## 4 Multiple Non-Linear Regression Analysis of Strength
### Comparison of Forecast Results  
All specimens in this study were prepared using laboratory-grade water and cured in a controlled environment at a constant temperature of 25 \({}^{\circ}\)C and relative humidity of 80% using a standard curing chamber. While these conditions provide consistency in test results, it is important to note that actual underground environments often exhibit variable temperatures and moisture levels, which may influence the hydration kinetics, strength development, and durability of cemented backfill. These factors should be considered in future field-oriented research.  
The strength of the cemented backfill can be predicted by the multiple non-linear regression equation established above, and the prediction results  
matching conditions. To investigate the influence law of single influencing factors on the strength of the filler, we found that under the same grading conditions, the strength of the cement body increased linearly with the increase in the maintenance age, increased exponentially with the increase in the mortar ratio, and changed as a negative power function with the increase in the water-cement ratio. Furthermore, when the slurry mass concentration increased, the greater the growth rate of the strength of the filler body, the greater the influence on the strength of the filler body (i.e., within the same age, the greater the value of the increase in the strength). The effect of slurry mass concentration on the strength of the filling body was in the form of a significantly exponential function.
* By conducting an analysis of the variance table on the test results, we found that the F-values of the influencing factors in descending order were as follows: age of curing > cement content > slurry concentration. This indicated that the age of curing and cement content were the main factors influencing the strength of the filler, and slurry mass concentration was a secondary factor affecting the strength of the filler.
* According to the results of the filling strength optimization, the ratio that can meet the requirements of mine filling strength is 70% to 72% by mass with 300 kg/m\({}^{3}\) of cement content and 72% by mass with 270 kg/m\({}^{3}\) of cement addition.
* A multi-factor non-linear model of the strength was developed based on the law of influence of the influencing factors on the strength of the filler. The regression equations of the strength at different curing ages were determined using MATLAB software. The three-dimensional visualization model of the strength and the relative error values revealed that the fitting coefficients of the strength models were larger than 0.93 and the absolute values of the relative errors did not exceed 4%, suggesting that the strength models established in this research are effective and valid.  
Conceptualization, W.Z. and Y.X.; methodology, Y.X.; software, Y.X.; validation, M.W., S.L. and Q.K.; investigation, M.W.; writing--original draft preparation, W.Z.; writing--review and editing, W.Z., M.W. and Y.X.; visualization, Q.K.; supervision, S.L.; funding acquisition, W.Z. and M.W. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Engineering and Technology Research Center for Development &Utilization of Phosphate Resources (NECP-2024-10) and the National Natural Science Foundation of China (No. 52174086).  
Not applicable.  
All data that support the findings of this study are included within this article.  
Authors [PERSON], [PERSON] and [PERSON] were employed by the company Yunnan Phosphate Group Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.