# Modeling the Trend of Credit Card Usage Behavior for Different Age Groups Based on Singular Spectrum Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. On Analysis Based on SCPC Data

#### 2.2. On Developments and Applications of Singular Spectrum Analysis

#### 2.3. On Trend Studies of Credit Card Usage

## 3. Singular Spectrum Analysis

#### 3.1. Decomposition

_{1}, …, x

_{N}) of length N with no missing value, a window of length L is chosen (2 < L < N/2) to embed the original time series. Then, the original time series X is mapped into L lagged vectors, X

_{i}= x

_{i}, …, x

_{i}

_{+L−1}for i = 1, …, K, where K = N − L + 1. Thus, T

_{X}is written as:

_{X}and the decomposed trajectory matrices T

_{i}are obtained. U

_{i}for 1 < i < L is a K

_{i}× L orthonormal matrix, D

_{i}for 1 < i < L is a diagonal matrix order of L, and V

_{i}for 1 < i < L is an L × L square orthonormal matrix. In this step, T

_{X}has L many singular values, which are:

_{i}to T

_{X}.

#### 3.2. Reconstruction

_{i}into subgroups according to the trend, the seasonal and semi-seasonal components, and residuals. The grouping step of the reconstruction stage is a partition of the set of indices 1, …, d into the collection of m disjoined subsets of I = I

_{1}, …, I

_{m}. Thus, T

_{i}corresponds to the group I = {I

_{1}, …, I

_{m}}. ${T}_{{I}_{i}}$ is a sum of T

_{j}, where j ∈ I

_{i}. So, T

_{X}can be expanded as

_{X}: T

_{L}and T

_{R}. The whole set will be I = {1, …, d}, R ∪ L = I. However, R is not a subset of L. T

_{I}is

_{L}= T

_{I}− T

_{R}under the assumption of weak separability. Thus, T

_{L}can be written as

## 4. Data Description

#### 4.1. The Survey of Consumer Payment Choice

#### 4.2. Credit Card Usage

## 5. Model Results

#### 5.1. Decomposition

_{1}, x

_{2}, …, x

_{n}} with no missing values of length N.

#### 5.2. Reconstruction

_{i}into several groups and then summing the matrices within each group. If I = i

_{1}, …, i

_{p}is one such group, then the matrix X

_{I}corresponding to the group I is defined as: X

_{I}= X

_{i}

_{1}+ … + X

_{ip}. For m such groups, X will be given as: X = X

_{I}

_{1}+ … + X

_{Im}. The contribution of component X

_{I}is measured by the share of the corresponding eigenvalues.

## 6. Summary, Conclusions, and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Nai, W.; Liu, L.; Wang, S.; Dong, D. Modeling the Trend of Credit Card Usage Behavior for Different Age Groups Based on Singular Spectrum Analysis. *Algorithms* **2018**, *11*, 15.
https://doi.org/10.3390/a11020015

**AMA Style**

Nai W, Liu L, Wang S, Dong D. Modeling the Trend of Credit Card Usage Behavior for Different Age Groups Based on Singular Spectrum Analysis. *Algorithms*. 2018; 11(2):15.
https://doi.org/10.3390/a11020015

**Chicago/Turabian Style**

Nai, Wei, Lu Liu, Shaoyin Wang, and Decun Dong. 2018. "Modeling the Trend of Credit Card Usage Behavior for Different Age Groups Based on Singular Spectrum Analysis" *Algorithms* 11, no. 2: 15.
https://doi.org/10.3390/a11020015