# Deep Regression Neural Networks for Proportion Judgment
## 1 Introduction  
People have the ability to distinguish between non-symbolic numerical magnitudes without counting, which is derived from the approximate number system (ANS) [1]. At the same time, various tasks require people to estimate ratios and proportions, comparing the magnitudes of two quantities [2]. Determining the proportion of open flowers to all the flower buds and flowers on the plant, estimating the ratio between a marked area and the total area of an image, and judging the share of a certain object in relation to the total number of objects on an image all serve as everyday examples. In this paper, we focus on the proportion judgment, where, by definition [2], an observer estimates the magnitude of one stimulus relative to another, larger stimulus, and based on that responds with a value between 0 and 1 (or between 0% and 100%). Proportion judgment can be seen as a special case of ratio judgment, where the observer estimates the ratio of two stimulus magnitudes. It should also be mentioned that, in everyday life, the terms proportion, ratio, fraction, or percentage are used interchangeably. Although proportion estimation can be an important prerequisite for decision-making (for example, plant protection with chemical or biological products, within optimal time-lines), numerous studies have shown that bias is systematically present in the assessment. Several authors have shown that small proportions are usually overestimated, and large proportions underestimated [3, 4]. Notably, there are fewer studies that discuss the reverse pattern, i.e., underestimation of small proportions and overestimation of largeproportions. Therefore, the problem of proportion estimation represents a suitable area for the application of artificial intelligence (AI) techniques. In computer vision, regression techniques can be applied in many fields, such as crowd counting, pose estimation, facial landmark detection, age and demographic analysis estimation, or image registration [5]. Nowadays, the convolutional neural network (CNN) is considered to be one of the best learning algorithms for understanding image content and has shown exemplary performance in a number of applications [6]. The common mode of implementation is deep regression, i.e., CNN with a (linear) regression top layer. For example, the authors in [7] proposed deep regression forests for age estimation. In [8], the authors used a ResNet-based deep regression model to learn the optimal repulsive pose for the safe collaboration between humans and robots. Deep learning methods modified for regression problems [9] were also applied to estimate gross tonnage, as a nonlinear measure of a ship's overall internal volume. [PERSON] et al. [10] used a deep regression framework based on manifold learning for manufacturing quality prediction. In [11], authors proposed a part-to-target tracker based on a deep regression model. [PERSON] et al. [12] applied an attention-guided deep regression architecture for cephalometric landmark detection. Single poultry tracking is demonstrated in [13], using a deep regression architecture based on the Alexnet network. [PERSON] et al. [14] proposed a deep regression framework for automatic pneumonia screening, which jointly learns the multi-channel images and multi-modal information to simulate the clinical pneumonia screening process. In [15], hierarchical deep regression with a network designed for hierarchical semantic feature extraction is used for traffic congestion detection, as an important aspect of vehicular management. The authors proposed in [16] the use of a regression convolutional neural network to find the 3-D position of arbitrarily oriented subjects or anatomy in a canonical space based on slices or volumes of medical images. With these examples, it becomes clear that deep regression algorithms are used in a wide variety of applications in very different domains.