PCA principal component analysis standardized data

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SUMMARY

The discussion centers on the advantages of using standardized data with a correlation matrix for Principal Component Analysis (PCA) compared to simply converting all measurements to the same units. Participants highlight that normalizing data using z-scores is equivalent to employing the correlation matrix instead of the covariance matrix in PCA. The conversation emphasizes that the method of normalization can lead to different results, and the choice between these methods depends on the mathematical definition of "better." Ultimately, the discussion reveals that the approach taken can significantly influence the outcomes of PCA.

PREREQUISITES
  • Understanding of Principal Component Analysis (PCA)
  • Knowledge of data normalization techniques, specifically z-scores
  • Familiarity with correlation and covariance matrices
  • Basic concepts of dimensionality reduction in data analysis
NEXT STEPS
  • Research the mathematical implications of using correlation matrices versus covariance matrices in PCA
  • Learn about the process of data normalization and its impact on analysis outcomes
  • Explore the differences between z-score normalization and unit conversion in data preprocessing
  • Investigate case studies where PCA results differ based on normalization methods
USEFUL FOR

Data analysts, statisticians, and researchers involved in data preprocessing and dimensionality reduction techniques, particularly those utilizing PCA in their analyses.

cutesteph
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Why is better to use the standardized data using the correlation matrix than say converting data into just similar units. Like say I had data that measured car speeds measured in seconds for some data and the other data measured in minutes. Why would it be better just to measure the data using the correlation matrix to normalize data than to just covert all the times to say meters traveled per second.
 
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Why would it be better just to measure the data using the correlation matrix to normalize data than to just covert all the times to say meters traveled per second.
I don't understand this sentence, but in general data analysis requires all data to have the same units.
 
I mean like say we are looking are car race data like from 1/4 a mile 1 mile are in seconds, while data for a 10 mile and a 50 mile race are in minutes. Can't you use normalize data using the correlation matrix within each group like 1/4 mile race even though it is in seconds to a 10 mile race even though it is in minutes? My professor analyzed data that way in a lecture and compared it to a method to just covert all units to meters per second and just take the covariance matrix of that.
 
cutesteph said:
Why would it be better just to measure the data using the correlation matrix to normalize data than to just covert all the times to say meters traveled per second.

What is your definition of "normalizing" the data? Does it amount to replacing the data on each axis by the "z-score" of the data?
 
Yes. It would be which would be equivalent to using the correlation matrix in lieu of the covariance matrix for PCA. I just not sure exactly why would it be better to use that method than to just chance the units to the same units of say in my example meters per second for each different race length.
 
cutesteph said:
I just not sure exactly why would it be better to use that method than to just chance the units to the same units of say in my example meters per second for each different race length.

We'd have to define what "better" means mathematically to investigate that question.

Perhaps the professor was illustrating that you can get different answers if you convert units and do PCA than if you do PCA and convert the units in the principal components afterwards. That difference doesn't mean that one way is always better or worse than the other.
 

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