6.6.1 Principal Component Analysis
Principal Component Analysis is useful for reducing and interpreting large multivariate data sets with underlying linear structures, and for discovering previously unsuspected relationships.
We will start with data measuring protein consumption in twenty-five European countries for nine food groups. Using Principal Component Analysis, we will examine the relationship between protein sources and these European countries.
Selecting Principal Methods
To determine the number of principal components to be retained, we should first run Principal Component Analysis and then proceed based on its result:
- Open a new project or a new workbook. Import the data file \samples\Statistics\Protein Consumption in Europe.dat
- Select the entire worksheet and then select Statistics: Multivariate Analysis: Principal Component Analysis.
- Accept the default settings in the open dialog box and click OK.
- Select sheet PCA Report.
- In the Eigenvalues of the Correlation Matrix table, we can see that the first four principal components explain 86% of the variance and the remaining components each contribute 5% or less. We will keep four main components.
- A scree plot can be a useful visual aid for determining the appropriate number of principal components. The number of components depends on the "elbow" point at which the remaining eigenvalues are relatively small and all about the same size. This point is not very evident in the scree plot, but we can still say the fourth point is our "elbow" point.
- Click the lock icon in the results tree and select Change Parameters in the context menu. In the Settings tab, set Number of Components to Extract to 4. Do not close the dialog; in the next steps, we will retrieve component diagrams.
Request Principal Component Plots
In the Plots tab of the dialog, users can choose whether they want to create a scree plot or a component diagram.
- Scree Plot
- The scree plot is a useful visual aid for determining an appropriate number of principal components.
- Component Plot
- Component plots show the component score of each observation or component loading of each variable for a pair of principal components. In the Select Principal Components to Plot group, users can specify which pair of components to plot. The component plots include:
- Loading Plot
- The loading plot is a plot of the relationship between the original variables and the subspace dimension. It is used to interpret relationships between variables.
- Score Plot
- The score plot is a projection of data onto subspace. It is used to interpret relationships between observations.
- The biplot shows both the loadings and the scores for two selected components in parallel.
- In the dialog that was opened in the preceding steps, select the Plots tab. Make sure Scree Plot, Loading Plot, and Biplot are selected.
- The first two components are usually responsible for the bulk of the variance. This is why we are going to plot the component plot in the space of the first two principal components. In the Select Principal Components to Plot group, set Principal Component for X Axis to 1, and set Principal Component for Y Axis to 2. Click OK.
Interpreting The Results
- In the Correlation Matrix, we can see that the variables are highly correlated. Many values are greater than 0.3. Principal Component Analysis is an appropriate tool for removing the collinearity.
- The main component variables are defined as linear combinations of the original variables. The Extracted Eigenvectors table provides coefficients for equations.
- The Loading Plot reveals the relationships between variables in the space of the first two components. In the loading plot, we can see that Red Meat, Eggs, Milk, and White Meat have similar heavy loadings for principal component 1. Fish, fruit, and vegetables, however, have similar heavy loadings for principal component 2.
- The biplot shows both the loadings and the score for two selected components in parallel. It can reveal the projection of an observation on the subspace with the score points. It can also find the ratio of observations and variables in the subspace of the first two components. (Note: Double-click the graph to open and customize.)
- We can use the Data Reader tool to open the Data Info window and examine the plot in greater detail. Click on a data point to read component scores for each country. We can see that Spain and Portugal's protein sources differ from those of other European countries. Spain and Portugal rely on fruits and vegetables, while eastern European countries such as Albania, Bulgaria, Yugoslavia, and Romania prefer cereals and nuts.
|To display country information in the Data Info window, as in the image above:
- Right-click the Data Info window and select Preferences....
- Highlight Country in the left-panel, then click the Select button (the right-pointing arrow) to add Country to the Data Info display, then click OK.
Note: Since Origin 2019, you can simply hover on a data point to show a tooltip with data point coordinate information. Both the tooltip and the Data Info display are customizable. See The Data Info Window and Data Point Tooltip for more information.
To create an arbitrary 3D loading plot, such as PC1-PC2-PC4 or PC1-PC3-PC5 loading plot:
- Click the lock icon in the results tree and select Change Parameters in the context menu. In the Settings tab, set Number of Components to Extract as needed. For example, if you want to do a PC1-PC2-PC4 loading plot, Number of Components to Extract should be set as 4. Click OK to close the dialog.
- The Extracted Eigenvectors table now has 4 columns. To set up a worksheet to create our loading plot, we'll begin by duplicating the sheet PCA Plot Data1. Right-click on the tab of PCA Plot Data1 and select Duplicate. The new sheet is named as PCA Plot Data2.
- To create our 3D loading plot of PC1-PC2-PC4, we need to add "Z" values to our added sheet PCA Plot Data2. Begin by highlighting Col (E), right-clicking and choosing Insert to add a blank column to the worksheet. Secondly, highlight the last column, right-click and choose Insert to add a second blank column to the worksheet.
- Hold the CTRL key and select the two blank columns, then right-click and choose Set As Z.
- Add PC 4 to the Long Name field in the two blank "Z" columns.
- Set the all the Eigenvalues of column E(Z2) to 0 by double-clicking in the F(x)= cell and entering 0.
- Click on the sheet PCA Report and highlight and copy the Coefficents of PC4 in the Extracted Eigenvectors table and paste the values into columnn H(Z3) of sheet PCA Plot Data2.
- Highlight columns C through H, then click Plot>3D: Vector: 3D Vector XYZ XYZ.
- You can further customize your plot by changing vector color and adding labels. Double-click on the plot to open Plot Details, then highlight Original in the left panel.
- Select the 3D Vector tab, set Color as Blue and Width as 4.
- Click on the Label tab, check Enable and set Label form = Col(I) (your protein source column), Attach to = Arrow Head, Color = Auto, Position = Below and Size = 20.
- Finally, click the Graph window icon on the left panel of Plot Details (the highest level of the Plot Details tree), click the Legends/Titles tab and clear Show Units (when available) for Auto Axis Titles, %?(X) %(?Y). Click OK to close the dialog. The 3D loading plot will be created as follows.