atmosphere Article
Application of Intelligent Dynamic Bayesian Network with Wavelet Analysis for Probabilistic Prediction of Storm Track Intensity Index Ming Li and Kefeng Liu * College of Meteorology and Oceanography, National University of Defense Technology, Nanjing 211101, China;
[email protected] * Correspondence:
[email protected]; Tel.: +86-151-6146-1535 Received: 3 May 2018; Accepted: 8 June 2018; Published: 11 June 2018
Abstract: The effective prediction of storm track (ST) is greatly beneficial for analyzing the development and anomalies of mid-latitude weather systems. For the non-stationarity, nonlinearity, and uncertainty of ST intensity index (STII), a new probabilistic prediction model was proposed based on dynamic Bayesian network (DBN) and wavelet analysis (WA). We introduced probability theory and graph theory for the first time to quantitatively describe the nonlinear relationship and uncertain interaction of the ST system. Then a casual prediction network (i.e., DBN) was constructed through wavelet decomposition, structural learning, parameter learning, and probabilistic inference, which was used for expression of relation among predictors and probabilistic prediction of STII. The intensity prediction of the North Pacific ST with data from 1961–2010 showed that the new model was able to give more comprehensive prediction information and higher prediction accuracy and had strong generalization ability and good stability. Keywords: storm track; probabilistic prediction; dynamic Bayesian network; wavelet analysis
1. Introduction One of the primary features of mid- and high-latitude atmospheric circulation (AC) is transient variability, which is closely related to the growth and decay of daily weather systems. In the 1970s, Blackmon [1] found sub-weekly (2.5–6 days) transient eddies over the North Pacific and North Atlantic with filtering data. He defined the two zonal-extended regions with the most intensive transient variability as “storm track” (ST), which can be divided, respectively, into North Pacific ST (NPST) and North Atlantic ST (NAST). ST corresponds significantly with cyclone and anticyclone activities, which can be the indication of the development of weather systems. Moreover, as a contacting link of heat and kinetic energy between ocean and atmosphere, ST plays an important role in the maintenance of AC and climate change [2]. ST is crucial to the short-term anomaly of AC with interactions between ST and low-frequency circulation. So far, many studies have revealed the interaction. Lau [3] studied the seasonal variation of ST and pointed out that the main mode of the variation was related to the teleconnection pattern of the low-frequency circulation in the northern hemisphere. Straus et al. [4] discovered that the ST anomaly was closely related to the sea surface temperature (SST) anomaly in the Kuroshio area. Zhu et al. [5] summarized the correlation between the winter NPST and the Pacific-North America teleconnection pattern (PNA) and Western Pacific teleconnection pattern (WP). Ren et al. [6] used the empirical orthogonal function (EOF) to analyze the temporal and spatial variability of the winter NPST and explained its coupled pattern with the mid-latitude air-sea system. Liu et al. [7] determined the correlation and potential influencing mechanisms between the Polar vortex intensity and NPST.
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Both observational research and theoretical studies have indicated the symbiotic relationship between ST and large-scale AC in the Northern Hemisphere. However, most studies are just diagnostic analysis about ST variability and correlation. To grasp the evolvement role of ST, prediction is becoming an urgent area of research. However, ST is a highly nonlinear system due to nonlinear processes in the air-sea system. There is relatively little research on the numerical forecasting or statistical forecasting of ST both at home and abroad. That may result from the diversity of influencing factors and the complexity of correlation mechanisms. In addition, strong transients and uncertain rules have also caused difficulties in ST prediction. In meteorological prediction, climate indexes are often used as predictands and predictors to explain the behavior of future climate. Therefore, how to quantify the intensity and spatial-temporal variation as indexes is the premise of ST prediction. At present, there are several indexes that can indicate the possible evolution of ST, whose calculation methods with filtering variance includes the central point representation [8,9], regional average [10], and EOF [11]. The above studies achieve the quantitative description of the nonlinear ST system by establishing an index. Thus, we can predict the temporal and spatial variation of ST with the ST index. The prediction of ST index belongs to the prediction of nonlinear time-series. In the field of meteorology and oceanology, data-driven models (i.e., statistical models) are suitable predicting tools due to their rapid development times, as well as low information requirements compared to physical-based models. Hong et al. [12] introduced the inversion idea and used genetic algorithm to reconstruct the nonlinear forecasting model of the subtropical high index from historical data. Liu et al. [13] integrated the EOF, wavelet decomposition and support vector machine (SVM) method to predict the 500 hPa geopotential height in summer. Zhu et al. [14] conducted a short-term forecast experiment of the tropical atmospheric seasonal oscillation (MJO) index, using both the singular spectrum analysis and auto-regression model. Jia et al. [15] applied the correlation analysis and optimal subset regression to select predictors and established a statistical prediction model for the subtropical high index. The above statistical methods require a large amount of historical data, but their efficiency on processing big data is low. Most importantly, these methods have weak ability to mine and express the internal relations from data quantitatively. Therefore, the above models are still flawed for prediction of ST index. With the rapid development of computer technology and information acquisition technology, machine learning (ML) and data mining (DM) have opened a new era—artificial intelligence. Breakthroughs have been made by the application of ML and DM in the fields of biology, finance, and medicine [16–18], and they have also brought opportunities for the development of predicting technology in meteorology and oceanology. Many scholars have applied ML and DM to meteorological prediction: Yang et al. [19] used the association rules mining to analyze the data set of North Atlantic hurricane history trace and predicted the intensity of the North Atlantic hurricane based on the mining results. Royston et al. [20] applied the semantic decision tree to conduct regular mining and forecast modeling with water level and meteorological data, to forecast the storm surge of Thames Estuary. Gordon et al. [21] constructed a meteorological prediction model using neural network (NN) and frequency domain algorithm to implement 24-hour refined prediction. Teng [22] extracted highly relevant factors and used the stepwise regression and SVM to establish the medium-term prediction model of the tropical cyclone path in the Western Pacific. To a certain extent, ML and DM can overcome the shortcomings of the above statistical methods and achieve data mining and reasoning with rapid development times. However, the above ML algorithms are all deterministic methods, that is, give a certain value for a certain predicting moment. Please note that ST is affected by the nonlinear action of various weather systems and has strong uncertainties. When the intensity and position of ST fluctuate greatly, deterministic single-point prediction may not achieve the desired accuracy. In contrast, the probabilistic prediction method could give the result in the form of probability distribution, covering more complete prediction information.
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As a new branch of ML theory, Bayesian network (BN) makes it feasible for the probabilistic prediction of ST index, which has been initially used in the field of meteorology and hydrology [23,24]. The emerging dynamic Bayesian Network (DBN) adds time information to the classical BN, which becomes a new probabilistic expression and reasoning tool owing to the ability to deal with uncertainties. Correspondingly, ST is affected by many factors in the mid-latitude air-sea system. There are random and non-linear interactions between these factors at same and different time. The features coincide exactly with the DBN, thus DBN is a powerful theoretical tool for probabilistic prediction of ST index. Additionally, note that time-series of the ST index is non-stationary. This limitation with non-stationary data has led to the recent formation of hybrid models, where data is preprocessed for non-stationary characteristics and then run through a predicting method such as ML algorithms to cope with the nonlinearity. Wavelet analysis (WA), an effective tool to deal with non-stationary data, has recently been applied to meteorological forecast. We will combine WA with DBN to achieve scientific and accurate prediction of ST index. In this paper, we constructed the WA-DBN model to predict the winter PST intensity index. To deal with the non-stationarity, nonlinearity, and uncertainty, we introduced DBN theory innovatively and combined WA to establish a data-driven model for predicting the monthly STII using large-scale climate indexes as the predictors. We first selected the climate indexes significantly related to ST as predictors. Then based on wavelet decomposition, a WA-DBN probabilistic prediction model was constructed through structure learning, parameter learning and probabilistic reasoning. Finally, a deeper comparative analysis of model performance is conducted with key statistical indicators. 2. Theory and Method 2.1. Definition of Storm Track Intensity Index To quantitatively describe the intensity of ST and its spatial-temporal variation, we refer to the existing definition method to calculate the intensity index. We identify the ST as the sub-weekly transient of the 500 hPa geopotential height. First sub-weekly transient eddies are derived from the geopotential height based on 31 symmetrical digital filter [25]. Then we calculate the monthly average band-pass filtering variance, selecting all grid points with the filtering variance greater than a certain fixed threshold, of which the mean value is defined as the ST intensity index (STII). The fixed threshold is usually taken as 20 dagpm2 . 2.2. Dynamic Bayesian Network Theory Bayesian Network (BN) was proposed by Judea Pearl in 1988, including the static Bayesian Network (SBN) and the dynamic Bayesian Network (DBN). Based on probability theory and graph theory, DBN integrates the time dimension into SBN to represent the temporal correlation, which forms a dynamic reasoning model with dynamic analysis and prediction of temporal information [26]. According to Bayesian theory, BN is a directed acyclic graph expressing the probabilistic relation between variables. It is mainly composed of nodes, directed arcs, and conditional probability distribution (CPT). DBN is an extension of SBN in the time dimension, and can be explained by a bigram < B0 , B→ >:
• •
B0 denotes the initial network, that is the SBN of each time slice. It contains the network structure and probability distribution of nodes at the same time; B→ denotes the transition network. It contains the causal link and the transition probability distribution of nodes in different time slice.
Define a variable set X = [ X1 , X2 , · · · , Xn ] and a finite time segment [0, 1, · · · , T ], then the joint probability distribution of X 0 , . . . , X T is T N P( X 0 , . . . , X T ) = P( X 0 ) · ∏ ∏ P Xit π Xit . t =1 i =1
(1)
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Atmosphere 9, x FOR where X t2018, denotes thePEER nodeREVIEW i located in t time slice, π
4 of 19 Xit denotes the parent of Xit . Formula (1) denotes the probabilistic reasoning of different time slices and different node states. The Theconstruction constructionof ofDBN DBNincludes includesstructure structure learning learningand and parameter parameter learning: learning: the theformer formerneeds needs 0 to construct and ; the latter needs to determine the initial probability ( ), the → ; the latter needs to determine the initial probability P ( X ), the observation to construct B0 and B→ observation conditional conditional probability probability P [Xit |π( Xit)], ,and andthe thetransition transitionconditional conditional probability probability P((X t | X t−1).) .There Thereare are two common learning technologies for DBN: manual construction based on expert knowledge and two common learning technologies for DBN: manual construction based on expert knowledge and automatic automaticlearning learningbased basedon onintelligent intelligentalgorithms algorithms[27]. [27].We Weadopt adoptaa combination combinationof of subjective subjectiveand and objective methods for DBN learning. Expert knowledge is used for structural learning while objective objective methods for DBN learning. Expert knowledge is used for structural learning while objective data dataisisused usedfor forparameter parameterlearning. learning. i
2.3. 2.3.Wavelet WaveletAnalysis Analysis Wavelet Wavelet analysis analysis (WA) (WA) is is aa mathematical mathematical function function that that can can be be used usedfor forthe theanalysis analysisof oftime-series time-series that thatcontain containnon-stationarities non-stationarities[28]. [28].WA WAof ofthe theinput inputvariables variablescan cananalyze analyzevarious varioussimilarities similaritieswithin within the the dataset dataset by by decomposing decomposing data data into into different different levels. levels. Large-scale Large-scale frequencies frequencies are are checked checked with with approximation series,while while small-scale frequencies are checked levels of approximation series, small-scale frequencies are checked by details by (4–5 details levels of(4–5 decomposition). decomposition). Waveletgives decomposition givesrepresentation time frequency of a temporal signal at different Wavelet decomposition time frequency of arepresentation signal at different domains, temporal domains, providing considerable information about the physical structure of the data. providing considerable information about the physical structure of the data. Wavelet reconstruction Wavelet reconstruction can frequency synthesizesignals the different frequency signals to achieve can synthesize the different to achieve information integration. Theinformation application integration. The application of WA in ismeteorology and oceanology is relatively mature [29,30], of WA in meteorology and oceanology relatively mature [29,30], therefore this paper will not give therefore this paper will not give unnecessary details. unnecessary details. ProbabilisticPrediction PredictionModel ModelBased Basedon onWA WA and and DBN DBN 3.3.Probabilistic TheSTII STIIprediction predictionmodel modelbased basedon onWA WAand andDBN DBN(WA-DBN) (WA-DBN)was wasproposed proposedfor fortwo twoproblems: problems: The First,the thetime-series time-seriesof ofintensity intensityindex indexisisnonlinear nonlineardue dueto tostrong strongtransient; transient;second, second,both bothST STintensity intensity First, andpredictors predictorscontain containprobabilistic probabilistic uncertainties. Figure 1 displays the technical structure of and uncertainties. Figure 1 displays the technical structure of WAWA-DBN model. DBN model.
Figure 1. Technical structure of WA-DBN probabilistic prediction model. Figure 1. Technical structure of WA-DBN probabilistic prediction model.
Seen from Figure 1, the WA-DBN prediction model includes two modules: WA module and from Figure 1, WA the WA-DBN model includes two modules:ofWA module and time DBN DBN Seen prediction module. is used forprediction the decomposition and reconstruction non-stationary prediction is used for theprediction decomposition andstructure reconstruction of non-stationary time series. series. DBNmodule. is used WA for probabilistic through learning, parameter learning and DBN is used for probabilistic prediction through structure learning, parameter learning and inference inference calculation, which is the core of this prediction model. calculation, which is the core of this prediction model. 3.1. Structural Learning The DBN structure describing the casual relation between various weather systems and the STII is the basis of intensity index prediction. The structural learning includes the selection of node variables and the determination of the dependencies among nodes. We adopt an expert-constructed method for structural learning. Based on professional knowledge, the predictors are selected as child
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3.1. Structural Learning The DBN structure describing the casual relation between various weather systems and the STII is the basis of intensity index prediction. The structural learning includes the selection of node variables and the determination of the dependencies among nodes. We adopt an expert-constructed method Atmosphere 2018, 9, x FOR PEER REVIEW 5 of for 19 structural learning. Based on professional knowledge, the predictors are selected as child nodes of the DBN and a causal is constructed, the initial networkthe andinitial the transition nodes of the DBNtopology and a causal topologyincluding is constructed, including networknetwork. and the transition network. 3.1.1. Node Determination—Predictor Selection
3.1.1. We Node Determination—Predictor choose key factors that have Selection significant influence on the ST as network nodes. Winter ST relates to many members in the North Pacific atmosphere-ocean Limpasuvan et al. [31] pointed We choose key factors that have significant influence on system. the ST as network nodes. Winter ST out that weakening of in thethe stratospheric polar vortex would affect the changes of the et STal. and jet; relates tothe many members North Pacific atmosphere-ocean system. Limpasuvan [31] Gao [32] conducted a preliminary exploration of the relationship between winter NPST and Arctic pointed out that the weakening of the stratospheric polar vortex would affect the changes of the ST Oscillation index, andadiscovered AO and NPST the samebetween phase ofwinter strongNPST and weak and jet; Gao (AO) [32] conducted preliminarythat exploration of the had relationship and variation; Gu et al. [33] determined the relationship between NPST anomaly in winter and the AC in Arctic Oscillation (AO) index, and discovered that AO and NPST had the same phase of strong and East Asia. In addition, the NPST intensity is closely related to the atmospheric system, such as the weak variation; Gu et al. [33] determined the relationship between NPST anomaly in winter and the WPin and PNA teleconnection patterns, flow anomaly, the Aleutian low pressure, AC East Asia. In addition, the NPST jet intensity is closelymonsoon related toactivity, the atmospheric system, such as Siberian high, and the ocean circulation and SST anomaly [34–39]. the WP and PNA teleconnection patterns, jet flow anomaly, monsoon activity, the Aleutian low A time-delayed analysis between AC [34–39]. indexes and the STII has been pressure, Siberian high,correlation and the ocean circulation andthe SSTabove anomaly made, and the 9 most relevant analysis indexes between are chosen predictors: AL index, PVImade, (polar A time-delayed correlation theas above AC indexes and theAO STIIindex, has been vortex intensity) index, PVA (polar vortex area) index, KI index (Kuroshio SST), NINO index and the 9 most relevant indexes are chosen as predictors: AL index, AO index, PVI (polar vortex (Niño-3.4index, SST), PNA SH (Siberian high pressure) and WP denoted as intensity) PVA index, (polar vortex area) index, KI index index (Kuroshio SST),index, NINOrespectively index (Niño-3.4 SST), AL, AO, PVI, PVA, KI, NINO, PNA, SH, and WP. The complete DBN node set is: PNA index, SH (Siberian high pressure) index and WP index, respectively denoted as AL, AO, PVI, PVA, KI, NINO, PNA, SH, and WP. The complete DBN node set is: Child node set (predictor) = {AL, AO, PVI, PVA, KI, NINO, PNA, SH, WP} Child = AL, AO, PVI, Parentnode nodeset set(predictor) (predictand) = {STII } PVA, KI, NINO, PNA, SH, WP Parent node set (predictand) = STII 3.1.2. Construction of Initial Network and Transition Network 3.1.2. Construction of Initial Network and Transition Network The definition of causality is the premise to express the transfer rules between different nodes. Based ondefinition the network nodes and we define thebetween following causality: The of causality is the the analysis premise in to Section express3.1.1, the transfer rules different nodes. Based on the network nodes and the analysis in Section 3.1.1, we define the following causality: ( P(AL, AO, PVI, PVA, KI, NINO, PNA, SH, WP|STII) (AL, AO, PVI, PVA, KI, NINO, PNA, SH, WP|STII) P(STII[t (STII[t++1]|STII[t]) 1]|STII[t]) Figure shows the theDBN DBNtopology topology structure between adjacent slices, including the Figure 22 shows structure between twotwo adjacent timetime slices, including the initial initial network and transition network. network and transition network.
Figure Figure2.2.DBN DBNstructure structurebetween betweentwo twoadjacent adjacenttime timeslices slicesof ofSTII STIIprediction. prediction.
3.2. Parameter Learning The aim of node parameters determination is to extract the probability distribution from the historical data that truly reflects the causality among variables. The learning steps includes determining the states taken by the node and training parameters by intelligent algorithms. Under the complete historical data, we choose the Expectation-Maximization (EM) algorithm to learn the
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3.2. Parameter Learning The aim of node parameters determination is to extract the probability distribution from the historical data that truly reflects the causality among variables. The learning steps includes determining the states taken by the node and training parameters by intelligent algorithms. Under the complete historical data, we choose the Expectation-Maximization (EM) algorithm to learn the parameters. 3.2.1. Determination of Node States As DBN is better at processing discrete data, the continuous data is required to be discretized to determine the number of states taken by the node. We analyze the historical data over a period and discretize the node states according to the maximum and minimum values. Consequently, discrete state space of each node is obtained used the equal interval division method [40]. 3.2.2. Calculation of Probability Distribution First initialize the probability distribution of each node, including prior probability, observation probability, and transition probability. Then, based on the inference mechanism and training data, use EM algorithm to learn parameters and correct the initial probability distribution, to get the probability distribution that matches the objective data. The idea of the EM algorithm is to replace the actual statistics with the expected statistics, whose learning process is iterative and involves two steps: • E step: Infer the distribution P( Z X, θ t ) of hidden variable Z with the current parameter θ t and observed variables X, and calculate the expectation of log likelihood LL(θ t Z, X ) for Z: Q θ θ t = EZ| X,θ t [ LL(θ | X, Z )],
•
(2)
M step: Find the parameter to maximize the expectation likelihood: θ t+1 = argmax Q θ θ t .
(3)
3.3. Probabilistic Inference of Prediction Distribution Based on the DBN structure and node parameters, the probability distribution calculation in the predicted time slice belongs to the probabilistic reasoning problem of BN. Bayesian inference algorithm includes exact algorithm and approximate algorithm. Approximate algorithm is more applied to large-scale network structure to solve the problem of excessive computation. Considering the scale of the network in our research, we apply the exact algorithm-joint tree inference algorithm to accurate reasoning [41]. Each predictor data is input as evidence into the DBN and the joint probability distribution is calculated, then it is marginalized to obtain the prediction distribution of STII. 4. Prediction Experiment of STII We use the WA-DBN probabilistic prediction model to predict STII and all experiments are performed with MATLAB (R2012a, The MathWorks, Natick, MA, USA). Both the wavelet decomposition and DBN construction are conducted with Wavelet Tool-Box and FullBNT Tool-Box (v1.0.4) [42]. 4.1. Data Introduction In this research, the study area is taken as [30◦ N–60◦ N, 120◦ E–120◦ W]. Winter data (November to March) of the 500 hPa geopotential height at a horizontal resolution of 2.5◦ × 2.5◦ are obtained from the National Center for Environment Prediction (NECP) and National Center for Atmospheric Research (NCAR) of United States of American for the period 1961–2010. Data sources of the predictors are shown in Table 1, and the coverage period is the same. We calculate the STII according to the definition
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in Section 2.1 and get a single time-series of each variable with 250 months. The first 240 months are chosen as training data and the last 10 months are test data. Atmosphere 2018, 9, x FOR PEER REVIEW
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Table 1. Data sources of 9 predictors in STII prediction. Table 1. Data sources of 9 predictors in STII prediction. Predictor
Predictor AO, AO,PNA, PNA, WP WP AL, SH AL, SH NINO NINO KI PVI, KI PVA PVI, PVA
Data Source
Data Source NOAA Climate Prediction Center NOAA Climate Prediction Center Calculation with sea level pressure data from NCEP/NCAR Calculation with sea level pressure data from NCEP/NCAR Hadley Center Center CalculationHadley with SST data from Hadley Center Calculation with SST data fromthe Hadley Center 74 circulation characteristics from National Climate Center 74 circulation characteristics from the National Climate Center
4.2. Construction of WA-DBN Prediction Model 4.2. Construction of WA-DBN Prediction Model 4.2.1. Wavelet Decomposition Module 4.2.1. Wavelet Decomposition Module We apply wavelet decomposition to original STII time-series (first 240 months) with Daubechies We apply wavelet decomposition to original STII time-series (first 240 months) with Daubechies orthogonal mother wavelet [43]. As a result, a total of seven detailed components and one level of orthogonal mother wavelet [43]. As a result, a total of seven detailed components and one level of approximation are acquired as shown in Figure 3. approximation are acquired as shown in Figure 3. Modes d1 − d3 contain the noise in the original sequence. d4 − d7 are Modes − contain the information noise information in the original sequence. −the detail are themodes detail with gradually increasing period and decreasing amplitude, containing the significant information of modes with gradually increasing period and decreasing amplitude, containing the significant the original sequence. a7 sequence. indicates the linear indicates trend of the sequence. information of theMode original Mode the original linear trend of the original sequence. 8
d1
3 -2 -7
d2
5 0
d3
-5 5
0
d4
-5 5
0
-5 2
d5
1 0 -1
Figure 3. Cont.
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d6
2
0
d7
-2 1
0
a7
-1 25 24 23 22 21 20 Year
Figure 3. Modes of wavelet decomposition STII time-series (First 240months). months). Figure 3. Modes of wavelet decomposition forfor STII time-series (First 240
4.2.2. DBN Prediction Module 4.2.2. DBN Prediction Module DBN is applied to predict each sub-mode, and the final probabilistic prediction of STII is DBN is applied to predict each sub-mode, and the final probabilistic prediction of STII is obtained obtained by integrating each prediction result with a reconstruction algorithm. by integrating each prediction result with a reconstruction algorithm. (1) Data Process (1) Data Process According to historical records, we select reasonable interval division steps for 9 predictors and According to and historical we select reasonable interval stepsstandard for 9 predictors and 8 sub-modes denoterecords, them with consecutive numbers. The division discretization is shown in 8 sub-modes and denote them with consecutive numbers. The discretization standard is shown in Table 2. Then we discretize the predictors and sub-modes with equal interval to obtain training data Table of 2. each Thenmodes we discretize theS1 predictors and sub-modes with equal interval to obtain training data of (See Table in supplementary material). each modes (See Table S1 in Supplementary Material). Table 2. Discretization standard for predictors and modes of STII. Table 2. Discretization standard for predictors and modes of STII.
Predictor Interval Step State Number AL 1 1–23 Predictor Interval Step State Number AO 0.1 1–78 AL 1 1–23 CVI 100.1 1–16 AO 1–78 CVI 10 1–16 CVS 10 1–29 CVS 1–29 KU 0.110 1–63 KU 0.1 1–63 NINO 0.10.1 1–48 NINO 1–48 PNA 0.10.1 1–46 PNA 1–46 1 1–24 SHSH 1 1–24 WP 0.1 1–59 WP 0.1 1–59
(2)
Sub-Mode Sub-Mode d1 d2 d3 d4 d5 d6 d7 a7 -
-
Interval Step State Number 1 1–15 Interval Step State Number 1 1–14 1 1–15 1 1 1–14 1–8 11 1–8 1–6 10.1 1–6 1–32 0.1 1–32 0.1 0.1 1–20 1–20 0.1 0.1 1–14 1–14 0.1 1–23 1–23 0.1 -
(2) Network Construction Parameter Learning Network Construction andand Parameter Learning
Based on node the node variables causality determinedininSection Section3.1, 3.1,the theDBN DBN structure structure is is Based on the variables andand causality determined generated MATLAB. In Figure 4, Node 1 denotes each sub-mode(d(1 −−d7 , ,a7 ) )ofofthe thepredictand predictand generated withwith MATLAB. In Figure 4, Node 1 denotes each sub-mode and nodes denote predictors. Nodes 1–10 previous timeslice slicewhile whilenodes nodes 11–20 11–20 (STII)(STII) and nodes 2–102–10 denote predictors. Nodes 1–10 areare in in thethe previous time are in the latter time slice. are in the latter time slice. constructing the network structure, algorithm is used learn parameters,i.e., i.e.the the prior prior AfterAfter constructing the network structure, EMEM algorithm is used to to learn parameters, probability, conditional probability, and transition probability of the nodes. Where the transition probability, conditional probability, and transition probability of the nodes. Where the transition probability is shown in Table A1 in Appendix A. probability is shown in Table A1 in Appendix A.
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Atmosphere 2018, 2018, 9, 9, xx FOR FOR PEER PEER REVIEW REVIEW Atmosphere
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55
15 15
66
16 16
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10 10
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Figure 4. 4. DBN DBN structure of STII STII prediction prediction under under MATLAB. MATLAB. DBN structure structure of Figure
4.3. Reasoning Reasoning Prediction Prediction and and Result Result Analysis Analysis 4.3. Following the thedetermination determination of structure structure and parameters, parameters, we discretize discretize the test testpredictors data of of Following the determination of and we the data Following of structure and parameters, we discretize the test data of predictors (later 10 months) according to Table 2 and input the discrete value into DBN to reason and predictors (later 10 months)to according to Table anddiscrete input the discrete valuetointo DBNand to reason (later 10 months) according Table 2 and input2the value into DBN reason predictand the predict the the probability probability distribution of each each in sub-mode in the the last 10 10The months. The results are shown in predict of sub-mode in last months. The in probability distributiondistribution of each sub-mode the last 10 months. results areresults shownare in shown Table A2 Table A2 in in Appendix Appendix A.median Take the the median ofprobable the most mostinterval probable interval as the the predictand predictand of each each Table A2 A. Take the probable as of in Appendix A. Take the of median the mostof as interval the predictand of each sub-mode, sub-mode, then apply the wavelet reconstruction to get the composite predictand of STII. Figure sub-mode, apply the wavelet reconstruction to get the composite of STII. Figure 55 then apply then the wavelet reconstruction to get the composite predictandpredictand of STII. Figure 5 plots the plots the monthly predicted and actual STII in the test period together with the prediction absolute plots the predicted monthly predicted in the test period together with the absolute prediction absolute monthly and actualand STIIactual in theSTII test period together with the prediction error (PAE) error (PAE) yield of the WA-DBN model for each tested month. error of (PAE) yield of the WA-DBN model formonth. each tested month. yield the WA-DBN model for each tested PAE PAE Predictand Predictand Actual value value Actual
30 30
10 10 88
STII STII
66
25 25 44
20 20 15 15
PAE PAE
35 35
22
11
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55
Month Month
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00
Figure 5. Monthly prediction prediction of of STII STII in Figure 5. Monthly in the the test test period. period.
At present, present, most most of of the the common common evaluation evaluation indicators indicators for for prediction prediction accuracy accuracy in in the the literature literature At are the following: average sum error, average absolute error, average relative error, root mean square Atfollowing: present, most of thesum common indicators foraverage prediction accuracy the mean literature are are the average error, evaluation average absolute error, relative error,inroot square error, etc. [44]. [44]. average All of of them them can measure theabsolute deviation between the relative predicted value and the actual actual value. the following: sumcan error, average error, average error, root mean squarevalue. error, error, etc. All measure the deviation between the predicted value and the To statistically statistically test measure the performance performance of WA-DBN WA-DBN model, three value prediction score metrics are etc. [44]. All of them test can the deviation between the predicted and the actual value.are To the of model, three prediction score metrics employed: root mean square error (RMSE), mean relative error (MRE) and correlation coefficient (R). To statistically the error performance WA-DBN threeand prediction score metrics (R). are employed: root meantest square (RMSE), of mean relativemodel, error (MRE) correlation coefficient The RMSE of the prediction result is 2.8954, the MRE is 0.0794, and the R value is 0.6579. The employed: root mean square error (RMSE), mean relative error (MRE) and correlation coefficient (R). The RMSE of the prediction result is 2.8954, the MRE is 0.0794, and the R value is 0.6579. The prediction variation of the the STII STII is is less and the the changing changing tendency agrees withisthat that in reality. reality. The RMSE of the prediction result 2.8954, MRE is 0.0794, and the R value 0.6579. The prediction prediction variation of is less and tendency agrees with in Different from previous prediction models, the WA-DBN WA-DBN model could provide aa casual casual graph graph variation of thefrom STII previous is less andprediction the changing tendency agrees with that could in reality. Different models, the model provide and conditional probability. Therefore, it can intuitively and quantitatively express the relationship Different from previous prediction models, the WA-DBN model could provide a casual graph and conditional probability. Therefore, it can intuitively and quantitatively express the relationship between STII and andprobability. climate indexes, indexes, whichitcould could deal with with the the uncertainty andexpress nonlinearity to improve improve and conditional Therefore, can intuitively anduncertainty quantitatively the relationship between STII climate which deal and nonlinearity to prediction accuracy. In contrast contrast with the certain certaindeal mapping relationship, the model could establish establish the between STII and climate indexes, which could with the uncertaintythe and nonlinearity to improve prediction accuracy. In with the mapping relationship, model could the probabilistic mapping between predictands and predictors and offer the comprehensive prediction prediction accuracy. In contrast with the certain mapping relationship, the model could establish the probabilistic mapping between predictands and predictors and offer the comprehensive prediction information with with probability distribution. probabilistic mapping between predictands and predictors and offer the comprehensive prediction information probability distribution. information with probability distribution.
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5. 5. Model ModelAnalysis Analysis and and Discussion Discussion To To make make aa further further test for the WA-DBN prediction model, we we conduct conduct another another prediction prediction experiment, experiment, the the regression regression fitting fitting experiment, experiment, and and the the comparison comparison experiment experiment with with NN NN and and Poisson Poisson regression (P-regression). Moreover, the error analysis of the prediction results is performed regression (P-regression). Moreover, the error analysis of the prediction results is performed and and discussed. discussed. 5.1. Model ModelExperiment Experiment and and Discussion Discussion 5.1. 5.1.1. Prediction Prediction Experiment Experiment 5.1.1. (1) Contrast Contrast experiment experiment with with the the Poisson Poisson regression regression (1) To test capacity of this model, a prediction experiment with thewith classic To test the theprediction prediction capacity of this model, a prediction experiment theP-regression classic Pis conducted for comparison [44]. We also use thealso firstuse 240the months formonths trainingfor and the lastand 10 the months regression is conducted for comparison [44]. We first 240 training last for predicting in Section 4. Figure 6 and Table 3 show the comparative results and error analysis. 10 months for predicting in Section 4. Figure 6 and Table 3 show the comparative results and error As evidence by higher and smaller RMSE, the WA-DBN model has better ability analysis. As evidence byRhigher R and smaller RMSE, the WA-DBN model has prediction better prediction than P-regression. ability than P-regression. 35
Actual value WA-DBN
30 STII
P-regerssion
25 20 15 1
2
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Month
6
7
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10
Figure6.6.Comparative Comparativeresults resultsof ofSTII STIIprediction predictionbetween betweenWA-DBN WA-DBN and and P-regression. P-regression. Figure Table 3. 3. Error Erroranalysis analysisof ofWA-DBN WA-DBN and and P-regression. P-regression. Table
Indicator Indicator MRE RMSE RMSE MRE Model 0.0794 WA-DBNWA-DBN 2.8954 2.8954 0.0794 P-regression P-regression 6.1703 6.1703 0.1778 0.1778
Model
R
R
0.6579 0.6579 0.1981 0.1981
(2) Prediction experiment with different training samples (2) Prediction experiment with different training samples In accordance with the experiment steps in section 4, we use 4 sets of data with different timeaccordance experiment steps in Section 4, we use 4 sets data withto different time-series seriesInlength (i.e., with 200 the months, 210 months, 220 months, and 230ofmonths) train the model length (i.e., 200 months, 210 months, 220 months, and 230 months) to train the model respectively, respectively, then successively predict for 10 months. The prediction result is shown in Figure 7 and then successively predict 10 months. The prediction result is shown in Figure 7 and error analysis error analysis is shown in for Table 4. is shown in Table 4. The RMSE of four groups of prediction results are all around 3.5, MRE is around 0.1, and R is The0.6, RMSE of four that groups prediction results are all around 3.5, MRE around 0.1,and andhigh R is around indicating the of model has good prediction accuracy, good iscorrelation, around 0.6, indicating that the model has good prediction accuracy, good correlation, and high stability. stability. More importantly, the prediction of extremums is more accurate, which is meaningful for More importantly, the prediction of extremums is more accurate, which is meaningful for ST prediction. ST prediction. However, there are also outliers of predictions, such as the large deviations in the However, there are also outliers of predictions, such as the large deviations in the prediction results prediction results from month 6 to 7 in Figure 7a and from month 1 to 3 in Figure 7c. from month 6 to 7 in Figure 7a and from month 1 to 3 in Figure 7c.
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31 29 27 25 23 21 19 17 15
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8 6
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7
(d) 230 months
Figure learned by by different different training training data. data. Figure 7. 7. STII STII Prediction Prediction with with WA-DBN WA-DBN model model learned
8
9
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PAE
4
23
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STII
6
PAE
25
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Table 4. Error analysis of WA-DBN prediction model with different training data. Atmosphere 2018, 9, x FOR PEER REVIEW
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210 Months
220 Months
230 Months
Table 4. Error analysis of WA-DBN prediction model with different training data.
RMSE MRE Training Data R Indicator
5.1.2. Fitting
3.6956 0.0818 200 Months 0.6341
3.8548 0.1119 210 Months 0.6806
3.9734 0.1152 220 Months 0.5245
2.7921 0.0881 230 Months 0.5346
3.6956 0.0818 0.6341
3.8548 0.1119 0.6806
3.9734 0.1152 0.5245
2.7921 0.0881 0.5346
RMSE MRE Experiment R
Mean 3.5789 0.0993 Mean 0.5935 3.5789 0.0993 0.5935
We train the model with the first 240-month data and input the corresponding predictor data for Fitting Experiment return5.1.2. fitting, comparing the fitting result with NN [45]. The comparison result and error analysis are We train the model shown in Figure 8 and Table with 5. the first 240-month data and input the corresponding predictor data for return fitting, comparing the fitting result with NN [45]. The comparison result and error analysis are shown in Figure 8 and Table Table 5. 5. Error analysis of WA-DBN and NN.
Model
Table 5. Error analysis of WA-DBN and NN. Indicator MRE RMSE Indicator RMSE MRE Model WA-DBN 3.7311 0.2315 NN 18.7454 2.1849 WA-DBN 3.7311 0.2315
NN
18.7454
2.1849
R
R
0.9771 0.2237 0.9771 0.2237
From Table 5, the single-point fitting accuracy with the WA-DBN probabilistic prediction model is From Table 5, the single-point fitting accuracy with the WA-DBN probabilistic prediction model significantly better than the deterministic NN method. When there are large fluctuations in the data, is significantly better than the deterministic NN method. When there are large fluctuations in the the error for NN, but the DBN canDBN givecan a probability distribution relatively data,increases the errorsignificantly increases significantly for NN, but the give a probability distribution close to the reality to the transition between different states of thestates STII in relatively closeaccording to the reality according to the transition between different of the the historical STII in thedata. All states of STII areAll presented in the without loss ofwithout results.loss of results. historical data. states of STII areprobability presented indistribution the probability distribution Fitting value Actual value
35 33 31 29
STII
27 25 23 21 19 17 15
Year (a) WA-DBN
Figure 8. Cont.
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Fitting value Actual value Fitting value
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Year (b) NN
(b) NN Figure 8. Comparative fitting results between WA-DBN and NN. Figure 8. Comparative fitting results between WA-DBN and NN. Figure 8. Comparative fitting results between WA-DBN and NN. To further test the time-validity of WA-DBN model over short- and long-term prediction, we input the test first month data and the first 10-month data ofover predictors performing regression To further of of WA-DBN model over shortand when long-term prediction, we input To further testthe thetime-validity time-validity WA-DBN model shortand long-term prediction, we fitting experiments. Figure 9 (a) indicates that only the first month data is input, the results of the the first data and first data ofdata predictors when performing regression fitting input themonth first month datathe and the10-month first 10-month of predictors when performing regression subsequent three months are that moreonly effective, but the other predictions have greater errors; (b) experiments. Figure 9Figure (a) indicates the first data is input, of the subsequent fitting experiments. 9 (a) indicates that onlymonth the first month datathe is results input, the results of the indicates that when input the first 10-month data, the results of the subsequent five months are more three monthsthree are more effective, but the other predictions have greater errors; (b) indicates that when subsequent months are more effective, but the other predictions have greater errors; (b) effective. Although the predictable time extends when the input time is increased from 1 to 10 input the that first when 10-month data, the results of the subsequent five months are more effective. Although indicates input the first 10-month data, the results of the subsequent five months are more months, the predictable time is still short. Thus, the model is not suitable for medium- and long-term the predictable time extends when input time is increased from 1DBN to 10 months, the predictable effective. Although predictable time extends when the input isdepends increased from 1 to 10 prediction. The the reason may be the that the probabilistic inference oftime on the priori time isprobability still the model isshort. not formodel mediumlong-term prediction. reason months, theshort. predictable time is still Thus,priori the is notand for mediumandThe long-term ofThus, nodes. When only the suitable finite probability issuitable given, such as one month or ten may be that The the inference DBN depends time oninference the prioriofprobability of nodes. only months, theprobabilistic reasoning error as the predicted increases. prediction. reason may beincreases that of the probabilistic DBN depends on When the priori
the finite priori probability is given, such as one month or ten months, the reasoning error increases as probability of nodes. When only the finite priori probability is given, such as one month or ten 35 Predictand the predicted time increases. months, the reasoning error increases as the predicted time increases. Actual value
STII
35
STII
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30
Predictand Actual value
25 20
25 15
20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Month (a) Prediction with one month
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Month (a) Prediction with one month Figure 9. Cont.
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Predictand Predictand Actual value Actual value
STII STII
30 30 25 25 20 20
15 15 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 1 3 5 7 9 11 13Month 15 17 19 21 23 25 27 29
Month (b) Prediction with ten months (b) Prediction with ten months
Figure 9. 9. Test Test experiment experiment of of predictable predictable time time with with different different inputting inputting data. data. (The (The effective effective predictions predictions Figure Figure 9. Test experiment of predictable time with different inputting data. (The effective predictions are highlighted highlighted with with dotted dotted line). line). are are highlighted with dotted line).
5.1.3. Expending Expending Experiment Experiment for for NAST NAST Intensity Intensity Index Index Prediction Prediction 5.1.3. 5.1.3. Expending Experiment for NAST Intensity Index Prediction To further further verify verifythe thegeneralization generalizationability abilityofofthis thismodel, model, perform a prediction experiment To wewe perform a prediction experiment of To further verify the generalization ability of this model, we perform a prediction experiment of ◦ ◦ ◦ ◦ of NAST N–70N, N, W–0W). W). TheNAST NASTintensity intensityindex indexisisalso alsocalculated calculatedwith with the the same same NAST (35°(35N–70° 90°90W–0° The NAST (35° N–70° N, 90° W–0° W). The NAST intensity index is also calculated with the same definition in in Section Section 2.1. 2.1. A A time-delayed time-delayed correlation correlation analysis analysisbetween betweenthe the AC AC indexes indexes [46] [46] and and the the definition definition in Section 2.1. A time-delayed correlation analysis between the AC indexes [46] and the NAST intensity index has also been made. Different from NPST, 6 most relevant indexes are chosen as NAST has also been made. Different from NPST, 6 most relevant indexes are chosen NAST intensity index has also been made. Different from NPST, 6 most relevant indexes are chosen predictors: AO index, Oscillation (ADO) (ADO) as predictors: AO index,North NorthAtlantic AtlanticOscillation Oscillation(NAO) (NAO)index, index, Atlantic Atlantic Decadal Oscillation as predictors: AO index, Atlantic Oscillation (NAO) index, Decadal Oscillation index, East Atlantic Atlantic (EA)North index, West Atlantic Atlantic index (WAI), andAtlantic North American American Jet Stream Stream(ADO) (NAJ) index, East (EA) index, West index (WAI), and North Jet (NAJ) index, East Atlantic (EA) index, West Atlantic index (WAI), and North American Jet Stream (NAJ) index, respectively denoted as AO, NAO, ADO, EA, WAI and NAJ. The DBN network with above index, respectively denoted as AO, NAO, ADO, EA, WAI The DBN network with above index,is asin NAO, nodes isrespectively constructeddenoted as shown shown inAO, Figure nodes constructed as Figure 10.ADO, EA, WAI and NAJ. The DBN network with above nodes is constructed as shown in Figure 10.
Figure 10. 10. DBN structure between two adjacent time slices of NAST Figure NAST prediction. prediction. Figure 10. DBN structure between two adjacent time slices of NAST prediction.
According According to to the the integrated integrated steps steps in in Section Section 4, 4, we we conduct conduct the the same same prediction prediction experiment experiment of of According to the integrated steps in Section 4, we conduct the same prediction experiment of NAST. intensity index index of of NAST. NAST. NAST. Figure Figure 11 11 displays displays the the monthly monthly predicted predicted intensity NAST. Figure 11 displays the monthly predicted intensity index of NAST. We calculate the evaluation indicators for prediction accuracy: the RMSE is 3.1717, the MRE is 0.1104, and the R value is 0.5568. Therefore, it is reasonable to generalize the WA-DBN model for other mid-latitude ST regions and the predicted results are reliable. This prediction model has adaptive ability owing to the flexible modeling features of DBN.
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Figure 11. Monthly Prediction of NAST intensity index in the test period.
We calculate the evaluation indicators for prediction accuracy: the RMSE is 3.1717, the MRE is 6. Conclusions 0.1104, and the R value is 0.5568. Therefore, it is reasonable to generalize the WA-DBN model for Effective short-term prediction of STII is significant for researches of mid-latitude weather systems, other mid-latitude ST regions and the predicted results are reliable. This prediction model has especially the analysis of abnormal changes. In this study, we have applied the state-of-the art adaptive ability owing to the flexible modeling features of DBN. artificial intelligence to predict the monthly intensity index of NPST with WA-DBN probabilistic prediction model. Considering the non-stationarity, nonlinearity, and uncertainty of the STII time-series, 6. Conclusions we first used the WA to decompose the intensity index into the sub-modes with different frequency Effective prediction of make STII isa probabilistic significant for researches mid-latitude domains. Thenshort-term we applied the DBN to prediction forof each sub-mode.weather Finally, systems, especially the analysis of abnormal changes. In this study, we have applied the state-of-the the independent prediction results of each mode were integrated with the wavelet reconstruction. art artificial intelligence the monthly intensity of NPST with WA-DBN probabilistic To further illustrateto thepredict advantages of the model, weindex conducted multiple sets of STII prediction prediction model. Considering the non-stationarity, nonlinearity, and uncertainty of the STII timeexperiments, fitting experiments, and comparison experiments. The results show that predicting series, we first used the WA to decompose the intensity index into the sub-modes with different correlation coefficient reached about 0.6 and fitting correlation coefficient reached 0.97. Moreover, frequency we applied the DBN to make the a probabilistic prediction for relatively each sub-mode. this modeldomains. is good atThen predicting extremums. Therefore, WA-DBN model exhibits better Finally, the independent prediction results of each mode were integrated with the wavelet performance in prediction of nonlinear uncertainties, as evidence by higher R and smaller RMSE. reconstruction. The improved performance of the WA-DBN model is attributable to two aspects: To further illustrate the advantages of the model, we conducted multiple sets of STII prediction 1. The input dataset of predictand decomposed into separate components based different experiments, fitting experiments, andiscomparison experiments. The results show thatonpredicting frequencies withreached WA, allowing of noisy data coefficient and revealing the 0.97. quasi-periodic correlation coefficient about 0.6removal and fitting correlation reached Moreover, components originalextremums. time-series. Therefore, the WA-DBN model exhibits relatively better this model is goodin atthe predicting Both theinrelationship between theuncertainties, predictand and the predictors at the same time and thatThe in 2. performance prediction of nonlinear as evidence by higher R and smaller RMSE. adjacent time slices are considered with DBN model. The expression of casual relationship with improved performance of the WA-DBN model is attributable to two aspects: network structure and probability distribution can better deal with the uncertainty of prediction. 1. The input dataset of predictand is decomposed into separate components based on different frequencies withthat WA,the allowing removal noisy data and and tested revealing the quasi-periodic We summarize WA-DBN modelofdeveloped in this study has goodcomponents prediction in the original time-series. skills of monthly STII, which is of great scientific guidance to study the abnormal changes of ST and 2. mechanisms. Both the relationship the predictand and theprediction predictorsmodel at the based same on time and that in its Above all,between we propose a new intelligent graph theory adjacent time sliceswhich are considered DBN model. The with expression casual relationship and probability theory, has wide with application prospects strongof generalization abilitywith and goodnetwork stability.structure and probability distribution can better deal with the uncertainty of prediction. Although the WA-DBN probabilistic predicting model works there are has stillgood someprediction problems. We summarize that the WA-DBN model developed and testedwell, in this study First, the selection of the predictors of the ST intensity index needs to be further improved. The skills of monthly STII, which is of great scientific guidance to study the abnormal changes ofexisting ST and studies indicate that if the of predictors exceeds 10, the predicting will be complex, its mechanisms. Above all,number we propose a new intelligent prediction modelcalculation based on graph theory and and the accuracy will not increase more with predictors. fewer predictors areand selected probability theory, which has widesignificantly application with prospects strong If generalization ability such 5, the accuracy will become poor due to loss of information. In this research, we chose 9 most goodas stability. relevant indicators as predictors. However, the selection predictors is crucial to some prediction, and Although the WA-DBN probabilistic predicting modelof works well, there are still problems. we need to improve this work. Second, the accuracy of the long-term prediction in this model is low. First, the selection of the predictors of the ST intensity index needs to be further improved. The These arestudies also the focus ofthat future work. existing indicate if the number of predictors exceeds 10, the predicting calculation will
be complex, and the accuracy will not increase significantly with more predictors. If fewer predictors are selected such as 5, the accuracy will become poor due to loss of information. In this research, we chose 9 most relevant indicators as predictors. However, the selection of predictors is crucial to
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Supplementary Materials: The following are available online at http://www.mdpi.com/2073-4433/9/6/224/s1, Table S1: Discrete training data (Take d1 as an example). Author Contributions: M.L. and K.L. conceived and designed the experiments; M.L. performed the experiments; M.L. and K.L. analyzed the data; M.L. wrote the paper. Funding: This research was funded by the Chinese National Natural Science Fund (Nos. 41375002, 51609254) and the Natural Science Fund (BK20161464) of Jiangsu Province. Conflicts of Interest: The authors declare no conflict of interest.
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Appendix A This section contains Tables A1 and A2 supplemental to the main text. Table A1. Transition probability of DBN (Take d1 as an example). State of Node d1
d1 [t + 1] d1 [t]
State of node d1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.0216 0.0487 0.0842 0.0421 0.1011 0.0479 0.0451 0.0191 0.0211 0.0119 0.0581 0.0567 0.0348 0.0531 0.0544
0.0945 0.0265 0.1204 0.1065 0.0469 0.0060 0.0016 0.0483 0.0464 0.0605 0.0558 0.0078 0.0121 0.0196 0.0269
0.1261 0.0237 0.0061 0.0759 0.0006 0.0366 0.0262 0.0018 0.0224 0.0650 0.0032 0.0159 0.0473 0.0071 0.0920
0.1034 0.0810 0.0595 0.1251 0.0646 0.0047 0.0149 0.0752 0.0443 0.0593 0.0264 0.0021 0.0070 0.0205 0.0293
0.0300 0.0257 0.0674 0.0434 0.1125 0.0732 0.0298 0.0501 0.0304 0.0031 0.0573 0.0802 0.0004 0.0525 0.0875
0.0595 0.0496 0.0458 0.0118 0.0172 0.0311 0.0298 0.0122 0.0675 0.0474 0.0028 0.0649 0.0506 0.0559 0.0058
0.0333 0.0363 0.0340 0.0462 0.0694 0.0829 0.0019 0.0939 0.0985 0.0364 0.0267 0.0204 0.0356 0.0140 0.0046
0.0457 0.0754 0.0480 0.0096 0.0489 0.0797 0.0822 0.0515 0.0492 0.0084 0.0368 0.0409 0.0518 0.0470 0.0054
0.0209 0.0622 0.0532 0.0063 0.0483 0.0438 0.0874 0.1213 0.0097 0.0348 0.1084 0.0775 0.0356 0.0966 0.0501
0.0958 0.1089 0.1058 0.0342 0.0318 0.0317 0.1250 0.1363 0.0161 0.0095 0.0731 0.0919 0.0809 0.0154 0.1055
0.0333 0.1322 0.0574 0.0422 0.1656 0.0646 0.0742 0.0483 0.0639 0.0440 0.0208 0.0118 0.0473 0.1670 0.0522
0.1433 0.2414 0.1593 0.0354 0.1431 0.1056 0.1462 0.1239 0.0672 0.2152 0.0334 0.0424 0.0013 0.2083 0.1092
0.1069 0.0000 0.0159 0.2644 0.0470 0.1533 0.1862 0.0036 0.2843 0.2097 0.1295 0.0538 0.3053 0.0627 0.2252
0.0584 0.0566 0.0008 0.0366 0.0471 0.1685 0.1423 0.0770 0.1767 0.0148 0.0685 0.4025 0.1681 0.0790 0.0271
0.0273 0.0874 0.1423 0.1205 0.0559 0.0702 0.0072 0.1375 0.0024 0.1800 0.2992 0.0312 0.1219 0.1014 0.1248
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Table A2. Monthly predicted probability distribution of d1 (The maximum probability of probability distribution for each month is highlighted). Month State 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1
2 10−31
1.77 × 4.30 × 10−15 1.23 × 10−8 3.71 × 10−5 1.15 × 10−9 0.00017127 0.99968979 0.00010175 2.23 × 10−8 5.79 × 10−8 9.39 × 10−24 1.82 × 10−18 2.66 × 10−27 1.77 × 10−31 1.75 × 10−26
3 10−22
1.84 × 6.15 × 10−25 2.95 × 10−7 0.96607572 0.00024291 0.03368107 9.37 × 10−12 1.07 × 10−12 1.89 × 10−15 9.98 × 10−27 6.20 × 10−31 3.21 × 10−35 1.18 × 10−30 2.35 × 10−35 2.35 × 10−35
4 10−38
2.58 × 9.92 × 10−30 1.30 × 10−26 1.86 × 10−10 1.51 × 10−8 0.99670829 3.18 × 10−6 2.85 × 10−19 0.00328851 1.97 × 10−15 6.15 × 10−30 1.52 × 10−32 1.63 × 10−37 2.58 × 10−38 2.29 × 10−29
5 10−28
7.61 × 1.16 × 10−9 1.78 × 10−12 0.04711185 3.08 × 10−5 8.29 × 10−5 0.12481569 7.14 × 10−5 0.82788733 1.45 × 10−12 1.18 × 10−31 4.61 × 10−29 4.44 × 10−23 1.50 × 10−32 1.50 × 10−27
6 10−33
4.06 × 5.74 × 10−23 1.42 × 10−14 1.99 × 10−8 0.61559523 0.35914973 0.00022297 1.30 × 10−5 0.0250191 2.37 × 10−13 2.45 × 10−25 2.24 × 10−21 8.72 × 10−31 1.43 × 10−35 1.43 × 10−30
7 10−32
1.96 × 1.62 × 10−21 3.85 × 10−22 4.68 × 10−12 1.19 × 10−7 0.13090291 0.00131173 0.86778512 1.14 × 10−7 3.61 × 10−19 2.17 × 10−18 1.40 × 10−31 1.06 × 10−31 8.70 × 10−35 8.70 × 10−35
8 10−31
5.13 × 2.34 × 10−13 1.48 × 10−15 3.71 × 10−9 1.18 × 10−7 2.02 × 10−5 0.99986409 6.07 × 10−5 8.20 × 10−9 5.49 × 10−5 2.93 × 10−27 3.40 × 10−29 5.38 × 10−30 1.17 × 10−31 1.17 × 10−21
9 10−31
2.42 × 1.86 × 10−30 3.50 × 10−15 1.30 × 10−11 0.00059951 0.99737456 4.70 × 10−9 2.12 × 10−15 0.00202593 1.01 × 10−14 1.45 × 10−33 8.46 × 10−35 1.75 × 10−33 7.75 × 10−35 7.87 × 10−30
10 10−34
2.75 × 2.09 × 10−21 1.23 × 10−5 9.51 × 10−8 0.00021533 0.99907177 0.00070003 1.59 × 10−14 4.30 × 10−7 1.10 × 10−18 6.63 × 10−33 4.39 × 10−21 2.64 × 10−24 2.75 × 10−34 5.63 × 10−29
1.44 × 10−35 1.18 × 10−27 4.82 × 10−8 6.19 × 10−6 0.7366376 0.18456279 0.00034761 4.45 × 10−13 0.07844577 1.92 × 10−15 4.81 × 10−31 8.99 × 10−28 9.63 × 10−31 5.76 × 10−36 5.76 × 10−31
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