# Multiclass Confusion Matrix Reduction Method and Its Application on Net Promoter Score Classification Problem

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Data Classification Problem Analysis

#### 1.2. Customer Experience and Associated Metrics

- Market surveys: The survey is performed typically in a random sample of the market population. This method has the advantage of measuring CX for all market competitors. Moreover, apart from the main CX metric the survey also measures the customer satisfaction for a number CX attributes, such as the product experience, the value perception, the touchpoint experience (call center, website, mobile app, shops, etc.), or key customer journeys (e.g., billing, product purchase);
- Customer feedback during or right after a transaction: this method is used so as to measure the customer satisfaction at different stages of a transaction (or more generally a customer journey) or to measure the CX of a specific touchpoint (e.g., shop, call center, website, mobile app, etc.) Such measurements capture only the feedback of own customers, however, they can reveal significant customer insights.

#### 1.3. The CX Metric Classification Problem

## 2. Classification Algorithm Performance Analysis

#### 2.1. Algorithm Performance for Binary Classification Problems

#### 2.2. Multiclass Confusion Matrix and Metrics

## 3. Multiclass Confusion Matrix Reduction Methods

#### 3.1. Class Grouping Options

**Relaxed Grouping of Classes**${\left[G\right]}^{R}$: As shown in Figure 3a, in this case any prediction of a class ${C}_{i}\in G$ with actual class ${C}_{j}\in G$ is considered to be a true positive instance. In the example of NPS classification, assuming that we are interested in the group of detractors (score from 0 to 6) then a prediction of score 1 with an actual score 3 is considered to be true positive since both the predicted and the actual group is “detractor”;**Strict Grouping of Classes**${\left[G\right]}^{S}$: As shown in Figure 3b, in this case only the predictions ${C}_{i}\in G$ which are identical to the actual class are considered to be true positive instances, i.e., only the instances ${C}_{i,i}$ with ${C}_{i}\in G$. For example, assuming the grouped class of detractors in NPS problem, if the predicted class is 3 then a TP instance occurs only if the actual class is 3;**Hybrid-RS Grouping of Classes**${\left[G\right]}^{RS}$: As shown in Figure 3c, in this case apart from the instances where the predicted class is identical to the actual class, there is an additional set of combinations of predicted and actual classes which are considered to be a TP instance. For example, assuming the group of detractors in NPS problem, assume that we are interested in an algorithm that predicts the scores that are equal of better than the actual scores.

#### 3.2. The Grouped Class Formal Definition

#### 3.3. The Intragroup Mismatch Instances

## 4. The Concept of the Reduced Confusion Matrix

_{j}a grouping option H

_{j}is being selected $(R,S,RS)$ leading to a set of grouped classes G

_{A}defined as follows:

_{j}so as to calculate the number of actual or the number of predicted instances accordingly.

#### 4.1. Consecutive Confusion Matrix Reduction Steps

#### 4.2. Performance Metrics for a Reduced Confusion Matrix

#### 4.3. Receiver Operating Characteristic for a Reduced $2\times 2+IM$ Confusion Matrix

- Step 1: As in the ordinary binary classification, the prediction of positive vs. negative grouped class is based on a threshold $\theta $;
- Step 2: The prediction of a specific class from the set of grouped positive or negative classes is based on maximum likelihood.

## 5. Confusion Matrix Reduction for NPS Classification

#### 5.1. NPS Classification Dataset

#### 5.2. Machine Learning Algorithms for NPS Classification

#### 5.3. Confusion Matrix for the NPS Classification Problem

**Step 1:**Relaxed grouping based on the definition of customer categories: detractor, passive and promoter (as shown in Figure 6a):

**Step 2:**In order to compare the performance of different classification algorithms we are going to consider the following grouping step:

#### 5.4. Performance Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Mathews Correlation Coefficient for Reduced Multiclass Confusion Matrix

## Appendix B. The Applied Machine Learning Algorithms

#### Appendix B.1. Decision Trees

Parameter | Values |
---|---|

Function measuring quality of split | Entropy |

Maximum depth of tree | 3 |

Weights associated with classes | 1 |

#### Appendix B.2. k-Nearest Neighbors

Parameter | Values |
---|---|

Number of neighbors | 5 |

Distance metric | Minkowski |

Weight function | uniform |

#### Appendix B.3. Support Vector Machines

Parameter | Values |
---|---|

Kernel type | Linear |

Degree of polynomial kernel function | 3 |

Weights associated with classes | 1 |

#### Appendix B.4. Random Forest

Parameter | Values |
---|---|

Number of trees | 100 |

Measurements of quality of split | Gini index |

#### Appendix B.5. Artificial Neural Networks

Parameter | Values |
---|---|

Number of hidden neurons | 6 |

Activation function applied for the input and hidden layer | ReIU |

Activation function applied for the output layer | Softmax |

Optimizer network function | Adam |

Calculated loss | Sparse categorical cross-entropy |

Epochs used | 100 |

Batch size | 10 |

#### Appendix B.6. Convolutional Neural Networks

Parameter | Values |
---|---|

Model | Sequential (array of Keras Layers) |

Kernel size | 3 |

Pool size | 4 |

Activation function applied | ReIU |

Calculated loss | categorical cross-entropy |

Epochs used | 100 |

Batch size | 128 |

#### Appendix B.8. Logistic Regression

Parameter | Values |
---|---|

Maximum number of iterations | 300 |

Algorithm used in optimization | L-BFGS |

Weights associated with classes | 1 |

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**Figure 1.**Confusion matrix examples. (

**a**) Binary classification problem confusion matrix. (

**b**) Multiclass classification problem confusion matrix.

**Figure 3.**True positive instances definition examples. (

**a**) TP for relaxed grouping. (

**b**) TP for strict grouping. (

**c**) Example TP for hybrid RS-grouping.

**Figure 6.**The reduction in the $11\times 11$ NPS classification confusion matrix into a $3\times 3+IM$ reduced confusion matrix. (

**a**) The $11\times 11$ NPS confusion matrix. (

**b**) The reduced $3\times 3+IM$ confusion matrix.

(a) The NPS customer catergorization | |

NPS Response | NPS Label |

9–10 | Promoter |

7–8 | Passive |

0–6 | Detractor |

(b) The CSAT customer catergorization | |

CSAT Response | CSAT Label |

5 | Very Satisfied |

4 | Satisfied |

3 | Neutral |

2 | Dissatisfied |

1 | Very Dissatisfied |

Metric | Formula |
---|---|

Accuracy | $Acc={\displaystyle \frac{TP+TN}{TP+TN+FP+FN}}$ |

True Positive Rate (Recall) | $TPR={\displaystyle \frac{TP}{TP+FN}}$ |

True Negative Rate (Specificity) | $TNR={\displaystyle \frac{TN}{TN+FP}}$ |

Positive Predictive Value (Precision) | $PPV={\displaystyle \frac{TP}{TP+FP}}$ |

Negative Predictive Value | $NPV={\displaystyle \frac{TN}{TN+FN}}$ |

${F}_{1}$-Score | ${F}_{1}=2\xb7{\displaystyle \frac{TPR\xb7PPV}{TPR+PPV}}$ |

False Negative Rate (Miss Rate) | $FNR={\displaystyle \frac{FN}{TP+FN}}$ |

False Positive Rate (Fall Out Rate) | $FPR={\displaystyle \frac{FP}{TN+FP}}$ |

False Discovery Rate | $FDR={\displaystyle \frac{FP}{TP+FP}}$ |

False Omission Rate | $FOR={\displaystyle \frac{FN}{TN+FN}}$ |

Fowlkes-Mallows index | $FM=\sqrt{PPV\xb7TPR}$ |

Balanced Accuracy | $BA={\displaystyle \frac{TPR+TNR}{2}}$ |

Mathews Correlation coefficient | $MCC={\displaystyle \frac{TP\xb7TN-FP\xb7FN}{\sqrt{(TP+FN)(TP+FP)(FN+TN)(FP+TN)}}}$ |

Prevalence Threshold | $PT={\displaystyle \frac{\sqrt{TPR(1-TNR)}+TNR-1}{TPR+TNR-1}}$ |

Informedness | $BM=TPR-FPR$ |

Markedness | $MK=PPV-FOR$ |

Threat Score (Critical Success Index) | $TS={\displaystyle \frac{TP}{TP+FN+FP}}$ |

Metric | Formula |
---|---|

Accuracy | $Acc\left({A}_{reduced}\right)={\displaystyle \frac{{\sum}_{i=1}^{N}TP\left({C}_{i}\right)}{{\sum}_{i=1}^{N}{\sum}_{j=1}^{N}{C}_{i,j}}}$ |

Recall of Class ${C}_{i}$ | $TPR\left({C}_{i}\right)={\displaystyle \frac{TP\left({C}_{i}\right)}{TP\left({C}_{i}\right)+FN\left({C}_{i}\right)}}$ |

Precision of Class ${C}_{i}$ | $PPV\left({C}_{i}\right)={\displaystyle \frac{TP\left({C}_{i}\right)}{TP\left({C}_{i}\right)+FP\left({C}_{i}\right)}}$ |

${F}_{1}$-Score of Class ${C}_{i}$ | ${F}_{1}\left({C}_{1}\right)=2\xb7{\displaystyle \frac{TPR\left({C}_{i}\right)\xb7PPV\left({C}_{i}\right)}{TPR\left({C}_{i}\right)\xb7PPV\left({C}_{i}\right)}}$ |

Recall (macro average) | $TPR\left(macro\right)={\displaystyle \frac{1}{N}}{\sum}_{i=1}^{N}TPR\left({C}_{i}\right)$ |

Precision (macro average) | $PPV\left(macro\right)={\displaystyle \frac{1}{N}}{\sum}_{i=1}^{N}PPV\left({C}_{i}\right)$ |

${F}_{1}$-Score (macro average) | ${F}_{1}\left(macro\right)=2\xb7{\displaystyle \frac{TPR\left(macro\right)\xb7PPV\left(macro\right)}{TPR\left(macro\right)+PPV\left(macro\right)}}$ |

Recall (micro average) | $TPR\left(micro\right)={\displaystyle \frac{{\sum}_{i=1}^{N}TP\left({C}_{i}\right)}{{\sum}_{i=1}^{N}[TP\left({C}_{i}\right)+FP\left({C}_{i}\right)]}}$ |

Precision (micro average) | $PPV\left(micro\right)={\displaystyle \frac{{\sum}_{i=1}^{N}RP\left({C}_{i}\right)}{{\sum}_{i=1}^{N}[TP\left({C}_{i}\right)+FP\left({C}_{i}\right)]}}$ |

${F}_{1}$-Score (micro average) | ${F}_{1}\left(micro\right)=2\xb7{\displaystyle \frac{TPR\left(micro\right)\xb7PPV\left(micro\right)}{TPR\left(micro\right)+PPV\left(micro\right)}}$ |

Metric | Formula |
---|---|

Accuracy of Reduced Confusion Matrix | $Acc\left({A}_{reduced}\right)={\displaystyle \frac{{\sum}_{j}TP\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)}{{\sum}_{j}TP\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)+{\sum}_{i}{\sum}_{j,i\ne j}{G}_{i,j}+{\sum}_{j}IM\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)}}$ |

Recall of Group ${G}_{j}$ | $TPR\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)={\displaystyle \frac{TP\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)}{TP\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)+FN\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)+IM\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)}}$ |

Precision of Group ${G}_{j}$ | $PPV\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)={\displaystyle \frac{TP\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)}{TP\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)+FP\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)+IM\left({\left[{G}_{j}\right]}^{{H}_{j}}\right)}}$ |

Metric | Formula |
---|---|

Accuracy | $Acc={\displaystyle \frac{TP+TN}{TP+TN+FP+FN+IMP+IMN}}$ |

True Positive Rate (Recall) | $\frac{TP}{(TP+FN+IMP)}$ |

True Negative Rate (Specificity) | $\frac{TN}{TN+FP+IMN}$ |

Positive Predictive Value (Precision) | $PPV={\displaystyle \frac{TP}{TP+FP+IMP}}$ |

Negative Predictive Value | $NPV={\displaystyle \frac{TN}{TN+FN+IMN}}$ |

False Negative Rate (Miss Rate) | $FNR={\displaystyle \frac{FN}{TP+FN+IMP}}$ |

False Positive Rate (Fall Out Rate) | $FPR={\displaystyle \frac{FP}{TN+FP+IMN}}$ |

False Discovery Rate | $FDR={\displaystyle \frac{FP}{TP+FP+IMP}}$ |

False Omission Rate | $FOR={\displaystyle \frac{FN}{TN+FN+IMN}}$ |

Logistic Regr. | SVM | k-NN | Decision Trees | Random Forest | Naïve Bayes | CNN | ANN | |
---|---|---|---|---|---|---|---|---|

Accuracy | 0.37 | 0.38 | 0.34 | 0.39 | 0.33 | 0.31 | 0.38 | 0.37 |

Precision | 0.16 | 0.17 | 0.17 | 0.23 | 0.19 | 0.17 | 0.21 | 0.14 |

Recall | 0.15 | 0.17 | 0.16 | 0.22 | 0.17 | 0.15 | 0.19 | 0.17 |

F1-score | 0.13 | 0.14 | 0.16 | 0.21 | 0.18 | 0.15 | 0.19 | 0.15 |

**Table 7.**The performance metrics of the “$3\times 3+IM$” reduced confusion matrix for each applied algorithm.

Logistic Regr. | SVM | k-NN | Decision Trees | Random Forest | Naïve Bayes | CNN | ANN | |
---|---|---|---|---|---|---|---|---|

Accuracy | 0.58 | 0.56 | 0.60 | 0.51 | 0.54 | 0.56 | 0.63 | 0.57 |

Precision | 0.68 | 0.65 | 0.62 | 0.50 | 0.55 | 0.54 | 0.66 | 0.69 |

Recall | 0.51 | 0.48 | 0.56 | 0.50 | 0.50 | 0.58 | 0.60 | 0.52 |

F1-score | 0.50 | 0.53 | 0.58 | 0.50 | 0.51 | 0.57 | 0.62 | 0.51 |

**Table 8.**The performance metrics of the “$2\times 2+IM$” reduced confusion matrix for each one of the applied algorithms.

Logistic Regr. | SVM | k-NN | Decision Trees | Random Forest | Naïve Bayes | CNN | ANN | |
---|---|---|---|---|---|---|---|---|

Accuracy | 0.58 | 0.56 | 0.60 | 0.51 | 0.54 | 0.56 | 0.63 | 0.57 |

Precision | 0.57 | 0.55 | 0.60 | 0.52 | 0.54 | 0.62 | 0.62 | 0.56 |

Recall | 0.64 | 0.61 | 0.63 | 0.52 | 0.58 | 0.54 | 0.65 | 0.62 |

F1-Score | 0.60 | 0.58 | 0.61 | 0.52 | 0.56 | 0.58 | 0.63 | 0.59 |

Specificity | 0.09 | 0.07 | 0.16 | 0.15 | 0.11 | 0.27 | 0.19 | 0.10 |

Miss Rate | 0.01 | 0.01 | 0.05 | 0.10 | 0.05 | 0.19 | 0.04 | 0.02 |

Negative Predictive Value | 0.11 | 0.09 | 0.18 | 0.15 | 0.13 | 0.21 | 0.21 | 0.13 |

Fall Out Rate | 0.14 | 0.10 | 0.10 | 0.10 | 0.12 | 0.06 | 0.09 | 0.13 |

False Discovery Rate | 0.12 | 0.10 | 0.10 | 0.10 | 0.12 | 0.07 | 0.09 | 0.12 |

False Omission Rate | 0.03 | 0.01 | 0.11 | 0.18 | 0.11 | 0.32 | 0.08 | 0.04 |

Fowlkes-Mallows index | 0.60 | 0.58 | 0.61 | 0.52 | 0.56 | 0.58 | 0.63 | 0.59 |

Mathews Correlation Coefficient | 0.33 | 0.37 | 0.35 | 0.37 | 0.36 | 0.34 | 0.38 | 0.36 |

PIMR | 0.35 | 0.39 | 0.32 | 0.38 | 0.37 | 0.27 | 0.31 | 0.36 |

PPIMR | 0.31 | 0.35 | 0.30 | 0.38 | 0.35 | 0.31 | 0.30 | 0.32 |

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**MDPI and ACS Style**

Markoulidakis, I.; Rallis, I.; Georgoulas, I.; Kopsiaftis, G.; Doulamis, A.; Doulamis, N.
Multiclass Confusion Matrix Reduction Method and Its Application on Net Promoter Score Classification Problem. *Technologies* **2021**, *9*, 81.
https://doi.org/10.3390/technologies9040081

**AMA Style**

Markoulidakis I, Rallis I, Georgoulas I, Kopsiaftis G, Doulamis A, Doulamis N.
Multiclass Confusion Matrix Reduction Method and Its Application on Net Promoter Score Classification Problem. *Technologies*. 2021; 9(4):81.
https://doi.org/10.3390/technologies9040081

**Chicago/Turabian Style**

Markoulidakis, Ioannis, Ioannis Rallis, Ioannis Georgoulas, George Kopsiaftis, Anastasios Doulamis, and Nikolaos Doulamis.
2021. "Multiclass Confusion Matrix Reduction Method and Its Application on Net Promoter Score Classification Problem" *Technologies* 9, no. 4: 81.
https://doi.org/10.3390/technologies9040081