category probabilities. In this study extended logistic regression is compared to the closely related ordered and censored logistic regression models. Ordered logistic regression avoids the logistic distribution assumption but does not yield full probability distribution forecasts, whereas censored regression directly fits the full conditional predictive distributions. The performance of these and other ensemble postprocessing methods is tested on wind speed and precipitation data from several European locations and ensemble forecasts from the European Centre for Medium-Range Weather Forecasts (ECMWF). Ordered logistic regression performed similarly to extended logistic regression for probability forecasts of discrete categories whereas full predictive distributions were better predicted by censored regression.

Highway Accident Modeling and Forecasting in Winter

Environmental attributes are critical risk factors that have proven to affect collision rates. Associated driving risks can be reduced by better maintenance of roadway infrastructure, enforcement of speed limits or other traffic laws. Given the preventive nature of these policies and regulations, accurate predictions of environmental attributes are needed. Currently, most of road safety prediction models are based on deterministic weather forecasts which are not able to capture changes in the likelihood of collision occurrence. As a result, probabilistic forecast is required to improve decision making, mainly in winter. In this paper, a stochastic approach to modeling highway collisions in winter time is considered which enables better assessment of driving conditions and a more accurate prediction. A logistic regression model with covariates is applied to crash data where environmental characteristics are modeled as a finite state space homogeneous multivariate discrete time Markov chain. After fitting the model, weather prediction as well as the conditional predictive probability distribution of collision occurrence are obtained. As the application, the ability of the proposed model to predict hourly environmental attributes and collision occurrence is examined using real highway crash data. The performance of the developed stochastic model is compared with several existing models in the literature using actual collision data. The results demonstrate that the proposed stochastic model outperforms existing models and it accurately predicts collision occurrence in the presence of stochastically changeable winter weather conditions. As a result, the proposed probabilistic forecast model can be used as a valuable tool in a decision support system.

Prospective surveillance of multivariate spatial disease data

Surveillance systems are often focused on more than one disease within a predefined area. On those occasions when outbreaks of disease are likely to be correlated, the use of multivariate surveillance techniques integrating information from multiple diseases allows us to improve the sensitivity and timeliness of outbreak detection. In this article, we present an extension of the surveillance conditional predictive ordinate to monitor multivariate spatial disease data. The proposed surveillance technique, which is defined for each small area and time period as the conditional predictive distribution of those counts of disease higher than expected given the data observed up to the previous time period, alerts us to both small areas of increased disease incidence and the diseases causing the alarm within each area. We investigate its performance within the framework of Bayesian hierarchical Poisson models using a simulation study. An application to diseases of the respiratory system in South Carolina is finally presented.

A multiple imputation approach for clustered interval‐censored survival data

Multivariate interval-censored failure time data arise commonly in many studies of epidemiology and biomedicine. Analysis of these type of data is more challenging than the right-censored data. We propose a simple multiple imputation strategy to recover the order of occurrences based on the interval-censored event times using a conditional predictive distribution function derived from a parametric gamma random effects model. By imputing the interval-censored failure times, the estimation of the regression and dependence parameters in the context of a gamma frailty proportional hazards model using the well-developed EM algorithm is made possible. A robust estimator for the covariance matrix is suggested to adjust for the possible misspecification of the parametric baseline hazard function. The finite sample properties of the proposed method are investigated via simulation. The performance of the proposed method is highly satisfactory, whereas the computation burden is minimal. The proposed method is also applied to the diabetic retinopathy study (DRS) data for illustration purpose and the estimates are compared with those based on other existing methods for bivariate grouped survival data.

Estimating stochastic volatility models using daily returns and realized volatility simultaneously

Realized volatility, which is the sum of squared intraday returns over a certain interval such as a day, has recently attracted the attention of financial economists and econometricians as an accurate measure of the true volatility. In the real market, however, the presence of non-trading hours and market microstructure noise in transaction prices may cause bias in the realized volatility. On the other hand, daily returns are less subject to noise and therefore may provide additional information on the true volatility. From this point of view, modeling realized volatility and daily returns simultaneously based on the well-known stochastic volatility model is proposed. Empirical studies using intraday data of Tokyo stock price index show that this model can estimate realized volatility biases and parameters simultaneously. The Bayesian approach is taken and an efficient sampling algorithm is proposed to implement the Markov chain Monte Carlo method for our simultaneous model. The result of the model comparison between the simultaneous models using both naive and scaled realized volatilities indicates that the effect of non-trading hours is more essential than that of microstructure noise and that asymmetry is crucial in stochastic volatility models. The proposed Bayesian approach provides an estimate of the entire conditional predictive distribution of returns under consideration of the uncertainty in the estimation of both biases and parameters. Hence common risk measures, such as value-at-risk and expected shortfall, can be easily estimated.

A Monte Carlo Method for Estimating Prediction Limit for the Arithmetic Mean of Lognormal Sample

A Monte Carlo (MC) method is suggested for calculating an upper prediction limit for the mean of a future sample of small size N from a lognormal distribution. This is done by obtaining a Monte Carlo estimator of the limit utilizing the future sample generated from the Gibbs sampler. For the Gibbs sampler, a full conditional posterior predictive distribution of each observation in the future sample is derived. The MC method is straightforward to specify distributionally and to implement computationally, with output readily adapted for required inference summaries. In an example, practical application of the method is described.

MCMC methods to approximate conditional predictive distributions

Sampling from conditional distributions is a problem often encountered in statistics when inferences are based on conditional distributions which are not of closed-form. Several Markov chain Monte Carlo (MCMC) algorithms to simulate from them are proposed. Potential problems are pointed out and some suitable modifications are suggested. Approximations based on conditioning sets are also explored. The issues are illustrated within a specific statistical tool for Bayesian model checking, and compared in an example. An example in frequentist conditional testing is also given.