This paper proposes a new approach to estimating the distribution of a response variable conditioned on observing some factors. The proposed approach possesses desirable properties of flexibility, interpretability, tractability and extendability. The conditional quantile function is modeled by a mixture (weighted sum) of basis quantile functions, with the weights depending on factors. The calibration problem is formulated as a convex optimization problem. It can be viewed as conducting quantile regressions for all confidence levels simultaneously while avoiding quantile crossing by definition. The calibration problem is equivalent to minimizing the continuous ranked probability score (CRPS). Based on the canonical polyadic (CP) decomposition of tensors, we propose a dimensionality reduction method that reduces the rank of the parameter tensor and propose an alternating algorithm for estimation. Additionally, based on Risk Quadrangle framework, we generalize the approach to conditional distributions defined by Conditional Value-at-Risk (CVaR), expectile and other functions of uncertainty measures. Although this paper focuses on using splines as the weight functions, it can be extended to neural networks. Numerical experiments demonstrate the effectiveness of our approach.
翻译:本文提出一种新的方法来估计以观察某些因素为条件的响应变量的分配情况。 提议的方法具有灵活性、 可解释性、可移动性和可扩展性等适当特性。 有条件的微量函数由基本微量函数的混合(加权和)制成, 其重量取决于各种因素。 校准问题被表述为一个二次曲线优化问题。 它可以被视为同时对所有信任水平进行量性回归, 同时根据定义避免孔径交叉。 校准问题相当于将连续的分级概率评分( CRPS)最小化。 基于高压分解的罐形多元( CP), 我们建议了一种降低参数拉力等级的维度削减方法, 并提议了一种交替的估算算法。 此外, 根据风险二次曲线框架, 我们普遍采用条件值- at- Risk( CVaR)、 预测性和其他不确定性计量功能界定的有条件分布方法。 虽然本文侧重于使用质谱作为重量函数, 但它可以扩展到神经网络。</s>