On the Correlation between Random Variables and their Principal Components

• Autorzy: Gniazdowski Zenon

Identyfikatory

DOI

10.26348/znwwsi.28.41

• Języki publikacji: PL

Abstrakty

EN

The article attempts to find an algebraic formula describing the correlation coefficients between random variables and the principal components representing them. As a result of the analysis, starting from selected statistics relating to individual random variables, the equivalents of these statistics relating to a set of random variables were presented in the language of linear algebra, using the concepts of vector and matrix. This made it possible, in subsequent steps, to derive the expected formula. The formula found is identical to the formula used in Factor Analysis to calculate factor loadings. The discussion showed that it is possible to apply this formula to optimize the number of principal components in Principal Component Analysis, as well as to optimize the number of factors in Factor Analysis.

Słowa kluczowe

EN

correlation; random components; principal components; factors; common variance

• Wydawca: Warszawska Wyższa Szkoła Informatyki
• Czasopismo: Zeszyty Naukowe Warszawskiej Wyższej Szkoły Informatyki
• Rocznik: 2023
• Tom: nr 28
• Strony: 41--55
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Gniazdowski Zenon

• Warsaw School of Computer Science

Bibliografia

• [1] Z. Gniazdowski, “New Interpretation of Principal Components Analysis,” Zeszyty Naukowe WWSI, vol. 11, no. 16, pp. 43-65, 2017. [Online]. http://doi.org/10.26348/znwwsi.16.43
• [2] ——, “Principal Component Analysis versus Factor Analysis,” Zeszyty Naukowe WWSI, vol. 15, no. 24, pp. 35-88, 2021. [Online]. http://doi.org/10.26348/znwwsi.24.35
• [3] P. Francuz and R. Mackiewicz, Liczby nie wiedz ˛a, sk ˛ad pochodz ˛a. Przewodnik po metodologii i statystyce nie tylko dla psychologów. Lublin: Wydawnictwo KUL, 2007.
• 4] Z. Gniazdowski, “On the Analysis of Correlation Between Nominal Data and Numerical Data,” Zeszyty Naukowe WWSI, vol. 16, no. 27, pp. 57-82, 2022. [Online]. http://doi.org/10.26348/znwwsi.27.57
• [5] ——, “Geometric interpretation of a correlation,” Zeszyty Naukowe WWSI, vol. 7, no. 9, pp. 27–35, 2013. [Online]. http://doi.org/10.26348/znwwsi.9.27
• [6] S. Banerjee and A. Roy, Linear Algebra and Matrix Analysis for Statistics. CRC Press, 2014. [Online]. https://students.aiu.edu/submissions/profiles/resources/onlineBook/C9M2d6_Linear_Algebra_and_Matrix_Analysis_for_Statistics.pdf
• [7] C. R. Johnson and R. A. Horn, Matrix analysis. Cambridge university press Cambridge, 2013. [Online]. http://www.anandinstitute.org/pdf/Roger_A.Horn._Matrix_Analysis_2nd_edition(BookSee.org).pdf
• [8] E. Million, “The Hadamard product,” Course Notes, vol. 3, no. 6, pp. 1-7, 2007. [Online]. http://buzzard.pugetsound.edu/courses/2007spring/projects/million-paper.pdf