On the Continuity of Monotonic Functions

1967 ◽  
Vol 74 (3) ◽  
pp. 314 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Monotonic Functions

scholarly journals Monotonic Functions Method and Its Application to Staging of Patients with Prostate Cancer According to Pretreatment Data

2021 ◽  
Vol 11 (9) ◽  
pp. 3836
Author(s):  
Valeri Gitis ◽  
Alexander Derendyaev ◽  
Konstantin Petrov ◽  
Eugene Yurkov ◽  
Sergey Pirogov ◽  
...  
Keyword(s):  
Prostate Cancer ◽  
Learning Algorithm ◽  
Binary Classification ◽  
Preoperative Staging ◽  
Threshold Value ◽  
Logical Function ◽  
Adequate Treatment ◽  
Monotonic Functions ◽  
Selection Of

Prostate cancer is the second most frequent malignancy (after lung cancer). Preoperative staging of PCa is the basis for the selection of adequate treatment tactics. In particular, an urgent problem is the classification of indolent and aggressive forms of PCa in patients with the initial stages of the tumor process. To solve this problem, we propose to use a new binary classification machine-learning method. The proposed method of monotonic functions uses a model in which the disease’s form is determined by the severity of the patient’s condition. It is assumed that the patient’s condition is the easier, the less the deviation of the indicators from the normal values inherent in healthy people. This assumption means that the severity (form) of the disease can be represented by monotonic functions from the values of the deviation of the patient’s indicators beyond the normal range. The method is used to solve the problem of classifying patients with indolent and aggressive forms of prostate cancer according to pretreatment data. The learning algorithm is nonparametric. At the same time, it allows an explanation of the classification results in the form of a logical function. To do this, you should indicate to the algorithm either the threshold value of the probability of successful classification of patients with an indolent form of PCa, or the threshold value of the probability of misclassification of patients with an aggressive form of PCa disease. The examples of logical rules given in the article show that they are quite simple and can be easily interpreted in terms of preoperative indicators of the form of the disease.

Improving correlation accuracy of crashworthiness applications by combining the CORA and MADM methods

2021 ◽  
pp. 095440702110696
Author(s):  
Christophe Bastien ◽  
Alexander Diederich ◽  
Jesper Christensen ◽  
Shahab Ghaleb
Keyword(s):  
Numerical Models ◽  
Monotonic Functions ◽  
Virtual Test ◽  
Industry Standard ◽  
Discrepancy Method ◽  
Original Equipment Manufacturers ◽  
The Relationship ◽  
Correlation Accuracy ◽  
New Framework ◽  
Selection Of

With the increasing use of Computer Aided Engineering, it has become vital to be able to evaluate the accuracy of numerical models. This research poses the problem of selection of the most accurate and relevant correlation solution to a set of corridor variations. Specific methods such as CORA, widely accepted in industry, are developed to objectively evaluate the correlation between monotonic functions, while the Minimum Area Discrepancy Method, or MADM, is the only method to address the correlation of non-injective mathematical variations, usually related to force/acceleration versus displacement problems. Often, it is not possible to differentiate objectively various solutions proposed by CORA, which this paper proposes to answer. This research is original, as it proposes a new innovative correlation optimisation framework, which can select the best CORA solution by including MADM as a subsequent process. The paper and the methods are rigorous, having used an industry standard driver airbag computer model, built virtual test corridors and compared the relationship between different CORA and MADM ratings from 100 Latin Hypercube samples. For the same CORA value of ‘1’ (perfect correlation), MADM was capable to objectively differentiate between 13 of them and provide the best correlation possible. The paper has recommended the MADM settings n = 1; m = 2 or n = 3; m = 2 for a congruent relationship with CORA. As MADM is performed subsequently, this new framework can be implemented in already existing industrial processes and provide automotive manufacturers and Original Equipment Manufacturers (OEM) with a new tool to generate more accurate computer models.

An Interesting Class of Monotonic Functions

1909 ◽  
Vol 16 (1) ◽  
pp. 4
Author(s):  
Arthur R. Schweitzer
Keyword(s):  
Monotonic Functions ◽  
Interesting Class
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