Revisiting Akaike’s Final Prediction Error and the Generalized Cross Validation Criteria in Regression from the Same Perspective: From Least Squares to Ridge Regression and Smoothing Splines

In regression, despite being both aimed at estimating the Mean Squared Prediction Error (MSPE), Akaike’s Final Prediction Error (FPE) and the Generalized Cross Validation (GCV) selection criteria are usually derived from two quite different perspectives. Here, settling on the most commonly accepted definition of the MSPE as the expectation of the squared prediction error loss, we provide theoretical expressions for it, valid for any linear model (LM) fitter, be it under random or non random designs. Specializing these MSPE expressions for each of them, we are able to derive closed formulas of the MSPE for some of the most popular LM fitters: Ordinary Least Squares (OLS), with or without a full column rank design matrix;Ordinary and Generalized Ridge regression, the latter embedding smoothing splines fitting. For each of these LM fitters, we then deduce a computable estimate of the MSPE which turns out to coincide with Akaike’s FPE. Using a slight variation, we similarly get a class of MSPE estimates coinciding with the classical GCV formula for those same LM fitters.

Extended Range(10–30 Days) Heavy Rain Forecasting Study Based on a Nonlinear Cross-Prediction Error Model

Extended range （10-30 d） heavy rain forecasting is difficult but performs an important function in disaster prevention and mitigation. In this paper, a nonlinear cross prediction error （NCPE） algorithm that combines nonlinear dynamics and statistical methods is proposed. The method is based on phase space reconstruction of chaotic single-variable time series of precipitable water and is tested in 100 global cases of heavy rain. First, nonlinear relative dynamic error for local attractor pairs is calculated at different stages of the heavy rain process, after which the local change characteristics of the attractors are analyzed. Second, the eigen-peak is defined as a prediction indicator based on an error threshold of about 1.5, and is then used to analyze the forecasting validity period. The results reveal that the prediction indicator features regarded as eigenpeaks for heavy rain extreme weather are all reflected consistently, without failure, based on the NCPE model; the prediction validity periods for 1-2 d, 3-9 d and 10-30 d are 4, 22 and 74 cases, respectively, without false alarm or omission. The NCPE model developed allows accurate forecasting of heavy rain over an extended range of 10-30 d and has the potential to be used to explore the mechanisms involved in the development of heavy rain according to a segmentation scale. This novel method provides new insights into extended range forecasting and atmospheric predictability, and also allows the creation of multi-variable chaotic extreme weather prediction models based on high spatiotemporal resolution data.

Impact of observational MJO forcing on ENSO predictability in the Zebiak-Cane model: PartⅠ.Effect on the maximum prediction error

With the observational wind data and the Zebiak-Cane model, the impact of Madden-Iulian Oscillation （MJO） as external forcing on El Nino-Southern Oscillation （ENSO） predictability is studied. The observational data are analyzed with Continuous Wavelet Transform （CWT） and then used to extract MJO signals, which are added into the model to get a new model. After the Conditional Nonlinear Optimal Perturbation （CNOP） method has been used, the initial errors which can evolve into maximum prediction error, model errors and their join errors are gained and then the Nifio 3 indices and spatial structures of three kinds of errors are investigated. The results mainly show that the observational MJO has little impact on the maximum prediction error of ENSO events and the initial error affects much greater than model error caused by MJO forcing. These demonstrate that the initial error might be the main error source that produces uncertainty in ENSO prediction, which could provide a theoretical foundation for the adaptive data assimilation of the ENSO forecast and contribute to the ENSO target observation.

Robust Beamforming Under Channel Prediction Errors for Time-Varying MIMO System

The accuracy of acquired channel state information(CSI)for beamforming design is essential for achievable performance in multiple-input multiple-output(MIMO)systems.However,in a high-speed moving scene with time-division duplex(TDD)mode,the acquired CSI depending on the channel reciprocity is inevitably outdated,leading to outdated beamforming design and then performance degradation.In this paper,a robust beamforming design under channel prediction errors is proposed for a time-varying MIMO system to combat the degradation further,based on the channel prediction technique.Specifically,the statistical characteristics of historical channel prediction errors are exploited and modeled.Moreover,to deal with random error terms,deterministic equivalents are adopted to further explore potential beamforming gain through the statistical information and ultimately derive the robust design aiming at maximizing weighted sum-rate performance.Simulation results show that the proposed beamforming design can maintain outperformance during the downlink transmission time even when channels vary fast,compared with the traditional beamforming design.

Investigate Targeted Factors to Achieve Prediction Goal in Stroke Convalescence in Terms of Causal Relationships of Prediction Error

Background and Purpose: To investigate target functional independence measure (FIM) items to achieve the prediction goal in terms of the causal relationships between prognostic prediction error and FIM among stroke patients in the convalescent phase using the structural equation modeling (SEM) analysis. Methods: A total of 2992 stroke patients registered in the Japanese Rehabilitation Database were analyzed retrospectively. The prediction error was calculated based on a prognostic prediction formula proposed in a previous study. An exploratory factor analysis (EFA) then the factor was determined using confirmatory factorial analysis (CFA). Finally, multivariate analyses were performed using SEM analysis. Results: The fitted indices of the hypothesized model estimated based on EFA were confirmed by CFA. The factors estimated by EFA were applied, and interpreted as follows: “Transferring (T-factor),” “Dressing (D-factor),” and “Cognitive function (C-factor).” The fit of the structural model based on the three factors and prediction errors was supported by the SEM analysis. The effects of the D- and C-factors yielded similar causal relationships on prediction error. Meanwhile, the effects between the prediction error and the T-factor were low. Observed FIM items were related to their domains in the structural model, except for the dressing of the upper body and memory (p < 0.01). Conclusions: Transfer, which was not heavily considered in the previous prediction formula, was found in causal relationships with prediction error. It is suggested to intervene to transfer together with positive factors to recovery for achieving the prediction goal.

A Meaningful Image Encryption Algorithm Based on Prediction Error and Wavelet Transform

Image encryption(IE)is a very useful and popular technology to protect the privacy of users.Most algorithms usually encrypt the original image into an image similar to texture or noise,but texture and noise are an obvious visual indication that the image has been encrypted,which is more likely to cause the attacks of enemy.To overcome this shortcoming,many image encryption systems,which convert the original image into a carrier image with visual significance have been proposed.However,the generated cryptographic image still has texture features.In line with the idea of improving the visual quality of the final password images,we proposed a meaningful image hiding algorithm based on prediction error and discrete wavelet transform.Lots of experimental results and safety analysis show that the proposed algorithm can achieve high visual quality and ensure the security at the same time.

Characterizing prediction errors of a new tree height model for cut-to-length Pinus radiata stems through the Burr TypeⅫdistribution

Unlike height-diameter equations for standing trees commonly used in forest resources modelling,tree height models for cut-to-length(CTL)stems tend to produce prediction errors whose distributions are not conditionally normal but are rather leptokurtic and heavy-tailed.This feature was merely noticed in previous studies but never thoroughly investigated.This study characterized the prediction error distribution of a newly developed such tree height model for Pin us radiata(D.Don)through the three-parameter Burr TypeⅫ(BⅫ)distribution.The model’s prediction errors(ε)exhibited heteroskedasticity conditional mainly on the small end relative diameter of the top log and also on DBH to a minor extent.Structured serial correlations were also present in the data.A total of 14 candidate weighting functions were compared to select the best two for weightingεin order to reduce its conditional heteroskedasticity.The weighted prediction errors(εw)were shifted by a constant to the positive range supported by the BXII distribution.Then the distribution of weighted and shifted prediction errors(εw+)was characterized by the BⅫdistribution using maximum likelihood estimation through 1000 times of repeated random sampling,fitting and goodness-of-fit testing,each time by randomly taking only one observation from each tree to circumvent the potential adverse impact of serial correlation in the data on parameter estimation and inferences.The nonparametric two sample Kolmogorov-Smirnov(KS)goodness-of-fit test and its closely related Kuiper’s(KU)test showed the fitted BⅫdistributions provided a good fit to the highly leptokurtic and heavy-tailed distribution ofε.Random samples generated from the fitted BⅫdistributions ofεw+derived from using the best two weighting functions,when back-shifted and unweighted,exhibited distributions that were,in about97 and 95%of the 1000 cases respectively,not statistically different from the distribution ofε.Our results for cut-tolength P.radiata stems represented the first case of any tree species where a non-normal error distribution in tree height prediction was described by an underlying probability distribution.The fitted BXII prediction error distribution will help to unlock the full potential of the new tree height model in forest resources modelling of P.radiata plantations,particularly when uncertainty assessments,statistical inferences and error propagations are needed in research and practical applications through harvester data analytics.

Generating Synthetic Data to Reduce Prediction Error of Energy Consumption

Renewable and nonrenewable energy sources are widely incorporated for solar and wind energy that produces electricity without increasing carbon dioxide emissions.Energy industries worldwide are trying hard to predict future energy consumption that could eliminate over or under contracting energy resources and unnecessary financing.Machine learning techniques for predicting energy are the trending solution to overcome the challenges faced by energy companies.The basic need for machine learning algorithms to be trained for accurate prediction requires a considerable amount of data.Another critical factor is balancing the data for enhanced prediction.Data Augmentation is a technique used for increasing the data available for training.Synthetic data are the generation of new data which can be trained to improve the accuracy of prediction models.In this paper,we propose a model that takes time series energy consumption data as input,pre-processes the data,and then uses multiple augmentation techniques and generative adversarial networks to generate synthetic data which when combined with the original data,reduces energy consumption prediction error.We propose TGAN-skip-Improved-WGAN-GP to generate synthetic energy consumption time series tabular data.We modify TGANwith skip connections,then improveWGANGPby defining a consistency term,and finally use the architecture of improved WGAN-GP for training TGAN-skip.We used various evaluation metrics and visual representation to compare the performance of our proposed model.We also measured prediction accuracy along with mean and maximum error generated while predicting with different variations of augmented and synthetic data with original data.The mode collapse problemcould be handled by TGAN-skip-Improved-WGAN-GP model and it also converged faster than existing GAN models for synthetic data generation.The experiment result shows that our proposed technique of combining synthetic data with original data could significantly reduce the prediction error rate and increase the prediction accuracy of energy consumption.

Adaptive Hammerstein Predistorter Using the Recursive Prediction Error Method

The digital baseband predistorter is an effective technique to compensate for the nonlinearity of power amplifiers （PAs） with memory effects. However, most available adaptive predistorters based on direct learning architectures suffer from slow convergence speeds. In this paper, the recursive prediction error method is used to construct an adaptive Hammerstein predistorter based on the direct learning architecture, which is used to linearize the Wiener PA model. The effectiveness of the scheme is demonstrated on a digital video broadcasting-terrestrial system. Simulation results show that the predistorter outperforms previous predistorters based on direct learning architectures in terms of convergence speed and linearization. A similar algorithm can be applied to estimate the Wiener PA model, which will achieve high model accuracy.

What Kind of Initial Errors Cause the Severest Prediction Uncertainty of E1 Nino in Zebiak-Cane Model

With the Zebiak-Cane （ZC） model, the initial error that has the largest effect on ENSO prediction is explored by conditional nonlinear optimal perturbation （CNOP）. The results demonstrate that CNOP-type errors cause the largest prediction error of ENSO in the ZC model. By analyzing the behavior of CNOPtype errors, we find that for the normal states and the relatively weak E1 Nifio events in the ZC model, the predictions tend to yield false alarms due to the uncertainties caused by CNOP. For the relatively strong E1 Nino events, the ZC model largely underestimates their intensities. Also, our results suggest that the error growth of E1 Nifio in the ZC model depends on the phases of both the annual cycle and ENSO. The condition during northern spring and summer is most favorable for the error growth. The ENSO prediction bestriding these two seasons may be the most difficult. A linear singular vector （LSV） approach is also used to estimate the error growth of ENSO, but it underestimates the prediction uncertainties of ENSO in the ZC model. This result indicates that the different initial errors cause different amplitudes of prediction errors though they have same magnitudes. CNOP yields the severest prediction uncertainty. That is to say, the prediction skill of ENSO is closely related to the types of initial error. This finding illustrates a theoretical basis of data assimilation. It is expected that a data assimilation method can filter the initial errors related to CNOP and improve the ENSO forecast skill.

Timing Prediction Error Volatility and Dynamic Asset Allocation

We solve a portfolio selection,problem in which,return predictability,risk predictability and transaction cost are incorporated.In the problem,both expected return,prediction error volatility,and transaction cost are time-varying.Our optimal strategy suggests trading partially toward a dynamic aim portfolio,which is a weighted average of expected future tangency portfolio and is highly influenced by the common fluctuation of prediction error volatility(CPE).When CPE is high,the investor would invest less and trade less frequently to avoid risk and transaction cost.Moreover,the investor trades more closely to the aim portfolio with a more persistent CPE signal.We also conduct an empirical analysis based on the commodities futures in Chinese market.The results reveal that by timing prediction error volatility,our strategy outperforms alternative strategies.

Reversible data hiding based on histogram and prediction error for sharing secret data

With the advancement of communication technology,a large number of data are constantly transmitted through the internet for various purposes,which are prone to be illegally accessed by third parties.Therefore,securing such data is crucial to protect the transmitted information from falling into the wrong hands.Among data protection schemes,Secret Image Sharing is one of the most popular methods.It protects critical messages or data by embedding them in an image and sharing it with some users.Furthermore,it combines the security concepts in that private data are embedded into a cover image and then secured using the secret-sharing method.Despite its advantages,this method may produce noise,making the resulting stego file much different from its cover.Moreover,the size of private data that can be embedded is limited.This research works on these problems by utilizing prediction-error expansion and histogram-based approaches to embed the data.To recover the cover image,the SS method based on the Chinese remainder theorem is used.The experimental results indicate that this proposed method performs better than similar methods in several cover images and scenarios.

Errors Prediction for Vector-to-Raster Conversion Based on Map Load and Cell Size

Vector-to-raster conversion is a process accompanied with errors.The errors are classified into predicted errors before rasterization and actual errors after that.Accurate prediction of the errors is beneficial to developing reasonable rasterization technical schemes and to making products of high quality.Analyzing and establishing a quantitative relationship between the error and its affecting factors is the key to error prediction.In this study,land cover data of China at a scale of 1:250 000 were taken as an example for analyzing the relationship between rasterization errors and the density of arc length(DA),the density of polygon(DP) and the size of grid cells(SG).Significant correlations were found between the errors and DA,DP and SG.The correlation coefficient(R2) of a model established based on samples collected in a small region(Beijing) reaches 0.95,and the value of R2 is equal to 0.91 while the model was validated with samples from the whole nation.On the other hand,the R2 of a model established based on nationwide samples reaches 0.96,and R2 is equal to 0.91 while it was validated with the samples in Beijing.These models depict well the relationships between rasterization errors and their affecting factors(DA,DP and SG).The analyzing method established in this study can be applied to effectively predicting rasterization errors in other cases as well.

ENSO Predictions in an Intermediate Coupled Model Influenced by Removing Initial Condition Errors in Sensitive Areas： A Target Observation Perspective

Previous studies indicate that ENSO predictions are particularly sensitive to the initial conditions in some key areas（socalled ＂sensitive areas＂）. And yet, few studies have quantified improvements in prediction skill in the context of an optimal observing system. In this study, the impact on prediction skill is explored using an intermediate coupled model in which errors in initial conditions formed to make ENSO predictions are removed in certain areas. Based on ideal observing system simulation experiments, the importance of various observational networks on improvement of El Ni n？o prediction skill is examined. The results indicate that the initial states in the central and eastern equatorial Pacific are important to improve El Ni n？o prediction skill effectively. When removing the initial condition errors in the central equatorial Pacific, ENSO prediction errors can be reduced by 25%. Furthermore, combinations of various subregions are considered to demonstrate the efficiency on ENSO prediction skill. Particularly, seasonally varying observational networks are suggested to improve the prediction skill more effectively. For example, in addition to observing in the central equatorial Pacific and its north throughout the year,increasing observations in the eastern equatorial Pacific during April to October is crucially important, which can improve the prediction accuracy by 62%. These results also demonstrate the effectiveness of the conditional nonlinear optimal perturbation approach on detecting sensitive areas for target observations.

Stock Price Prediction Using Predictive Error Compensation Wavelet Neural Networks

:Machine Learning(ML)algorithms have been widely used for financial time series prediction and trading through bots.In this work,we propose a Predictive Error Compensated Wavelet Neural Network(PEC-WNN)ML model that improves the prediction of next day closing prices.In the proposed model we use multiple neural networks where the first one uses the closing stock prices from multiple-scale time-domain inputs.An additional network is used for error estimation to compensate and reduce the prediction error of the main network instead of using recurrence.The performance of the proposed model is evaluated using six different stock data samples in the New York stock exchange.The results have demonstrated significant improvement in forecasting accuracy in all cases when the second network is used in accordance with the first one by adding the outputs.The RMSE error is 33%improved when the proposed PEC-WNN model is used compared to the Long ShortTerm Memory(LSTM)model.Furthermore,through the analysis of training mechanisms,we found that using the updated training the performance of the proposed model is improved.The contribution of this study is the applicability of simultaneously different time frames as inputs.Cascading the predictive error compensation not only reduces the error rate but also helps in avoiding overfitting problems.

Ship motion extreme short time prediction of ship pitch based on diagonal recurrent neural network

A DRNN (diagonal recurrent neural network) and its RPE (recurrent prediction error) learning algorithm are proposed in this paper .Using of the simple structure of DRNN can reduce the capacity of calculation. The principle of RPE learning algorithm is to adjust weights along the direction of Gauss-Newton. Meanwhile, it is unnecessary to calculate the second local derivative and the inverse matrixes, whose unbiasedness is proved. With application to the extremely short time prediction of large ship pitch, satisfactory results are obtained. Prediction effect of this algorithm is compared with that of auto-regression and periodical diagram method, and comparison results show that the proposed algorithm is feasible.

The“Spring Predictability Barrier” Phenomenon of ENSO Predictions Generated with the FGOALS-g Model

Using the sea surface temperature （SST） predicted for the equatorial Pacific Ocean by the Flexible Global Ocean-Atmosphere-Land System Model-gamil （FGOALS-g）, an analysis of the prediction errors was performed for the seasonally dependent predictability of SST anomalies both for neutral years and for the growth/decay phase of El Nino/La Nina events. The study results indicated that for the SST predictions relating to the growth phase and the decay phase of El Nino events, the prediction errors have a seasonally dependent evolution. The largest increase in errors occurred in the spring season, which indicates that a prominent spring predictability barrier （SPB） occurs during an El Nino-Southern Oscillation （ENSO） warming episode. Furthermore, the SPB associated with the growth-phase prediction is more prominent than that associated with the decay-phase prediction. However, for the neutral years and for the growth and decay phases of La Nifia events, the SPB phenomenon was less prominent. These results indicate that the SPB phenomenon depends extensively on the ENSO events themselves. In particular, the SPB depends on the phases of the ENSO events. These results may provide useful knowledge for improving ENSO forecasting.

Reversible Data Hiding in Encrypted Images Based on Prediction and Adaptive Classification Scrambling

Reversible data hiding in encrypted images(RDH-EI)technology is widely used in cloud storage for image privacy protection.In order to improve the embedding capacity of the RDH-EI algorithm and the security of the encrypted images,we proposed a reversible data hiding algorithm for encrypted images based on prediction and adaptive classification scrambling.First,the prediction error image is obtained by a novel prediction method before encryption.Then,the image pixel values are divided into two categories by the threshold range,which is selected adaptively according to the image content.Multiple high-significant bits of pixels within the threshold range are used for embedding data and pixel values outside the threshold range remain unchanged.The optimal threshold selected adaptively ensures the maximum embedding capacity of the algorithm.Moreover,the security of encrypted images can be improved by the combination of XOR encryption and classification scrambling encryption since the embedded data is independent of the pixel position.Experiment results demonstrate that the proposed method has higher embedding capacity compared with the current state-of-the-art methods for images with different texture complexity.

Additive predictions of aboveground stand biomass in commercial logs and harvest residues for rotation age Pinus radiata plantations in New South Wales,Australia

Two systems of additive equations were developed to predict aboveground stand level biomass in log products and harvest residue from routinely measured or predicted stand variables for Pinus radiata plantations in New South Wales,Australia.These plantations were managed under three thinning regimes or stand types before clear-felling at rotation age by cut-to-length harvesters to produce sawlogs and pulpwood.The residue material following a clear-fell operation mainly consisted of stumps,branches and treetops,short off-cut and waste sections due to stem deformity,defects,damage and breakage.One system of equations did not include dummy variables for stand types in the model specification and was intended for more general use in plantations where stand density management regimes were not the same as the stand types in our study.The other system that incorporated dummy variables was for stand type-specific applications.Both systems of equations were estimated using 61 plot-based estimates of biomass in commercial logs and residue components that were derived from systems of equations developed in situ for predicting the product and residue biomass of individual trees.To cater for all practical applications,two sets of parameters were estimated for each system of equations for predicting component and total aboveground stand biomass in fresh and dry weight respectively.The two sets of parameters for the system of equations without dummy variables were jointly estimated to improve statistical efficiency in parameter estimation.The predictive performances of the two systems of equations were benchmarked through a leave-one-plot-out cross validation procedure.They were generally superior to the performance of an alternative two-stage approach that combined an additive system for major components with an allocative system for sub-components.As using forest harvest residue biomass for bioenergy has increasingly become an integrated part of forestry,reliable estimates of product and residue biomass will assist harvest and management planning for clear-fell operations that integrate cut-to-length log production with residue harvesting.

Error Compensation of Thin Plate-shape Part with Prebending Method in Face Milling

Low weight and good toughness thin plate parts are widely used in modem industry, but its flexibility seriously impacts the machinability. Plenty of studies locus on the influence of machine tool and cutting tool on the machining errors. However, few researches focus on compensating machining errors through the fixture. In order to improve the machining accuracy of thin plate-shape part in face milling, this paper presents a novel method for compensating the surfacc errors by prebending the workpiece during the milling process. First, a machining error prediction model using finite element method is formulated, which simplifies the contacts between the workpiece and fixture with spring constraints. Milling fbrces calculated by the micro-unit cutting force model arc loaded on the error prediction model to predict the machining error. The error prediction results are substituted into the given formulas to obtain the prebending clamping forces and clamping positions. Consequently, the workpiece is prebent in terms of the calculated clamping forces and positions during the face milling operation to reduce the machining error. Finally, simulation and experimental tests are carried out to validate the correctness and efficiency of the proposed error compensation method. The experimental measured flatness results show that the flatness improves by approximately 30 percent through this error compensation method. The proposed mcthod not only predicts the machining errors in face milling thin plate-shape parts but also reduces the machining errors by taking full advantage of the workpiece prebending caused by fixture, meanwhile, it provides a novel idea and theoretical basis for reducing milling errors and improving the milling accuracy.