Principal Component Analysis (PCA) is a powerful technique used to reduce the dimensionality of datasets while retaining most of the variance. When working with multiple datasets, a common task is to compare them. Often, instead of comparing the raw data directly, researchers and analysts prefer to Compare Components Pc, focusing on the principal components derived from each dataset. This approach can reveal underlying similarities or differences in the data structure, especially when dealing with high-dimensional data. This article explores methods for comparing datasets based on their principal components, focusing on scenarios where datasets share the same observations but may have different variables.
Let’s consider a situation where you have two datasets, say Dataset V and Dataset W. Both datasets have the same number of observations (rows), meaning each row in Dataset V corresponds to a row in Dataset W. However, the variables (columns) in each dataset might be different, although they could be conceptually related or entirely distinct.
A straightforward approach to compare components PC is to first perform PCA on each dataset separately. Let’s assume after performing PCA on Dataset V, we obtain principal components vPC1, vPC2, vPC3, and for Dataset W, we get wPC1, wPC2, wPC3. PCA is typically performed on the covariance matrix, resulting in PC scores that are centered (mean=0) and have variances equal to the eigenvalues of the principal components.
To illustrate, consider the following example with numerical data.
v1 v2 v3 w1 w2 w3 vPC1 vPC2 vPC3 wPC1 wPC2 wPC3
1.0 1.0 7.0 1.0 3.0 4.0 -2.6987 3.65782 .00800 -3.11221 -1.76367 -.19015
2.0 2.0 6.0 2.0 2.0 3.0 -1.6755 2.26692 -.12792 -2.03625 -3.12071 -.21763
3.0 3.0 5.0 6.0 3.0 4.0 -.6523 .87601 -.26385 1.85967 -1.51558 .27774
2.0 2.0 4.0 2.0 4.0 5.0 -2.1171 .60370 -1.14705 -2.19941 -.30739 .02448
1.0 3.0 5.0 1.0 5.0 6.0 -2.4920 .88391 .52056 -3.27537 1.04965 .05196
2.0 4.0 6.0 2.0 4.0 5.0 -1.0272 1.15623 1.40376 -2.19941 -.30739 .02448
3.0 5.0 5.0 3.0 3.0 4.0 -.0040 -.23468 1.26784 -1.12346 -1.66443 -.00299
4.0 6.0 4.0 4.0 4.0 5.0 1.0192 -1.62559 1.13191 -.21066 -.20816 .21164
3.0 5.0 3.0 7.0 5.0 6.0 -.4457 -1.89790 .24871 2.69088 1.34735 .61343
2.0 4.0 2.0 2.0 6.0 7.0 -1.9105 -2.17021 -.63449 -2.36257 2.50593 .26660
1.0 3.0 1.0 2.0 5.0 6.0 -3.3754 -2.44253 -1.51769 -2.28099 1.09927 .14554
2.0 4.0 2.0 2.0 6.0 5.0 -1.9105 -2.17021 -.63449 -2.15537 1.22728 -1.25725
3.0 5.0 3.0 3.0 7.0 4.0 -.4457 -1.89790 .24871 -1.03536 1.40490 -2.56647
4.0 6.0 4.0 2.0 5.0 5.0 1.0192 -1.62559 1.13191 -2.17739 .45994 -.61638
4.0 4.0 5.0 4.0 3.0 6.0 .5917 .31671 .10978 -.33629 -.33617 1.61444
5.0 5.0 6.0 5.0 4.0 7.0 2.0566 .58903 .99299 .57651 1.12011 1.82907
6.0 3.0 7.0 15.0 5.0 3.0 2.5490 2.52738 -.42135 10.95669 -.17369 -.92371
5.0 2.0 7.0 8.0 6.0 5.0 1.3050 3.08668 -.79498 3.81088 1.52498 -.69578
10.0 5.0 5.0 6.0 1.0 2.0 6.4351 -.26234 -1.47762 2.02283 -4.32890 .03563
7.0 6.0 4.0 7.0 5.0 7.0 3.7788 -1.63744 -.04471 2.58728 1.98668 1.37536A natural approach is to concatenate the principal components from each dataset into single vectors. For instance, to compare the first two principal components, we can create vectors {vPC1, vPC2} and {wPC1, wPC2} for each observation. We can then calculate the Pearson correlation between these concatenated vectors. In our example, the correlation between {vPC1;vPC2} and {wPC1;wPC2} is approximately 0.30552. Since PC scores are centered, this correlation is equivalent to cosine similarity.
However, this simple concatenation approach can be influenced by the unequal variances of the principal components. Typically, PC1 has a higher variance than PC2, and so on. Therefore, in the above correlation, the similarity between vPC1 and wPC1 will have a greater impact than the similarity between vPC2 and wPC2.
To address this, we can equalize the variances of the principal components before concatenation. This is achieved by z-standardizing each PC score column. After standardization, all principal components will have a variance of 1 and a mean of 0. Recalculating the correlation with standardized PC scores in our example yields a value of approximately 0.09043. Again, this is also the cosine similarity because z-standardization maintains the centered nature of the data.
Notice that this variance equalization gives each principal component equal “weight” in the overall comparison. In our example, the correlation between vPC1 and wPC1 is about 0.61830, while the correlation between vPC2 and wPC2 is approximately -0.43745. The correlation of 0.09043 with equal variance weighting is the average of these two correlations: (0.61830 + (-0.43745)) / 2 ≈ 0.09043.
The Issue of Sign in Principal Components
An important consideration when comparing principal components is the sign ambiguity. The sign of PC scores is arbitrary. For any principal component, we can reverse the sign of all its scores without changing the PCA solution itself. For example, we could reverse the sign of vPC2 and wPC2 without affecting the validity of our PCA.
However, if we reverse the sign of only one of them, say just vPC2, the calculated similarity will change. If we reverse the sign of vPC2 in our example, the correlation between vPC2 and wPC2 becomes +0.43745. The correlation between the concatenated (non-standardized) vectors {vPC1; vPC2} and {wPC1; wPC2} becomes approximately 0.55626, and with equal weighting (standardized PCs), it becomes around 0.52788. This value is again the average: (0.61830 + 0.43745) / 2 ≈ 0.52788.
This raises the question: when is it legitimate to reverse the sign of a PC in one dataset but not the other? If the original variables in Dataset V and Dataset W are completely unrelated, then perhaps adjusting the sign to maximize similarity could be justifiable. However, this should be done cautiously and with a clear understanding of the potential impact on interpretation.
Alternative Comparison Methods
Besides simple correlation of concatenated PCs, other methods can be used to compare components PC. For instance, one could average the squared correlations (to ignore the sign) or use Fisher’s z-transformation before averaging correlations.
Furthermore, if you want to directly assess the relationship between two sets of variables, Canonical Correlation Analysis (CCA) provides a more direct approach. CCA is specifically designed to find linear combinations of variables from two sets that are maximally correlated with each other. PCA focuses on variance within a single dataset, while CCA focuses on covariance between two datasets. The choice between PCA-based comparison and CCA depends on the specific research question and the nature of the datasets.
Comparing PCA Loadings for Datasets with Different Cases
If your datasets have different observations (rows) but share the same variables (features), the approach to compare components PC shifts. In this scenario, comparing PC scores is less relevant. Instead, we should focus on comparing the PCA loadings. Loadings represent the weights of the original variables in each principal component and reveal the contribution of each variable to each PC.
To compare PCA loadings from two datasets with the same variables but different cases, we can use cosine similarity between the loading vectors. This measure is also known as Tucker’s coefficient of congruence in the context of factor analysis. Additionally, Procrustes rotation can be applied to one loading matrix to align it optimally with the other before calculating the similarity. This rotation helps to remove rotational ambiguity and can provide a more accurate comparison of the underlying factor structures.
Conclusion
Comparing principal components is a valuable technique for analyzing and contrasting datasets, especially when dealing with complex, high-dimensional data. When you compare components PC, you can choose from several methods, including concatenating PC scores and calculating correlations, considering variance equalization, and addressing sign ambiguity. For comparing sets of variables, CCA offers a more direct approach. When datasets differ in observations but share variables, comparing PCA loadings using cosine similarity and Procrustes rotation becomes the appropriate strategy. The best method depends on the specific characteristics of your data and the research questions you aim to address.
