How to read the charts¶
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Principal Component Analysis¶
Theoretical introduction¶
This section is for information only and may be disregarded. For an interpretation of the graphs, see Interpretation of PCA
General concept and goal of PCA¶
In exploratory data analysis and visualization, a Principal Components Analysis is a method of summarizing the data to reduce the number of features (variables), often referred to as the dimensionality of a data set, in order to ease its visualisation and interpretation.
In practice, we aim at defining a new set of features (called the principal components), built as linear composition of the original features, and which describe as much as possible of the variance, or variation, within the dataset. This in turn will help understand which features are redundant, related to each other, if they are positively or negatively related, or how well they differentiate between each point of the dataset.
Of course the goal is to have a number of principal components smaller set than the original one variables. These new variables summarize the data, each one comprising a proportion of the original features of the dataset. The higher the proportion, the more weight the original variable holds in explaining the variation.
Choosing the features
Let's imagine we are trying to describe a number of wines. We could theoretically use any "feature" to characterize them: color, alcohol content, region, producer... But characterizing them should allow us to differentiate them. The temperature for example, doesn't seem to be any relevant in our characterization of the wine, if they are all at room temperature: only knowing the temperature of the wine is not enough for us to identify them.
When performing a sensory analysis, we choose a set of criteria we hope will allow us to best differentiate the samples between them, and if not, will prove the samples are very similar.
Applying a PCA is a way to understand if these criteria are indeed relevant to differentiate the wines.
The principal components are sorted by order of importance: the first principal components is the one explaining the biggest part of the variation among the dataset, the second principal component explains the second highest variation, and so on. The charts are obtained by plotting principal components against each other (PC1 VS PC2, PC1 VS PC3, PC2 VS PC3, etc), each dot representing the contribution of one feature (tasting criteria or taster) to the considered principal components.
Applying PCA to descriptive sensory analysis: defining features¶
Applying the dimensionality reduction of a PCA to descriptive sensory analysis is not as straightforward as it could seem at first. Conceptually, the results are described by two sets of features: the sensory criteria on one side, and the assessors themselves on the other side. Both should be looked at in parallel, albeit expecting opposites results for each case:
- When applying a PCA with the sensory criteria as features (usually then only keeping the consensus averaged scores), we want to have criteria which are as independent from one another as possible; in other words, we want each of the criteria to have the strongest contribution possible to the variance in the dataset; the opposite would mean that said criteria does not allow to differentiate properly between samples.
- On the other hand, when applying a PCA with the assessors as features (usually then only keeping the total score of each sample), the results reflect the agreement between the tasters. Considering two assessors A and B, if they completely agree on a set of samples, i.e. if both of them give the same score to each individual sample, we are not able to tell which results are from taster A and which are from taster B. The opposite is also true, if they are strongly disagreeing, looking at a single set of scores will allow us to identify the taster.
The dashboards are therefore displaying the results of two PCA: one using the tasting criteria as features, using the scores of each sample averaged on the whole panel, and a second one using the tasters as features, using the total scores given be each taster for every sample.
Interpretation of PCA¶
Goals of the PCA¶
Applying a Principal Component Analysis to sensory evaluation can be particularly helpful to interpret and understand the results, whether it is to compare products, assessors, or both. It aims at finding patterns and relationships in the data, between tasting criteria and between tasters.
It can therefore, on one side, show how tasting criteria are related (for example, it might proves that two sensory characteristics are closely related, evolving concurrently, or on the contrary, negatively correlated, the increase of one leading to the decrease of the other). It can also proves the relevance of a sensory criteria, i.e. if a given criteria allows to differentiate between samples or is actually rather meaningless.
Example PCA on tasting criteria
Consider the following example: a sample of rum has been stored in wood barrels for 2 years, and monthly tasted by the same group of experts. The goal was to evaluate the evolution of the primary aromas versus tertiary aromas. Four tasting criteria were rated:
- Sugar can presence in the nose
- Wood contribution in the nose
- Sugar cane presence on palate
- Wood contribution on palate
For the sake of illustration, let's considered that the experts agreed every time. The scores are averaged over the whole panel for each criteria and every tasting date. We therefore obtain a dataset of 24 lines (one per tasting date) and 4 columns (one per criteria), on which is performed a PCA. The results are given below, with PC1 represented in the x-axis and PC2 on y-axis.
From the chart, we can conclude that:
- Sugar cane presence and wood contribution are negatively correlated, as the wood parts are situated on the positive side of PC1, while sugar cane presence is on its negative side. While the wood contribution gets stronger, the sugar cane presence tends to decrease, albeit more slowly as their absolut values are smaller along PC1;
- The wood contributions on nose and palate are very similar, although it seems to be stronger in nose than on palate;
- Finally, the sugar cane presence on palate, given its very small value, has very little impact on the overall evaluation: it is in that case not evolving much over time, has it would hardly be enough to distinguish between the dates of tasting.
On the other side, a PCA is useful to find patterns between groups of tasters, as a measure of their agreement. Tasters with similar opinions will be close to each others on the charts, with a positive correlation, while disagreeing assessors will display a negative correlation, being opposed on the charts. Similarly, an assessor who provides very similar evaluation from one sample to the other, in other words who is not really able to differentiate between sample, will display a very small contribution to the principal components.
Explained variance¶
As all principal components are uncorrelated, each explains a proportion of the variation measured in the dataset, with PC1 explaining the highest variation, PC2 the second highest variation, and so on. The proportion of variation is known as the explained variance, and expressed as a percentage. It can be understood as an indication on how important a principal component is in explaining the variation of data.
The cumulative explained variance is drawn on top of the PCA charts, as an indication on how much of the variation of the data is captured by the components represented. The cumulative explained variance corresponding to the above chart is given below.
It is important to keep in mind the value of the explained variance when analyzing the PCA plots: how much of the variation is captured is an indication of how relevant the component is for further interpretation. In the example above, the first component already explains 93% of the variation, while the second only explains 4% (cumulative explained variance of 93 + 4 = 97%). It is then evident that the contribution of the second component is insignificant compared to the first one; when reading the PCA charts, the differences or correlations along the x-axis (PC1) should be given much more attention than those along the y-axis (PC2).
General rules for interpretation¶
As a general rule of thumb, the relevance of a feature is given by how far away from the origin it lies: characteristics which do not allow to differentiate samples will fall close to the origin, while characteristics with values close to one are good descriptors of the products. Two concentric circles are drawn on the charts: features falling between the two circles can be safely considered relevant (and the closer to the external circle the better), while features found within the smaller inside circle are most probably not very important, either because the panelists disagree strongly, or because they don't allow for a clear distinction between the products.
Take into account the explained variance
Pay attention to the explained variance of the components: the higher the component's explained variance, the higher the contribution of the feature to the differences between samples. As the components' explained variance decreases with the number of components (the first component explaining the biggest variation, second one explaining the second biggest and so on), the x-axis usually matters more than the y-axis.
Features or characteristics which are similar to each other will appear close to each other on the plots; characteristics which are opposed will appear on opposite sides of the origin.
- Sensory characteristics (the tasting criteria used to evaluate the samples) which are related, or evolving concurrently (i.e. an increase in one leading to an increase in the other and vice versa) will lie close to each other on the charts;
- In contrast, characteristics which are opposed (i.e. an increase in one will lead to a decrease of the other and vice versa) will appear on opposite sides of the origin.
The same conclusion can be drawn when looking at the assessors:
- Two assessors which are strongly agreeing will appear close to each other on the chart; they share the same opinion and give similar scores to the samples;
- On the contrary, two panelists strongly disagreeing will be displayed on opposite sides of the origin axis.
On more general terms, panelists agreeing will appear grouped together.
Example¶
The following charts illustrates the above concepts. The points from A to E represent 5 different features: they can be seen as panelists or sensory characteristics of the products.
What conclusions can be drawn from the chart.
- Features A and B are strongly relevant and correlated
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They are very close to each other, and both comprised between the two circles, with a high x value (along PC1).
If the considered features are sensory characteristics, they allow to differentiate between samples quite well and therefore of quite high importance for further analysis. There are also two aspects to consider:
- First, the two characteristics are evolving concurrently: an increase (respectively a decrease) of the sensory characteristic A always drive an increase (respectively a decrease) of the characteristic B, and vice versa. They are therefore strongly linked according to the panelists.
- However, fundamentally, both these attributes might actually measure the same characteristic!
If the considered features are actually the panelists, A and B share very similar opinions, and are able to clearly differentiate between samples
- Feature C is also strongly relevant, but opposed to A and B
-
C also shows a high x value (along PC1), but negative. A, B and C are close to a straight line through the origin, with C on opposite side of A and B.
If the considered features are sensory characteristics, C is also an attribute that allows to differentiate samples. However, it evolves in opposite direction than A and B: an increase in the feature C will lead to a decrease to both A and B and vice versa.
If the considered features are panelists, there again C is a panelist which can clearly differentiate between samples. However, it shows opinions strongly opposed to panelists A and B. Special attention should then be given to understand where the differences might come from: trainings, sensitivities, social background...
- Feature D might or might not be relevant depending on the explained variance of PC2, and is independent from A, B and C
-
D shows a very low x value (along PC1, the strongest in terms of explained variance), but a very high y value along PC2
In this case, it is important to have a look at the proportion of explained variance of PC2. Should that value be very low, the feature D is rather irrelevant. If a significant part of the variation is explained by PC2 however, the feature D is relevant, although less than A, B and C (as the explained variance of PC1 is always higher than the explained variance of PC2).
If the explained variance on PC2 is significant, feature D is considered relevant. It shows very little relation to A, B and C, as D contributes almost exclusively to PC2 (very low value along the x axis but high value along the y axis), while A, B and C contribute almost exclusively to PC1.
In other words, if the features are sensory characteristics, D allows for a clear differentiation between products but is independent of features A, B and C. If the features are panelists, D shows a third opinion and different sensitivities than panelists A, B and C. They are neither agreeing nor disagreeing with either A and B or C.
- Feature E is the less relevant
-
E shows a very low contribution to both PC1 and PC2 as it falls close to the origin of the chart and is within to inner circle
If the features are sensory characteristics, the feature E doesn't really allow to differentiate between products, or reflects a disagreement between panelists. It should be discarded for further analysis. Special attention should be given to that criteria in future tastings: maybe it is misunderstood by the assessors, or simply not relevant to the considered category of products.
If the features are panelist, assessor E doesn't seem to be able to clearly differentiate products, and their evaluation does not really influence the overall results. This could come from an improper use of the scales (the panelist might give very similar scores to all products), or a lack of training.