diff --git "a/69E4T4oBgHgl3EQfCAuy/content/tmp_files/load_file.txt" "b/69E4T4oBgHgl3EQfCAuy/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/69E4T4oBgHgl3EQfCAuy/content/tmp_files/load_file.txt" @@ -0,0 +1,1090 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf,len=1089 +page_content='Neural Spline Search for Quantile Probabilistic Modeling Ruoxi Sun1*, Chun-Liang Li1*, Sercan Ö.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Arık1, Michael W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Dusenberry2, Chen-Yu Lee1, Tomas Pfister 1 1Google Cloud AI 2Google Research, Brain Team {ruoxis, chunliang, soarik, dusenberrymw, chenyulee, tpfister}@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='com Abstract Accurate estimation of output quantiles is crucial in many use cases, where it is desired to model the range of possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Modeling target distribution at arbitrary quantile levels and at arbitrary input attribute levels are important to offer a compre- hensive picture of the data, and requires the quantile function to be expressive enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The quantile function describing the target distribution using quantile levels is critical for quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Althought various parametric forms for the distri- butions (that the quantile function specifies) can be adopted, an everlasting problem is selecting the most appropriate one that can properly approximate the data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In this paper, we propose a non-parametric and data-driven approach, Neural Spline Search (NSS), to represent the observed data distribution without parametric assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS is flexible and expressive for modeling data distributions by transform- ing the inputs with a series of monotonic spline regressions guided by symbolic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We demonstrate that NSS out- performs previous methods on synthetic, real-world regression and time-series forecasting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Introduction For many machine learning applications, modeling the pre- diction intervals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' estimating the ranges all individual predictions observation fall), beyond point estimates, is cru- cial (Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Tagasovska and Lopez-Paz 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The prediction intervals can help with decision making for retail sales optimization (Simchi-Levi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2008), medi- cal diagnoses (Begoli, Bhattacharya, and Kusnezov 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Mhaskar, Pereverzyev, and van der Walt 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2012), information safety (Smith, Dinev, and Xu 2011), fi- nancial investment management (Engle 1982), robotics and control (Buckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018), autonomous transformation (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2014) and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' To estimate prediction intervals, we would need to estimate different levels of quantiles for the target distribution using quantile regression (Koenker and Regression 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Wald- mann 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' A real-world challenge is to select the paramet- ric forms of target distributions, which is specified by the quantile function (also known as the inverse CDF function), These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 X 2 1 0 1 2 Y X=X0 P(Y|X) True Quantile 25% Quantile 75% Figure 1: Modeling multiple quantiles at different condition-levels with a universal quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The goal is to model target data distribution y at any arbitrary quantile level and attribute level X, using one versatile quan- tile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gray dots are observed data points, while green and blue lines indicate 25% and 75% quantile levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The data distribution y varies at different levels of X, say variance of y increases when X is away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Red dots are data points at X = X0, p(Y |X0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' to properly align with observed data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Different choices for the target distribution (Gaussian, Poisson, Neg- ative Binomial, Student-t etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=') may yield different quantile predictions, and misalignment of the assumption with the real distribution may hinder the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' There- fore, such heuristic or empirical hand-picking based paramet- ric assumptions for the distribution can be sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' An approach based on learning from the data in an automated way, would be highly desirable, from both foundational and practical perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' For learnable parametric modeling, one challenge is how to model all quantiles for all input attributes level in a com- putationally efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' First, modeling an any arbitrary quantile, as opposed to a couple of pre-defined quantile levels, offers a more comprehensive view on the target distribution, and provides convenience to use the quantile model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' no need to re-train the model when quantiles at testing are dif- ferent from the ones at training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Second, real-world data can have complex distributions beyond what simple assumptions can model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 1 shows different input attribute X levels arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='04857v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='AI] 12 Jan 2023 0 1 2 3 4 5 Y|X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='020 Probability Density PDF(Y|x) 0 1 2 3 4 5 Y|X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 Probability Density CDF(Y|x) Figure 2: An example target distribution with a complex shape, in PDF and CDF space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Black lines are observed tar- get distributions, in the form of mixture of the other three dis- tributions shown with color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fitting the black line accurately would be extremely difficult for most of the commonly-used single parametric splines, motivating for the use of learnable spline family composed of multiple splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' have different dependency dynamics with target y level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' the variance of y increases when X apart from 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2 shows that the observed distribution cannot trivially fit well with one single distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Therefore, in order to model all quantiles at all X, we need a quantile function with a com- plexity that does not increase significantly with number of input attributes and the number of quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This necessitates a versatile and highly-expressive quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' There has been many efforts on improving various aspects of quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (2019) proposes linear spline interpolation between knots in the inverse CDF space to model the target distribution in time-series forecasting setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This is proposed to avoid the assumption on paramet- ric form of the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (2022) and Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (2021) focus on learning a valid quantile function with- out quantile crossing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' quantiles violate monotonically increasing property), via special design of the neural network architecture or first-order inequality constraint optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Despite being distribution agnostic, these approaches for de- scribing the target distribution (specified by quantile function) are restricted to one function family (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' linear spline), which may limit the expressiveness to represent the target distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In this paper, with the goal of designing an expressive quantile function for various quantiles and input levels, we propose a data-driven approach Neural Spline Search (NSS), which transforms the inputs with a series of monotonic spline regressions guided by symbolic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The contributions of our paper can be summarized as: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We propose an efficient search space and mechanism to find an expressive quantile function to model the data distribution, avoiding specifying a parametric form of the observed distribution as prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We propose a novel approach to generate an expressive quantile function using a combination of different distri- butions and operators guided by symbolic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The proposed method can be incorporated into other tasks (including but not limited to time series forecasting) as their quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We demonstrate significant accuracy improvements across numerous regression or time series forecasting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' For example, on UCI benchmarks, we show 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5%-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0% im- provement compared to next best methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Related Work Quantile regression is used to estimate the target distribu- tion at different quantile levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The α-quantile estimator is the solution when minimizing quantile loss at level α (Koenker and Bassett Jr 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Another quantile regression related loss is continuous ranked probability score (CRPS) (Gneiting and Raftery 2007), which is the averaging over all quantile levels, instead of one single quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Neural network quantile forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' To model sequential dependency of time series, several forecasting models pro- pose a hidden state-emission framework ((Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' de Bézenac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019)), where the dynamics of hidden states are modeled by auto-regressive recurrent neural works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' LSTM), which takes previous hidden states and current ob- servations as input and outputs current observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Different from modeling the likelihood with parametric distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gaussian (Salinas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2020)), emission models for quantile estimation is to learn the parameters of quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The overall framework is optimized by employing a quantile (Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2017) or CRPS (Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Symbolic regression has shown great success in many fields, including program synthesis (Parisotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2016), mathe- matical expressions extraction (Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2020), physics- based learning (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' As the search space is enormous and scaled exponentially with the length of operators, symbolic regression rule operators are usually set to be a small number and are learned by Monte Carlo Tree Search guided evolutionary strategies (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019) or reinforcement learning (Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Methods Learning quantile function in quantile regression Let the input data attributes X and the target variable y are jointly distributed as p(X, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The conditional cumulative distribution function (CDF) is F(Y = y|X) = P(Y ≤ y|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The quantile function, which is also called the inverse CDF function, takes quantile level as inputs and returns a threshold value Y below which random draws from the given CDF would fall quantile percent of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Specifically, the α-th quantile function of y|X = x is denoted as: q(α, x) = F −1 y|X=x(α) = inf{y : F(y|X = x) ≥ α} (1) Here we can think the quantile function is to perform a transformation on a uniform-distributed random variable α ∼ U(0, 1) to the target distribution p(y|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Quantile function is able to fully specify a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' So specifying the quantile function is describing the target distribution p(y|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Quantile regression estimates different conditional quantile levels of the target variable given a certain level of input P(y|X) Inverse CDF alpha y P(y|X) Inverse CDF alpha y P(y|X) Inverse CDF alpha y P(y|X) Inverse CDF alpha y Spline Basis P(y|X) Inverse CDF alpha y S C + P(y|X) Inverse CDF alpha y Spline Basis P(y|X) Inverse CDF alpha y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. P(y|X) Inverse CDF alpha y P(y|X) Inverse CDF alpha y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. P(y|X) Inverse CDF alpha y P(y|X) Inverse CDF alpha y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. NSS-sum Initial distribution target distribution Spline Basis Spline Basis NSS-chain P(y|X) Inverse CDF alpha y Operators + Sum S Scale C Chaining .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='. Figure 3: Overview of Neural Spline Search (NSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Modeling the target data distribution can be done by learning the quantile function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' inverse CDF), which maps a [0, 1]-variable (quantile) to a target value y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Unlike parametric methods which specify a distribution family and learn the parameters, NSS can generate the target distribution through a set of transformations on the inverse CDF space (quantile space), where the transformation is guided by a series of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Here, the bottom gray box shows possible operators (denoted as circles), including but not limited to summation (“+”), scale (“S”), and chaining (“C”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The basis splines are shown with color-shaded squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The initial distribution is a uniform distribution, as shown in the leftmost panel (blue shaded), and the target distribution is the rightmost distribution (purple shaded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' There is no obvious parametric distribution to achieve this transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Therefore, NSS is used to search for the suitable transformation through simple operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In the first row of the middle panel, we show operators for NSS-sum, where the initial uniform distribution is transformed by the red- and the yellow-shaded splines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' c-spline) through sum (“+”) and scale (“S”) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The second row shows the chaining transformation of the initial distribution, where the orange and cyan splines are used to transform the initial spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The parameters of the splines are learned by a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In general, the operators and transformations in NSS are not limited to two splines (we represent them as the gray splines next to the yellow and cyan shaded splines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' attributes, as opposed to regression, which estimates the con- ditional mean of the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In quantile regression, a particular quantile level α of the conditional distribution of y given X = x, q(α, x) is estimated by minimizing the pinball loss ρ (or quantile loss), as the the quantile function q is shown to be the minimizer of the expected pinball loss (Koenker and Bassett Jr 1978): ρα(y, q) = (y − q)(α − 1(y < q)), (2) q(α, x) = arg min q Ey[ρα(y, q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (3) where 1 is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' One shortcoming of pinball loss is only measuring the loss at a single quantile level, which hinders the estimated q for a global picture of the distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' other α levels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' On contrast, the continuous ranked probability score (CRPS) considers all quantile levels by integrating the pinball loss over α = [0, 1] (Matheson and Winkler 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gneiting and Raftery 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' CRPS(y, q) = � 1 0 2ρα(y, q)dα (4) As a proper scoring rule (Gneiting and Raftery 2007), CRPS is minimized when the quantile function is q = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' That is, F −1 y = arg min q Ey[CRPS(y, q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (5) Please refer (Koenker and Regression 2005) for detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Improving the expressiveness of quantile function Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2 demonstrate the need of an expressive quantile func- tion for modeling target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Inspired from neural architecture search (NAS) (Elsken, Metzen, and Hutter 2019), we propose an approach to search for the suitable combina- tion of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The search is over different operations and basis distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We first introduce parametrization of quantile function, and the two non-parametric spline-based distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Parameterizing quantile functions We propose to param- eterize the quantile function qθ(α, x) using a deep neural network with parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The quantile function is aimed to be accurate for any quantile levels α and input attributes level X = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' X is high dimensional in real data, not as the one dimensional in the toy examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' C-spline distribution The c-spline (yα = qcsplie θ (α, x)) describes the CDF (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2, Right Panel) of a probability distribution Fy|X by setting K anchor points (denoted as knots) on the CDF curve and performing linear interpolation to fill in the gap between the knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Specifically, the knots split CDF curve into bins and c-spline learns the width wi and height hi of bins by neural networks NN that depend on the input attributes level X = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' {wi, hi}K = NNθ(x) yα = r({wi, hi}K, α) ∀α ∈ [0 : 1] where hi and wi are non-negative delta values imposed by non-negative activation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Relu or Sigmoid), and the loca- tion of each bin (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Y|X) is Li = �i k=0 wk and quantile level αi = �i k=0 hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The accumulation sum design is to en- sure that quantile function is monotically increasing and there is no quantile crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' r is a function to convert knots to output of quantile function: for quantile level αi that is on the knots, we can directly read from li , for quantile levels that are off the knots, quantile values can be computed through linear algebra operations on the two nearby knots r(α) = � li + (α−αi)(lj−li) αj−αi , if αi ≤ α ≤ αj 0 ≤ i, j ≤ K lk, if hk = α P-spline distribution The difference between p-spline from c-spline is having anchor knots in PDF space, instead of CDF space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Similarly with C-spline, P-spline also per- form linear interpolation over knots, and the quantile level is achieved by integration over pdf via polynomial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Neural Spline Search (NSS) We describe our proposed method, Neural Spline Search (NSS), which is overviewed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Similar to symbolic regression (Parisotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019), NSS effec- tively searches in the space of discrete symbolic operators and distribution space for a candidate that can better fit the target data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Specifically, let T(O, S, k) denote the space of all transformations, via operators O on all distribu- tion S with a maximum sequence length k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS aims to find the function f(x) selecting operators and distributions in the space T such that {f(x) ∈ T(O, S, k) : ℓ(f(x), xtrain) ≤ δ }, where ℓ denotes loss function CRPS, xtrain is training data and δ is the acceptance threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Given the large search space composed of combinations of numerous splines and operators, we restrict to use spline-based distribution as the basis distribution, and limit the operator search space to sum- mation and chaining operations upon the transformation basis spline regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Note that this work can be easily extend to other operations and distributions, which we leave to fu- ture work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We describe the following NSS transformations as they are observed to work well consistently across different datasets: NSS with summation (NSS-sum) and NSS with chaining (NSS-chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Algorithm 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 4(b) NSS-sum NSS-sum performs transformations using the scale and sum- mation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We represent this scenario with two splines: Spline 1: c-spline and Spline 2: p-spline, and two operators: scale O1 : O(a) = λa and summation O2 : O(a, b) : a + b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' therefore, the overall transformation is (Spline 1-Operator 1) - (Spline 2-Operator 2), which yields: f = c-spline + λ p-spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Essentially, NSS-sum performs weighted sum of different splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The motivation behind is that c-spline with fewer parameters can be more robust against overfitting, whereas p-spline increases the expressiveness of the splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS-chain Another proposed NSS design is NSS-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We focus on the chaining operator due to its expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This design is inspired by the success of normalizing flow (Rezende and Mohamed 2015), where a sequence of bijector transforms is utilized to transform distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Different from normaliz- ing flow which has practical applicability challenges, NSS- chain only requires the forward pass of the transformation, not the inverse as normalizing flow does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This significantly reduces the computational complexity and broadens the fea- sibility of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' As mentioned, quantile function takes input attributes level (X) to predict the target value (y) at quantile level (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' y = qθ(X, α), (6) where X ∈ Rm and α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We present two designs to chain different transformations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 4 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We note that chaining of transformation is not limited to the two designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Algorithm 1: Neural Spline Search Operators = {+, ×, Scale, Chain, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='} Splines = {c-spline, p-spline, Gaussian, Cauchy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='} Data: Quantile level α ∈ [0, 1], N data points {X ∈ Rd, y ∈ R1}N, d ≥ 1, with chain depth k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Transform indicates the transformation using the input spline Sθ and operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Result: p(y|X) and F −1 y|X(α) k ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' while k ≤ K do Select O = {Oi}no ∈ Operators ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Select S = {Sj}ns ∈ Splines ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' θ ← MLP(X) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' ypred ← Transform(Sθ, O, α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' if α NSS-chain then Normalize ypred to [0, 1] as y′ pred ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' α ← y′ pred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' else X ← Y ▷ if X-NSS-chain ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' end k ← k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' end α-chaining The α-chaining is when we consider the condition level (X) unchanged during the chain of transformation, and the output of each transformation is a scaled version of quantile level for the next transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In particular, after each transformation, we normalize the output y to be in the range [0, 1], and then the normalized output is re-input as the new α to the next transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This is repeated until the maximum depth is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' This design is more similar with normalizing flow methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' y = qθK(X, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='fn(qθ2(X, fn(qθ1(X, α))))) (7) θk for k=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='.K are parameters for different splines in K-length chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' fn is the normalization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=" X-chaining X-chaining is when we consider quantile level α level is unchanged during chaining, as each transformation learns alpha P(y|X) y_alpha Inverse CDF alpha y MLP X Spline's parameters alpha P(y|X) y_alpha Inverse CDF alpha y MLP X Spline's parameters x-chaining alpha P(y|X) y_alpha Inverse CDF alpha y MLP X Spline's parameters alpha-chaining NNS Chain Figure 4: (a) Illustration of NSS-chain methods." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The dia- gram demonstrates chaining for NSS-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Left: α-chaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The output y of the spline, after re-scaling to [0, 1], is re- inputted to the quantile spline at quantile level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Right: X- chaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The output y is instead re-inputted to the quantile spline as X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Both rely on input attributes X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' a suitable condition level (or feature) for next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Similarly with α-chaining in the iterative manner, except that the output y of each transformation is projected to generate X for the next iteration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' y = qθK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='qθ2(qθ1(X, α), α), α) (8) The advantage of this approach, compared tp α-chaining, is that we keep quantile levels α unchanged, and re- normalizing output is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Remarks on NSS: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (1) why a simple spline-based algo- rithm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' C-spline, is not enough?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Although in theory spline-based algorithms can represent any arbitrary distribu- tions with sufficiently high number of knots K, in practice, we find a large K often lead to unstable training, as also studied in (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In contrast, we find the combina- tion (combined or chained) over a relatively restricted splines are more robust in capturing the overall of the target distribu- tion (2) Include both spline-based distribution and classic parametric distribution In addition to spline-based distribu- tion, we also encourage incorporating parametric distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gaussian) as basis distribution for NSS, especially when prior knowledge (say Gaussian noise) is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Because, it is challenging for spline based methods to reconstruct Gaus- sian distribution even with infinite number of knots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' and , the benefits of combining the two are the parametric distribution offers advantage of classic statistics and robust to noise, and the non-parametric spline offers flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Training Once we select the operators and splines, the parameters of the splines are trained in an end-to-end way by optimizing CRPS (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Specifically, during training, we fit parameters by optimizing over with the empirical mean of CRPS over N data points: θ∗ = arg min θ 1/N N � i=1 Ey[CRPS(y, qθ(Xi, α))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' (9) Algorithm 2 overviews the training of NSS for spline parame- ter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Because of the form of the transformations, the analytical solution of CRPS integration is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Thus, we use a Monte Carlo estimation for the CRPS loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In par- ticular, we sample m number of α values from the range of [0, 1] and average them for the corresponding pinball loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Algorithm 2: Training with CRPS Data: N data points {Xi ∈ Rd, yi ∈ R1}N i=1, m quantile levels, T transformation, which takes selected splines Sselect and selected operators Oselect from NSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' lr is learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Result: Neural network weights θ e ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' while e ≤ Nepoch do f = Transform(Sselect, Oselect) ℓ ← 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' for α in [0, 1 m, 2 m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='.1] do ypred α = fθ(X, α) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' ℓ ← ℓ + pinball_loss (ypred α , y, α) end CRPS = ℓ/m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' θ ← θ − lr · ∇θ CRPS ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' e ← e + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' end Experiments Comparison methods QD (Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018) generates prediction intervals (PIs) for estimating uncertainty for regression tasks with the as- sumption that high-quality PIs should be as narrow as possi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Deep Quantile Aggregation (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2021) proposes weighted ensembling strategies where aggregation weights vary over both individual models and feature values plus (pairs of) quantile levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The monotonization layer in the network is applied to avoid crossing of quantile estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' RQspline (Durkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019) proposes a fully-differentiable module based on monotonic rational-quadratic splines, which enhances the flexibility of coupling and autoregressive trans- forms while retaining analytic invertibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Global-Coarse (Ratcliff 1979) provides an analysis of distribution statis- tics of group reaction time distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' MLE (NB) and Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' MLE are Negative Binomial and mixture likelihood based methods (Awasthi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' C-spline is proposed in (Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019), where C-spline is used as the quantile function in time-series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Metrics For point predictions, we focus on the following metrics: Mean absolute error (MAE): 1 n �n t=1 |Tt − Pt| where Tt and Pt are true and predicted value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Mean Absolute Percentage Error (MAPE): 1 n �n t=1 | Tt−Pt Tt |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Weighted Average Percent- age Error (WAPE): �n t=1 |Tt−Pt| �n t=1 |Tt| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' and Root Mean Square Error (RMSE): � �N t (Tt−Pt)2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' For quantile predictions, we use the Pinball Loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2), with 50%-th, Q50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 90%-th, Q90;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' and 10%-th Q10 quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Methods Boston Concrete kin8nm Power Protein Wine Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0978 QD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='3945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='3688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4456 RQspline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0912 p-sline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0570 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0453 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0966 c-spline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0447 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0947 NSS-X-chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0588 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0962 NSS-α-chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0976 NSS-sum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0442 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0957 Gain percentage 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='7% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='7% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6% Table 1: Mean Absolute Error (MAE) on UCI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Test performance of the proposed method (NSS) and existing methods on UCI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We use the 50th quantile estimator as our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The dash indicates unavailability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The shaded area is the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Bold is the top one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gaussian: Gaussian kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' QD is quantity-driven methods proposed in (Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' RQ spline proposed in (Durkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' c-spline proposed in (Gasthaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Boston, Concrete, Power is short for Boston Housing, Concrete Strength, Power Plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Gain percentage is computed as (best nss - best baseline)/best baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Training For simplicity, the proposed NSS methods use depth- 2 splines, which contain {(c-spline, p-spline), (c- spline, p-spline), (c-spline, c-spline), (p-spline, p- spline)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS-sum is tuned with λ in the range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS-chain nor- malizing of y in α chaining can be achieved by applying sigmoid layer or scaling by max value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' As splines are monotonically-increasing functions, the spline value y with α = 0 is the minimum value of y and α = 1 yields the maximum value of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Scale is yscale = y−ymin ymax−ymin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We use a batch size=128 and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='005 for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Results To demonstrate the effectiveness of proposed methods, we conduct experiments on synthetic, real-world tabular regres- sion, and time series forecasting datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Synthetic data Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We generate 2000 data points (X ∈ R1 and y ∈ R1), where X is in the range of [−2, 2] and y has Gaus- sian distribution y ∼ N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='3 sin(3x), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2x2), where sin is the sinusodial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We construct the validation and test sets to come from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Unlike real-world data, the synthetic data would have known quantile levels, that can be used for evaluating the accuracy of quantile estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We make the task more challenging by setting a data-dependent variance for the Gaussian noise to evaluate the ability of learn- ing condition-specific quantile values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 5 shows that the proposed NSS-chain and NSS-sum can capture the true under- lying quantiles, whereas QD (Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018) struggles on the varying variance locations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' around x = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The upper and lower black lines are the predicted 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5%-th and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5%-th quantiles for the observed data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' red dots), shown along with the ground truth quantiles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' shaded red area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The results indicate that more expressive NSS transformations are superior in more challenging scenarios, where true data points are distributed differently (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=', distributions depend on the value of the inputs").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 6 shows the calibration plot of the predicted vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' true distributions at different quantile 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 x 2 1 0 1 2 y QD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 x 2 1 0 1 2 y NSS-SUM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 x 2 1 0 1 2 y NSS-CHAIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Figure 5: NSS on Synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We compare the per- formance of proposed NSS against existing methods QD (Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The red dots are observed data points, shaded red area is the ground truth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5% quantile levels, and the dark black lines are the predicted 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5% quantile levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 True percentile 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 Predicted percentile QD NSS-SUM NSS-CHAIN 2 0 2 X y X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 True percentile Calibration plot QD NSS-SUM NSS-CHAIN 2 0 2 X y X=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 True percentile QD NSS-SUM NSS-CHAIN 2 0 2 X y X=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5 Figure 6: Calibration plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Predicted vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' ground truth per- centiles at condition levels: X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The perfect calibration would correspond to the diagonal (red dotted) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Here, we show the true percentile p as the fraction of data in the dataset such that the p percentile of the predictive distribution is larger than the ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The perfect prediction would be the diagonal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 6 indicates that the proposed methods NSS-sum and NSS-chain can capture the proposed true distribution at various levels by close to the red line, whereas QD does not fit as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Real-world tabular regression We use UCI benchmarks (Asuncion and Newman 2007) that contain tabular data from diverse domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' real estate and physics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Following (Salem, Langseth, and Ramampiaro 2020), the datasets are normalized with z-score standardiza- Methods Boston Concrete kin8nm Power Protein Wine Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0357 Global-Coarse∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1321 — Deep Quantile Aggregation∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1253 — QD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='1164 RQspline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0418 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0424 p-sline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0160 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0358 c-spline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0351 NSS-X-chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0707 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0358 NSS-α-chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0363 NSS-sum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0357 Gain percentage 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='5% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='6% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0% Table 2: Average pinball loss on UCI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The test pinball loss (the lower, the better) is over 99 quantile levels, α = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='02, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='99}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The compared methods are Global-Coarse proposed in (Ratcliff 1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' QD (Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Deep Quantile Aggregation (DQA) (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' RQspline (Durkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' ∗ indicates entries are from (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2021) (under the same experiment setup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Methods MAPE WAPE RMSE Q50 Q90 Q10 MLE (NB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='27240 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='70958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='27240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='15275 Mix MLE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26838 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='22556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14508 c-spline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26635 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='06332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14241 p-spline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26834 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14333 NSS-sum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26545 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='96697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14266 NSS-chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='44883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26420 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='91726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='26420 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='10243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='14149 Table 3: Performance comparisons for time series forecasting on M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Different evaluation metrics are included in this table for M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Detailed descriptions of the metrics are in Sec .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Qk indicates the pinball loss of k-th quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Q50 is the pinball loss of 50th quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We evaluate the accuracy for both point predictions and quan- tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' As the point predictions, we use the 50th quantile es- timator as our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Table 1 shows that the proposed NSS methods outperform the other existing methods on most datasets in mean absolute error (MAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In mean square error (MSE), the results are provided in Appendix Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We observe that the NSS-sum performs better than NSS-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' For quantile metrics, we use the pinball loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' 2) over 100 quantile levels α = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='02, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='99} in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The results indicate that NSS consistently outperforms other al- ternatives across different UCI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' In pinball loss, NSS-sum performs better than NSS-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We attribute the superiority of NSS-sum for regression to make balance be- tween different transformation, which is helpful in explaining the variance in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Retail demand forecasting For time series forecasting, we focus on the M5 dataset, which contains time-varying sales data for retail goods, along with other relevant covariates like price, promotions, day of the week, special events etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' It represents an important real-world scenario, where the accurate estimation of the output distribution is crucial, as retailers use them to optimize prices or promotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The time series forecasting experiments are conducted by performing one-step ahead prediction, yielding predictions in an autoregressive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Table 3 shows the results of our method compared to other alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We observe consistent outperformance of NSS in various forecasting evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Different from regression tasks, we observe that NSS-chain is better than NSS-sum, indicating its benefit in capturing time-dependent relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Remarks on NSS-sum vs NSS-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The results show that NSS-sum is superior on regression, while NSS-chain has advantages on time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The observations may indicate NSS-sum is suitable for more constrained tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' regression, one time step time series-forecasting), where be- ing moderately expressive would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' NSS-sum is also more robust and easier to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' On the other hand, NSS-chain may be more expressive, which is beneficial to fit tasks re- quires more complex distributions at different time steps of the time series, but for individual step NSS-chain is not as accurate as NSS-sum in fitting the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' Conclusion We propose a novel approach for modeling uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' The proposed Neural Spline Search (NSS) method employs a se- ries of monotonic spline regression transformations, guided by symbolic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We demonstrate the effectiveness of NSS for superior modeling of output distributions, on both synthetic and real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' We leave the extensions to different operators and splines, including parametric distribu- tion transformations to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=' References Asuncion, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E4T4oBgHgl3EQfCAuy/content/2301.04857v1.pdf'} +page_content=';' metadata={'source': 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