## 1 Introduction
Fractal geometry has been demonstrated to be a powerful tool for solving important problems in applied science [12]. Much more physical fractal systems are being rapidly known. The use of concepts of fractal geometry in life sciences has contributed significantly to understanding the complexity and topological properties of networks that characterize DNA sequences. Pattern recognition [13] involves the evolution of systems that are oriented to find a solution to a given problem by using a set of example instances represented by a number of features. Statistical pattern recognition basically treats the problem of the automatic identification of objects belonging to one of the possible classes. Pattern recognition, dealing with identifying regularities in data with algorithms that learn to solve a problem using a limited set of measurement data, is closely related to informatics education. Statistical DNA pattern recognition is a technique based on finding genetic code in the DNA sequence and has applications in forensics, genetic engineering [14], bioinformatics [15], DNA nanotechnology, history, and so on. Hypoxia-inducible factor 1\(\alpha\) (HIF1A) [16] is the master transcriptional regulator of the cellular and systemic homeostatic response to hypoxia ruling the execution of the program that makes cells capable of running hypoxic stress [17]. As a protein-coding gene, HIF1A is the oxygen-sensitive subunit of the hypoxia-inducible factor (HIF), which, active under hypoxic conditions, plays an essential role in tumors. It functions by mediating transcription of over 40 genes involved in survival, glucose metabolism, invasion, metastasis, and angiogenesis (e.g., VEGF). Two subunits constitute the hypoxia-inducible factor 1 (HIF1) transcription factor as a heterodimer: Hypoxia-inducible factor 1, \(\alpha\) subunit (basic helix-loop-helix transcription factor) (HIF1A/HIF-1\(\alpha\)), and aryl hydrocarbon receptor nuclear translocator (ARNT/HIF-1\(\beta\)) 1,2; though, during ongoing synthesis, the HIF1A protein degrades swiftly by the ubiquitin-proteasome system under normal oxygen concentrations (normoxia) [18]. An adaptive response is driven to hypoxia by activating the expression of genes in which regulate erythropoiesis, angiogenesis, and glycolysis [19]. HIF1 is overexpressed in several cancers, often connected to poor prognosis, and is considered an interesting target for pharmacological manipulation [20; 21; 22]. Globally, lung cancer [23] has shown to be one of the most common types of cancer since 1985, in relation to incidence and mortality rates. Basically, two general types of lung cancer are recognized and are classified as non-small cell lung cancer (NSCL) [24; 25; 26; 27; 28; 29; 30] and small cell lung cancer (SCLC). They start growing in the lungs: The most common type is NSCL, while SCLC grows and spreads faster than NSCL. The first application of massively parallel sequencing in lung cancer research was published in 2008; most recently, an examination of paired NSCLC and normal lung tissue from a never-smoking patient with adenocarcinoma by [31]. Hypermethylation of cytosine guanine (CpG) islands [32] located in the promoter regions of tumor suppressor genes is now assumed as an important mechanism for gene inactivation. CpG island hypermethylation has resulted in almost every tumor type. Before 1994, CpG island promoter hypermethylation was conceived as a mechanism to inactivate genes in cancer fully restored as a result of the discovery that the von Hippel-Lindau (VHL) gene also undergoes methylation-associated inactivation [33]. Therefore, in this study, the application of a novel method is proposed, focusing on network and graph theory, fractal geometry, and statistical pattern recognition, to define the prognostic value of HIF-1\(\alpha\) expression in surgically treated lung cancer patients.