# Univariate Outlier Detection: Precision-Driven Algorithm for Single-Cluster Scenarios
## 1 Introduction  
Anomaly detection is pivotal across various sectors, such as finance, healthcare, and industrial monitoring, where accurately identifying anomalies is vital for operational efficiency and risk mitigation [1; 2]. In many applications, achieving high precision in anomaly detection is paramount, particularly in scenarios with a single cluster of data points where anomalies are less common but their detection holds significant consequences. In such contexts, false positives can result in costly disruptions or erroneous decisions, highlighting the importance of algorithms prioritizing precision. Univariate outlier detection techniques, such as the three-sigma rule and boxplot, rely on statistical assumptions regarding the normal distribution of data a [3]. While these methods offer a simple approach, their effectiveness can be limited in real-world scenarios where data often deviate from perfect normality. Additionally, these techniques are overly sensitive to outliers, leading to an increased number of false positives [4]. Alternative tests or criteria for univariate outlier detection, such as Grubbs' test or Chauvenet's criterion [5], offer more effective solutions in our context. These methods are less reliant on the assumption of a perfectly normal distribution and might provide better specificity, meaning they are less likely to flag normal data points as anomalies. As the complexity and dimensionality of data increase, even the improved techniques mentioned above encounter challenges in capturing the intricacies of the underlying data distribution [6]. Unsupervised anomaly detection algorithms emerge as a powerful solution in such cases. These algorithms offer a fundamentally different approach compared to traditional univariate methods. Unlike techniques relying on statistical assumptions or predefined thresholds, unsupervised algorithms can learn the inherent structure of the data itself. This allows them to identify anomalies that deviate significantly from thenormal patterns within the data, making them particularly well-suited for scenarios with complex and multifaceted anomalies (e.g., Support Vector Data Description (SVDD) [7], autoencoders [8], and isolation forests [9]). In this paper, we focus on anomaly detection in univariate data with a singular positive cluster. This refers to data that has a single, distinct group of positive values. Examples of such data include sensor readings, financial transactions, or customer engagement metrics. We will use the terms \"outlier\" and \"anomaly\" interchangeably throughout this paper to denote these data points. Anomaly detection in univariate data with a singular positive cluster is crucial for various applications, particularly where high precision is paramount. Existing methods like simple thresholding or statistical techniques often struggle with skewed distributions and lack sensitivity to subtle outliers. This can lead to a high number of false positives, hindering decision making and operational integrity. To address these challenges, we propose a novel unsupervised outlier detection algorithm specifically designed for this scenario, focusing on the precise identification of lower outliers. Our approach leverages three key innovations:  
* Dynamic Threshold-Based Filtration: This technique utilizes a threshold derived from the median and mean of the normalized data, offering greater robustness in handling skewed distributions compared to static thresholds. * Modified Hyperbolic Tangent Transformation: This non-linear transformation amplifies the differences between normal data points and outliers, enhancing the algorithm's ability to identify outliers with high precision, minimizing false positives, and ultimately improving decision making in real-world applications. * Empirical Evaluation: We extensively evaluate our algorithm on real-world data, showcasing its superior precision and effectiveness in detecting anomalies in univariate data with a single positive cluster.