A Hand-Computed PCA Example for Cells

From a Centered Matrix to Cell Coordinates

Principal component analysis (PCA) turns a matrix of measurements into new axes that summarize directions of variation. In single-cell RNA sequencing, a row can represent a cell and a column a gene. Once the matrix has been centered as required for PCA, the calculation has a concrete sequence: form a covariance matrix, find its principal directions, and project each cell onto those directions. This tutorial carries out that sequence for a small hypothetical example.

Assume the entries below are quantitative expression values already on a suitable common scale for this illustration. They are invented solely to make the arithmetic transparent; they are not real sequencing data. The introductory material on centering cells and genes explains the centering convention. Here we use gene-wise centering across cells and focus on the resulting PCA calculation.

The eigenvectors are chosen orthonormally, so each has length one and different component directions are perpendicular. Collecting the first two directions as columns of \(Q_2\), the score matrix is \(Z_2=XQ_2\). A score is a coordinate in the new axis system; it is not a raw expression measurement for any one gene.

Worked Calculation: Centering and Covariance

Finding the Principal Directions

For a symmetric covariance matrix, the principal directions are its unit eigenvectors. The eigenvalues identify the variance along their corresponding directions. In this example, the eigenvalues can be found from the characteristic equation:

Solving gives \(\lambda_1=16/3\) and \(\lambda_2=4/3\). For the first eigenvalue, the two rows of \(S-\lambda_1 I\) are proportional to \((-2,2)\), so an eigenvector is proportional to \((1,1)^\mathsf{T}\). For the second, an eigenvector is proportional to \((1,-1)^\mathsf{T}\). Normalizing both gives

These directions are perpendicular: \(q_1^\mathsf{T}q_2=(1-1)/2=0\), and each has squared length one. A direct check confirms the eigenvector calculations:

The order is by decreasing eigenvalue, so \(q_1\) is the first principal direction and \(q_2\) is the second. Reversing the sign of either eigenvector would describe the same axis and produce scores with the opposite sign; it would not change the PCA geometry.

Projecting Cells Onto the Two Axes

A cell’s score on an axis is its centered expression vector dotted with that axis’s unit direction. For the matrix above, \(Q_2\) has columns \(q_1\) and \(q_2\), so \(Z_2=XQ_2\). The two score coordinates for cell A, whose row is \((2,2)\), are

The same dot products for all four cells give the following coordinates. The decimal values are rounded to three significant figures; the exact forms show the underlying arithmetic.

Why the Eigenvalues Match Score Variances

The connection between the covariance calculation and the cell coordinates can be checked directly. For a unit direction \(q\), the vector of cell scores is \(Xq\). Its sample variance is

If \(q\) is an eigenvector with eigenvalue \(\lambda\), then \(q^\mathsf{T}Sq=q^\mathsf{T}(\lambda q)=\lambda\), because \(q^\mathsf{T}q=1\). Therefore the sample variance of the scores on that axis is \(\lambda\). This verifies, for the example, that the first-axis score variance is \(16/3\) and the second-axis score variance is \(4/3\). This calculation explains what the eigenvalues quantify; deciding how many components to retain is a separate question.

Here there are two genes and two orthonormal component directions, so retaining both axes reconstructs every centered row exactly. For cell B, its coordinates are \((0,-\sqrt{2})\); converting these coordinates back to the gene axes gives

which is the original centered row for B. The same identity holds for each cell because the two directions form an orthonormal basis of the two-gene coordinate space. If only the first direction were retained, this exact reconstruction would generally not hold; discarded directions contribute reconstruction error.

Hand Calculation Checks and Alternatives

A useful way to audit a small PCA calculation is to check each stage before interpreting the coordinates. The checks below are algebraic; PCA as a descriptive calculation does not require the cell values to be normally distributed. If the goal is to generalize conclusions to a population of cells, the sampling design and dependence among cells need separate consideration.

A common mistake is to interpret a score of zero on one component as evidence that the cell has no expression for the genes involved. That is incorrect: zero means the cell’s centered row has zero projection on that direction. For example, cell B has a first-axis score of zero because its centered values \((-1,1)\) cancel when projected onto the equal-weight direction; its expression values themselves are not zero.

For a small matrix, eigendecomposition of the covariance matrix makes the directions and their eigenvalues explicit. An equivalent route is the singular value decomposition \(X=U\Sigma V^\mathsf{T}\): the columns of \(V\) are the gene-space directions, and the cell scores are \(U\Sigma=XV\). The eigenvalues of \(S\) equal the squared singular values divided by \(n-1\). This equivalence can be useful computationally, but it does not change the meaning of the scores. Forming a covariance matrix is convenient by hand; for large or numerically challenging matrices, directly computing singular vectors can avoid some numerical issues associated with forming \(X^\mathsf{T}X\).

These calculations depend on using the matrix intended for PCA: the rows must be cells, the columns must be the chosen features, and centering must be applied consistently. A mismatch in row and column orientation can produce coordinates for genes instead of cells. Also, dividing by \(n\) rather than \(n-1\) rescales the covariance eigenvalues but leaves the eigenvectors and projected scores unchanged. Mixing those conventions can therefore produce plausible-looking directions alongside inconsistent reported variance values.