Maximum Likelihood Estimation Of Correlation Coefficients For Left-Censored Observational Data Of Agricultural Environmental Medi

Trace heavy metals and exotic substances in agriculture and the environment media(such as soil,water,etc.)are unevenly distributed.It is very common that the concentration of trace substances in soil and water is often below the limit of detection(LOD)of the detection method,and their content cannot be quantified.We defined such data as censored data.At the same time,the distribution of such data is often skewed,which makes it difficult to estimate the parameters of one variable and correlations accurately between variables.The is often used for parameter estimation of censored data.Therefore,this study will first deduce the probability density function and apply maximum likelihood method to estimate the parameters for left-censored data of two typical biased.Through the simulation study,the accuracy of the likelihood estimates under different scenarios was evaluated,and a new multi-factor evaluation system was constructed.The accuracy comparing the MLE with the substitution method(usually replacing the censored part with LOD or LOD/2)and the deletion method(deleting the censored part directly was evaluated.The model was applied to the correlation study between the Australian soil census data and the heavy metal data in the Huangshui River Basin in Qinghai.Building web services through the shiny package in R language to provide an interactive graphical interface for parameter estimation models.The results show that the maximum likelihood method can accurately estimate the mean and standard deviation of the log-normal distribution,and the estimated mean and standard deviation often have an opposite trend with the change of the censoring ratio.When the censoring ratio reaches 60 %,the accuracy of the estimated value is affected to a certain extent,but overall,the maximum censoring ratio which could provide effective estimates can reach 80%.The maximum likelihood method has a small deviation of the shape parameters of the gamma distribution,only when the overall correlation coefficient is low and the censoring ratio reaches 70%,there is a certain deviation,and overall,effective parameter estimation can be performed.The maximum likelihood method is relatively stable in the estimation of the inverse scale parameters,and the estimation effect is the best when the overall correlation coefficient is 0.5,which can be accurately estimated up to 80% of the censoring ratio.The larger the sample size,the more accurate correlation coefficient estimated by MLE in the study for two-dimensional log-normal distribution.When the sample size reaches 2000,the estimated value of the correlation coefficient of the two-dimensional log-normal distribution is basically convergence.The estimated value of the correlation coefficient of the MLE in the study varies with the censoring ratio(0% to 90%,10% interval)and the overall correlation coefficient(-1 to 1 interval is 0.1),the degree of deviation is small,indicating that the MLE has a certain asymptotic properties.The addition of distractors had a small effect on the accuracy of the MLE in the study,indicating its strong robustness.With the increase of the censoring ratio,the accuracy of the results of the deletion method and the replacement method deteriorates.The accuracy of the MLE method in the study is significantly better than the above two methods.Even if the censoring ratio reaches 80%,it can still be effectively estimated(1000 times RMSE < 0.1 for repeated study estimates).MLE can perform relatively unbiased estimation of skewed censored data parameters and their correlations simply and effectively,but it has a strong dependence on the estimation of single-dimensional statistics,and there is still some space for optimization.The correlation coefficient estimates of the MLE in the study for the left-censored bivariate gamma distribution were more stable than the left-censored bivariate log-normal distribution samples.When the sample size reaches 500,the estimated value of the correlation coefficient of the two-dimensional gamma distribution converges to the true value.The estimated value of the correlation coefficient varies with the censoring ratio(0% to 90%,with an interval of 10%)and the overall correlation coefficient(0.1 from 0 to 0.9 with an interval of 0.1),and there is no large deviation,indicating that the estimation results are accurate and consistent.By comparing the results of MLE with the substitution method and deletion method in repeated studies,it can be seen that the variance of the estimated value of MLE in repeated study has a small deviation from the true value,which is obviously better than the substitution method and the deletion method.Strong accuracy and stability.The graphical results of the model are as follows: based on shiny to provide a graphical interface for human-computer interaction of the model,so that the application of the model is more self-explanatory,the use of data is more flexible,the function module is more organized,the output results are clearer,which greatly increases the usability of the model and improves the efficiency of statistical work.The maximum likelihood method proposed in the study can make relatively unbiased estimates of the parameters and related coefficients of left-censored data in soil and water,but it has strong dependence on the estimation of single-dimensional statistics and still has some room for optimization.The graphical interface constructed on this basis provides convenience for the use of the model and provides a reference for the parameter problem of left censored data in agricultural environment.