Understanding Principal Component Analysis (PCA)

Agricultural and agribusiness data is rarely simple. A researcher studying crop performance across different regions might collect measurements on soil pH, rainfall, temperature, nitrogen levels, irrigation frequency, input costs, and market prices – all at once. Analysing each variable in isolation misses the bigger picture, but handling them all simultaneously creates what statisticians call the “curse of dimensionality.” This is where Principal Component Analysis (PCA) becomes an essential tool. It is a dimensionality reduction technique that cuts through complexity, preserving the patterns that matter most while discarding redundant noise.

What is Principal Component Analysis? 🔗

Principal Component Analysis is a multivariate statistical method that transforms a large set of possibly correlated variables into a smaller set of new, uncorrelated variables called principal components (PCs). These components are linear combinations of the original variables, and they are ordered so that the first component captures the highest amount of variance in the data, the second captures the next highest, and so on. Crucially, each PC is orthogonal – that is, completely uncorrelated – to all the others, which means each one describes an independent dimension of the data.

The core goal is straightforward: reduce the dimensionality of a dataset by forming new variables that are linear combinations of the original variables, while retaining as much of the original variance-covariance structure as possible. If a dataset originally has 20 variables, PCA might reveal that just 4 or 5 principal components capture 80-90% of the total variation – making the data far more manageable without losing meaningful insight.

Why variance matters in PCA 🔗

PCA is built on the idea that the most informative directions in a dataset are those along which the data varies the most. PCA assumes that the directions with the largest variances are the most “important” – that is, the most revealing about the underlying structure of the data. When variables are highly correlated, there is redundancy in the data, and PCA exploits this redundancy to compress information into fewer dimensions without significant loss.

For example, in an agronomy dataset, soil organic matter, nitrogen content, and phosphorus levels often move together – farms with rich organic matter tend to have higher nitrogen too. PCA can combine these correlated soil variables into a single principal component representing overall soil fertility, rather than treating each as a separate dimension.

How PCA works: the step-by-step process 🔗

Step 1: Standardise the data 🔗

Before running PCA, the dataset must be standardised. The aim of this step is to standardise the range of the continuous initial variables so that each one of them contributes equally to the analysis – because PCA is highly sensitive to the scale of the variables. A variable measured in tonnes will dominate one measured in percentages unless both are rescaled to a common standard. Standardisation is done by subtracting the mean and dividing by the standard deviation for each variable, giving every variable a mean of zero and a standard deviation of one.

Step 2: Compute the covariance matrix 🔗

The next step is computing a covariance matrix, which summarises how pairs of variables change together across observations. A high covariance means the two features are correlated, and PCA aims to eliminate redundancy by transforming data into uncorrelated principal components. The sign of a covariance tells us whether two variables increase together (positive) or move in opposite directions (negative).

Step 3: Calculate eigenvectors and eigenvalues 🔗

This is the mathematical heart of PCA. Eigenvectors of the covariance matrix are the directions of the axes where there is the most variance – the principal components – and eigenvalues are the coefficients attached to eigenvectors, giving the amount of variance carried in each principal component. In simpler terms, eigenvectors define the direction of each new axis, and eigenvalues tell us how much information that axis carries. By ranking eigenvectors from highest to lowest eigenvalue, we get the principal components ordered by importance.

Step 4: Select principal components 🔗

Not all principal components are equally useful. We rank the PCs in descending order based on their associated eigenvalues and select the most important components – often choosing the first k PCs that together explain a significant proportion of the total variance, typically 80-95%. Several methods help decide how many to retain. Under the widely used Kaiser criterion, only components with eigenvalues greater than 1 are retained, since they explain more variance than a single original variable. A common complementary tool is the scree plot.

Step 5: Transform the data 🔗

Finally, the original dataset is projected onto the selected principal components, creating a new, lower-dimensional dataset. This reorientation does not modify the original data itself but instead provides a new perspective to better represent it. The resulting dataset has fewer columns but retains the most critical patterns from the original variables.

Reading a scree plot 🔗

The scree plot is one of the most widely used tools for deciding how many principal components to keep. A scree plot displays the eigenvalues in a downward curve, ordering them from largest to smallest. The analyst looks for the “elbow” – the point where the curve bends and starts to flatten. The ideal pattern is a steep curve followed by a bend and then a straight line; components in the steep curve before the line trend starts are the ones to retain. Components beyond the elbow contribute very little additional variance and are typically dropped.

A study on Indian agricultural crop data – covering eight major crops including rice, wheat, maize, and bajra – demonstrated this efficiency well. PCA recommended that only two principal components were needed to explain 93% of the total variability in the dataset. This kind of compression is precisely what makes PCA so powerful for large-scale agricultural data.

Interpreting loadings and scores 🔗

Once the principal components are computed, two key outputs help interpret them: loadings and scores. Loadings (derived from eigenvectors) show the contribution of each original variable to a given principal component. The larger the absolute value of a coefficient, the more important the corresponding variable is in calculating the component. Scores represent the position of each observation in the new principal component space and are used to visualise patterns, clusters, and outliers.

In agronomic research, loading coefficients can be likened to a fingerprint, defining the contribution of each trait to successive principal components. A biplot – a combined plot of loadings and scores – allows researchers to see both how variables relate to each other and how individual farms, varieties, or treatments differ, all in a single visualisation.

Applications in agriculture and agribusiness 🔗

Crop and soil analysis 🔗

PCA is used to study the relationships between agronomic traits and efficiently visualise the similarities between individuals or treatments where various factors exert strong effects on growth, yield, and physiological traits. In crop breeding, PCA helps identify which genetic or phenotypic variables are most influential. In a study on blackgram, PCA reduced 13 quantitative characters – including grain yield, plant height, and chlorophyll content – into fewer components, with the first two principal components capturing approximately 67% of total variability.

Precision agriculture and yield prediction 🔗

Precision agriculture generates enormous volumes of data from sensors, satellites, and IoT devices. PCA simplifies datasets by identifying the most relevant features – if a dataset starts with 50 variables, PCA may reduce these to just 5 principal components without losing critical information. For yield prediction, PCA transforms correlated inputs like soil moisture, temperature, and historical performance into independent principal components that can be fed into regression or machine learning models, reducing computational complexity and improving forecast accuracy.

Agribusiness market and commodity analysis 🔗

Agribusiness analysts regularly work with multiple economic variables – weather forecasts, production estimates, export demand, input costs, and currency movements – all affecting commodity prices simultaneously. PCA helps identify the primary drivers of price volatility from this web of indicators, enabling traders and farm managers to focus on what matters most. In the food safety domain, dimensionality reduction techniques such as PCA have been widely used to analyse big datasets in food safety, helping companies detect spoilage patterns and quality deviations more efficiently across supply chains.

Food security and e-commerce research 🔗

PCA is also used at a policy and systems level. Researchers have applied it to develop composite indices for complex, multi-indicator phenomena. A novel PCA-based E-Commerce Food Security Contribution Index was developed using 26 food availability, access, utilisation, and stability indicators to assess how online agri-food sellers contribute to food security – a clear example of PCA enabling structured, evidence-based policy analysis from otherwise unwieldy data.

Limitations of PCA to keep in mind 🔗

PCA is a powerful technique, but it has important constraints. First, PCA only works with quantitative variables – counts, percentages, and numbers. Categorical or binary traits like disease presence or crop colour cannot be analysed directly. Second, because principal components are linear combinations of all original variables, they can be difficult to interpret intuitively – a component that blends soil pH, rainfall, and fertiliser cost into a single axis does not map neatly to any one concept a farmer or manager recognises. Third, PCA is sensitive to outliers; a few unusual observations can disproportionately distort the principal components, especially in high-dimensional datasets. Finally, if the data has no underlying correlated structure, there is nothing for PCA to compress – the dimensions simply cannot be meaningfully reduced.

Despite these limitations, when data is appropriately collected, standardised, and structured, PCA is a main tool for the analysis of genotype-by-environment interaction and pattern detection across complex agricultural datasets. Used alongside domain expertise, it transforms what would otherwise be an unworkable mass of variables into clear, actionable insights.

What do you think? If you were analysing crop performance data across 30 farms with 20 measured variables each, how would you decide which principal components are worth retaining – and how would you explain what those components mean to a farmer with no statistics background? Could the limitations of PCA – particularly its inability to handle categorical variables – pose a practical challenge in the kind of agribusiness data you work with?

References

1. https://www.sciencedirect.com/topics/agricultural-and-biological-sciences/principal-component-analysis
2. https://www.allsubjectjournal.com/assets/archives/2020/vol7issue1/6-12-67-979.pdf
3. https://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/112-pca-principal-component-analysis-essentials/
4. https://builtin.com/data-science/step-step-explanation-principal-component-analysis
5. https://medium.com/@sahin.samia/principal-component-analysis-pca-made-easy-a-complete-hands-on-guide-e26a3680c0bc
6. https://bioinformaticamente.com/2025/03/11/exploring-principal-component-analysis-pca/
7. https://www.datacamp.com/tutorial/pca-analysis-r
8. https://en.wikipedia.org/wiki/Scree_plot
9. https://support.minitab.com/en-us/minitab/help-and-how-to/statistical-modeling/multivariate/how-to/principal-components/interpret-the-results/all-statistics-and-graphs/
10. https://onlinelibrary.wiley.com/doi/full/10.1002/glr2.70003
11. https://arccjournals.com/journal/legume-research-an-international-journal/LR-5038
12. https://www.numberanalytics.com/blog/5-pca-techniques-revolutionizing-modern-agriculture
13. https://www.nature.com/articles/s41538-025-00394-y
14. https://link.springer.com/article/10.1007/s12571-023-01392-2
15. https://www.doriane.com/blog/pca-statistics-agronomy
16. https://link.springer.com/article/10.1007/s13253-019-00355-5