登录查看更多内容

Last updated on 2024年7月11日

How do you interpret the canonical correlation coefficients and canonical loadings in CCA?

由人工智能和领英社区提供技术支持

Canonical correlation analysis (CCA) is a statistical technique that allows you to explore the relationship between two sets of variables, such as personality traits and job performance. CCA can help you identify the linear combinations of variables that have the highest correlation with each other, and also measure the strength and significance of the overall relationship. In this article, you will learn how to interpret the canonical correlation coefficients and canonical loadings in CCA, and how they can help you understand the nature and importance of the variables involved.

此文章中的业界达人

由社区从 6 条内容中精选。了解更多

Raymond Wong

PhD Student at Imperial | JPMorgan Chase & Co. Quantitative Research Virtual Experience Participant
Philemon Kiprono

AI Engineer | Microsoft Certified | Azure | LlamaIndex | RAGs | DSPy

1 Canonical correlation coefficients

The canonical correlation coefficients are the values that indicate how strongly the two sets of variables are related. They range from 0 to 1, where 0 means no relationship and 1 means a perfect relationship. The first canonical correlation coefficient is the highest possible value, and the subsequent ones are lower and orthogonal to the previous ones. The number of canonical correlation coefficients is equal to the smaller number of variables in either set. You can use the canonical correlation coefficients to test the null hypothesis that there is no relationship between the two sets of variables, and to compare the relative importance of each pair of canonical variables.

添加您的观点

Philemon Kiprono

AI Engineer | Microsoft Certified | Azure | LlamaIndex | RAGs | DSPy
举报内容
Canonical correlation coefficients describe the relationships between two sets of variables. They maximize the correlation between the canonical variates while ensuring equal variance. These coefficients help understand how the variables in one set relate to those in the other set.

已翻译

赞
Raymond Wong

PhD Student at Imperial | JPMorgan Chase & Co. Quantitative Research Virtual Experience Participant
(已编辑)
举报内容
Canonical Correlation Analysis (CCA) finds the linear combinations that aim to maximise the correlation between a set of variables X and set of variables Y (typically your inputs and outputs for machine learning). This process generates a pair of components refered as Canonical Variates (CV), where maximum number of pairs generated is dictated by the smaller number of variables within set X or Y. The CCA correlation coefficients refer to the coefficients of the linear model when plotting the pairs of CV, such as plotting CV_Y against CV_X. The coefficients of the pairs of CV should gradually decrease the more it generates, such that the first CV contains the maximum amount of correlation and variance captured.

已翻译

赞
Fábio Rebou?as

Professor | Consultant | University Teacher | Epidemiologist | Health Information Analyst | Sanitarist
举报内容
Like ancient hieroglyphs etched into the annals of statistical lore, these enigmatic symbols reveal the underlying structures and patterns concealed within our data. With each loading, we unravel the mysteries of correlation and causation, deciphering the cryptic messages encoded within our variables. So let us venture forth with open minds and keen intellects, for in the realm of canonical correlation analysis, enlightenment awaits those bold enough to seek it.

已翻译

赞

2 Canonical loadings

The canonical loadings are the correlations between the original variables and the canonical variables, which are the linear combinations of variables that maximize the canonical correlation coefficients. The canonical loadings can help you interpret the meaning and direction of the canonical variables, and how they relate to each other. For example, if a variable has a high positive loading on a canonical variable, it means that it contributes positively to the canonical correlation, and that it is positively associated with the other variables that have high positive loadings on the same canonical variable. The canonical loadings can also be used to compute the canonical redundancy, which is the amount of variance in one set of variables that is explained by the other set of variables.

添加您的观点

Fábio Rebou?as

Professor | Consultant | University Teacher | Epidemiologist | Health Information Analyst | Sanitarist
举报内容
As we delve deeper into the labyrinthine depths of CCA, let us remember the sage counsel of our academic predecessors and proceed with caution. Though the path may be fraught with challenges and uncertainties, let us press onward with unwavering determination and scholarly resolve. Together, we shall navigate the complexities of canonical correlation analysis, emerging triumphant with newfound insights and discoveries. So hoist the flag of intellectual inquiry high, my esteemed colleagues, for our journey into the heart of statistical discovery has only just begun!

已翻译

赞
Raymond Wong

PhD Student at Imperial | JPMorgan Chase & Co. Quantitative Research Virtual Experience Participant
(已编辑)
举报内容
Canonical Correlation Analysis (CCA) finds the linear combinations that aim to maximise the correlation between a set of variables X and set of variables Y (typically your inputs and outputs for machine learning). This process generates a pair of components refered as Canonical Variates (CV), where maximum number of pairs generated is dictated by the smaller number of variables within set X or Y. The Canonical Loadings (structure coefficients) represent the correlation between the original variable and the generated CV. Indicating how strongly each original variable contributes towards the CV. Example: If an original variable has a small Weight but large Loading - Variable has little impact globally but contributes a lot to this CV.

已翻译

赞

3 Canonical cross-loadings

The canonical cross-loadings are the correlations between the original variables and the canonical variables of the other set. They can help you assess the degree of overlap or redundancy between the two sets of variables, and how they influence each other. For example, if a variable has a high positive cross-loading on a canonical variable of the other set, it means that it is highly correlated with that canonical variable, and that it shares some common variance with the variables that have high loadings on that canonical variable. The canonical cross-loadings can also be used to compute the canonical communality, which is the amount of variance in one set of variables that is shared with the other set of variables.

添加您的观点

4 Canonical weights

The canonical weights are the coefficients that are used to create the canonical variables from the original variables. They are similar to the regression coefficients in multiple regression, and they indicate how much each variable contributes to the formation of the canonical variable. The canonical weights can help you determine which variables are most important or influential in creating the canonical variables, and how they affect the canonical correlation. However, the canonical weights are not directly comparable across different canonical variables, because they depend on the scaling and ordering of the variables. Therefore, it is more common to use the standardized canonical weights, which are normalized by the standard deviations of the variables.

添加您的观点

Raymond Wong

PhD Student at Imperial | JPMorgan Chase & Co. Quantitative Research Virtual Experience Participant
(已编辑)
举报内容
Canonical Correlation Analysis (CCA) finds the linear combinations that aim to maximise the correlation between a set of variables X and set of variables Y (typically your inputs and outputs for machine learning). This process generates a pair of components refered as Canonical Variates (CV), where maximum number of pairs generated is dictated by the smaller number of variables within set X or Y. The Canonical Weights represent the coefficeints (weighting) used to create the canonical variates from the original variables, highlighting the importance of each original variable in defining the canonical relationship. As this shows the effect globally, this can be interpreted similarly to the regression coefficients in multiple regression.

已翻译

赞

5 Canonical scores

The canonical scores are the values of the canonical variables for each observation or case in the data. They are computed by multiplying the original variables by the canonical weights, and they represent the position of each case on the canonical dimensions. The canonical scores can help you visualize and cluster the cases based on their similarities or differences on the canonical variables, and to perform further analyses such as classification, regression, or discriminant analysis. You can also use the canonical scores to examine the outliers or extreme cases that have high or low values on the canonical variables.

添加您的观点

6 Canonical plots

The canonical plots are graphical representations of the canonical correlation analysis, where you can see the relationship between the two sets of variables and the cases. The most common types of canonical plots are biplots and scatterplots. A biplot shows both the original variables and the cases on the same plot, where the variables are represented by vectors and the cases are represented by points. The length and direction of the vectors indicate the loadings and weights of the variables on the canonical variables, and the position and distance of the points indicate the scores and correlations of the cases on the canonical variables. A scatterplot shows only the cases on the plot, where each axis represents a canonical variable. The shape and location of the points indicate the scores and clusters of the cases on the canonical dimensions. You can use the canonical plots to explore and summarize the results of the canonical correlation analysis, and to identify patterns or trends in the data.

添加您的观点

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

添加您的观点

Statistical Data Analysis

+ 关注

给文章评分

我们借助人工智能创建了此文章。您认为这篇文章怎么样？

很棒不太好

举报此文章

查看全部

How do you interpret the canonical correlation coefficients and canonical loadings in CCA?

1

2

3

4

5

6

7

1 Canonical correlation coefficients

2 Canonical loadings

3 Canonical cross-loadings

4 Canonical weights

5 Canonical scores

6 Canonical plots

7 Here’s what else to consider

Statistical Data Analysis

给文章评分

感谢您的反馈

更多Statistical Data Analysis相关文章

更多相关阅读内容