Joint Conditional Probability Research Articles

Predicting the Performance of Ensemble Classification Using Conditional Joint Probability

In many machine learning applications, there are many scenarios when performance is not satisfactory by single classifiers. In this case, an ensemble classification is constructed using several weak base learners to achieve satisfactory performance. Unluckily, the construction of the ensemble classification is empirical, i.e., to try an ensemble classification and if performance is not satisfactory then discard it. In this paper, a challenging analytical problem of the estimation of ensemble classification using the prediction performance of the base learners is considered. The proposed formulation is aimed at estimating the performance of ensemble classification without physically developing it, and it is derived from the perspective of probability theory by manipulating the decision probabilities of the base learners. For this purpose, the output of a base learner (which is either true positive, true negative, false positive, or false negative) is considered as a random variable. Then, the effects of logical disjunction-based and majority voting-based decision combination strategies are analyzed from the perspective of conditional joint probability. To evaluate the forecasted performance of ensemble classifier by the proposed methodology, publicly available standard datasets have been employed. The results show the effectiveness of the derived formulations to estimate the performance of ensemble classification. In addition to this, the theoretical and experimental results show that the logical disjunction-based decision outperforms majority voting in imbalanced datasets and cost-sensitive scenarios.

Estimating metocean environments associated with extreme structural response to demonstrate the dangers of environmental contour methods

Extreme value analysis (EVA) uses data to estimate long-term extreme environmental conditions for variables such as significant wave height and period, for the design of marine structures. Together with models for the short-term evolution of the ocean environment and for wave–structure interaction, EVA provides a basis for full probabilistic design analysis. Alternatively, environmental contours provide an approximate approach to estimating structural integrity, without requiring structural knowledge. These contour methods also exploit statistical models, including EVA, but avoid the need for structural modelling by making what are believed to be conservative assumptions about the shape of the structural failure boundary in the environment space. These assumptions, however, may not always be appropriate, or may lead to unnecessary wasted resources from over design. We demonstrate a methodology for efficient fully probabilistic analysis of structural failure. From this, we estimate the joint conditional probability density of the environment (CDE), given the occurrence of an extreme structural response. We use CDE as a diagnostic to highlight the deficiencies of environmental contour methods for design; none of the IFORM environmental contours considered characterise CDE well for three example structures.

Prediction for underground seismic intensity measures using conditional generative adversarial networks

With the escalating development and utilization of subterranean spaces, the seismic hazards faced by underground structures are progressively increasing. However, owing to the challenges associated with acquiring underground seismic data and historical seismic design norms, pertinent regulations and research in this domain are scarce. This study focused on three crucial intensity measures in the seismic design process of underground structures peak ground acceleration (PGA), peak ground velocity (PGV), and peak ground displacement (PGD). The research leverages seismic data obtained from the California Strong Motion Instrumentation Program (CSMIP) to train and evaluate a conditional generative adversarial network (CGAN) model. This model was employed to establish a multivariate joint conditional probability distribution among the intensity measures at varying depths, facilitating the stochastic prediction of shallow intensity measures. In contrast to empirical formulas, the CGAN model eliminates the need for a predefined equation structure and enables the simultaneous prediction of multiple intensity measures. The performance of the model was evaluated by comparing the predictive accuracy of the CGAN model and empirical fitting formulas across diverse site conditions and depth intervals using metrics such as relative error coefficients. It can be concluded that the proposed CGAN model can accurately predict shallow seismic intensity measures, and the predictions conform to a specific conditional distribution while retaining the stochastic nature of seismic motion. Compared with empirical formula models, the CGAN model exhibited an enhanced predictive capability.

Distilling dynamical knowledge from stochastic reaction networks

Stochastic reaction networks are widely used in the modeling of stochastic systems across diverse domains such as biology, chemistry, physics, and ecology. However, the comprehension of the dynamic behaviors inherent in stochastic reaction networks is a formidable undertaking, primarily due to the exponential growth in the number of possible states or trajectories as the state space dimension increases. In this study, we introduce a knowledge distillation method based on reinforcement learning principles, aimed at compressing the dynamical knowledge encoded in stochastic reaction networks into a singular neural network construct. The trained neural network possesses the capability to accurately predict the state conditional joint probability distribution that corresponds to the given query contexts, when prompted with rate parameters, initial conditions, and time values. This obviates the need to track the dynamical process, enabling the direct estimation of normalized state and trajectory probabilities, without necessitating the integration over the complete state space. By applying our method to representative examples, we have observed a high degree of accuracy in both multimodal and high-dimensional systems. Additionally, the trained neural network can serve as a foundational model for developing efficient algorithms for parameter inference and trajectory ensemble generation. These results collectively underscore the efficacy of our approach as a universal means of distilling knowledge from stochastic reaction networks. Importantly, our methodology also spotlights the potential utility in harnessing a singular, pretrained, large-scale model to encapsulate the solution space underpinning a wide spectrum of stochastic dynamical systems.

Probability causal inference of interest rate fluctuations: Evidence from private credit in emerging markets

Preventing risks in the private credit market is an important component of maintaining financial stability, especially in emerging markets. As loan amounts serve as both the cause and the effect of credit interest rates, through the weighted path convergence design of conditional probability and joint probability, this study establishes a probabilistic causal inference framework of loan amounts and interest rates to explore the mechanism of interest rate fluctuations in China's private credit market, market behaviour of the allocation of private loan amount, and the design of policies. The analysis provides several results. (1) The fluctuation of private credit interest rate occurred during the outbreak of the epidemic (2019Q1-2020Q1), the credit interest rate was mainly controlled by the loan amount, and the interest rate fluctuation mainly originated from private credit market behaviour. In the early stage of the post COVID-19 economic recovery (2020Q3-2021Q3), the credit interest rate is mainly controlled by non-loan amount factors, and the interest rate fluctuations occurred mainly because of the non-market behaviour of private credit, which is related to government policy support and the increase in low-interest loans from financial institutions. (2) As a result of market behaviour, for the 19.54 %–22.50 % of high interest rates on borrowing, the probability of high interest rates due to the loan amount increased by 40.48 %. (3) By policy design, reallocating the risky loan amount, the interest rate can be reduced by up to 7.36 % on a highlight-interest rate day. The reallocation of loan amounts for different security methods has varying effects on interest rates reductions, and there may even be ineffective allocations. The findings provide decision support for financial regulators in emerging markets to stabilise interest rate volatility in private credit markets.

Probability distributions for dynamic and extreme responses of linear elastic structures under quasi-stationary harmonizable loads

The non-stationary load models based on the evolutionary power spectral density (EPSD) may lead to ambiguous structural responses. Quasi-stationary harmonizable processes with non-negative Wigner-Ville spectra are suitable for modeling non-stationary loads and analyzing their induced structural responses. In this study, random environmental loads are modeled as quasi-stationary harmonizable processes. The Loève spectrum of a harmonizable load process contains several random physical parameters. An explicit approach to calculate the probability distributions for the dynamic and extreme responses of a linear elastic structure subjected to a quasi-stationary harmonizable load is proposed. Conditioned on the specific values of the load spectral parameters, the harmonizable load process is assumed to be Gaussian. The conditional joint probability density function (PDF) of structural dynamic responses at any finite time instants and the conditional cumulative distribution function (CDF) of the structural extreme response are provided. By multiplying these two conditional probability distributions with the joint PDF of the load spectral parameters, and then integrating these two products over the parameter sample space, the joint PDF of structural dynamic responses at any finite time instants and the CDF of the structural extreme response can be calculated. The efficacy of the proposed approach is numerically validated using two linear elastic systems, which are subjected to non-stationary and non-Gaussian wind and seismic loads, respectively. The merit of the harmonizable load process model is highlighted through a comparative analysis with the EPSD load model.

HOME CARE DYADIC COMMUNICATION PATTERNS OF FAMILY CAREGIVERS AND PERSONS LIVING WITH DEMENTIA: SEQUENTIAL ANALYSIS

Abstract Examining temporal relationships of communication patterns between family caregivers and persons living with dementia (care recipients) is useful to determine which caregiver communication patterns are more effective in engaging care recipients and prioritize the use of such communication patterns. This study examined temporal relationships between antecedent family caregiver communication and subsequent care recipient communication during 75 in-home care video observations. A 5-second timed-window analysis was conducted to examine the likelihood of the relationships between dyadic communication patterns. Joint frequency and conditional probability statistics demonstrated that care recipients used the same communication patterns at differing frequencies during and within a 5-second window following each communication pattern by family caregivers. Dyadic communication patterns yielded likelihood probabilities with an effect size greater than chance (Yule’s Q = 0.31-1.00). Care recipient’s engaging body language most likely occurred after the caregiver’s facilitative body language preceded (Yule’s Q = 0.31-.1.00). Care recipient’s relevant communication and verbal response most likely occurred after the caregiver’s simplified message preceded (Yule’s Q = 0.55-1.00). Specific caregiver facilitative communication exhibited subsequent care recipient engaging communication. Future research can expand on how characteristics, types, and frequencies of caregiver communication would impact care recipient engaging communication.

The significance of senior education in the Internet era for the construction of lifelong education system

Abstract This paper uses Bayesian networks for the state combination of random variables to describe the relationship between prior probability and posterior probability by likelihood. Meanwhile, the elimination of complex variables is achieved by combining the operation results of multivariate non-independent joint conditional probability decomposition. Based on this, Bayes’ theorem is used to calculate the learning efficiency, the comprehensive pass rate of any interactive courseware, and the growing value of the visualized learning process to improve the sharing of resources for senior education. To verify the feasibility of the model, a teaching effectiveness test was conducted, and the results showed that 66.6% of older adults believed that participation in senior education could strengthen their bodies and minds. The percentage of medical and health care knowledge courses was 63.20%. The direct effect of the learning environment on the participation behavior of senior education was 0.143, while the indirect effect of educational resources on the participation behavior of senior education was 0.083. In addition, 61.78% of seniors thought that attitude towards senior education was very important. Therefore, it can be concluded that the model of the elderly education system fills the gap in the structure of the lifelong education system, enriches the uppermost structure of the lifelong education system, and promotes the improvement and development of the lifelong education system.

Interval Forecasting Method of Aggregate Output for Multiple Wind Farms Using LSTM Networks and Time-Varying Regular Vine Copulas

Interval forecasting has become a research hotspot in recent years because it provides richer uncertainty information on wind power output than spot forecasting. However, compared with studies on single wind farms, fewer studies exist for multiple wind farms. To determine the aggregate output of multiple wind farms, this paper proposes an interval forecasting method based on long short-term memory (LSTM) networks and copula theory. The method uses LSTM networks for spot forecasting firstly and then uses the forecasting error data generated by LSTM networks to model the conditional joint probability distribution of the forecasting errors for multiple wind farms through the time-varying regular vine copula (TVRVC) model, so as to obtain the probability interval of aggregate output for multiple wind farms under different confidence levels. The proposed method is applied to three adjacent wind farms in Northwest China and the results show that the forecasting intervals generated by the proposed method have high reliability with narrow widths. Moreover, comparing the proposed method with other four methods, the results show that the proposed method has better forecasting performance due to the consideration of the time-varying correlations among multiple wind farms and the use of a spot forecasting model with smaller errors.

Quantitative interpretation of geoelectric inverted data with a robust probabilistic approach

The nonuniqueness of the solution to the geophysical inverse problem can lead to misinterpretation while characterizing the subsurface. To tackle this situation, ground-truth information from excavations and wells can be used to improve, calibrate, and interpret inverted models. We refer to quantitative interpretation as the decision analysis based on probability theory, which is focused on solving a classification problem. First, we present a probabilistic approach to classify the different types of materials or “categories” observed in borehole logs using multiple data sources: inverted 2D electrical resistivity tomography and induced polarization data and the positions ( x, z) of these boreholes. Then, using the Bayes’ rule and permanence of ratios, we compute the joint conditional probabilities of each category, given all data sources in the whole inverted model domain. We validate this approach with synthetic data modeling for a complex anthropogenic-geologic scenario and using real data from an old landfill. Afterward, we assess the performance of the probabilistic approach for classification and compare it with the machine learning algorithm of multilayer perceptron (MLP). In addition, we analyze the effect that the different data sources and the number of boreholes (and their distribution) have on both approaches with the synthetic case. Our results indicate that the MLP performance is better for delineating the different categories where the lateral contrasts in the synthetic resistivity model are small. Nevertheless, the classification obtained with the probabilistic approach using real data seems to provide a more geologically realistic distribution. We conclude that the probabilistic approach is robust for classifying categories when high spatial heterogeneity is expected and when ground-truth data are limited or not sparsely distributed. Finally, this approach can be easily extended to integrate multiple geophysical methods and does not require the optimization of hyperparameters as for MLP.

Double-Layered Dual-Syndrome Trellis Codes Utilizing Channel Knowledge for Robust Steganography

Robust steganography aims to hide message in cover data with high security and guarantee the success of its message extraction although it is disturbed in transmission channel. In this paper we propose a framework of coding scheme extended from Dual-Syndrome Trellis Codes (Dual-STCs) for robust adaptive steganography. We use the conditional probability distribution of correct stego bits conditioned on disturbed stego data as channel knowledge, and formulate error-correcting as maximizing this probability. By extending Dual-STCs to double-layered embedding, we design an iteratively decoding scheme for error-correcting two layer stego bits from their joint conditional probabilities, and strictly prove its convergence. Besides, we design a method to estimate these probability distributions from stego data pairs uploaded/downloaded from the lossy transmission channel. The channel knowledge can also be used by steganographer, and we propose a universal method to revise steganographic distortion values for higher robustness under the guidance of the channel knowledge. Compared with existing coding methods for robust steganography, our method can make use of channel knowledge to improve error correcting ability and meanwhile maintain high security, which is demonstrated by experimental results.

INFORMATION TECHNOLOGY OF CONDUCTING THE LEARNING PROCESS ON THE BASIS OF SELF-ORGANIZING MEANS OF KNOWLEDGE CONTROL

The need for effective organization of the educational process in institutions of higher education raises urgent problems of its informatization. The analysis of literary sources established that the existing systems with the possibility of adaptation are inefficient, developed without taking into account the methods and means of self-organization, do not meet the modern requirements of education. An approach to the tasks of creating knowledge control systems is proposed, which is based on scientific results obtained in the theory of information interaction. In order to ensure a sufficient level of research on methodological problems related to the learning process, the features of information interaction during automated learning were analyzed using structural schemes of traditional and computer-based learning processes, information flows between their components. The relevance of applying and improving methods related to measuring the level of knowledge and tracking their transformation in the learning process is considered. The transformation of information and its transformation into knowledge in the learning process is represented by an information scheme with defined stages. As a result of the analysis of the features of information interaction during automated training, it was determined that systems capable of adapting to the level and structure of learning knowledge can be truly effective. A description of the development and use of knowledge control tools in educational testing based on the scientific and methodological basis of the theory of information interaction is presented. It is proposed to develop the scientific foundations of student knowledge modeling in adaptive means of knowledge control systems through the reflection of information interaction processes in the subject area of study. The uncertainty of the state of knowledge and the formation of the survey technology is presented using the method of calculating the estimate of the joint conditional probability of knowing the answer to the question of the discipline according to individual conditional probabilities. A method of self-organization of the knowledge model was built in order to adapt the survey or training technology to the level and distribution of knowledge of the person studying in the subject area of the discipline. It performs the function of optimizing the sequence of questions asked, performs an analysis of the existing macro-informational knowledge model of the learner, and determines the question, the answer to which will maximally change the definition of this model. Schemes of information flows and information interaction in the self-organization module of the knowledge model are presented. The main dependencies of information processes have been implemented.

Conditional multivariate distributions of phase-type for a finite mixture of Markov jump processes given observations of sample path

This paper presents conditional joint probability distributions of first-exit times to absorbing sets of a finite mixture of right-continuous Markov jump processes given observations of its sample path. Each underlying Markov jump process comprising the mixture has its own intensity matrix and is defined on the same finite state space with overlapping absorbing sets. The main distinctive aspect of the mixture is that despite the underlying processes have the Markov property, the mixture itself in general does not. Conditional transition matrix and the joint probability distribution of the first-exit times form non-stationary functions of time with ability to capture path dependence when conditioning on available information (either full or partial) of the sample path. In particular, initial profile of the joint distribution forms a generalized mixture of the multivariate phase-type distributions of Assaf et al. (1984). When the underlying Markov processes have the same intensity matrix, in which case the mixture becomes a simple Markov jump process, the path dependence and non-stationary properties are removed from the transition matrix and the joint distribution, while the initial profile distribution reduces to Assaf et al. (1984). Some explicit and numerical examples are discussed to illustrate the results.

Comparative Study of Flood Coincidence Risk Estimation Methods in the Mainstream and its Tributaries

The coincidence of floods in the mainstream and its tributaries may lead to a large flooding in the downstream confluence area, and the flood coincidence risk analysis is very important for flood prevention and disaster reduction. In this study, the multiple regression model was used to establish the functional relationship among flood magnitudes in the mainstream and its tributaries. The mixed von Mises distribution and Pearson Type III distribution were selected to fit the probability distribution of the annual maximum flood occurrence dates and magnitudes, respectively. The joint distributions of the annual maximum flood occurrence dates and magnitudes were established using copula function, respectively. Fuhe River in the Poyang Lake region was selected as a study case. The joint probability, co-occurrence probability and conditional probability of flood magnitudes were quantitatively estimated and compared with the predicted flood coincidence risks. The results show that the selected marginal and joint distributions can fit observed flood dataset very well. The coincidence probabilities of flood occurrence dates in the upper mainstream and its tributaries mainly occur from May to early July. It is found that the conditional probability is the most consistent with the predicted flood coincidence risks in the mainstream and its tributaries, and is more reliable and rational in practice.

Image classification with multi-view multi-instance metric learning

Image classification is a critical and meaningful task in image retrieval, recognition and object detection. In this paper, three-side efforts are taken to accomplish this task. First, visual features with multi-instance representation are extracted to characterize the image due to the merits of bag-of-words representations. And a new distance function is designed for bags, which measures the relationship between bags more precisely. Second, the idea of multi-view learning is implemented since multiple views encourage the classifier to be more consistent and accurate. Last but not least, the metric learning technique is explored by optimizing the joint conditional probability to pursue view-dependent metrics and the importance weights of the newly-designed distance in multi-view scenario. Therefore, we propose a multi-view multi-instance metric learning method named MVMIML for image classification, which integrates the advantages of the multi-view multi-instance representation and metric learning into a unified framework. To solve MVMIML, we adopt the alternate iteration optimization algorithm and analyze the corresponding computational complexity. Numerical experiments verify the advantages of the new distance function and the effectiveness of MVMIML.

Joint probability of drought encounter among three major grain production zones of China under nonstationary climate

Better understanding for the likelihood of drought occur in the main grain production zones of China is key to minimize the risk for food security, especially in a changing climate. In an attempt to encompass the climate variability in the assessment of drought, a nonstationary standardized precipitation index (NSPI) was established for three major grain production zones (Songnen Plain (SN), North China Plain (NC), and Sichuan Basin (SC), hereinafter referred to GPZs) by means of using the climate indices as covariates. Then three commonly used Archimedean copulas (i.e., Clayton, Frank, and Gumbel-Hougaard) were adopted to determine the joint drought probability model. Using the joint probability distribution, the risk of drought in different degree occurring among the three GPZs, conditional joint probability, and conditional return period of different drought combinations of GPZs were analyzed. The results show that NSPI reflected the drought conditions well in three GPZs, and the copula-based joint probability distributions of different drought situations among the three GPZs were reasonable. For the situation of droughts encounter in two GPZs, synchronous droughts in different degree are more prone to encounter in NC & SC, and the conditional joint drought probability (return period) is also higher (lower) in the combination of SC and NC. For the situation of droughts encounter in three GPZs, the probability of synchronous droughts in the three GPZs is decreasing with increase of drought degree. If a specifically drought (NSPI ≤ -1) occurred in two GPZs, the conditional joint drought probability is highest in NC, then followed by SC, and SN. The findings of this study could provide important implications of decision-making for food security.

On the genotype frequencies and generating function for frequencies in a dyploid model

For a locus with two alleles ($I^A$ and $I^B$), the frequencies of the alleles are represented by $$ p=f(I^A)= \frac {2N_{AA} +N_{AB}} {2N} , q=f(I^B)= \frac {2N_{BB} +N_{AB}}{2N}$$ where $N_{AA},~N_{AB}$ and $N_{BB}$ are the numbers of $I^A I^A ,~I^A I^B$ and $I^B I^B$ respectively and $N$ is the total number of populations. The frequencies of the genotypes expected are calculated by using $p^2 ,~2pq$ and $q^2$. So in this paper, we consider the method of whether some genotypes is in Hardy-Weinburg equilibrium. Also we calculate the probability generating function for the offspring number of genotype produced by a mating of the $i$th male and $j$th female under a diploid model of $N$ population with $N_1$ males and $N_2$ females. Finally, we have conditional joint probability generating function of genotype frequencies.

Dynamic reliability analysis of a floating offshore wind turbine under wind-wave joint excitations via probability density evolution method

Floating offshore wind turbine (FOWT) towers are dynamically sensitive to wind and wave excitations. Since the wave tends to be harsher with the increase of wind speed, FOWT towers are likely to experience the most severe vibration operating at the cut-out wind speed. In the present study, the short-term dynamic reliability of a spar-type FOWT is evaluated based on the probability density evolution method (PDEM). For this purpose, an integrated coupled dynamics model for the FOWT is firstly established by incorporating the multibody dynamics with the finite element (FE) method. Next, the conditional joint probability distribution of the significant wave height and peak spectral wave period at the cut-out wind speed is constructed based on the copula model. Then, the stochastic dynamic response and reliability of the FOWT can be analyzed via PDEM. The numerical example of reliability analysis of a 5-MW spar-type FOWT operating at the cut-out wind speed is carried out, in which the long-term met-ocean data at a South China Sea site is utilized. Simulation results show that the reliability of the FOWT for normal operation is less than 0.2 when the acceleration at the tower top is adopted as the failure criterion.

Brain transcriptome wide association study (TWAS) implicates 8 genes across 6 loci in Alzheimer’s disease

AbstractBackgroundAlzheimer’s disease (AD), an incurable neurodegenerative disease, currently affects 1.75% of the United States population, with projected growth to 3.46% by 2050. Common genetic variation linked transcript expression differences which confer AD‐risk isl necessary to elucidate AD mechanism and develop therapeutic interventions. We modified the FUSION pipeline to ingest expression from multiple neocortical regions, provide a set of 6780 gene weights which is abstracatable across the neocortex, and leverage these to find 8 genes with associated AD risk validated through summary mendelian randomization (SMR) utilizing IGAP summary statistics.MethodAncestry matching based on 1000 genomes supervised clustering utilizing yielded 2003 Northern European (CEU) profiles, 888 of which had matched genotype‐RNASeq profiles. Expression Z‐scaling within‐tissue allowed weights to be trained from multiple Neo‐cortical tissues (TCX = 248, FP = 50, IFG = 41, STG = 34, PHG = 34, DLPFC = 461) after RNA‐Seq profiles were corrected for covariate and AD diagnosis stage within study.ResultCis‐variant architecture alone was informative to train weights for 6780 (49.67%) autosomal genes, the majority of which signficantly correlated with gene expression; FDR < 5%: N = 6775 (99.92%), Bonferroni: N = 6716 (99.06%). The median correlation to observed gene expression was 0.2176 (IQR = 0.179‐0.277). Validation rate in 621 matched genotype to RNASeq profiles from the CommonMind Consortium (CMC) was (71.08%) in DLPFC and (22.52%) in anterior cingulate cortex (ACC) profiles. After correction for joint conditional probability (JCP) and SMR validation, 8 genes were significantly associated with AD (FDR < 0.05); APOC1 (JCP = 2.22e‐22, SMR = 3.41e‐4), EED (JCP = 3.373e‐5, SMR = 2.50e‐4), CD2AP (JCP = 2.96e‐5, SMR = 2.66e‐4), CEACAM19 (JCP = 3.27e‐5, SMR = 1.00e‐2), CLPTM1 (JCP = 4.04e‐3, SMR = 2.58e‐3), MTCH2 (JCP = 0.011, SMR = 3.32e‐6), TREM2 (JCP = 0.021, SMR = 2.64e‐3), KNOP1(JCP = 0.039, SMR = 2.50e‐4).ConclusionWe provide evidence of cis‐genetic variation confering AD risk through 8 genes across six distinct genomic loci. Moreover, we provide a valuable resource to the community in the form of predictive gene expression weights that are abstractable to multiple neocortical regions and neuronal phenotypes.

Modeling the dependence pattern between two precipitation variables using a coupled copula

Hydrological process is very complex, so it is difficult for one copula to describe dependence patterns between two hydrological variables comprehensively (dependence pattern refers to the correlation and tail dependence between two random variables). This paper applied a linear weighted function of Gumbel copula, Clayton copula and Frank copula (coupled copula) to study dependence patterns between two hydrological variables and take precipitation as an example. Two experiments to study the joint probabilistic characteristics of the daily precipitation sequences in summer at two pairs of stations on the tributaries of Jinghe are performed to test our new method and compared with Gumbel copula, Clayton copula and Frank copula. Both experiments indicate that the coupled copula is superior to study the upper tail dependence, lower tail dependence and symmetric tail dependence between two precipitation sequences simultaneously. Moreover, the coupled copula is applied to estimate the joint return periods and conditional probabilities, and the joint return periods are 57.5 and 59.6 when the designed return period is 100. The result shows that there is a high probability of occurrence of precipitation extremes at the Huanxian and Xifeng stations when once in a 1000 or 100 years daily precipitation occur at the Guyuan and Pingliang stations. The coupled copula can also be applied in flood and drought frequency analysis.