David Hason
email: david.hasonrudd@student.uts.edu.au

Huan Huo

Guandong Xu

Improved Churn Causal Analysis Through Restrained High-Dimensional Feature Space Effects in Financial Institutions

Keywords: Churn analysis, Bayesian networks, Deep neural networks, Data mining, Data sampling Abbreviations ANNs Artificial

Customer churn describes terminating a relationship with a business or reducing customer engagement over a specific period. Customer acquisition cost can be five to six times that of customer retention, hence investing in customers with churn risk is wise. Causal analysis of the churn model can predict whether a customer will churn in the foreseeable future and identify effects and possible causes for churn. In general, this study presents a conceptual framework to discover the confounding features that correlate with independent variables and are causally related to those dependent variables that impact churn. We combine different algorithms including the SMOTE, ensemble ANN, and Bayesian networks to address churn prediction problems on a massive and high-dimensional finance data that is usually generated in financial institutions due to employing interval-based features used in Customer Relationship Management systems. The effects of the curse and blessing of dimensionality assessed by utilising the Recursive Feature Elimination method to overcome the high dimension feature space problem. Moreover, a causal discovery performed to find possible interpretation methods to describe cause probabilities that lead to customer churn. Evaluation metrics on validation data confirm the random forest and our ensemble ANN model, with %86 accuracy, outperformed other approaches. Causal analysis results confirm that some independent causal variables representing the level of super guarantee contribution, account growth, and account balance amount were identified as confounding variables that cause customer churn with a high degree of belief. This article provides a real-world customer churn analysis from current status inference to future directions in local superannuation funds.

Introduction

Businesses rely heavily upon retaining satisfied customers in the global marketplace, which account for a large portion of their revenue. As the market becomes increasingly saturated, businesses have learned to emphasise maintaining existing clients. Although obtaining new clients was critical for initial business success, retention policies should have equal weight. Many previous studies have identified retention rate's substantial effect on the market, but clients can always churn away from a business, resulting in potential losses for the organisation. However, customers usually offer some warning before being churned. Hence, churn prediction systems primarily focus on customer behaviour to identify specific customers who are likely to churn out and indicate reasons for the churn. Such factors would aid marketing to develop effective retention strategies, increasing overall customer lifetime value, and assisting in growing the company's market value. For example, applying incentive programs for customers who had fewer transactions/interactions over time. In general, churn rate is calculated by dividing the number of lost customers by initial customers [START_REF] Domingos | Experimental analysis of hyperparameters for deep learning-based churn prediction in the banking sector[END_REF].

Voluntary churn is when a consumer wants to leave the company on their own. This could be due to dissatisfaction with the product, or perhaps they feel they are not receiving that value they expected. In contrast, involuntary churn is where a customer leaves the company for unavoidable reason(s), such as payment problems due to expired account details, network issues, or inadequate cash, and incidental churn occurs due to location or financial situation changes. Deliberate churn arises due to customer desire to change innovation and price, where the most common reasons are poor service quality, non-competitive pricing, and missing customer expectations. Companies can utilise churn analysis to determine the individual user risk levels and develop appropriate, focused retention initiatives.

The proposed framework applies churn analysis for a local superannuation fund. Data preparation and analysis verified that most customer accounts were active for less than one year, for several possible reasons, including some customers being involuntary churners who were subsequently enrolled in a different superannuation fund(s) operated by their employers. In contrast, voluntary churn occurs when a customer decides to cease the account for personal reasons, including perceived quality, technology, and price. The findings from this study make several major contributions to the current literature, such as:

• This study is the only empirical investigation into the impact of a causal effect on churn in superannuation funds. • The present study has enhanced our understanding of exploiting correlation discovery tools within the causal inference of churn. First, we analysed 12-month time window data to predict churners for the next 6 months. Since mining data from the 12-month historical data allowed us to extract the latent factors that cause churn. Then, we scaled up the model by exploiting Bayesian networks to describe how deliberate churn occurs due to multiple causes and demonstrated that hypothesis tests on common features greatly influence prediction results. Finally, we proposed a specific causality analysis method that can be applied to other similar datasets employed in most superannuation funds.

It should be noted that this study only focuses on the performance of a DFF NNs to address churn prediction on a massive and high-dimensional sparse dataset that is usually created in financial institutions. These very high dimensions data are primarily generated in subscribed-base businesses, mostly in superannuation fund(s) due to employing interval-based features used in Customer Relationship Management (CRM) systems. The Recursive Feature Elimination method is employed to overcome the high dimension feature space problem, and then compared the results with proposed ensemble ANN and other classifiers.

The remainder part of the article proceeds as follows: Sect. 2 presents a comprehensive literature review of recent relevant works on customer churn problem, and Sect. 3 gives preliminary knowledge on the application of the deep learning (DL) method in churn prediction problems. Section 4 discusses specific research and analysis methods employed to predict underlying reasons for churn including problem definition and churn propensity modelling workflow. Section 5 setup experiment analyses on gathered data to address research questions: a causal analysis of churn through the financial data collected by superannuation funds and Sect. 6 presents results on prediction and causality analysis outcomes and discusses implications from the findings for real-world applications and limitations. Finally, Sect. 7 summarises and concludes the article.

Related Work

Customer Churn Prediction

Different churn prediction techniques have been evaluated to identify optimal approaches [START_REF] Tamaddoni | Comparing churn prediction techniques and assessing their performance: a contingent perspective[END_REF]. Most previous studies focused on determining churn variables for a particular dataset rather than customer churn causation analysis.

The state-of-the-art churn prediction framework is designed, which is predicated upon "deep neural models, time-to-next-event models, and Big Data processing" using large parallel computing with GPU units [START_REF] Simion-Constantinescu | Deep neural pipeline for churn prediction[END_REF]. Developing the predictive model of customer behaviour in order to plan for and handle such situations could be quite beneficial. Staff churn and staff loss would be similar to customer churn, however the effect of losing a significant customer for an organisation will almost certainly be even more stressful (since organisations have no physical sense of losing their 1 3 staff), whereas the implications of seeking well employees rather than missed employees, and also the expense of inservice coaching which should be provided for the new hires, might be significant [START_REF] Dolatabadi | Designing of customer and employee churn prediction model based on data mining method and neural predictor[END_REF]. Using the partial least squares (PLS)-based technique upon highly linked sets of data across variables, create accurate and compact predictive models for churn prediction [START_REF] Lee | Mining churning behaviors and developing retention strategies based on a partial least squares (PLS) model[END_REF]. The use of hybrid learning algorithms towards churn prediction in mobile networks allows for predictive modelling of buyer behaviour [START_REF] Yeshwanth | Evolutionary churn prediction in mobile networks using hybrid learning[END_REF]. A locally linear model tree (LOLIMOT) method combines the benefits of neural networks, fuzzy modelling, and tree models [START_REF] Ghorbani | The application of the locally linear model tree on customer churn prediction[END_REF]. Moreover, the RemsProp training approach outperformed "conventional stochastic gradient descent (SGD), Adam, AdaGrad, Adadelta, and AdaMax" algorithms in terms of accuracy for DL-based churn prediction [START_REF] Domingos | Experimental analysis of hyperparameters for deep learning-based churn prediction in the banking sector[END_REF].

The effectiveness of a typical random forests (RFs) strategy for predicting customer churn was studied, and the integration of sampling approaches and price learning into the method to outperform many proposed algorithms used a real bank dataset [START_REF] Xie | Customer churn prediction using improved balanced random forests[END_REF]. Based on Orange Belgium customer information, the real-life churn prediction case gets constructed. The first section of the paper involves the creation of such a precise prediction model. An Easy Ensemble technique uses a RF classifier to address a considerable class imbalance between two categories [START_REF] Ullah | A churn prediction model using random forest: analysis of machine learning techniques for churn prediction and factor identification in telecom sector[END_REF].

For non-subscription-oriented business sets, a method for predicting customer churn was suggested. The set of general features can be derived from practically all non-subscription-oriented firms' revenue and transaction information and used to forecast customer churn. For prediction, "a neural network-based Multilayer perceptron" is often used [START_REF] Shah | Prediction and causality analysis of churn using deep learning[END_REF].

Churn analysis, modelling, and prediction (CHAMP) is an integrated system for forecasting consumers cancelling their cellular phone service [START_REF] Pearl | Causal inference in statistics: an overview[END_REF]. Alyuda neurointelligence employs neural networks (NNs) for data mining to forecast customer churn at banks [START_REF] Zoric | Predicting customer churn in banking industry using neural networks[END_REF]. Integrating textual data using customer churn prediction (CCP) algorithms adds value [START_REF] De Caigny | Incorporating textual information in customer churn prediction models based on a convolutional neural network[END_REF], and combining different classifiers, e.g., gradient boost, oversampling, and contrast sequential pattern mining on single-year observation windows have been demonstrated to be a practical strategy to deal with highly skewed data collected from superannuation funds [START_REF] Culbert | Customer churn prediction in superannuation: a sequential pattern mining approach[END_REF].

Hidden churn is a common problem for superannuation funds, where customer accounts become dormant once mandatory employer payments cease. Various remedies for insufficient consumer interaction have been proposed, addressing unbalanced and fully leveraged data problems, and multiple classifiers have been developed from sampled datasets [START_REF] Culbert | Customer churn prediction in superannuation: a sequential pattern mining approach[END_REF]. In addition, DL techniques can handle large datasets compared with standard machine learning (ML) approaches and combining DL and convolutional neural networks (CNNs) has successfully forecasted churn [START_REF] Mishra | A novel approach for churn prediction using deep learning[END_REF].

Studies on Causal Inference of Churn

Recent causal inference developments highlight fundamental alterations required to move from standard statistical analysis toward multivariate causal analysis [START_REF] Pearl | Causal inference in statistics: an overview[END_REF]. New PC-stable approaches effectively learn causal structures using DFS data, allowing temporal causal modelling for enormous time information datasets [START_REF] Mohan | Temporal causal modelling on large volume enterprise data[END_REF]. Directed acyclic graph (DAG) has been proposed to represent causal relationships in Bayesian networks [START_REF] Huang | Identifiability in causal Bayesian networks: a sound and complete algorithm[END_REF]. A likelihood of churn anticipates, as are the driving factors on banking data. Subsequently, Shah et al. [START_REF] Shah | Prediction and causality analysis of churn using deep learning[END_REF] trained a model to produce suitable weights for features that predict whether a customer would churn. They contrasted churn definitions commonly used in business administration, marketing, IT, telecommunications, newspapers, insurance, and psychology. Many studies have described churn loss, feature engineering, and prediction models using this approach [START_REF] Ahn | A Survey on Churn Analysis in Various Business Domains[END_REF]. The text content from daily Twitter posts was analysed using a convolution neural network to determine if the sentiment was good, bad, or indifferent. Granger causality analysis has been used to cross-validate that generated mood data set [START_REF] Lattimore | A primer on causal analysis[END_REF].

According to our literature review, no research has been linked with causal analysis on client attrition in superannuation fund(s). Moreover, the above approaches have been variously adapted for current churn prediction methods. However, most researchers evaluate models based on telco, media, retail, medical, and insurance data. In contrast, the proposed approach was tested using a massive financial dataset with high-dimensional sparse data associated with a local superannuation fund. Motivated by the studies mentioned above, we proposed a new approach that combines different algorithms including RFE, SMOTE, ANNs DL and Bayesian Networks to improve churn propensity modelling.

Preliminary Knowledge

Churn prediction models will provide earlier churner identification and hence assist customer intervention program development. This is basically a classification problem, i.e., to categorise each customer as a potential churner or nonchurner. Several ML techniques for churn prediction have produced verifiable results from interpretable models, such as boosting technique, a non-parametric method, and logistic regression. However, this approach is only valid for predictability under circumstances where the customer database size is minimal, varying sample size, and fails for larger datasets [START_REF] Tamaddoni | Comparing churn prediction techniques and assessing their performance: a contingent perspective[END_REF]. Therefore, we propose a DL algorithm to deal with massive financial data volumes, since DL (feature transformation) differentially weights features using historical data. DL algorithms follow human brain architecture using artificial neural networks (ANNs) [START_REF] Zaıane | Principles of knowledge discovery in databases[END_REF]. Feed-forward NNs use the 1 3 output from one layer as input for the next layer, with no loops between layers [START_REF] Nielsen | Neural networks and deep learning[END_REF]. Therefore, the deep learning feedforward (DFF) or ANNs algorithm is employed to deal with high-dimensional sparse data. General advantages for ANNs DL can be summarised as follows:

1. superior accuracy from DL; 2. includes more variables than classical ML; 3. DL algorithms can extract patterns while avoiding blind spots from extensive customer demographics, behavioural variables, and billions of customer engagement logs; 4. reduces time-consuming feature engineering and manual financial data analysis.

One significant weakness of DL is that it remains a blackbox model, i.e., DL does not express uncovered patterns in the underlying data in easily understandable ways. However, to address the complexity of the ANNs model, a causal inference model is proposed to identify churn effects of treatments and describe cause probabilities that lead to churn.

Methodology

Data Mining and Classification Problem

Data mining can identify useful knowledge in terms of pattern extraction from different sources, and various feature engineering tools can extract hidden patterns from massive datasets [START_REF] Zaıane | Principles of knowledge discovery in databases[END_REF][START_REF] Karp | Using logistic regression to predict customer retention[END_REF]. Figure 1 demonstrates that this study included 12 datasets from a superannuation company for real-world experiments to verify the proposed model's effectiveness.

This study addresses the classification problem of minority class and overfitting due to the massive dataset. The classification problem is limited to two classes. A vector includes input data as each member has n-components or features. The pattern of each member's data is P with n-dimensional feature space in class 1 (minority class) or class 0 (majority class). So, a training set of vectors {x1, x2, x3, … xk, … xn} with class label of {y1, y2, y3, … yk, … yn} , so that yk ∈ {0, 1} is defined to recognise the vector of n-components or patterns a decision boundary is defined in a discriminant function named D(x) into decision boundary of D(x) > 0 and D(x) < 0 to assign each sample to a churn (1) or non-churn (0) class as demonstrated in Eq. ( 1): where x is an input pattern and t w denotes a 6-month time window, acc_close_ t w represents current time window, and acc_close_t w -1 demonstrates previous 6-month time win- dow. Thus, we built a simple linear discriminant function by calculating the sum of the training patterns and bias as shown in (2):

where w represents weight of pattern and b symbolises bias. Since a singular criterion for decision boundary is defined, a linear discrimination function can separate the classes without error.

As mentioned in Eq. 1, we define a customer as a churner if they closed their account during the subsequent 6-month time window. Therefore, we use a binary outcome for each customer [0 or 1], where 1 means the account closed and 0 that it was not close in the subsequent 6-month time window.

In churn analysis, the data mining method should fit the problem structure based on the definition of active and inactive account holders in financial institutions. Two main inclusion criteria are defined to satisfy dimensionality reduction by eliminating massive noisy data. First, customers with more than six months of account tenure are only retained. Second, account balances below $1500 are considered low engagement members and removed. Figure 2 demonstrates features in the observation window to predict which user will be churned or non-churned in the next 6-month outcome window.

D(x) > 0 ∃ x ∈ class(0) ⟺ acc_close_tw > acc_close_tw -1 + t w D(x) < 0 ∃ x ∈ class(1) ⟺ acc_close_t w ≤ acc_close_tw -1 + t w (1) D(x) = 0 or acc_close_t w = t w decision boundary, (2) D(x) = w.x + b

High Dimension Feature Space

The dataset included more features than observations for each member results risk of overfitting in the model and occurring in high dimensional feature space, which is common in finance data. On the other hand, many features have low variance and correlation with the target variable. The model performance of causal analysis and causal effect of treatment outcomes depends on observing all available causal variables, possible covariates, and cofounders against dropping low importance feature increase bias in causal model. Therefore, we should tradethe effect of the curse of dimensionality [START_REF] Donoho | High-dimensional data analysis: the curses and blessings of dimensionality[END_REF] and the blessing of dimensionality [START_REF] Kainen | Utilising geometric anomalies of high dimension: when complexity makes computation easier[END_REF]. To achieve this, different dimensionality reduction methods are investigated including feature dropping, wrapper methods and feature importance with RF. Finally, the recursive feature elimination RFE [START_REF] Guyon | Gene selection for cancer classification using support vector machines[END_REF] obtained robust results to overcome high dimensional data while retaining the possible influential cofounders and causal variables. The RFE is a popular feature-ranking algorithm to remove low-weight features since it gives us control to set threshold and fixed number of top-ranked features.

RFE removes the feature with the smallest ranking criterion using the cost function DJ(i) is demonstrated in (3). The feature weight assesses changing cost function output in weight Dω i = ω i by eliminating a given feature i in an iterative procedure called RFE [START_REF] Kohavi | Wrappers for feature subset selection[END_REF].

where DJ(i) in (3) ranks weight criterion to expand J in Taylor series to second order, and J in (4) denotes cost function in classifier [START_REF] Guyon | Gene selection for cancer classification using support vector machines[END_REF].

The features remove in into subset Fm each iteration from the lowest to the highest ranked features. Therefore, we proposed a method to combine SMOTE and RFE algorithm to overcome minority class and high dimension feature space problem so that features with low variance value and predictive power are removed from the data without significant impact on the causal inference model.

(3)

DJ(i) = (1∕2) 2 J w 2 i Dw i 2 (4) J = ∑ x∈X |w ⋅ x -y| 2

Churn Propensity Modelling

We compared the proposed DFF NNs algorithm approach with seven current best-practise classifiers to evaluate the proposed framework's effectiveness on the datasets with 12-month observation window and 6-month test window, as discussed in problem definition section. The workflow of the proposed model demonstrates in Fig. Furthermore, a majority voting ensemble from the Scikitlearn library is employed [START_REF] Kramer | Scikit-learn[END_REF] to combine predictions from multiple models, which can be helpful to improve model performance for classification tasks. Ensemble hard and soft voting were both applied for the comparison supervised models. Hard voting counts each individual classifier votes and the majority wins; whereas soft voting weights each prediction by classifier importance, and the target label with the greatest sum of weighted probabilities wins [START_REF] Kramer | Scikit-learn[END_REF]. In addition, a specific ANNs architecture is built to address the research problem using ensemble ANNs networks. Table 1 shows the ensemble ANN's architecture optimised based on the hyperparameter tuning analysis. Finally, in the last step of the churn propensity model, we considered features that have the most predictive power on classifier outcome as possible causal effect by analysing model output with two popular plots named SHapley Additive Explanation (SHAP) and partial dependence plot (PDP). Therefore, the initial assumption is generated by observing how a feature affects predictions and what is the relationships between influential features and prediction accuracy. Then, our assumptions are represented in a DAG based on the previous knowledge obtained from correlation analysis and evaluating influential predictors. Therefore, we take advantage of both ML and Bayesian systems in this study. In ML, we predicted the target or independent variable by observing dependent variables, selected the most influential features to enhance prediction accuracy, and used them in our proposed causal inference model. Therein after, a novel approach for causal analysis with robust outcome is applied in this study. We built a dependency structure between dependent features, i.e., a causal graph, based on statistical relationships between independent features. Different methods to identify causal traces from large datasets enabled predicting more flexible causal relations [START_REF] Lopez-Paz | Towards a learning theory of cause-effect inference[END_REF].

Experiment Setup

Datasets

Experiments were conducted on 12 datasets for members holding accounts provided by a local finance company. All 12 datasets included customer accounts, demographics, customer engagement, and financial data, with approximately 250,000 examples with 88 features (71 numerical and 17 nominal) in each dataset. Figure 4 visualises customer and account tenure distribution data that are almost identical in all datasets. Therefore, it can be inferred that approximately 90% of customers had 5-12 months account tenure, with an average tenure of ≈ 7.6 months. Thus, most customers closed their accounts in less than 1 year. Therefore, we adopted the rational approach to only retain customer accounts that were open for more than six months, i.e., we eliminated all recently opened accounts.

Evaluation Metrics

Measuring model performance is essential for ML, and several techniques evaluate model effectiveness for regression and classification tasks. We used area under the curve (AUC) and recall visualising overall performance of a classifier. AUC represents how capable the model is to discriminate classes [START_REF] Bradley | Use of the area under the ROC curve in the evaluation of machine learning algorithms[END_REF], with larger AUC indicating better distinguishability between churned and non-churned customers, and AUC ≈ 1 represents good separability. AUC is defined as [START_REF] Culbert | Customer churn prediction in superannuation: a sequential pattern mining approach[END_REF] and shown below equation, where i denotes the whole data points from 1 to m when the churn label 1 in churned class pi and j denotes all data points that represent from 1 to n when the churn label 0 in non-churned class. Both pi and pj are probabilities out- comes related to each class here; 1 is an indicator when the condition pi > pj is true.

Correlation Analysis

Relationships between variables can be measured as their correlation defined as [START_REF] James | An introduction to statistical learning-with applications in R-Gareth James[END_REF]:

The correlation analysis identifies potentially meaningful connections between variables and is applied to select highly correlated relationships for subsequent causal discovery.

(5)

AUC = 1 mn n ∑ i=1 m ∑ j=1 1.pi > pj (6) Cor(X, Y) = Σ n i=1 x i -x y i -y √ Σ n i=1 x i -x y i -y

Causality Analysis Experiment Setup

We employed Bayesian causal graphs to encode assumptions and determine dependency levels between features, using the DoWhy python package [START_REF] Sharma | DoWhy: an end-to-end library for causal inference[END_REF]. DoWhy performs causal discovery on all potential ways to identify a desired effect based on the Bayesian causal graph model, exploiting graphbased criteria to find possible interpretation methods [START_REF] Sharma | DoWhy: an end-to-end library for causal inference[END_REF].

Results and Discussion

Prediction Results

Figure 5 shows experiment outcomes conducted on the most updated observation and outcome windows. The RF outperformed the other algorithms with AUC = 80%, whereas the proposed ensemble ANNs and RF have almost the same performance, with improved AUC by 7.5% compared with logistic regression. Thus, ensemble ANNs outcomes were comparable with current best-practise classifiers and achieved maximum prediction accuracy on test data. These findings provide a solid evidence base for exploiting DL reduced time-consuming feature engineering requiring expert knowledge on these specific financial datasets. Furthermore, as shown in Table 2, the low Cohen Kappa Score of 0.86 for the proposed model well proves that there is no big difference between the null error and test accuracy results. Moreover, reliability of model output was measured by the Matthews Correlation Coefficient (MCC) at an acceptable level of 0.45 for the proposed algorithm. Descriptive approaches in statistical analysis define feature weights, reflecting their contribution to pushing model output from its base (average output on the training dataset) to more meaningful outputs. Figure 6 demonstrates SHAP, representing feature impacts on model output, where red features increase and blue reduce prediction outcomes relative to the base value, e.g., feature acc_balance_change_amount reduced and sg_recency increased prediction outcomes. Furthermore, the PDP plot shown in Fig. 7 is employed to assess the impact of the feature assumed as a causal effect The impact of the most potent predictors on the model performance is shown in Fig. 7. The result of PDP and SHAP plots prove that acc_ballance_change_ratio, login_recency, acc_tenure, cust_tenure, and account_growth_change can be a robust causal estimands in the DoWhy under our assumptions.

Causality Analysis Results

Figure 8 demonstrates empirical results and related causal graphs based on assumptions that would affect churn as follows:

• High-limit account balance might affect the churn as users with a lower limit (account balance) might not be loyal to the superannuation funds compared to customers with a high account balance. • The account balance change amount could affect the customer tenure. The customer tenure is often based on account balance, after all. The account balance amount itself might affect the churn. An account balance should not be below $1500 in most superannuation funds, and a low account balance generally represents an inactive customer who may be willing to churn. • A cascade relation between balance and gender shows that gender would affect account balance and then indirectly affect churn. We identified treatment causal effects on churn outcome based on the initial assumptions by holding other potential effects constant while changing the target treatment. For instance, Linear regression estimation indicates that estimated effect = -0.033853 corresponds to churn probability reducing by ≈ 3% when the customer has lower account growth rate.

To test our assumption so that if the assumption is correct, the new estimation effect should not significantly change. Therefore, we applied Data Subset Refuter (DSR) [START_REF] Sharma | DoWhy: an end-to-end library for causal inference[END_REF] to refute the above estimates by rerunning them on a random subset of the original dataset. Outcome from the refuting method = -0.033920, almost identical to the estimation result. Thus, we can confirm that the assumption Fig. 6 The above SHAP graph illustrates the feature's impact on prediction outcomes. The promotion_ pref and acc_tenure are the most influential features pushing the prediction higher, unlike feature acc_ balance_change_amount and sg_recency, which pulls the model's prediction lower Fig. 7 The six features are identified as the highest predictive power variables on the model's predicted outcome and visualised in the above PDP plot Fig. 8 Causal graph exhibiting assumptions encoded in the causal model was correct that high account tenure was a causal feature for churn outcome.

The treatment's causal effect on the outcome is based on the change in the value of the treatment variable. How strong the effect is a matter of statistical estimation. There are many methods for the statistical estimation of the causal effect. In this study, we used the "Propensity Score-Based Inverse Weighting" method [START_REF] Rosenbaum | The central role of the propensity score in observational studies for causal effects[END_REF] and concluded the estimations and churn probability results in Table 3. For instance, the mean estimation for variable sg recency is ∼ 0.15, which is equivalent to saying that the probability of churn is increased by ∼ 15% when the customer has higher days since the last day of super guarantee SG contribution. The mean estimate of ∼ 0.03 for the account growth variable can be concluded that churn probability increases by ∼ 3% when the customer has a negative account growth rate. Although the causal analysis result has successfully demonstrated that our assumptions to identify confounding factors are correct with a high degree of belief, it has certain limitations in terms of analysing the identified confounding effects with other popular causal inference methods like the counterfactual analysis.

Conclusion

Losing customers is inevitable for most businesses, but churn can be managed at acceptable levels by investing in customers with the risk of churn. A novel churn propensity model was built and integrated with the causal Bayesian networks. Unbalanced churned and non-churned classes were levelled in pre-processing with SMOTE sampling methods, and then accuracy was compared between the proposed ensemble ANNs and ten best-practise classifiers. Although RF achieved superior AUC, ensemble ANNs obtained comparable AUC with the highest accuracy of all considered models on test data. We analysed possible customer churn causes for a particular financial dataset created at superannuation fund(s). Causal analysis results confirmed variables representing recent SG contribution, annual report and statement preference changed, account growth rate, and balance amount were identified as confounding factors for customer churn with a high degree of belief. The churn rate can be reduced by ∼ 3% for customers with active account > 1 year, consistent with expert knowledge. Furthermore, the probability of churn is decreased by ∼ 9% when the customer has a high account balance of over $100 k. A natural progression of this work is extending pattern mining techniques with smaller outcome windows that should be investigated to obtain more efficient prediction results in future studies. Furthermore, different methods to identify causes of churn based on counterfactual causal analysis should be investigated. 1 3 included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.

Fig. 1

 1 Fig. 1 Closed and non-closed customer accounts in periods of 6-month

Fig. 2

 2 Fig. 2 Extracting feature values by sliding observation window on datasets. Observation and Outcome time-windows set to 12-months and 6-months respectively

3 .

 3 After feature selection, the proposed algorithms were employed for training (80%), and test (20%) sets to derive 193 features from 124,363 total examples. Binary classification generally produces severely skewed data distributions. Hence, we employed the synthetic minority oversampling technique (SMOTE) [17] in pre-processing to synthesise new examples for the minority class to ensure equal sample count (14,031) for each class (1 or 0) in the training set. Experimental results confirmed improved performance for the proposed model using the sampling method.

Fig. 3

 3 Fig. 3 Churn propensity modelling

Fig. 4

 4 Fig. 4 Distribution of customer and account tenure

Fig. 5

 5 Fig. 5 Several evaluation metrics are applied to the proposed ensemble ANNs model and best-practise classifiers. The ensemble ANNs obtained the highest test accuracy

Table 1

 1 Network architecture of ensemble ANNs

	Network type	Deep ANN-1	Deep ANN-2
	Hidden layers	4	4
	Dense activation 1,2,3	tanh, and 3 × relu	tanh, and 3 × relu
	and 4
	Dropout 1, 2,3 and 4	0.2, 0.2, 0.2, and 0.2 0.4, 0.4, 0.4, and 0.4
	Output activation func-	Sigmoid	Sigmoid
	tion
	Learning rate	0.000474718	0.000012
	Epochs	100	100
	Batch size	512	512
	Optimisation algorithm ADAM	ADAM

Table 2

 2 The result of test accuracy, Cohen Kappa Score, and Matthew correlation coefficient are presented in ten different classifiers

	Model	Test ACC Cohen Kappa Matthew Coeff
	Naive Bayes	0.74	0.36	0.31
	Logistic regression	0.80	0.43	0.40
	Decision tree	0.71	0.32	0.26
	Random forest	0.86	0.51	0.50
	AdaBoost	0.71	0.32	0.27
	ExtraTrees	0.86	0.49	0.49
	GradientBoosting	0.85	0.51	0.49
	XGboost	0.85	0.51	0.50
	Stack ensemble (H.V)	0.81	0.44	0.41
	Stack ensemble (S.V)	0.72	0.36	0.30
	Ensemble ANNs (Pro-	0.86	0.45	0.45
	posed)

• Account growth and high balance change could directly affect churn. No variation in account growth indicates stopped business for most accounts with consequential increased churn propensity.

Acknowledgements

This work is partially supported by the Australian Research Council under Grant numbers: DP22010371, LE220100078, DP200101374, and LP170100891.

Availability of data and materials

The python implementation for the proposed framework and causal analysis result, and visualisation in detail, is available on GitHub (https:// github. com/ David Hason/ Causal Analysis), to simplify reproducing and improving this study experiment results.

Author contributions DHR made substantial contributions to conceptualisation, investigation, and methodology, analysis and interpretation of data. GX and HH helped in the revision and gave final approval of the version to be published. Project administration, GX and HH.

Funding Not applicable.

Declarations

Conflict of interest

The authors declare that this research work has non-financial academic and intellectual competing interests..

Consent for publication

We hereby grant and assign all rights to Human-Centric Intelligence Systems for publication.

Ethics approval and consent to participate

The dataset used in this study has been unidentified by the company and has no data ethical or privacy issues.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are


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Causal Analysis of Customer Churn Using Deep Learning 1 st

Keywords: Churn Analysis, Causality Analysis, Deep Neural Network, Data Mining

Customer churn describes terminating a relationship with a business or reducing customer engagement over a specific period. Two main business marketing strategies play vital roles to increase market share dollarvalue: gaining new and preserving existing customers. Customer acquisition cost can be five to six times that for customer retention, hence investing in customers with churn risk is smart. Causal analysis of the churn model can predict whether a customer will churn in the foreseeable future and assist enterprises to identify effects and possible causes for churn and subsequently use that knowledge to apply tailored incentives. This paper proposes a framework using a deep feed-forward neural network for classification accompanied by a sequential pattern mining method on high-dimensional sparse data. We also propose a causal Bayesian networks to predict cause probabilities that lead to customer churn. Evaluation metrics on test data confirm the XGBoost and our deep learning model outperformed previous techniques. Experimental analysis confirms that some independent causal variables representing the level of super guarantee contribution, account growth, and customer tenure were identified as confounding factors for customer churn with a high degree of belief. This paper provides a real-world customer churn analysis from current status inference to future directions in local superannuation funds.

I. INTRODUCTION

Businesses rely heavily upon retaining satisfied customers in the global marketplace, which account for a large portion of their revenue. As the market becomes increasingly saturated, businesses have learned to emphasize maintaining existing clients. Although obtaining new clients was critical for initial business success, retention policies should have equal weight. Many previous studies have identified retention rate's substantial effect on the market, but clients are always able to churn away from a business, resulting in potential losses for the organization. However, customers usually offer some warning before being churned, hence churn prediction systems primarily focus on customer behaviour to identify specific customers who are likely to churn out and indicate reasons for the churn. Such factors would aid marketing to develop effective retention strategies, increasing overall customer lifetime value, and assisting in growing the company's market value. For example, Bank or financial organization churn customers who close their accounts and had fewer transactions/interactions over time [START_REF] Domingos | Experimental Analysis of Hyperparameters for Deep Learning-Based Churn Prediction in the Banking Sector[END_REF]. Churn rate, ChurnRate = LostCustomers/InitialCustomers is the percentage of customers a company loses in a fixed period, and represents the business pulse [START_REF] Ahn | A Survey on Churn Analysis in Various Business Domains[END_REF]. Voluntary churn is where a consumer wants to leave the company on their own. This could be due to dissatisfaction with product, or perhaps they feel they are not receiving that value they expected. In contrast, involuntary churn is where a customer leaves the company for unavoidable reason(s), such as payment problems due to expired account details, network issues, or inadequate cash; and incidental churn occurs due to location or financial situation changes. Deliberate churn arises due to customer desire to change innovation and price, where the most common reasons are poor service quality, noncompetitive pricing, and missing customer expectations. Companies can utilize churn analysis to determine the individual user risk levels and develop appropriate focused retention initiatives.

The proposed framework applies churn analysis for a local superannuation fund. Data preparation and analysis verified that most customer accounts were active for less than one year, for several possible reasons, including some customers being involuntary churners who were subsequently enrolled in different superannuation fund(s) operated by their employers. In contrast, voluntary churn occurs when a customer decides to cease the account for personal reasons, including perceived quality, technology, and price.

As our main contribution to this study, we built a specific churn propensity modelling work-frame to analyse customer financial behaviour data, then generalised the model for any superannuation fund(s) businesses. First, we analysed 12-month time window data to predict churners for the next six months. Since mining data from the 12-month observation window allowed us to extract the latent factors cause of churn. Then, we scaled up the model by exploiting Bayesian networks to describe how deliberate churn occurs due to multiple causes and showed that hypothesis tests on common features greatly influence prediction results. finally, we proposed a specific causality analysis method that can be applied to other similar datasets employed in most superannuation funds.

It should be noted that this study only focuses on the performance of a DFF NN to address churn prediction on a massive and high dimensional sparse dataset that is usually created in financial networks. We conducted preprocessing phase in the proposed model for the baseline classifiers to address the sparse and skewed data and then compared the results with the output of proposed DFF NN. These very high dimensions data are primarily generated in subscribed-base businesses mostly in superannuation funds due to employing interval-based features used in customer relationship management (CRM) systems.

The remainder part of the paper proceeds as follows: Section II reviews recent relevant churn studies, and Section III gives preliminary knowledge on the application of the deep learning method in churn prediction problems. Section IV discusses specific research and analysis methods employed to predict underlying reasons for churn including problem definition and churn propensity modelling workflow. Section V analyses gathered data to address research questions: a causal analysis of churn through the financial data collected by superannuation funds and present experiments causality analysis outcomes. Finally, Section VI summarises and concludes the paper and discusses implications from the findings for real-world applications.

II. RELATED WORKS

A. Customer churn prediction and causal analysis

Most previous studies focused on determining churn variables for a particular dataset rather than customer churn causation analysis.

CHAMP (Churn analysis, modelling, and prediction) is an integrated system for forecasting consumers cancelling their cellular phone service [START_REF] Pearl | Causal inference in statistics: An overview[END_REF]. Alyuda Neu-roIntelligence employs neural networks (NNs) for data mining to forecast customer churn at banks [START_REF] Bilal Zorić | Predicting customer churn in banking industry using neural networks[END_REF]. Integrating textual data using customer churn prediction (CCP) algorithms adds value [START_REF] De Caigny | Incorporating textual information in customer churn prediction models based on a convolutional neural network[END_REF], and combining different classifiers, e.g. gradient boost, oversampling, and contrast sequential pattern mining on single year observation windows has been shown to be a practical strategy to deal with highly skewed data collected from superannuation funds [START_REF] Culbert | Customer churn prediction in superannuation: a sequential pattern mining approach[END_REF]. Different churn prediction techniques have been evaluated to identify optimal approaches [START_REF] Tamaddoni | Comparing churn prediction techniques and assessing their performance: A contingent perspective[END_REF].

Hidden churn is a common problem for superannuation funds, where customer accounts become dormant once mandatory employer payments cease. Various remedies for insufficient consumer interaction have been proposed, addressing unbalanced and fully leveraged data problems, and multiple classifiers have been developed from sampled datasets [START_REF] Culbert | Customer churn prediction in superannuation: a sequential pattern mining approach[END_REF]. Deep learning techniques can handle very large datasets compared with standard machine learning (ML) approaches, and combining deep learning and convolutional neural networks (CNNs) have successfully forecast churn [START_REF] Mishra | A novel approach for churn prediction using deep learning[END_REF].

Recent causal inference developments highlight fundamental alterations required to move from standard statistical analysis towards multivariate causal analysis [START_REF] Pearl | Causal inference in statistics: An overview[END_REF]. New PC-Stable approaches effectively learn causal structures using DFS data, allowing temporal causal modelling for enormous time information datasets [START_REF] Mohan | Temporal Causal Modelling on Large Volume Enterprise Data[END_REF]. Directed acyclic graphs have been proposed to represent causal relationships in Bayesian networks [START_REF] Huang | Identifiability in causal Bayesian networks: A sound and complete algorithm[END_REF]. A likelihood of churn anticipates, as are the driving factors on banking data. Subsequently, Shah et al [START_REF] Shah | Prediction and Causality analysis of churn using deep learning[END_REF] trained a model to produce suitable weights for features that predict whether a customer would churn. They contrasted churn definitions commonly used in business administration, marketing, IT, telecommunications, newspapers, insurance, and psychology. Many studies have and described churn loss, feature engineering, and prediction models using this approach [START_REF] Ahn | A Survey on Churn Analysis in Various Business Domains[END_REF].

The above approaches have been variously adopted for current churn prediction methods, but most are employed for collecting data from telco, banks, and retail markets. In contrast, the proposed approach was tested using a massive financial dataset with high-dimensional sparse data. To our best knowledge, no previous study considered causality analysis for customer churn in superannuation companies. Motivated by the studies mentioned above, We proposed a new approach that combines deep learning and Bayesian theory to improve churn prediction modelling systems.

III. PRELIMINARY KNOWLEDGE

Churn prediction models will provide earlier churner identification and hence assist customer intervention program development. This is basically a classification problem, i.e., to categorise each customer as potential churner or non-churner. Several machine learning techniques for churn prediction have produced verifiable results from interpretable models, such as boosting technique, a nonparametric method, logistic regression. However, this approach is only valid predictability under circumstances where the size of the customer database is very small, varying sample size, and fails for larger datasets [START_REF] Tamaddoni | Comparing churn prediction techniques and assessing their performance: A contingent perspective[END_REF]. Therefore, we propose a deep learning (DL) algorithm to deal with massive financial data volumes since deep learning (feature transformation) differentially weights features using historical data. Deep learning algorithms follow human brain architecture using artificial neural networks (ANNs) [START_REF] Zaïane | Principles of knowledge discovery in databases[END_REF]. Feed-forward NNs use output from one layer as input for the next layer with no loops between layers [START_REF] Nielsen | Neural networks and deep learning[END_REF]. Therefore, we employed the deep learning feed-forward (DFF) algorithm for experiments, trading-off between reduced computational load and highest accuracy for different algorithms. General advantages for DFF NNs can be summarized as 1) superior accuracy from DL; 2) includes more variables than classical ML; 3) DL algorithms can extract patterns while avoiding blind spots from extensive customer demographics, behavioural variables, and billions of customer engagement logs. 4) reduces time-consuming feature engineering and manual financial data analysis. We employed a DFF NN approach not only due to DL useful features, but also we only need changing the last layer(s) to update the proposed model for future data, skipping time-consuming training models. One significant weakness for deep learning is that is remains a black-box model, i.e., DL does not express uncovered patterns in the underlying data in easily understandable ways. However, we conducted experimental analyses to identify churn effects and possible causes using Bayesian networks to address NN model complexity.

IV. METHODOLOGY

A. Problem definition

Data mining can identify useful knowledge in terms of pattern extraction from different sources, and various feature engineering tools can extract hidden patterns from massive datasets [START_REF] Zaïane | Principles of knowledge discovery in databases[END_REF], [START_REF] Karp | Using logistic regression to predict customer retention[END_REF]. Figure 1 shows that this study included 12 datasets from a superannuation company for real-world experiments to verify the proposed model's effectiveness. We define a customer as a churner if they closed their account during the subsequent 6 month time window. Therefore, we use binary outcome for each customer [0 or 1], where 1 means the account closed and 0 that it was not close in the subsequent 6 month time window.

We applied two main inclusion criteria for account tenure and balance to satisfy model development. First, we only retained data for customers with more than 6 month account tenure; and second, we removed account balances below $1500 to improve prediction since predicting churn probability for inactive accounts has low value for superannuation funds. Figure 2 shows features in the observation window to predict which user will be churned or non-churned in the next 6 month outcome window. Furthermore, We employed a majority voting ensemble from the scikit-learn library [START_REF] Kramer | Scikit-learn[END_REF] to combine predictions from multiple models, which can helpful to improve model performance for classification tasks. Ensemble hard and soft voting were both applied on the comparison supervised models. Hard voting counts each individual classifier votes and the majority wins; whereas soft voting weights each prediction by classifier importance, and the target label with the greatest sum of weighted probabilities wins [START_REF] Kramer | Scikit-learn[END_REF]. We defined a specific ANN architecture to address the research problem. Table I shows the DFF ANN's architecture optimised based on the hyperparameter tuning analysis. Finally, in the last step of the churn propensity model, we considered features that have the most predictive power on classifier outcome as causal variables. Then, we used a directed acyclic graph (DAG) to trace the latent variables and unknown parameters. Therefore, we take advantage of both Machine learning (ML) and Bayesian systems in this study. In ML, we predicted the target or independent variable by observing dependent variables, selected the most influential features to enhance prediction accuracy, and used them in our proposed causal Bayesian network. Thereinafter we employ a different approach for causal analysis. We built a dependency structure between dependent features, i.e., a causal graph, based on statistical relationships between independent features. Different methods to identify causal traces from large, labelled datasets enabled predicting more flexible causal relations [START_REF] Lopez-Paz | Towards a learning theory of cause-effect inference[END_REF].

V. EXPERIMENT SETUP AND RESULTS

A. Datasets

Experiments were conducted on 12 datasets for members holding accounts provided by a local finance com- Figure 4 visualises customer and account tenure distribution data that are almost same on the 12 datasets. It can be inferred that approximately 90% customers had 5-12 month account tenure, with average tenure ≈ 7.6 month. Thus, most customers closed their accounts in less than one year. Therefore, we adopted the rational approach to only retain customer accounts that were open for more than six months, i.e., we eliminated all recently opened accounts.

B. Evaluation metrics

Measuring model performance is essential for ML, and several techniques evaluate model effectiveness for regression and classification tasks. We used area under the curve (AUC) and recall to visualise overall performance of a classifier. AUC represents how capable the model is to discriminate classes [START_REF] Bradley | Use of the area under the ROC curve in the evaluation of machine learning algorithms[END_REF], with larger AUC indicating better distinguishability between churned and non-churned customers, and AUC ≈ 1 represents good separability. AUC can be defined as [START_REF] Culbert | Customer churn prediction in superannuation: a sequential pattern mining approach[END_REF]

AU C = 1 mn n i=1 m j=1 1.pi > pj, (1)
where i denotes to whole data points from 1 to m when the churn label 1 in churned class p i and j denotes to all data points represent from 1 to n when the churn label 0 in non-churned class. Both p j and p j are probabilities outcome related to each class, here 1 is an indicator when the condition p i > p j is true.

C. Correlation analysis

Relationships between variables can be measured as their correlation defined as [START_REF] James | An Introduction to Statistical Learning -with Applications in R -Gareth James[END_REF]:

Cor(X, Y ) = Σ n i=1 (x i -x)(y i -ȳ) Σ n i=1 (x i -x)(y i -ȳ) , (2)
We used correlation analysis to identify potentially meaningful connections between variables, selecting Fig. 5: Proposed DFF NN and current best-practise model outcomes highly correlated relationships for subsequent causal discovery and hypothesis testing, as described in Section IV.

D. Causality analysis experiment setup

We employed Bayesian causal graphs to encode assumptions and determine dependency levels between features, using the DoWhy python package [START_REF] Sharma | DoWhy: An end-to-end library for causal inference[END_REF]. DoWhy performs causal discovery on all potential ways to identify a desired effect based on the Bayesian causal graph model, exploiting graph based criteria to find possible interpretation methods [START_REF] Sharma | DoWhy: An end-to-end library for causal inference[END_REF].

E. Prediction results

Figure 5 shows experiment outcomes conducted on the most updated observation and outcome windows. XGBoost outperformed the other algorithms with AUC = 80%; whereas the proposed DFF NN and Random Forest (RF) have almost the same performance, with improved AUC by 7.5% compared with logistic regression. Thus, DFF NN outcomes were comparable with current bestpractise classifiers and achieved maximum prediction accuracy on test data. These findings provide a solid evidence base for exploiting DL reduced time-consuming feature engineering requiring expert knowledge on these specific financial datasets.

Descriptive approaches in statistical analysis define feature weights reflecting their contribution to push model output from its base (average output on the training dataset) to more meaningful outputs. Figure 6 shows SHAP, representing feature impacts on model output, where red features increase and blue reduce prediction outcomes relative to the base value, e.g. feature acc balance change amount reduced and sg recency increased prediction outcomes. Fig. 6: SHAP visualisation on prediction outcomes. The age, acc tenure, and sg recency are the most influential features pushing the prediction higher, unlike feature acc balance change amount that is pulling the model's prediction lower.

F. Causality analysis results

Figure 7 shows empirical results and related causal graphs based on assumptions that would affect churn as follows:

• High limit account balance might affect the churn as users with a lower limit (account balance) might not be loyal to the superannuation funds compared to customers with a high account balance. • The account balance change amount could affect the customer tenure. The customer tenure is often We identified treatment causal effects on churn outcome based on the initial assumptions by holding other potential effects constant while changing the target treatment. Linear regression estimation indicates that effect = -0.026298 corresponds to churn probability reducing by ≈ 2.5% when the customer has higher account tenure. To test our assumption so that if the assumption is correct, the new estimation effect should not significantly change. Therefore, we applied random common cause (RCC) [START_REF] Sharma | DoWhy: An end-to-end library for causal inference[END_REF] to refute obtained estimates by adding independent random variable res random as a common cause in the dataset. Outcome from the refuting method = -0.026299, almost identical to the estimation result. Thus, we can confirm the assumption was correct that high account tenure was a causal feature for churn outcome.

The treatment's causal effect on the outcome is based on the change in the value of the treatment variable. How strong the effect is a matter of statistical estimation. There are many methods for the statistical estimation of the causal effect. In this study, we used the "Propensity Score-Based Inverse Weighting" method [START_REF] Rosenbaum | The central role of the propensity score in observational studies for causal effects[END_REF] and concluded the estimations and churn probability results in Table II. For instance, the mean estimation for variable sg recency is ∼ 0.15, where it is equivalent to saying that the probability of churn is increased by ∼ 15% when the customer has higher days since the last day of super guarantee SG contribution. The mean estimate of ∼ 0.03 for the account growth variable can be concluded that churn probability increases by ∼ 3% when the customer has a negative account growth rate. The python implementation for the proposed framework and result visualisation, including hyper-parameter sensitivity analysis of DFF NN in hiplot, and our causal analysis in detail, is available in GitHub (https: //github.com/DavidHason/Causal Analysis), to simplify reproducing and improving this study experiment results.

VI. CONCLUSION

Losing customers is inevitable for most businesses, but churn can be managed at acceptable levels by investing in customers with churn risk. A new churn propensity model was designed and integrated with the Bayesian causal network. Unbalanced churned and non-churned classes were levelled in pre-processing, and then accuracy was compared between the proposed DFF NN approach and ten current best-practise churn prediction classifiers. Although XGBoost achieved superior AUC, DFF NN obtained comparable AUC with highest accuracy of all considered models on test data.

We analysed possible customer churn causes for a particular financial dataset created at superannuation fund(s) corporations. Causal analysis results confirmed variables representing recent SG contribution, annual report preference changed, account growth and balance amount were identified as confounding factors for customer churn with high degree of belief. Churn rate can be reduced by ∼ 3% for customers with active account > 1year, consistent with expert knowledge.

A natural progression of this work is extending pattern mining techniques with smaller outcome windows that should be investigated to obtain more efficient prediction results in future studies. Also, different methods to identify causes of churn based on counterfactual causal analysis should be investigated.

VII. ACKNOWLEDGMENT

Fig. 1 :

 1 Fig. 1: Closed and non-closed customer accounts in time periods of 6-month.

Fig. 2 :

 2 Fig. 2: Extracting feature values by sliding observation window on datasets

Fig. 3 :

 3 Fig. 3: Churn propensity modelling

Fig. 4 :

 4 Fig. 4: Distribution of customer and account tenure

Fig. 7 :

 7 Fig. 7: Causal graph exhibiting assumptions encoded in the causal model

TABLE I :

 I Network Architecture

	Network type	Deep feed-forward neural network
	Hidden layers	4
	Dense activation 1,2,and 3	tanh, and 2 x relu
	Dropout 1 and 2	0.3 and 0.0
	Output activation function	Sigmoid
	Learning rate	0.000474718
	Epochs	100
	Optimization algorithm	ADAM

TABLE II :

 II Causal analysis of customer churn

	Variable	Estimate Mean Value	Pobability of churn
	acc tenure	-0.0262	decreased by ∼3%
	annualrpt pref	0.1444	increased by ∼14%
	stmt pref	-0.1427	decreased by ∼14%
	acc balance	-0.0916	decreased by ∼9%
	account growth	0.0331	decreased by ∼3%
	cust tenure	-0.0279	decreased by ∼3%
	sg recency	0.1563	increased by ∼15%
	promotional pref	-0.0864	decreased by ∼8%

This work is partially supported by the Australian Research Council (ARC) under Grant No. DP200101374 and LP170100891.


Ikechukwu Udema
email: udemaikechukwu75@yahoo.com

Desirability of Reduced mass Concept, Conflicting CODATA Values of some Constants: Variants of Fine structure constant and Rydberg constant Equations Subtitle: Reduced mass, fine structure constant and Rydberg constant

Keywords: Average ionisation energy, fine structure constant, reduced mass, rest mass, Rydberg constant, variants of Rydberg constant and fine constant equations

Background:

The desirability of the use of the reduced mass concept as opposed to the rest mass seems to give conflicting results. Objectives: To derive variants of the equation of the Rydberg constant, fine structure constant, and wave number that relate with the Rydberg constant; Methods: The research is theoretical in nature. Results: The Rydberg constant (R∞) equations derived in this research are variants of each other and of those known in the literature; this is also applicable to the fine structure constant (a). The values of R∞ range between 10973731.6 and 1097373.89 /m for the rest mass case, while for the reduced mass case, the range is between 10966253.06 and 1096784.63 /m; regarding a, the derived equations gave results that are the same. The wave number was related to R∞ at the atomic level for the first time. The number showed a trend with the square of the atomic number, similar to the square of the average ionisation energy. Conclusion: The use of rest mass and reduced mass gives conflicting values for physical constants. Other variants of the equation for R∞ may be determined in the future.

Introduction

The desirability of the use of reduced mass concept as opposed to the rest mass seems to give conflicting results. "Reduced mass" is a term for the appropriate mass corresponding to a linear combination of coordinates involving two or more individual particles such as atoms in the context of the investigation carried by the authors [START_REF] Truhlar | Reduced mass in the one-dimensional treatment of tunneling[END_REF]; a simpler definition is that in physics the reduced mass is the effective inertial mass appearing in the two-body problem of Newtonian mechanics (wikipedia).

In deriving the equation of reduced mass concept, the equations of motion of two mutually interacting bodies is reduced to a single equation describing the motion of one body in a reference frame centred in the other body. (Britannica.com). Barbosa [START_REF] Barbosa | The noninertial origin of the reduced mass 2006[END_REF] who sees the reduced mass as a quantity studied in a two-particle system problem in mechanics have attempted to explain the origin of the reduced mass concept. According to the author's note [START_REF] Barbosa | The noninertial origin of the reduced mass 2006[END_REF], the reduced mass of a two-particle system has its origin in the combination of noninertial nature of an observer attached to particle 1 leading to an inertial force (Fin); the force (F1(2)) which drives the observer is intimately related to the force applied to particle 2 (F2(1)) through the Newtonian 3 rd law of motion as they arise from the mutual interaction.

The reduced mass is known to cause the stark broadening of the spectral lines of the hydrogen atoms; it has also been implicated in the deformation of the central part of the line shape of the spectrum [START_REF] Barbosa | The noninertial origin of the reduced mass 2006[END_REF]. Meanwhile, the Rydberg constant has a role in spectroscopy and is related to the fine structure constant. Hence the desirability of the reduced mass concept, the conflicting CODATA [START_REF] Mohr | CODATA recommended values of the fundamental physical constants[END_REF][START_REF] Mohr | CODATA recommended values of the fundamental physical constants[END_REF] values of some constants, and the variants of the fine structure constant and Rydberg constant equations have to be examined or investigated. The goal is to reveal a possible conflict in some fundamental physical constants by embarking on the following objectives: 1) Derive variants of the equation of Rydberg constant and compare the resulting value with literature; 2) derive variants of the equation relating wave number with Rydberg constant at the atomic level; 3) show that there could be multiple variants of the equation of fine structure constant, which gives values that are the mass of an electron invariant.

Materials and Methods

The research is theoretical without materials and experiments of any kind.

Results

Worthy of note in this research is the observation that the physical variables, the wave number, average ionisation energies, and the square of the effective nuclear charge vary with the square of nuclear charge in the same way (Figure 1). The square of the nuclear charge is chosen because it best correlates with the physical variable determined based on the average ionisation energy. The values of the Rydberg constant calculated in this research ranges between 10973731.6-10973733.89/m for the rest mass case while, for the reduced mass case, the range is between 10966253.06-1096784.63/m (Table 1) which compares with the CODATA [START_REF] Mohr | CODATA recommended values of the fundamental physical constants[END_REF] value of 10973731.568/m.

Rest mass values are therefore, more similar to the 2016, CODATA [START_REF] Mohr | CODATA recommended values of the fundamental physical constants[END_REF] value. This is against the backdrop of higher values of the average ionisation energy of hydrogen and Bohr's radius calculated using rest mass and reduced mass respectively.

Variants of the equations for the fine structure constant (), Eq. (33) and Eq. ( 34 Values are approximations to the 3 rd decimal places.

The values of the wave number calculated based on Eq. (40) are not influenced by the mass of the electron if the ionisation based radii were calculated using experimentally determine average ionisation energies. An exception may apply to hydrogen if either the rest mass or reduced mass was applied for the calculation of Bohr's radius. The calculations based on Eq. ( 37) are influenced by the use of either rest mass or reduced mass. The values ranges between 2.735 and 4.036 exp. (+9)/m for Eq.

(37); with Eq. ( 40) it is between 3.006 and 4.038 exp. (+9)/m (Table 2).

Theory

The ratio of the product of the ionisation energy dependent atomic radius and the momentum of the orbital electron to the energy level as a function of the composite fundamental constant

In this section, the relationship between the ratio of the product of the momentum and the ionisation energy-dependent radius of the atom to the energy level and a composite fundamental physical constant are derived. In line with pedagogical principle, one begins from the known to the unknown. The Rydberg constant (R) is given as:

𝑅 ∞ = 𝛼 2 𝑚 𝑒 𝑐 2 ⁄ ℎ, (1)
where α, c, and h are the fine structure constant, rest mass of an electron, velocity of light in a vacuum, and Planck constant respectively.

𝛼 = 𝑒 2 2ℎ 0 𝑐 ⁄ , ( 2
)
where 0 is the permittivity in free space; α is the fine structure constant. Substitute Eq. ( 2) into Eq. (1) to give:

𝑅 ∞ = 𝑚 𝑒 𝑒 4 8 ℎ 3  0 2 𝑐 ⁄ , (3)
Equation ( 3) is definitely not a novelty, but with Eqs (1) and ( 2), it serves an immediate reference.

It has been shown that the effective nuclear charge (Zeff) is inversely proportional to α, directly proportional to the principal quantum number (n), and directly proportional to the square root of the average ionisation energy (Ei) [7]; thus,

𝑍 𝑒𝑓𝑓 = 𝑛 ( 2𝐸 𝑖 𝑚 𝑒 ) ½ 𝛼𝑐 ⁄ , (4)
Therefore,

𝑐 = 𝑛 (2𝐸 𝑖 𝑚 𝑒 ⁄ ) ½ 𝑍 𝑒𝑓𝑓 𝛼 ⁄ , (5)
Substitute Eq. ( 5) into Eq. ( 5) into Eq. ( 3) to give:

𝑅 ∞ = (𝑍 𝑒𝑓𝑓 𝑚 𝑒 𝑒 4 𝛼 8 ℎ 3  0 2 𝑛 ⁄ ) ( 𝑚 𝑒 2𝐸 𝑖 ) ½ , (6)
From Eq. ( 6) is given:

𝐸 𝑖 = (𝑍 𝑒𝑓𝑓 𝑚 𝑒 𝑒 4 𝛼 8 ℎ 3  0 2 𝑛 ⁄ 𝑅 ∞ ) 2 2 ⁄ , (7)
Substitute Eq. ( 2) into Eq. ( 7) to get

𝐸 𝑖 = 𝑚 𝑒 (𝑍 eff 𝑚 𝑒 𝑒 6 𝛼 16 ℎ 4  0 3 𝑛 ⁄ 𝑐 𝑅 ∞ ) 2 2 ⁄ , (8)
Solving for Zeff gives:

𝑍 eff = ( 512𝐸 𝑖 𝑚 𝑒 3 ) ½  0 3 ℎ 4 𝑛 𝑖 𝑅 ∞ 𝑐 𝑒 6 ⁄ , (9)
Substitute Eq. ( 9) into Bohr's equation for ionisation energy to give:

( 512𝐸 𝑖 𝑚 𝑒 3 ) 2 ( 0 3 ℎ 4 𝑛 𝑖 𝑅 ∞ 𝑐) 2 𝑒 4 𝑚 𝑒 8 0 2 ℎ 2 𝑛 2 𝑒 12 = 𝑍 eff 8   0 𝑎 𝑖 ⁄ , ( 10a
)
where ai is the average ionisation energy-dependent radius of any atom.

Simplification and solving for Zeff gives:

𝑍 eff = 512𝐸 𝑖  0 5 ℎ 6 𝑅 ∞ 2 𝑐 2 𝑎 𝑖  𝑚 𝑒 2 𝑒 10 , ( 10b
) From Eq. (10b) is given  0 2 ℎ 4  2  2 ⁄ 𝑚 𝑒 2 𝑒 4 which is  𝑎 0 2  2
where a0 is the Bohr's radius for hydrogen; Therefore, Eq. (10b) becomes:

𝑍 eff = 512𝐸 𝑖 𝑎 𝑖 (  0  𝑒 2 ) 3 (ℎ𝑅 ∞ 𝑐 𝑎 0 ) 2 , ( 11
)
Since the atomic radius is inversely proportional to the average ionisation energy-a kinetic energy-on the basis of Coulomb's law, one can multiply both sides of Eq. ( 11) by ai and convert Ei to its relation to momentum (P) to give:

𝑍 𝑒ff 𝑎 𝑖 = 512𝑃 2 a 𝑖 2 (  0  𝑒 2 ) 3 (ℎ𝑅 ∞ 𝑐 𝑎 0 ) 2 2 𝑚 𝑒 ⁄ , (12)
In Eq. ( 12), ai Zeff is equal to n 2 h 2 0/e 2 me = a0n 2 ; thus,

𝑝 2 𝑎 𝑖 2 = 𝑎 0 𝑛 2 2 8 ( 𝑒 2  0  ) ( 1 ℎ𝑅 ∞ 𝑐 𝑎 0 ) 2 , ( 13
) 𝑝𝑎 𝑖 𝑛 = 1 16 ℎ 𝑐 𝑅 ∞ ( 𝑚 𝑒 𝑎 0 𝑒 6  0 3  3 ) ½ , (14)
Given that (8EHa0) 3 = (e 2 /0) 3 , one can rewrite, Eq. ( 14) for the purpose of calculational simplicity, as follows:

𝑝𝑎 𝑖 = 𝑛𝑎 0 (2𝐸 H 3 𝑚 𝑒 ) ½ ℎ𝑅 ∞ 𝑐 ⁄ , (15)
If ai = a0 and n = 1, then, hydrogen is referred to:

𝑅 ∞ = (2𝐸 H 3 𝑚 𝑒 ) ½ ℎ 𝑐 ⁄ 𝑝, (16a)
Equation (16a) can be simplified to obtain equation similar to a literature version [7]:

𝑅 ∞ = 𝐸 H ℎ ⁄ 𝑐, (16b)
There are two clear corollaries:

𝑅 ∞ = 𝑛𝑎 0 (2𝐸 H 3 𝑚 𝑒 ) ½ ℎ 𝑐 𝑝𝑎 𝑖 ⁄ , (17a)
Equation (17a) can be simplified further if one solves mathematically p (i.e. 2me Ei) ½ and ai (= 𝑒 2 𝑛 8  0 (𝐸 𝐻 𝐸 𝑖 ) ½ ); after rearrangement the results is:

𝑅 ∞ = 8   0 𝑎 0 𝐸 H 2 ℎ 𝑐 𝑒 2 , ( 17b
)
Like Eq. ( 2), one can derive a similar equation for effective nuclear charge as follows: First recall that [7]:

1 α = ( 𝑚 𝑒 𝑐 2 2𝐸 𝐻 ) ½ , (18)
Also, EH can be expresses as [7-9]:

𝐸 𝐻 = 𝑛 2 𝐸 𝑖 𝑍 eff 2 ⁄ , (19)
Substitute Eq. ( 19) into Eq. ( 18) to give after rearrangement:

𝑚 𝑒 𝑐 2 = 2 α 2 𝑛 2 𝐸 𝑖 𝑍 eff 2 , (20)
Solving for Zeff gives [7]:

𝑍 eff = 𝑛 𝛼𝑐 ( 2𝐸 𝑖 𝑚 𝑒 ) ½ , (21)
Substitute Eq. (2) into Eq. ( 21) to give:

𝑍 eff = 2𝑛𝑣ℎ 0 𝑒 2 , ( 22
)
where v is the velocity of the electron and n may be ≥ 1; this is determined as the result of (2𝐸 𝑖 𝑚 𝑒 ⁄ ) ½ .

Thus, given the reciprocal of Eq. ( 2), one sees the similarity in the mathematical structure as follows:

1 α ⁄ = 2 𝑛 0 𝑐 ℎ  0 𝑒 2 (23)
where n0 is = 1. The obvious corollary is that:

𝑍 eff 𝑛 𝑣 = 1 𝑛 0 𝑐 𝛼 = 2ℎ  0 𝑒 2 , ( 24
)
𝑍 eff = 𝑛 𝑣 𝑛 0 𝑐 𝛼 , (25)
Equation ( 14) can be rewritten to give two equations.

𝑣 = 𝑛 16 𝑎 𝑖 ℎ 𝑐 𝑅 ∞ ( 𝑒 6 𝑚 𝑒 𝑎 0  0 3  3 ) ½ , (26)
The values of ai can range between a0 and a0n 2 /Zeff where Zeff is between 1 and < 137 and n can range between 1 and 7.

Fine structure constant-based equations for the determination of the radius of any atom

As in the literature [START_REF] Udema | Calculation of the radius for any atom based on the Planck constant-free equation, mathematical complexity, and atomism[END_REF], the radius of any atom is expressed as:

𝑎 𝑖 = 𝑒 2 𝑛 8  0 (𝐸 𝐻 𝐸 𝑖 ) ½ , (27)
Note that if hydrogen is referred to EH is = Ei and n = 1; if it is multi-proton hydrogenic ion, EH is  Ei, though n remains = 1. One can then re-express ai in terms of v as follows:

Given that Ei is = to EH 𝑍 eff 2 𝑛 2 ⁄ ,
Eq. ( 27) becomes:

𝑎 𝑖 = 𝑒 2 𝑛 2 8  0 𝐸 𝐻 𝑍 eff , (28a)
Then substituting Eq. (25) into Eq. (28a) gives:

𝑎 𝑖 = 𝑒 2 𝑛 𝑛 0 𝑐 𝛼 8   0 𝐸 𝐻 𝑣 , (28b)
Taking v as (2Zeff e 2 /8  0 ai me) ½ and substitute into Eq. (28b) to give:

𝑎 𝑖 = 𝑒 2 𝑛 𝑛 0 𝑐 𝛼 (8 0 𝑎 𝑖 𝑚 𝑒 ) ½ 8   0 𝐸 𝐻 (2𝑒 2 𝑍 eff ) ½
Simplification gives:

𝑎 𝑖 = 𝑒 2 𝑛 0 2 𝑐 2 α 2 𝑚 𝑒 𝑛 2 16𝐸 𝐻 2   0 𝑍 eff , (29)
Re-substitution of Eq. (25) into Eq. (29) gives:

𝑎 𝑖 = 𝑒 2 𝑛 0 3 𝑐 3 α 3 𝑚 𝑒 𝑛 16𝐸 𝐻 2   0 𝑣 , (30)
One can repeat the operation leading to Eq. ( 29) to give:

𝑎 𝑖 = (𝑐 𝛼 𝑛 0 ) 6 𝑚 𝑒 3 𝑛 2 𝑒 2 64𝐸 𝐻 4   0 𝑍 eff , (31)
Ultimately, one can further re-substitute Eq. (25) into Eq. (31) as in Eq. (29) to give:

𝑎 𝑖 = (𝑐 𝛼 𝑛 0 ) 7 𝑚 𝑒 3 𝑛 𝑒 2 64𝐸 𝐻 4   0 𝑣 , (32)
Clearly, it can be seen that none of the equations is final because there is no end to substitutions.

Experimental information is needed in order to proceed with Eqs (28a), (28b), ( 29), (30), (31), and (32) all of which give the same result.

Deductions with respect to fine structure constant

Beginning from Eqs (28b), (30), and (32), there are possibilities of reproducing Eq. ( 18) in terms of the fine structure constant if all the fundamental constants are experimental variables, the average ionisation energies, are substituted into the equations. Beginning from Eqs (28b), (30), and (32), there are possibilities of reproducing Eq. ( 18) in terms of the fine structure constant if all the fundamental constants are experimental variables, the average ionisation energies, are substituted into the equations. The resulting fine structure constant is approximately 7.2935 exp. (-3). In all cases the equation is:

𝛼 = 1 𝑐 𝑛 0 ( 2𝐸 𝐻 𝑚 𝑒 ) ½
where n0 is always equal to one. The equation became a reality after substituting Eq. ( 27) and (2Ei/me) ½ into the respective equations and solving for . Rederiving the equation based on Eqs (29) and

(31) requires substituting Bohr's equation. This is despite information about ionisation energies, a deterministic variable first discovered by a Russian scientist, yet most western scientists (or rather not to be too general, "my west") but not limited to them, have been vehement in criticising Bohr's equation

(𝑎 𝑖 = 𝑛 2 ℎ 2  0 𝑚 𝑒 𝑒 2 𝑍 eff ⁄
): It appears no motivation is seen in the Russian scientist, yet there is no better alternative that could be seen as a motivation for rejecting the possibility that ionisation energies that are experimentally generated anywhere in the global space can be used to determine the effective nuclear charge. How effective nuclear charge can be calculated based on ionisation energies and enabling equations are available in the literature [7]. With Eqs (29) and (31), the new equations after the substitution and solving for the fine structure constant are respectively:

 = 4 ℎ  0 𝐸 H 𝑒 2 𝑚 𝑒 𝑐 , ( 33
)
If Bohr's equation for ionisation energy is substituted into Eq. (33), it gives:  = e 2 /20hc.

 = 2 𝑐 𝑛 0 ( 𝐸 H 2 ℎ  0 𝑚 𝑒 2 𝑒 2 ) ⅓ , (34)
It must be made known that other variants of the equation can be derived if the series beyond Eq. ( 32) is continued. The values the fine structure constant expected from Eqs (33) and ( 34) is approximately 7.2945 exp. (-3). Equation ( 34) is a major innovation as far as the variants of the equation of fine structure constant is concerned.

Deductions with respect to wave number.

The core atomic physicists, Bohr, Rydberg-Ritz etc have concern for wave number though such concern is restricted to hydrogenic atoms and ions only. Moreover it is all about mainly emission spectrum. It is pointless being in doubt because, ai and p can be separately calculated as along as the average ionisation energy and energy level are known; this is usually so for most elements. Here, deductively or as a corollary, one can relate the wave number of multi-electron atoms to the Rydberg constant. In this regard, refer to Eq. ( 26) where the momentum p is equal to me v which, in turn, is equal to h/λ, the well-known de Broglie equation where the reciprocal of the wave length (1/λ) is equal to the wave number (∇). Therefore, given Eq. ( 15), derivation is as follows:

ℎ 𝑎 𝑖 𝜆 = 𝑛𝑎 0 (2𝐸 H 3 𝑚 𝑒 ) ½ ℎ𝑅 ∞ 𝑐 ⁄ , (35)
"No beating the gun", just step-by-step, until one obtains the following:

𝑎 𝑖 𝑛 𝜆 = 𝑎 0 (2𝐸 H 3 𝑚 𝑒 ) ½ ℎ 2 𝑅 ∞ 𝑐 ⁄ , (36)
∇ = 𝑛𝑎 0 (2𝐸 H 3 𝑚 𝑒 ) ½ ℎ 2 𝑅 ∞ 𝑐 ⁄ 𝑎 𝑖 , (37)
Two equations attributed to Bohr are: 1) 𝐸 i = 𝑍 eff 2 𝑒 4 𝑚 𝑒 8 0 2 𝑛 2 ℎ 2 ⁄ and 2) 𝑎 𝑖 = 𝑛 2 ℎ 2  0  𝑚 𝑒 𝑒 2 𝑍 eff ⁄ .

Substituting the latter into the former and solving for Ei gives:

𝐸 𝑖 = 𝑛 2 ℎ 2  0 8  2 𝑚 𝑒 𝑎 𝑖 2 ⁄ , (38)
Based on Eq. ( 38), the momentum of the electron is

p = nh/2ai, ( 39
)
Consequently, the wave number is:

∇ = 𝑛 2  𝑎 𝑖 ⁄ , ( 40
)
With Eqs (17a) and (40) the following can be derived. The equations contain n/ai; therefore,

1 2  = a 0 (2𝐸 H 3 𝑚 𝑒 ) ½ 𝑅 ∞ ℎ 2 𝑐 , ( 41a
)
𝑅 ∞ = a 0  (8𝐸 H 3 𝑚 𝑒 ) ½ ℎ 2 𝑐 , (41b)

Discussion

Various equations were derived, notable of which is the equation unknown in the literature for the calculation of the Rydberg constant (Ra) and fine structure constant () as well. With the various equations, it has been possible to show that the use of reduced mass as opposed to rest mass leads to lower values of physical constants compared to CODATA values. Higher values of Ra were obtained from all equations in which Bohr's radius and the average ionisation energy of hydrogen were calculated using the rest mass of the electron. These revelations are made possible by the application of the derived equations, which are therefore analysed and discussed as follows: Given Eq. (17a), it could be seen that Ra is inversely proportional to the product of the momentum, p, of the orbital electron and its distance from the nucleus, ai, for multi-electron atoms, whereas it is inversely proportional to the p (Eq. (16a)) for hydrogen and hydrogenic ions as well. Incidentally, Ra is also directly proportional to the square of the average ionisation energy (EH) of a hydrogen atom (Eq. 17b) for hydrogen atom only. Also, Eq. (

shows that Ra is inversely proportional to the product of the wave number and the ionisation energydependent atomic radius and directly proportional to the square root of the 3 rd power of the EH for all multi-electron atoms; it is strictly directly proportional to the square root of the 3 rd power of the EH for the hydrogen atom only. With this apparent analysis, one can now x-ray the physical implications.

It is not certain whether or not the concept of reduced mass is applied in the spectroscopic determination of ionisation energies. If not, all the equations where R∞ is EH-dependent should give the same results. On the contrary, if theoretically calculated hydrogenic average ionisation energy is the case, the application of either the rest mass or the reduced mass in Bohr's equation for the equation of ) for ionisation energy, different results are expected. One is higher than the other (2.179872176 exp. (-18) and 2.178690844 exp. (-18) J). If these values are substituted again into the Coulomb equation for the hydrogen atom, the radii above, the first and the second are expected. Yet all calculations of R∞ involving "physical constants" where rest mass featured gave values that were higher than those where reduced mass was the case. Besides, the results from the former (the rest mass case) are very similar to the CODATA value. Therefore, it seems the rest mass played a role in the CODATA value of R∞. Then the question is: what is the desirability of the use of the reduced mass concept as opposed to the rest mass, which seems to give a conflicting result? In any case, if EH is experimentally generated and gives a consistent result, then R∞ may be regarded as a fundamental physical constant, considering the fact that the rest mass (an expression of the quantity of matter in any specific object) is also constant.

With a clear presence of the ratio of average ionisation energy to any kind of electron mass in all the variants of the equation of fine structure constant (Eqs (33) and ( 34)), the results of calculation should be electron mass of any kind invariant. A recent experimental study showed the value (7.297352563 exp.

(-3)) of the fine structure constant with an accuracy of 81 PPT [START_REF] Morel | Determination of the fine-structure constant with 81 parts-per-trillion accuracy[END_REF], which is lower than the calculated value in this study and the CODATA value, giving the impression that one or more of the fundamental physical terms in the equation of the fine structure constant may be less accurate. However, one may wish to know if a lower value from any future experiment results in higher accuracy. This has become a pertinent question because the CODATA value or any other unknown in the course of this research seems not to be a standard reference for comparison. Be it as it may, the fine structure constant is directly proportional to EH (Eq. ( 33)), and it is directly proportional to the cube root of the square of EH.

Equation (36), once again, implies that the ratio of the ionisation energy-dependent radius of any atom to the product of the wave length and principal quantum number is equal to the ratio of composite fundamental constants. Ultimately, the wave number for any multi-electron atom is now related to the Rydberg. Note that Eq. ( 37) is generalisable to all atoms, both hydrogenic and nonhydrogenic atoms and ions. The only experimental information needed, for instance, is the average ionisation energy coupled with the enabling equation, Eq. ( 27) for instance. In order to evaluate Eq. (37), the first members of the groups in the first period of the period table were used as examples whose first average ionisation energies are known. If there is any way by which ai can be determined either experimentally or theoretically, it is not just the average ionisation energy that can be calculated according to the conventional Bohr's equation but the wave number as well. The wave number is directly proportional to the square root of the cube of EH and the principal quantum number and inversely proportional to the radius, as shown by equations ( 37) and (40).

There seems to be more interest in fine structure constant than Rydberg constant despite the fact that the latter is related to the former in the equation given as follows [START_REF] Mohr | CODATA recommended values of the fundamental physical constants[END_REF][START_REF] Mohr | CODATA recommended values of the fundamental physical constants[END_REF]: R∞= α 2 mec /2h. The  has been studied for various reasons including reevaluation of the standard model [START_REF] Gabrielse | Towards an improved test of the standard Model's most precise prediction[END_REF][START_REF] Parker | Measurement of the finestructure constant as a test of the Standard Model[END_REF]. Some studies are also devoted to measurements in other to achieve better results in terms of higher precision [START_REF] Yu | Atom-interferometry measurement of the fine structure constant[END_REF][START_REF] Sherbon | Quintessential nature of the fine-structure constant[END_REF][START_REF] Sherbon | Fundamental physics and the fine-structure constant[END_REF]; what looks like theoretical method entailed the application of quartic equations to which is related the much talked about (but yet to be fully elucidated) golden ratio and a-not-too clear classical "harmonic proportions" [START_REF] Sherbon | Fine-structure Constant from Sommerfeld to Feynman[END_REF][START_REF] Heyrovska | Golden Ratio based fine structure constant and Rydberg constant for hydrogen spectra[END_REF]. In this research, however, variants of the equation of  were derived giving results that are always not more than 1/137. These are in addition to the derived equation relating effective nuclear charge to the  [7] and the equations that confirm the universality of R∞; this is so because such equation can be used to either determine it or be used to determine nuclear property such as mass radius of any nucleon [START_REF] Udema | The universality of Rydberg constant is investigated with concern for ethics[END_REF]. Recent mention of R∞ is in connection with Rydberg atom explored in the determination of the angle-of-arrival of an incident radio-frequency (RF) wave or signal [START_REF] Robinson | Determining the angle-of-arrival of an radio-frequency source with a Rydberg atom based sensor[END_REF]; this seems to be an application of Rydberg concept in technology besides spectroscopic facilities. The  has application in the derivation of the equation of the mass radius of the nucleon and larger subatomic particles [START_REF] Udema | Theoretical determination of the mass radii of the nucleons and heavier subatomic particles[END_REF].

It is interesting to note that the physical variables exhibit a similar trend with an increasing value of the square of the atomic number. It is akin to the chemical periodicity of elements. Different atoms are characterised by different average ionisation energies, which is another name for the average kinetic energy needed to expel an electron from an uncertain position in its ground state to infinity. It is not unlikely that an electron may, under unnatural conditions (or such natural conditions within the stars, the sun being the nearest of such), move to unspecified lower energy with the release of an amount of energy equal to its ionisation energy at its ground-state energy level. Therefore, wave number may not be restricted to the spectra of different elements; any elementary particle in motion has a de Broglie wave length and, as such, a wave number.

Conclusion

The use of rest mass and reduced mass gives conflicting values for physical constants. The question as to the desirability of the reduced mass concept is valid considering the observed value calculated in this research based on rest mass is very similar to the CODATA value. Other variants of the equations for the fine structure constant and, consequently, the Rydberg constant may be determined in the future.

Figure 1 :

 1 Figure 1: The plot illustrating the trend of all related physical variables with the square of atomic number. The legends, IE, WN, and ENC-Sqd represent the average ionisation energy, wave number, effective nuclear charge squared. The IE, WN, and ENC-Sqd values were seperately plotted versus the square of atomic number/10. Division by ten was to enhance resolution of the curves despite sum points of contact or intersections.

  ), in particular, are not known in the literature. The values are exactly the same for the reduced mass case (7.294715912 exp. (-3)) and 7.297352577 exp. (-3), for the rest mass case. The  achieved with accuracy of 81 PPT is equal to 7.297352563 exp. (-3) [2] and the CODATA [4] value (7.2973525664 exp. (-3)) (

  average ionisation energy would obviously give different results. Recall that Bohr's equation for the radius is inversely proportional to the mass of the electron; using rest mass gives a lower value of the radius (≈5.289643 exp. (-11) m), while the reduced mass (9.104424487 exp. (-31) kg) calculated by deploying all 2012 CODATA [3] values of relevant physical constants gives 5.292523949 exp. (-11) m, which is slightly higher than the 2012 CODATA value of 5.2917721092 exp. (-11) m [3]. If these values are substituted into the Coulomb equation for hydrogen atoms, different values of the kinetic energies (EH values) are obtainable. In a similar way, if the rest mass and the reduced mass are separately substituted into Bohr's equation (𝐸 i = 𝑍 eff 2 𝑒 4 𝑚 𝑒 8 0 2 𝑛 2 ℎ 2 ⁄

Table

Table 1 : Values of the Rydberg constant and the fine structure constant based on derived equations S/N Equations Based on rest mass Based on reduced mass

 1

	1	𝑅 ∞ = 8   0 𝑎 0 𝐸 H 2 ℎ 𝑐 𝑒 2 ⁄	10973731.6	10966253.06
	2	𝑅 ∞ = 𝐸 H ℎ ⁄ 𝑐 (Here & in the lit.)	10973731.6	10967784.63
	3.	𝑅 ∞ =  2 𝑚 𝑒 𝑐 2 ℎ ⁄	(CODATA-I)	10973731.57	10967758.834
	4.	𝑅 ∞ = 𝑒 4 𝑚 𝑒 8  0 2 𝑐ℎ 3 ⁄	(CODATA-II)	10973733.89	10967760.69
	5.	Eq. (33)			7.297352577 exp. (-3)	7.294715912 exp.(-3)
	6. Eq. (34)			7.297352577 exp. (-3)	7.294715912 exp.(-3)
	Serial number (S/N) 1 equation was derived in this research and S/N 2 equation, was a rederivation in
	this research, which reproduced the literature (lit.) version. The S/N 3 (CODATA-I) and 4 (CODATA-II)
	equations are CODATA records and derived equation respectively, in which fine structure constant
	equation was substituted into S/N 3 equation to give S/N 4 equation. The ionisation energy-based radii of
	atoms used can be found in the literature [5].
		The values of the wave number calculated based on Eq. (40) are not influenced by the mass of
	the electron if the ionisation based radii were calculated using experimentally determine average

). The rest mass value is higher than CODATA and Morel et al values.

Table 2 : Calculated wave numbers based on reduced mass and rest mass of the electron

 2 ∇* is the wave number affected by the use of rest mass. ∇ is the wave number affected by the use of reduced mass.

	Elements	H	He	Be	B	C	N	O	F
	∇* (Eq. (37))	3.006	4.038 2.490	2.348	2.736	2.983	3.007	3.403
	exp. (9)/m
	∇ (Eq. (37))	3.005	4.036 2.495	2.347	2.735	2.982	3.006	3.401
	exp. (9)/m
	∇* (Eq. (40)	3.006	4.038 2.490	2.348	2.736	2.983	3.007	3.402
	exp. (9)/m
	𝛁(Eq. (40)	3.008
	exp. (9)/m

Acknowledgments:

The management of the Royal Court Yard Hotel in Agbor, Delta State, Nigeria, is immensely appreciated for the supply of electricity during the preparation of the manuscript. The provider of the QuillBot grammar checker is thanked for improving the English language quality of the manuscript.

Author contributions:

The sole author designed, analysed, interpreted and prepared the manuscript.

Funding: "This research received no external funding." However, Science International, under the aegis of LiveDNA, promised a waiver of the article processing charge if the manuscript was accepted for publication.

Conflicts of Interest:

There is no competing interest; specifically, there is no financial interest with any government, corporate body, or individual except the awaited unpaid retirement benefits by the Delta state government of Nigeria for about four years after retirement.