Cubic q-Rung Orthopair Hesitant Exponential Similarity Measures for the Initial Diagnosis of Depression Grades
Abstract
:1. Introduction
- The main challenge is the urgent need to establish a scientific method of decision making for depression assessment with fuzzy information. Existing initial diagnostic methods for depression are usually based on precise information and use objective evaluations without considering fuzzy information diagnostic methods. As a result, incomplete, uncertain, and inaccurate information may be lost in clinical investigations and initial diagnoses. The depression data of some patients may belong to different depression levels, which subsequently produce unreasonable and uncertain assessment results and lead to difficulties in the assessment of depressive symptoms.
- As mentioned above, there is no literature on the combination of Cq-ROHFS with the diagnosis of depression. Thus, the idea of expressing ambiguous information based on Cq-ROHFS is just beginning and there is further room for exploration. Due to the uncertainty and hesitancy of test and assessment data in assessing depression levels in depressed patients, there may be mixed information with symmetry between membership function values and non-membership function values. In addition, existing scoring systems for depressed patients cannot express mixed information.
- Moreover, using hesitancy-containing preference information based on Cq-ROHFS to identify depression levels is another interesting challenge, which requires better decision making under uncertainty. Finally, a comprehensive comparison of the proposed method with other methods to understand the strengths and weaknesses of the proposed method is an attractive challenge to explore.
- This research contributes to the scientific community by helping visualize real-world clinical diagnostic problems and treatment options. To alleviate the main challenges, a new assessment method is proposed in the context of Cq-ROHFS to exploit the potential ability of Cq-ROHFS. In order to overcome the shortcomings of existing evaluation methods, ambiguous information of uncertainty and hesitation needs to be appropriately expressed in the composite score of depression diagnosis. However, in some special cases, decision support systems may have hesitations in determining membership and non-membership. In this case, several interval values and exact values are needed to represent the membership and non-membership degrees. In this study, we first propose Cq-ROHFS, which consists of IVq-ROHFS and q-ROHFS, for expressing the mixed information of both types.
- We propose an exponential distance and similarity measure with hesitation by means of an extension of the least common multiple (LCM) method. In the decision-making process, the proposed method allows the adjustment of the parameters according to the decision makers’ preferences to more accurately represent their evaluation information.
- Fifteen clinical cases are provided as examples of depression rating assessment in depressed patients to demonstrate the validity and applicability of the proposed depression rating assessment method in the Cq-ROHFS setting. Compared with existing methods [38,39,40], the proposed evaluation method can effectively and flexibly deal with depression assessment in a hierarchical setting, showing its advantages of flexibility, applicability, and practicality.
2. Preliminaries
- Membership-internal (briefly, M-Internal) if the following inequality holds:
- 2.
- Non-membership-internal (briefly, N-Internal) if the following inequality is valid:
- If , then , and , i.e., , , , , and , , for , , , and ;
- If , then, andfor.
3. Novel Distance and Similarity Measures
3.1. Novel Exponential Distance Measures between Cq-ROHFSs
- ;
- , if ;
- .
- ;
- , if;
- .
3.2. Novel Exponential Similarity Measures between Cq-ROHFSs
- ;
- , if ;
- ;
- If , then , .
4. Depression Grade Assessment Method Using the Cq-ROHFS Similarity Measure
- The PHQ-9 Scale scores are divided into three levels: “never” is 0 points; “a few days” is 1 point; “more than a week” is 2 points; and “almost every day” is 3 points [51,52]. Scores on the scale may not exceed 27 points. Mild depression is rated at points, moderate depression at points, and major depression at points.
5. Discussion of the Results and Comparisons
5.1. Numerical Illustration (15 Clinical Cases)
5.2. Result and Discussion
5.3. Comparison Analysis
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Depression Grade | HAMD Score | PHQ-9 Score |
---|---|---|
(Mild Depression) | ||
(Moderate Depression) | ||
(Major Depression) |
Depression Grade | ||
---|---|---|
Patient Pi | HAMD Score | Membership of HAMD | Non-Membership of HAMD |
---|---|---|---|
P1 | [18,21] | {0.6,0.75} | {0.25,0.35} |
P2 | [8,14] | {0.6,0.8} | {0.4,0.5} |
P3 | [20,22] | {0.3,0.65,0.85} | {0.4,0.5} |
P4 | [22,24] | {0.6,0.7,0.87} | {0.21,0.4} |
P5 | [15,18] | {0.5} | {0.5,0.6} |
P6 | [8,14] | {0.35} | {0.7,0.8} |
P7 | [21,24] | {0.4} | {0.5,0.6} |
P8 | [41,46] | {0.65,0.68,0.8} | {0.32,0.44} |
P9 | [22,25] | {0.64,0.79} | {0.35,0.65,0.68} |
P10 | [28,37] | {0.37,0.79} | {0.23,0.29} |
P11 | [46,48] | {0.59,0.75} | {0.2,0.51} |
P12 | [15,16] | {0.56,0.77} | {0.25,0.35} |
P13 | [30,34] | {0.54,0.81} | {0.23,0.39} |
P14 | [18,19] | {0.53,0.68} | {0.5,0.6,0.65} |
P15 | [40,42] | {0.61,0.82} | {0.25,0.35} |
Patient Pi | PHQ-9 Score | Membership of PHQ-9 | Non-Membership of PHQ-9 |
---|---|---|---|
P1 | [15,17] | {0.66,0.71} | {0.22,0.36} |
P2 | [5,6] | {0.3,0.55} | {0.35,0.55,0.81} |
P3 | [12,16] | {0.5,0.8} | {0.27,0.31} |
P4 | [18,21] | {0.8,0.88} | {0.2,0.35} |
P5 | [10,14] | {0.34,0.75} | {0.27,0.63} |
P6 | [6,7] | {0.41,0.62} | {0.57,0.61,0.87} |
P7 | [8,11] | {0.46,0.61} | {0.45,0.52,0.9} |
P8 | [18,20] | {0.55,0.83} | {0.29,0.33} |
P9 | [21,22] | {0.61,0.76} | {0.22,0.27} |
P10 | [23,24] | {0.61,0.76} | {0.27,0.34} |
P11 | [18,21] | {0.56,0.76} | {0.35,0.65,0.68} |
P12 | [5,6] | {0.3,0.55} | {0.28,0.46} |
P13 | [21,23] | {0.51,0.7} | {0.22,0.29,0.6} |
P14 | [5,9] | {0.41,0.73} | {0.23,0.51} |
P15 | [24,25] | {0.45,0.85} | {0.22,0.65} |
Patient Pi | C1: HAMD | C2: PHQ-9 |
---|---|---|
P1 | <[0.35,0.4],[0.6,0.65],{0.6,0.75},{0.25,0.35}> | <[0.56,0.63],[0.37,0.44],{0.66,0.71},{0.22,0.36}> |
P2 | <[0.15,0.27],[0.73,0.85],{0.6,0.8},{0.4,0.5}> | <[0.19,0.22],[0.78,0.81],{0.3,0.55},{0.35,0.55,0.81}> |
P3 | <[0.38,0.42],[0.58,0.62],{0.3,0.65,0.85},{0.4,0.5}> | <[0.44,0.59],[0.41,0.56],{0.5,0.8},{0.27,0.31}> |
P4 | <[0.42,0.46],[0.54,0.58],{0.6,0.7,0.87},{0.21,0.4}> | <[0.67,0.78],[0.22,0.33],{0.8,0.88},{0.2,0.35}> |
P5 | <[0.29,0.35],[0.65,0.71],{0.5},{0.5,0.6}> | <[0.37,0.52],[0.48,0.63],{0.34,0.75},{0.27,0.63}> |
P6 | <[0.15,0.27],[0.73,0.85],{0.35},{0.7,0.8}> | <[0.22,0.26],[0.74,0.78],{0.41,0.62},{0.57,0.61,0.87}> |
P7 | <[0.4,0.46],[0.54,0.6],{0.4},{0.5,0.6}> | <[0.3,0.41],[0.59,0.7],{0.46,0.61},{0.45,0.52,0.9}> |
P8 | <[0.79,0.88],[0.12,0.21],{0.65,0.68,0.8},{0.32,0.44}> | <[0.67,0.74],[0.26,0.33],{0.55,0.83},{0.29,0.33}> |
P9 | <[0.42,0.48],[0.52,0.58],{0.64,0.79},{0.35,0.65,0.68}> | <[0.78,0.81],[0.19,0.22],{0.61,0.76},{0.22,0.27}> |
P10 | <[0.54,0.71],[0.29,0.46],{0.37,0.79},{0.23,0.29}> | <[0.85,0.89],[0.11,0.15],{0.61,0.76},{0.27,0.34}> |
P11 | <[0.88,0.92],[0.08,0.12],{0.59,0.75},{0.2,0.51}> | <[0.67,0.78],[0.22,0.33],{0.56,0.76},{0.35,0.65,0.68}> |
P12 | <[0.29,0.31],[0.69,0.71],{0.56,0.77},{0.25,0.35}> | <[0.19,0.22],[0.78,0.81],{0.3,0.55},{0.28,0.46}> |
P13 | <[0.77,0.81],[0.19,0.23],{0.54,0.81},{0.23,0.39}> | <[0.78,0.85],[0.15,0.22],{0.51,0.7},{0.22,0.29,0.6}> |
P14 | <[0.35,0.37],[0.63,0.65],{0.53,0.68},{0.5,0.6,0.65}> | <[0.19,0.34],[0.66,0.81],{0.41,0.73},{0.23,0.51}> |
P15 | <[0.77,0.81],[0.19,0.23],{0.61,0.82},{0.25,0.35}> | <[0.89,0.93],[0.07,0.11],{0.45,0.85},{0.22,0.65}> |
Patient Pi | |||
0.8509, 0.9288, 0.8349 | 0.8974, 0.8208, 0.7074 | 0.8598, 0.9202, 0.8091 | |
0.7778, 0.8841, 0.7209 | 0.8415, 0.7595, 0.5800 | 0.8013, 0.8716, 0.7094 | |
0.7312, 0.8572, 0.6363 | 0.8003, 0.7164, 0.4947 | 0.7683, 0.8378, 0.6417 | |
0.6985, 0.8385, 0.5758 | 0.7711, 0.6847, 0.4355 | 0.7469, 0.8142, 0.5923 | |
0.6740, 0.8246, 0.5319 | 0.7496, 0.6596, 0.3926 | 0.7318, 0.7972, 0.5545 | |
0.6550, 0.8136, 0.4989 | 0.7332, 0.6393, 0.3604 | 0.7206, 0.7844, 0.5249 | |
0.6399, 0.8046, 0.4734 | 0.7204, 0.6224, 0.3355 | 0.7119, 0.7744, 0.5009 | |
0.6276, 0.7973, 0.4532 | 0.7101, 0.6082, 0.3159 | 0.7049, 0.7664, 0.4814 | |
0.6173, 0.7910, 0.4367 | 0.7017, 0.5962, 0.3000 | 0.7008, 0.7610, 0.4679 | |
0.6087, 0.7858, 0.4231 | 0.6947, 0.5859, 0.2869 | 0.6945, 0.7542, 0.4513 | |
Patient Pi | |||
0.7971, 0.8863, 0.8858 | 0.8616,0.8764,0.7330 | 0.8114,0.7377,0.6276 | |
0.7081, 0.8261, 0.7865 | 0.7864, 0.7917, 0.6281 | 0.7197, 0.6642, 0.5293 | |
0.6482, 0.7860, 0.7066 | 0.7291, 0.7258, 0.5614 | 0.6610, 0.6180, 0.4689 | |
0.6062, 0.7569, 0.6453 | 0.6812, 0.6744, 0.5140 | 0.6207, 0.5846, 0.4252 | |
0.5758, 0.7348, 0.5990 | 0.6414, 0.6345, 0.4784 | 0.5911, 0.5590, 0.3915 | |
0.5531, 0.7174, 0.5639 | 0.6088, 0.6034, 0.4508 | 0.5683, 0.5387, 0.3649 | |
0.5355, 0.7036, 0.5368 | 0.5826, 0.5785, 0.4319 | 0.5501, 0.5222, 0.3435 | |
0.5215, 0.6923, 0.5153 | 0.5611, 0.5583, 0.4105 | 0.5351, 0.5086, 0.3261 | |
0.5128, 0.6847, 0.5007 | 0.5436, 0.5416, 0.3955 | 0.5226, 0.4971, 0.3117 | |
0.5008, 0.6752, 0.4837 | 0.5290, 0.5276, 0.3828 | 0.5120, 0.4873, 0.2997 | |
Patient Pi | |||
0.8115, 0.8216, 0.7098 | 0.7676, 0.8400, 0.8931 | 0.7821, 0.8464, 0.8432 | |
0.7367, 0.7323, 0.6150 | 0.6418, 0.7527, 0.8278 | 0.6754, 0.771, 0.7576 | |
0.6843, 0.6721, 0.5577 | 0.5659, 0.6939, 0.7843 | 0.6036, 0.7164, 0.6913 | |
0.6448, 0.6286, 0.5190 | 0.5157, 0.6523, 0.7536 | 0.5535, 0.6756, 0.6393 | |
0.6139, 0.5961, 0.4910 | 0.4798, 0.6218, 0.7309 | 0.5172, 0.6447, 0.5993 | |
0.5889, 0.5710, 0.4697 | 0.4525, 0.5988, 0.7134 | 0.4899, 0.621, 0.5684 | |
0.5683, 0.5511, 0.4530 | 0.4309, 0.5808, 0.6997 | 0.4685, 0.6026, 0.5443 | |
0.5511, 0.5351, 0.4394 | 0.4131, 0.5665, 0.6885 | 0.4513, 0.5879, 0.525 | |
0.5365, 0.5219, 0.4282 | 0.4015, 0.5572, 0.6809 | 0.4402, 0.5783, 0.512 | |
0.5240, 0.5108, 0.4187 | 0.3856, 0.5451, 0.6714 | 0.4254, 0.5662, 0.4963 | |
Patient Pi | |||
0.7541, 0.8344, 0.8847 | 0.7469, 0.8165, 0.8848 | 0.9207, 0.8656, 0.7308 | |
0.6237, 0.7482, 0.8244 | 0.5987, 0.6962, 0.7803 | 0.8641, 0.7845, 0.5793 | |
0.5479, 0.6899, 0.7854 | 0.5123, 0.6286, 0.7242 | 0.828, 0.7285, 0.4878 | |
0.4981, 0.6456, 0.7574 | 0.4558, 0.5824, 0.6831 | 0.8027, 0.6876, 0.428 | |
0.4624, 0.6109, 0.7364 | 0.4157, 0.5494, 0.6524 | 0.7836, 0.6569, 0.3861 | |
0.4354, 0.5835, 0.72 | 0.3854, 0.5249, 0.629 | 0.7685, 0.6331, 0.3551 | |
0.4141, 0.5617, 0.707 | 0.3615, 0.5058, 0.6109 | 0.7561, 0.6143, 0.3312 | |
0.3969, 0.5441, 0.6965 | 0.3421, 0.4906, 0.5964 | 0.7458, 0.5992, 0.3124 | |
0.3826, 0.5297, 0.6879 | 0.326, 0.4782, 0.5848 | 0.737, 0.5868, 0.297 | |
0.3707, 0.5178, 0.6807 | 0.3125, 0.4678, 0.5751 | 0.7295, 0.5764, 0.2843 | |
Patient Pi | |||
0.7920, 0.8619, 0.8785 | 0.9077, 0.8595, 0.7375 | 0.7219, 0.8023, 0.8816 | |
0.6741, 0.7923, 0.8194 | 0.8464, 0.7958, 0.6182 | 0.5723, 0.6948, 0.8211 | |
0.6013, 0.7439, 0.7732 | 0.8025, 0.7568, 0.5416 | 0.4903, 0.6313, 0.7833 | |
0.5531, 0.7064, 0.7362 | 0.7695, 0.7289, 0.4894 | 0.4403, 0.5895, 0.7574 | |
0.5188, 0.6761, 0.7071 | 0.7439, 0.7073, 0.4521 | 0.4066, 0.5593, 0.7382 | |
0.493, 0.6514, 0.6843 | 0.7232, 0.6899, 0.4243 | 0.3823, 0.5363, 0.7231 | |
0.4729, 0.6311, 0.6664 | 0.7061, 0.6755, 0.4029 | 0.3637, 0.518, 0.7109 | |
0.4566, 0.6144, 0.6521 | 0.6917, 0.6633, 0.3858 | 0.349, 0.5031, 0.7008 | |
0.4431, 0.6005, 0.6405 | 0.681, 0.6547, 0.3753 | 0.337, 0.4905, 0.6922 | |
0.4317, 0.5889, 0.6309 | 0.6687, 0.6438, 0.3605 | 0.327, 0.4799, 0.6849 |
Patient Pi | ||
---|---|---|
0.375 | 0.595 | |
0.22 | 0.205 | |
0.4 | 0.515 | |
0.44 | 0.725 | |
0.32 | 0.445 | |
0.22 | 0.24 | |
0.43 | 0.355 | |
0.835 | 0.705 | |
0.45 | 0.795 | |
0.625 | 0.87 | |
0.9 | 0.725 | |
0.3 | 0.205 | |
0.79 | 0.805 | |
0.36 | 0.265 | |
0.79 | 0.91 |
Patient Pi | Depression Grade of the Proposed Evaluation Method Use | Grade of Depression According to Tang and Zhang [50] | Grade of Depression According to Kurt et al. [51] | Grade of Depression According to Guo et al. [52] |
---|---|---|---|---|
or | ||||
or | ||||
or | or | |||
or | ||||
or | ||||
Patient Pi | |||
0.8286, 0.9015, 0.7911 | 0.8556, 0.8006, 0.6722 | 0.8436, 0.8851, 0.7717 | |
0.831, 0.9051, 0.7923 | 0.8624, 0.8063, 0.6758 | 0.8466, 0.8903, 0.7754 | |
0.8387, 0.9169, 0.7959 | 0.8858, 0.825, 0.6868 | 0.8562, 0.9079, 0.7872 | |
0.8413, 0.9213, 0.7972 | 0.895, 0.8318, 0.6906 | 0.8596, 0.9146, 0.7913 | |
0.8468, 0.931, 0.7997 | 0.9167, 0.8465, 0.6985 | 0.8666, 0.9303, 0.7998 | |
0.8496, 0.9364, 0.8009 | 0.9304, 0.8545, 0.7025 | 0.8703, 0.9399, 0.8043 | |
Patient Pi | |||
0.7743, 0.8602, 0.8395 | 0.8516, 0.8459, 0.719 | 0.7604, 0.7151, 0.6176 | |
0.7772, 0.8636, 0.8409 | 0.8499, 0.8467, 0.7204 | 0.7684, 0.7238, 0.6245 | |
0.7862, 0.8745, 0.8452 | 0.8451, 0.8491, 0.7246 | 0.795, 0.7526, 0.6464 | |
0.7893, 0.8784, 0.8467 | 0.8436, 0.8499, 0.726 | 0.8049, 0.7632, 0.6542 | |
0.7957, 0.8865, 0.8497 | 0.8405, 0.8515, 0.7289 | 0.827, 0.7867, 0.6706 | |
0.799, 0.8909, 0.8512 | 0.839, 0.8523, 0.7304 | 0.8394, 0.7998, 0.6793 | |
Patient Pi | |||
0.7913, 0.7783, 0.6924 | 0.7301, 0.8134, 0.8691 | 0.7486, 0.8117, 0.8018 | |
0.7953, 0.7845, 0.6977 | 0.7314, 0.815, 0.8707 | 0.751, 0.8173, 0.8067 | |
0.8078, 0.8045, 0.7144 | 0.7352, 0.8199, 0.8758 | 0.7612, 0.8336, 0.8236 | |
0.8122, 0.8118, 0.7202 | 0.7365, 0.8216, 0.8775 | 0.7645, 0.8397, 0.8296 | |
0.8215, 0.8275, 0.7325 | 0.7392, 0.8251, 0.8811 | 0.7713, 0.8527, 0.8424 | |
0.8264, 0.8361, 0.739 | 0.7405, 0.8268, 0.883 | 0.7748, 0.8598, 0.8493 | |
Patient Pi | |||
0.7157, 0.809, 0.8576 | 0.6961, 0.7667, 0.822 | 0.8799, 0.8279, 0.6776 | |
0.7171, 0.8109, 0.8613 | 0.6974, 0.7693, 0.8264 | 0.8851, 0.8317, 0.6798 | |
0.7211, 0.8165, 0.8733 | 0.7016, 0.7771, 0.8405 | 0.9024, 0.8437, 0.6863 | |
0.7225, 0.8185, 0.8776 | 0.703, 0.7798, 0.8456 | 0.909, 0.848, 0.6885 | |
0.7253, 0.8224, 0.8867 | 0.7059, 0.7853, 0.8564 | 0.9241, 0.8569, 0.693 | |
0.7267, 0.8244, 0.8917 | 0.7074, 0.7882, 0.8622 | 0.9331, 0.8617, 0.6953 | |
0.7563, 0.844, 0.8609 | 0.8655, 0.827, 0.7018 | 0.6779, 0.7724, 0.8623 | |
0.7573, 0.8453, 0.863 | 0.8708, 0.8328, 0.7061 | 0.6786, 0.7733, 0.8643 | |
0.7602, 0.8495, 0.8696 | 0.8894, 0.8521, 0.7169 | 0.6808, 0.776, 0.8708 | |
0.7611, 0.8509, 0.8718 | 0.8964, 0.8592, 0.7209 | 0.6815, 0.7769, 0.873 | |
0.7631, 0.8538, 0.8765 | 0.9122, 0.8748, 0.729 | 0.683, 0.7788, 0.8777 | |
0.7641, 0.8553, 0.8789 | 0.9215, 0.8836, 0.7332 | 0.6837, 0.7797, 0.8801 |
Depression Grade | ||||||
---|---|---|---|---|---|---|
SN | References | Disadvantage | Grade Diagnosis |
---|---|---|---|
1 | Zedah [1] | Lack of non-membership | Not Possible |
2 | Mehmood et al. [43] | Lack of non-membership | Not Possible |
3 | Liu et al. [44] | Lack of Interval fuzzy values | Not Possible |
4 | Guo et al. [50] | Lack of fuzzy values | Partially undetermined |
5 | Tang and Zhang [51] | Lack of PHQ-9 scores | Partially undetermined |
6 | Kurt et al. [52] | Lack of HAMD scores | Partially undetermined |
7 | Proposed Method in this paper | Long and heavy calculations in grades diagnosis | This problem can be solved with the help of a computer program |
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Ying, C.; Slamu, W.; Ying, C. Cubic q-Rung Orthopair Hesitant Exponential Similarity Measures for the Initial Diagnosis of Depression Grades. Symmetry 2022, 14, 670. https://doi.org/10.3390/sym14040670
Ying C, Slamu W, Ying C. Cubic q-Rung Orthopair Hesitant Exponential Similarity Measures for the Initial Diagnosis of Depression Grades. Symmetry. 2022; 14(4):670. https://doi.org/10.3390/sym14040670
Chicago/Turabian StyleYing, Changyan, Wushour Slamu, and Changtian Ying. 2022. "Cubic q-Rung Orthopair Hesitant Exponential Similarity Measures for the Initial Diagnosis of Depression Grades" Symmetry 14, no. 4: 670. https://doi.org/10.3390/sym14040670
APA StyleYing, C., Slamu, W., & Ying, C. (2022). Cubic q-Rung Orthopair Hesitant Exponential Similarity Measures for the Initial Diagnosis of Depression Grades. Symmetry, 14(4), 670. https://doi.org/10.3390/sym14040670