Gaussian process model for vector-valued function has been shown to be useful for multi-output prediction. The existing method for this model is to re-formulate the matrix-variate Gaussian distribution as a multivariate normal distribution. Although it is effective in many cases, re-formulation is not always workable and is difficult to apply to other distributions because not all matrix-variate distributions can be transformed to respective multivariate distributions, such as the case for matrix-variate Student−t distribution. In this paper, we propose a unified framework which is used not only to introduce a novel multivariate Student−t process regression model (MV-TPR) for multi-output prediction, but also to reformulate the multivariate Gaussian process regression (MV-GPR) that overcomes some limitations of the existing methods. Both MV-GPR and MV-TPR have closed-form expressions for the marginal likelihoods and predictive distributions under this unified framework and thus can adopt the same optimization approaches as used in the conventional GPR. The usefulness of the proposed methods is illustrated through several simulated and real data examples. In particular, we verify empirically that MV-TPR has superiority for the datasets considered, including air quality prediction and bike rent prediction. At last, the proposed methods are shown to produce profitable investment strategies in the stock markets.

The hyperparameters in Gaussian process regression (GPR) model with a specified kernel are often estimated from the data via the maximum marginal likelihood. Due to the non-convexity of marginal likelihood with respect to the hyperparameters, the optimization may not converge to the global maxima. A common approach to tackle this issue is to use multiple starting points randomly selected from a specific prior distribution. As a result the choice of prior distribution may play a vital role in the predictability of this approach. However, there exists little research in the literature to study the impact of the prior distributions on the hyperparameter estimation and the performance of GPR. In this paper, we provide the first empirical study on this problem using simulated and real data experiments. We consider different types of priors for the initial values of hyperparameters for some commonly used kernels and investigate the influence of the priors on the predictability of GPR models. The results reveal that, once a kernel is chosen, different priors for the initial hyperparameters have no significant impact on the performance of GPR prediction, despite that the estimates of the hyperparameters are very different to the true values in some cases.

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The double-ring infiltrometer is widely used to measure soil unsaturated hydraulic conductivity in the field. The scale effect of the inner and outer ring size (especially the inner one) affects the measurement results. In the semi-arid steppe, where water is scarce and transportation is inadequate, studying the scale effect caused by the inner-ring diameter of the infiltrometer can reduce the test consumption on the premise of ensuring the test accuracy. In this paper, a total of 190 double-ring infiltration tests with different inner-ring diameters (15,20,25,30, and 40 cm) and 0.33 times outer buffer index were carried out at 38 sites with five soil types in the Xilin river basin, China. Results showed that: (1) When comparing the simulated parameters of six infiltration models, parameters increased with the increase of the infiltrometer inner diameter, but the trend gradually slowed down, indicating that the increase of the infiltrometer inner diameter would weaken the influence of the infiltrometer scale effect. However, the infiltrometer with an inner diameter of 40 cm is not enough to completely overcome the scale effect. (2) Through principal component analysis, the infiltration process is mainly affected by the particle size and the initial moisture content. (3) The soil infiltration map based on infiltration tests was more practical than the soil type map, which can provide a theoretical basis for ecological and soil restoration in the future. species diversity does not reduce, but rather improves, the soil erosion resistance [13,14], and finally allows the whole ecosystem to achieve a positive cycle [15].To achieve the goal of obtaining the soil water retention characteristics of the study area, it is necessary to accurately evaluate this capacity, which can be estimated by the final infiltration rate (FIR) and infiltration time when the infiltration rate first reaches the final infiltration time (FIT). A slower infiltration rate indicates that the soil has better water retention [10]. The process of soil infiltration can be simulated in the laboratory using the ring-knife or a soil column [16,17]. However, such studies are influenced by disturbances of the original soil bulk density, porosity, and other soil physical properties. Instead, an infiltration test based on in situ undisturbed soil is capable of reflecting the original state of soil more effectively.Compared with a single-ring infiltrometer, a double-ring infiltrometer can be used to construct an infiltration buffer, which reduces the main external interference factors on the infiltration process of the inner ring [18]. Previous studies have shown that the average infiltration rate increases with the infiltrometer diameter [19,20]. Lai et al. [21] found through a large number of simulations that, compared with the outer buffering index (bi), the inner-ring diameter (di) plays a more important role in obtaining a relatively stable and representative measurement result. When the bi is equal or larger than 0.33, the buffer may meet the measureme...