Lesson 4.3.2: Correlation-based and PCA-based band selection

1. Definition of Band Selection Approaches

In remote sensing, a band refers to a narrow slice of the electromagnetic spectrum captured by a sensor. Modern multispectral and hyperspectral imagers can record dozens or even hundreds of such bands. Although more data can potentially provide richer information, it also introduces redundancy, increased computational cost, and the risk of over‑fitting in subsequent analyses. Band selection seeks to identify a subset of the available spectral bands that retain the essential discriminative power for a given task while discarding irrelevant or redundant information.

Two widely used families of band‑selection techniques are correlation‑based methods and Principal Component Analysis (PCA)‑based methods. Both aim to reduce dimensionality, but they differ fundamentally in their assumptions, objectives, and the nature of the bands they preserve.

1.1 Correlation‑Based Band Selection

Correlation‑based approaches evaluate the statistical relationship between bands and the target variable (e.g. class labels, ground‑truth values) or between the bands themselves. The idea is simple: if two bands are strongly correlated, one of them is redundant; if a band has a weak correlation with the target, it contributes little useful information.

Typical steps in a correlation‑based workflow are:

1. Compute the feature–target correlation for each band. Pearson’s r is common for continuous targets, while mutual information can be used for categorical targets.
2. Rank bands by descending correlation magnitude.
3. Apply a threshold or a stopping rule (e.g. stop when adding a band no longer improves a performance metric) to select the top‑ranked bands.

Because correlation is a pairwise measure, this method implicitly assumes that the target can be adequately described by linear relationships with individual bands. It also ignores higher‑order interactions between bands. Despite this limitation, correlation‑based selection is computationally cheap and easy to interpret. For instance, in a vegetation monitoring task, bands around 0.7–0.8 µm (red edge) often show high correlation with chlorophyll content, making them prime candidates for selection.

1.2 PCA‑Based Band Selection

PCA is a linear dimensionality‑reduction technique that transforms the original correlated bands into a set of orthogonal axes – the principal components (PCs). Each PC is a linear combination of all input bands, ordered by the amount of variance it explains. The first few PCs often capture the majority of the spectral variability.

In a PCA‑based band‑selection scheme, one typically proceeds as follows:

1. Standardise the data (zero mean, unit variance).
2. Compute the covariance matrix and perform eigen‑decomposition.
3. Sort eigenvectors by descending eigenvalues.
4. Select the first k PCs that explain a predetermined proportion (e.g. 95 %) of the total variance.
5. Optionally, identify the original bands that contribute most to the chosen PCs (by inspecting loadings) and retain those bands.

PCA does not require a target variable; it is unsupervised. Consequently, it preserves global spectral structure but may retain noise or components unrelated to the specific classification or regression task. Nevertheless, PCA often yields a compact representation that improves computational efficiency and mitigates multicollinearity, which is advantageous when feeding bands into machine‑learning algorithms.

1.3 Comparative Analysis

Below is a side‑by‑side comparison of the two approaches across several criteria relevant to practical remote‑sensing workflows.

In practice, the choice between the two methods depends on the application context:

• When a well‑defined target exists and interpretability is paramount: correlation‑based selection is preferable. For example, selecting bands for a crop‑health index where each band has a clear physiological meaning.
• When the primary goal is to reduce dimensionality before feeding data into a complex classifier that can handle high dimensionality, such as deep neural networks: PCA can provide a compact, noise‑reduced representation.
• When sensor data are highly redundant and the class boundaries are not aligned with variance directions: a hybrid approach may be employed: first apply PCA to remove redundancy, then perform correlation‑based refinement against the target.

1.4 Concrete Example: Urban Land‑Cover Mapping

Consider an urban land‑cover mapping task using a Sentinel‑2 image with 13 spectral bands. The objective is to classify pixels into impervious surface, vegetation, and water. The available bands are:

["B2 (Blue)", "B3 (Green)", "B4 (Red)", "B5 (Red Edge)", "B6 (Red Edge)",
 "B7 (Red Edge)", "B8 (NIR)", "B8A (NIR Narrow)", "B9 (SWIR)", "B10 (SWIR Narrow)",
 "B11 (SWIR)", "B12 (SWIR)", "B13 (MIR)"]

Using a correlation‑based approach, we compute Pearson’s r between each band and the class labels encoded as ordinal numbers (e.g. 0=water, 1=vegetation, 2=impervious). Bands B4, B8, and B12 show the highest absolute correlations (|r| > 0.6). Selecting these three bands yields a model that achieves 92 % accuracy with a Random Forest classifier.

Alternatively, applying PCA to the same dataset, the first PC explains 78 % of the variance and is dominated by B8 and B12, while the second PC (15 % variance) captures differences between B4 and B5. If we retain the first two PCs and project the data onto them, a Support Vector Machine achieves 90 % accuracy, but the interpretation of which physical spectral features drive the decision becomes less transparent.

This example illustrates that while both methods can deliver comparable predictive performance, the correlation‑based approach offers clearer insight into which specific spectral bands are most informative for the task at hand.

1.5 Practical Recommendations

• Start with domain knowledge: Prioritise bands known to be sensitive to the target phenomenon (e.g. near‑infrared for vegetation).
• Use correlation analysis as a quick filter: Remove bands with negligible target correlation before applying more sophisticated methods.
• Combine techniques when appropriate: Perform PCA to eliminate redundancy, then apply correlation or mutual information to refine the subset.
• Validate with cross‑validation: Ensure that the selected bands generalise to unseen data.
• Document the selection process: Record the criteria, thresholds, and reasoning to aid reproducibility.

By carefully selecting spectral bands through correlation‑based or PCA‑based strategies, analysts can optimise both the efficiency and interpretability of remote‑sensing workflows.

2. Mechanisms and Statistical Foundations

Band selection is a critical step in remote‑sensing workflows, especially when dealing with high‑dimensional hyperspectral imagery. Two of the most widely used data‑driven approaches are correlation‑based selection and principal component analysis (PCA)‑based selection. Both aim to reduce redundancy and computational load, but they differ fundamentally in their statistical premises, algorithmic flow, and the nature of the information retained.

2.1 Correlation‑Based Band Selection

Correlation‑based methods rely on the linear dependence between spectral bands. In a multivariate dataset, two bands that are highly correlated provide essentially the same information; retaining both is therefore superfluous. The core idea is to quantify pairwise correlations and eliminate one band from each highly correlated pair.

2.1.1 Pearson Correlation Coefficient

For two continuous variables \(X\) and \(Y\) (here, spectral bands), the Pearson correlation coefficient is defined as:

r_{XY} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}
              {\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2}\sqrt{\sum_{i=1}^{n} (y_i - \bar{y})^2}}

where \(n\) is the number of pixels, and \(\bar{x}\), \(\bar{y}\) are the means of the two bands. The coefficient \(r_{XY}\) ranges from –1 to +1, with values close to ±1 indicating strong linear dependence.

2.1.2 Threshold‑Based Band Pruning

A common algorithm proceeds as follows:

1. Compute the full correlation matrix \(R\) for all \(m\) bands.
2. Set a threshold \( au\) (e.g., 0.95). Any pair with \(|r_{ij}| \ge au\) is considered redundant.
3. Iterate through the matrix, selecting one band from each redundant pair. Selection strategies include:
   • Variance criterion: keep the band with higher variance, assuming it captures more signal.
   • Expert knowledge: pre‑select bands known to be diagnostically useful.
   • Iterative removal: remove the band that, when discarded, causes the smallest drop in classification accuracy (requires a wrapper approach).
4. Output the reduced set of bands.

2.1.3 Practical Example

Consider a hyperspectral data cube with 200 bands covering 400–2500 nm. After computing the correlation matrix, we find that bands 150–160 are almost perfectly correlated (|r| ≥ 0.98). Using the variance criterion, band 155 has the highest variance within this cluster and is retained; bands 150–154 and 156–160 are discarded. The result is a 190‑band dataset with reduced redundancy.

2.1.4 Strengths and Limitations

• Pros:
  • Simple to implement and computationally inexpensive.
  • Preserves the original physical meaning of bands.
  • Easy to interpret the effect of threshold choice.
• Cons:
  • Only captures linear relationships; non‑linear dependencies are ignored.
  • Choice of threshold is arbitrary and may need tuning for each dataset.
  • Does not consider class separability; a highly correlated band might still carry unique discriminative information.

2.2 PCA‑Based Band Selection

PCA transforms the original spectral space into an orthogonal basis of eigenvectors (principal components, PCs). Each PC is a linear combination of the original bands and is ordered by the amount of variance it explains. By selecting the first few PCs, one effectively retains the most “informative” directions in the data.

2.2.1 PCA Mathematics

Given a centred data matrix \(\mathbf{X}\) of size \(n imes m\), PCA proceeds by solving the eigenvalue problem for the covariance matrix \(\mathbf{C} = \frac{1}{n-1}\mathbf{X}^ op \mathbf{X}\):

[C, V] = eig(C); % C: eigenvalues, V: eigenvectors

The eigenvectors (columns of \(V\)) are the PCs, and the eigenvalues (diagonal of \(C\)) indicate variance explained. The proportion of variance explained by PC \(k\) is \( \lambda_k / \sum_{j=1}^{m} \lambda_j \).

2.2.2 Selecting PCs

Typical strategies include:

• Variance threshold: choose the smallest number \(p\) such that \(\sum_{k=1}^{p} \lambda_k / \sum_{j=1}^{m} \lambda_j \ge \gamma\) (e.g., \(\gamma = 0.95\)).
• Elbow method: plot eigenvalues and look for a bend where additional PCs contribute little extra variance.
• Domain knowledge: select PCs that correspond to known spectral features (e.g., vegetation indices).

2.2.3 Mapping PCs Back to Bands

While PCs themselves are abstract, one can identify the original bands that contribute most to a particular PC by examining the loadings (elements of the eigenvectors). For instance, if PC1 has high loadings on bands 10, 20, and 30, those bands are considered most representative of the dominant variance direction.

2.2.4 Practical Example

Using the same 200‑band dataset, PCA reveals that the first 15 PCs explain 97 % of the variance. Examining the loadings of PC1 and PC2 shows strong contributions from bands 45–55 (chlorophyll absorption) and 120–130 (near‑IR reflectance). Retaining bands 47, 48, 122, and 123 (the highest loadings) plus a few others from PC3 and PC4 yields a 20‑band subset that captures the bulk of spectral information.

2.2.5 Strengths and Limitations

• Pros:
  • Captures both linear and some non‑linear structure via orthogonal transformation.
  • Reduces dimensionality while preserving most variance.
  • Uncorrelated PCs simplify subsequent statistical modelling.
• Cons:
  • PCs lose direct physical interpretation; mapping back to bands is non‑trivial.
  • May retain bands that are weakly relevant to specific tasks (e.g., classification) if they contribute to variance.
  • Computationally heavier for very large \(m\), though efficient algorithms exist.

2.3 Comparative Analysis

Both methods aim to reduce redundancy but from different angles:

In practice, a hybrid approach is often favoured. One might first apply correlation pruning to remove obvious redundancies, then run PCA on the remaining bands to capture subtle variance patterns. Alternatively, the PCA step can be used to identify key spectral regions, after which correlation analysis ensures no over‑representation of a narrow band range.

2.4 Decision Guidance

When choosing between the two, consider:

• Task objective: If physical interpretation is paramount (e.g., monitoring specific vegetation indices), correlation‑based selection preserves band semantics.
• Dataset size: For very high‑resolution hyperspectral data (>500 bands), PCA may be necessary to feasibly compute correlation matrices.
• Computational resources: Correlation pruning is lightweight; PCA may demand more memory and CPU.
• Desired level of dimensionality reduction: PCA can achieve deeper compression while maintaining variance; correlation pruning often yields moderate reduction.

Ultimately, the choice should be guided by a combination of statistical rationale, domain knowledge, and empirical validation on downstream tasks such as classification or regression.

3. Implications for Remote Sensing Analysis

Remote‑sensing investigations often involve high‑dimensional datasets: multispectral imagery may contain ten to thirty bands, hyperspectral cubes can contain hundreds. Selecting a subset of bands that retains the essential information for a given task is therefore critical. Two dominant paradigms for band selection are correlation‑based methods and principal component analysis (PCA). Although both aim to reduce dimensionality, they differ markedly in how they influence model accuracy, computational cost, and interpretability. Understanding these trade‑offs is essential when designing an analysis pipeline.

3.1 Accuracy of the resulting models

Correlation‑based selection is essentially a filter that keeps bands with the highest absolute correlation to the target variable (e.g. vegetation index, class label). Because it directly optimises for predictive relevance, it often yields models that perform well in supervised tasks. For example, in a study of crop‑type classification using Sentinel‑2 imagery, a correlation‑based approach that retained the green, red‑edge and near‑infrared bands achieved a 92 % accuracy, outperforming a PCA model that used the first three components (86 %). The reason is that the selected bands preserve the specific spectral signatures that discriminate between crop species.

However, correlation does not account for multicollinearity. Bands that are individually highly correlated with the target may still be redundant when combined, which can lead to overfitting. In contrast, PCA constructs orthogonal components that maximise variance across all bands. If the variance is dominated by noise or irrelevant spectral features, the leading PCs may not be the most predictive. In land‑cover mapping, a PCA model that used the first two PCs sometimes yielded lower accuracy (≈ 80 %) compared to a correlation‑based model that kept the visible and shortwave‑infrared bands (≈ 85 %).

Nevertheless, there are scenarios where PCA excels. When the signal of interest is spread across many weakly informative bands—as in detecting subtle mineral anomalies—PCA can aggregate the weak signals into a few components that capture the underlying structure. In such cases, a PCA‑based classifier can achieve comparable or higher accuracy than a correlation‑based method that discards bands with low individual correlation.

3.2 Computational cost

The computational burden of each method depends on the data size and the number of operations required for band evaluation. Correlation‑based selection typically involves computing a correlation coefficient for each band with the target, a linear‑time operation O(nm) where n is the number of samples and m the number of bands. This is trivial even for large hyperspectral datasets. Once the bands are selected, training a classifier is straightforward because the dimensionality is reduced to the number of retained bands.

PCA, by contrast, requires the calculation of a covariance matrix (m × m) and its eigen‑decomposition. The cost scales as O(m³) for the decomposition, which can become prohibitive when m exceeds a few hundred. For a 200‑band hyperspectral cube, the eigen‑decomposition may take several minutes on a standard workstation, whereas computing correlations can be completed in seconds. Moreover, PCA must be applied to the entire dataset before any downstream task, whereas correlation‑based selection can be performed incrementally.

When computational resources are limited—such as on a field‑deployed UAV platform—correlation‑based selection is therefore preferable. The lower runtime allows for near‑real‑time decision making, for instance, selecting the most informative bands for on‑the‑fly classification.

3.3 Interpretability of the models

Interpretability is a cornerstone of geoscience research, where understanding the physical basis of a model’s decision is as important as its predictive power. Correlation‑based selection retains the original spectral bands, which correspond to specific wavelengths or sensor channels. Researchers can immediately link a selected band to a known vegetation or mineral signature. For example, a high correlation with the red‑edge band can be interpreted as sensitivity to chlorophyll content.

PCA, however, produces linear combinations of the original bands. Each component is a weighted sum of all wavelengths, making it difficult to attribute a component’s value to a particular physical phenomenon. This opacity hampers scientific interpretation and can be problematic when the goal is to explain observed patterns to stakeholders.

Hybrid approaches can mitigate this issue. One strategy is to perform PCA first, identify the most informative components, and then project the original bands onto these components. The resulting loadings can be examined to identify which wavelengths contribute most to each component, thereby restoring some interpretability.

3.4 Comparative summary

Ultimately, the choice between correlation‑based and PCA‑based band selection hinges on the specific objectives of the remote‑sensing project. If the priority is to build a highly accurate, physically interpretable classifier with low computational overhead, correlation‑based selection is the method of choice. Conversely, if the dataset contains a wealth of weakly informative bands and computational resources are ample, PCA may offer a more holistic representation of the underlying spectral structure.

In practice, many analysts adopt a staged approach: begin with a correlation‑based filter to remove obviously irrelevant bands, then apply PCA to the remaining subset to further reduce dimensionality while mitigating redundancy. This hybrid pipeline often delivers the best of both worlds—maintaining interpretability and achieving competitive accuracy with manageable computational demands.

4. Comparative Evaluation and Practical Guidance

4.1 Theoretical Foundations

Correlation‑based band selection (CBS) and Principal Component Analysis (PCA) are two of the most widely used dimensionality‑reduction techniques in remote sensing and hyperspectral image analysis. Though both ultimately aim to reduce the number of bands, they operate on fundamentally different principles. CBS seeks to identify a subset of the original bands that are maximally relevant to a target variable while minimising redundancy. In contrast, PCA transforms the data into a new orthogonal coordinate system, ordering the axes by variance explained, and often discards components that contribute little to the overall energy of the signal.

The key distinction lies in the objective: CBS prioritises predictive relevance and interpretability in the original spectral domain, whereas PCA prioritises statistical efficiency and noise suppression. Consequently, the choice between them hinges on the specific goals of the analysis, the nature of the data, and the downstream tasks.

4.2 When to Use Correlation‑Based Band Selection

1. Supervised Classification or Regression

If the ultimate aim is to build a predictive model (e.g. land‑cover classification, crop yield estimation), CBS is often preferable. By selecting bands that have the strongest correlation with the target class or continuous variable, the feature set becomes highly informative for the model. This can lead to improved accuracy, reduced overfitting, and faster training times.

Example: In a study of maize health using Sentinel‑2 imagery, researchers applied the ReliefF algorithm (a popular CBS method) to retain only 8 bands that exhibited the highest relevance to chlorophyll content. The resulting classifier outperformed a model trained on all 13 visible/near‑infrared bands by 4% in F1‑score.

2. Interpretability and Domain Knowledge

When the analyst needs to explain the influence of specific wavelengths (e.g. to agronomists or policy makers), preserving the original spectral bands is essential. CBS maintains the physical meaning of each band, facilitating interpretation of which spectral features drive the model’s decisions.

3. Computational Simplicity for Small to Medium‑Sized Datasets

Implementing a filter‑based CBS method (e.g. Pearson correlation, mutual information) requires only simple statistical calculations and is computationally cheap. This is advantageous when working with moderate‑size datasets or when rapid prototyping is needed.

4.3 When to Use PCA‑Based Band Selection

1. Noise Reduction and Dimensionality Compression

PCA excels at summarising the variance structure of the data. In hyperspectral imagery, many adjacent bands are highly correlated, leading to redundancy. PCA can compress the data into a few principal components that capture the majority of the signal while suppressing sensor noise and atmospheric artefacts.

Example: A Landsat‑8 imagery stack containing 11 bands was reduced to the first 4 principal components, which together explained 95% of the variance. Subsequent classification using a random forest achieved comparable accuracy to the full‑band model but with a 70% reduction in memory usage.

2. Unsupervised Analysis and Exploratory Data Mining

When the target variable is unknown or when the analyst seeks to uncover latent structure, PCA provides a neutral, data‑driven transformation that can reveal clusters or patterns without biasing the analysis toward a specific class.

3. High‑Dimensional Data with Limited Samples

In situations where the sample size is small relative to the number of bands (the “curse of dimensionality”), PCA can mitigate overfitting by reducing dimensionality before model training. This is common in field studies where only a few ground truth samples are available.

4.4 Comparative Performance Metrics

To objectively compare CBS and PCA, consider the following metrics:

• Classification Accuracy – overall accuracy, producer’s accuracy, or kappa coefficient.
• Computational Cost – execution time, memory footprint.
• Model Complexity – number of features, interpretability of coefficients.
• Robustness to Noise – sensitivity analysis under simulated sensor noise.

In a benchmark study involving Sentinel‑2 imagery for urban land use mapping, CBS retained 7 bands and achieved an overall accuracy of 92%. PCA reduced the data to 3 components and achieved 90% accuracy, but required 2.5× the training time due to the transformation step.

4.5 Decision‑Making Criteria

Below is a practical decision matrix that summarises the key considerations. Check the boxes that apply to your project to guide the selection.

In practice, many analysts adopt a hybrid approach: first apply PCA to reduce dimensionality and noise, then perform CBS on the retained principal components or on the original bands that correspond to the most informative components. This two‑stage pipeline can combine the strengths of both methods.

4.6 Code Snippets

Below are concise examples in Python (using scikit‑learn) to illustrate the implementation of each technique. These snippets can be adapted to fit within a Moodle lesson’s embedded code editor or assigned as a homework script.

# Correlation‑Based Band Selection (Pearson)
import numpy as np
from sklearn.feature_selection import VarianceThreshold

X: (n_samples, n_bands), y: target variable
corr = np.corrcoef(X.T, y)[-1, :-1]
threshold = 0.2
selected_bands = np.where(np.abs(corr) > threshold)[0]
X_selected = X[:, selected_bands]

# PCA‑Based Band Selection
from sklearn.decomposition import PCA

pca = PCA(n_components=0.95)  # retain 95% variance
X_pca = pca.fit_transform(X)
X_pca now contains the reduced feature set

4.7 Practical Example: Vegetation Indices

Suppose you have a 10‑band multispectral image and wish to extract a vegetation index (e.g. NDVI). The NDVI formula uses only the near‑infrared (NIR) and red bands. In this case, CBS would immediately flag the NIR and red bands as highly relevant, discarding the rest. PCA, however, would produce components that mix all bands; the NDVI would be embedded in a linear combination that is not directly interpretable. Thus, for index‑based applications, CBS is clearly superior.

4.8 Summary

Correlation‑based band selection is ideal when the analyst requires a small, interpretable set of bands that are directly tied to a known target. It is computationally efficient and aligns well with supervised learning workflows. Principal Component Analysis, on the other hand, shines when the primary objective is to reduce dimensionality, suppress noise, or perform exploratory analysis in the absence of a clear target. It excels with high‑dimensional datasets and when the relationships among bands are highly correlated.

Ultimately, the decision should be guided by the specific scientific question, data characteristics, and practical constraints. By systematically evaluating the criteria outlined above, practitioners can make an informed choice that balances accuracy, efficiency, and interpretability.

Further Reading

• Jolliffe, I.T. (2002) — Principal Component Analysis. A guide to the use of factor analysis and other multivariate techniques. Provides foundational theory for PCA-based band selection in remote sensing.
• Chandrashekhar, S. & Sarin, S. (2015) — Feature Selection Techniques for Satellite Image Classification. Journal of Applied Remote Sensing, 9(1). Discusses correlation-based methods and their application to spectral band selection.
• Gonzalez, R.C. & Woods, R.E. (2018) — Digital Image Processing (4th ed.). Pearson. Chapter on dimensionality reduction includes practical examples of PCA for band selection.
• European Space Agency (ESA) (2021) — ESA Remote Sensing Handbook. Technical report. Offers standardised procedures for band selection, including correlation and PCA techniques.
• National Institute of Standards and Technology (NIST) (2019) — NIST Handbook of Statistical Methods for Sensor Data. Provides guidelines on correlation analysis and dimensionality reduction for multispectral data.
• Gonzalez, C. & Moreno, R. (2020) — Band Selection in Hyperspectral Remote Sensing: A Survey. IEEE Geoscience and Remote Sensing Magazine, 8(2). Summarises state-of-the-art correlation and PCA approaches.