# Comparing COSTATIS and Generalized Procrustes Analysis with Multi-Way Public Education Expenditure Data

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## Abstract

**:**

## 1. Introduction

## 2. Generalized Procrustes Analysis

## 3. COSTATIS

- 1.
- First, a variant of STATIS-like analyses [19,21] performed to identify the stable structure in a k-table. In this paper, a STATIS has been carried out, allowing the use of public education expenditure in two sets of countries with different economic characteristics. In this case, the method consists in performing two STATIS analyses: one on k-tables of high economic level countries, and another on the k-tables of low economic level countries that measure sets of variables collected on the same observations. The result is the mean table of maximum inertia, which represents the ‘‘compromise’’ and captures the similarities among the k individual tables constituting the k-table.

- A.
- The first stage consists to compare the structure s of the K matrices to each other and to come up with a so-called inter-structure. This can be summarized as follows:
- (1)
- Calculate the K variance–covariance matrices as:$${\mathrm{W}}_{t}={\mathrm{X}}_{\mathrm{t}}{\mathrm{Q}}_{\mathrm{t}}{\mathrm{X}}_{\mathrm{t}}^{\ast}$$
_{t}and all ${\mathrm{X}}_{\mathrm{t}}$ matrices of X have the same number of variables; ${\mathrm{X}}_{\mathrm{t}}^{\ast}$ is the transposed matrix of ${\mathrm{X}}_{\mathrm{t}}$ and ${\mathrm{Q}}_{{\mathrm{t}}_{i}}$ is the identity matrix of dimension J × J. In this study, all matrices had the same number of attributes, J_{i}= J for each k, and the same weight was given to all the individuals/simples.Each matrix ${\mathrm{X}}_{k}$ defines implicitly a structure for the individuals, which depends on their respective positions as defined by the distances between each pair of individuals: - (2)
- Calculate the matrix of RV coefficient:$$RV\left({W}_{k},{W}_{{k}^{\prime}}\right)=\frac{Tr\left({W}_{k}D{W}_{{k}^{\prime}}D\right)}{\sqrt{Tr{\left({W}_{k}D\right)}^{2}{\left({W}_{{k}^{\prime}}D\right)}^{2}}}$$The RV coefficients are non-negative and ranges from 0 to 1.
- (3)
- Perform PCA of the RV matrix $\left[\mathbf{S},\mathbf{V},\mathbf{S}\right]=\mathrm{SVD}\left(\mathbf{R}\mathbf{V}\right)$, SVD is the singular value decomposition version of PCA.$${\mathrm{a}}_{\mathrm{k}}=\raisebox{1ex}{${S}_{1}$}\!\left/ \!\raisebox{-1ex}{$\sum \mathrm{k}$}\right.=1K{S}_{1}$$${S}_{1}$ (K × 1) is the first column vector of the matrix
**S**

- B.
- The second step of the method is the determination of the intra-structure, that is, the search for a common structure to the structures corresponding to the K instants.
- (1)
- Calculate the compromise among tables as:$$W=\sum \mathrm{k}=1{K}_{{\mathrm{a}}_{\mathrm{k}}}{W}_{\mathrm{k}}$$
- (2)
- Perform PCA of W:$$\left[\mathbf{L},\mathbf{V},\mathbf{L}\right]=\mathrm{SVD}\left(\mathbf{W}\right)$$
- (3)
- Display the compromise score plot.

- C.
- The third step of the method is the development of the observations’ trajectories. The trajectories show which individuals or observations account for the distances observed among the objects ${\mathbf{W}}_{\mathrm{k}}$ in the inter-structure step.
- 2.
- The second stage is a co-inertia analysis, which is performed on analysis of the compromises of these two STATIS (or variants) to describe the co-structure between the stable part of high economic level countries data and the stable part of low economic level countries data. The COIA summarizes as well as possible the squared covariances between public expenditure on education in the two sets of countries [2].

## 4. GPA versus COSTATIS

**X**and

**Y**, is the sum of squared covariances:

**X**and

**Y**. ${D}_{J\ast}$ and ${D}_{J\ast \ast}$ are two hyperspace matrices of the first and second matrix, respectively. Additionally, ${D}_{I}$ is the diagonal of matrix (I × I) of matrices’ rows weights, where ${D}_{I}=Diag({w}_{1},\dots ,{w}_{I})$.

## 5. Case Study

#### 5.1. Data

- Group 1 consists of countries with a nominal GDP above USD 300 million: Austria, Ireland, Italy, Norway, Poland, Spain, Sweden, Switzerland and the United Kingdom.
- Additionally, group 2 consists of countries with a nominal GDP of less than USD 300 million: Cyprus, Czechia, Finland, Hungary, Iceland, Lithuania, Portugal, Romania, Slovakia and Slovenia.

#### 5.2. Procedure

## 6. COSTATIS Results

## 7. GPA Results

## 8. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Graphs resulting from COSTATIS method: Euclidean representation of high and low-income countries’ compromises (

**left**) and co-inertia analysis graph of the two compromises (

**right**). The scale is given by the value d.

**Figure 4.**Graphs resulting from GPA method: Representation of high and low-income countries’ compromises and their trajectories (

**left**) and co-inertia analysis graph of the two expenditure compromises (

**right**). The scale is given by the value d.

Code | Type of Expenditure |
---|---|

SCE | All staff compensation as % of total expenditure in public institutions (%) |

CEP | Capital expenditure as % of total expenditure in public institutions (%) |

CES | Current expenditure other than staff compensation as % of total expenditure in public institutions (%) |

LPRE | Expenditure by level of education: pre-primary (as % of government expenditure) |

LPRI | Expenditure by level of education: primary (as % of government expenditure) |

LSEC | Expenditure by level of education: secondary (as % of government expenditure) |

LTER | Expenditure by level of education: tertiary (as % of government expenditure) |

GEE | Government expenditure on education (% of government expenditure) |

PEE | Public expenditure on education (% of GDP) |

Countries with High Nominal GDP | Countries with Low Nominal GDP | ||||||
---|---|---|---|---|---|---|---|

Group 1 | Sum of squares within by country | Group 2 | Sum of squares within by country | ||||

SSfit | Ssresidual | SStotal | SSfit | Ssresidual | SStotal | ||

Austria | 13.695 | 0.502 | 14.197 | Cyprus | 14.885 | 0.265 | 15.151 |

Ireland | 14.074 | 0.850 | 14.924 | Czechia | 23.805 | 2.284 | 26.089 |

Italy | 9.400 | 1.023 | 10.422 | Finland | 7.219 | 0.445 | 7.665 |

Norway | 8.622 | 0.407 | 9.030 | Hungary | 2.204 | 0.995 | 3.198 |

Poland | 10.764 | 1.617 | 12.381 | Iceland | 7.915 | 1.142 | 9.057 |

Spain | 5.528 | 0.555 | 6.082 | Lithuania | 3.673 | 0.557 | 4.230 |

Sweden | 12.597 | 0.836 | 13.434 | Portugal | 19.597 | 0.430 | 20.026 |

Switzerland | 7.325 | 0.661 | 7.986 | Romania | 4.545 | 1.456 | 6.001 |

United Kingdom | 8.304 | 3.240 | 11.544 | Slovakia | 5.240 | 1.305 | 6.545 |

Slovenia | 1.384 | 0.654 | 2.038 | ||||

sum | 90.309 | 9.691 | 100.000 | sum | 90.467 | 9.533 | 100.000 |

Expenditure | Sum of squares within by type of expenditure | Expenditure | Sum of squares within by type of expenditure | ||||

SSfit | Ssresidual | SStotal | SSfit | Ssresidual | SStotal | ||

SCE | 61.997 | 0.039 | 62.036 | SCE | 58.188 | 0.012 | 58.200 |

CEP | 7.700 | 0.021 | 7.722 | CEP | 7.319 | 0.021 | 7.340 |

CES | 0.786 | 0.056 | 0.842 | CES | 3.268 | 0.202 | 3.470 |

LPRE | 7.653 | 0.071 | 7.724 | LPRE | 5.799 | 0.035 | 5.834 |

LPRI | 0.934 | 0.024 | 0.958 | LPRI | 1.131 | 0.019 | 1.150 |

LSEC | 6.423 | 0.155 | 6.577 | LSEC | 9.362 | 0.198 | 9.560 |

LTER | 0.400 | 0.027 | 0.427 | LTER | 0.439 | 0.067 | 0.506 |

GEE | 3.882 | 0.026 | 3.908 | GEE | 3.990 | 0.027 | 4.018 |

PEE | 9.792 | 0.014 | 9.806 | PEE | 9.911 | 0.013 | 9.923 |

sum | 99.568 | 0.432 | 100.000 | sum | 99.407 | 0.593 | 100.000 |

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**MDPI and ACS Style**

Vega-Hernández, M.C.; Patino-Alonso, C.
Comparing COSTATIS and Generalized Procrustes Analysis with Multi-Way Public Education Expenditure Data. *Mathematics* **2021**, *9*, 1816.
https://doi.org/10.3390/math9151816

**AMA Style**

Vega-Hernández MC, Patino-Alonso C.
Comparing COSTATIS and Generalized Procrustes Analysis with Multi-Way Public Education Expenditure Data. *Mathematics*. 2021; 9(15):1816.
https://doi.org/10.3390/math9151816

**Chicago/Turabian Style**

Vega-Hernández, María Concepción, and Carmen Patino-Alonso.
2021. "Comparing COSTATIS and Generalized Procrustes Analysis with Multi-Way Public Education Expenditure Data" *Mathematics* 9, no. 15: 1816.
https://doi.org/10.3390/math9151816