Background Pedigrees continue to be extremely important in agriculture and conservation genetics, with the pedigrees of modern breeding programmes easily comprising millions of records. This size can make visualising the structure of such pedigrees challenging. Being graphs, pedigrees can be represented as matrices, including, most commonly, the additive (numerator) relationship matrix, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12711_2025_994_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> </math></EquationSource> </InlineEquation>, and its inverse. With these matrices, the structure of pedigrees can then, in principle, be visualised via principal component analysis (PCA). However, the naive PCA of matrices for large pedigrees is challenging due to computational and memory constraints. Furthermore, computing a few leading principal components is usually sufficient for visualising the structure of a pedigree. Results We present the open-access R package <Emphasis FontCategory="NonProportional">randPedPCA</Emphasis> for rapid pedigree PCA using sparse matrices. Our rapid pedigree PCA builds on the fact that matrix-vector multiplications with the additive relationship matrix can be carried out implicitly using the extremely sparse inverse relationship factor, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12711_2025_994_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{L}^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="bold-italic">L</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, which can be directly obtained from a given pedigree. We implemented two methods. Randomised singular value decomposition tends to be faster when very few principal components are requested, and Eigen decomposition via the <Emphasis FontCategory="NonProportional">RSpectra</Emphasis> library tends to be faster when more principal components are of interest. On simulated data, our package delivers a speed-up greater than 10,000 times compared to naive PCA. It further enables analyses that are impossible with naive PCA. When only two principal components are desired, the randomised PCA method can half the running time required compared to <Emphasis FontCategory="NonProportional">RSpectra</Emphasis>, which we demonstrate by analysing the pedigree of the UK Kennel Club registered Labrador Retriever population of almost 1.5 million individuals. Conclusions The leading principal components of pedigree matrices can be efficiently obtained using randomised singular value decomposition and other methods. Scatter plots of these scores allow for intuitive visualisation of large pedigrees. For large pedigrees, this is considerably faster than rendering plots of a pedigree graph.

randPedPCA: rapid approximation of principal components from large pedigrees

期刊论文
出版模式：

开放获取
发表日期：

2025年8月28日

查看全文

Hanbin Lee,
Rosalind Françoise Craddock,
Gregor Gorjanc,
Hannes Becher

摘要

Background

Pedigrees continue to be extremely important in agriculture and conservation genetics, with the pedigrees of modern breeding programmes easily comprising millions of records. This size can make visualising the structure of such pedigrees challenging. Being graphs, pedigrees can be represented as matrices, including, most commonly, the additive (numerator) relationship matrix, $\varvec{A}$ $A$ , and its inverse. With these matrices, the structure of pedigrees can then, in principle, be visualised via principal component analysis (PCA). However, the naive PCA of matrices for large pedigrees is challenging due to computational and memory constraints. Furthermore, computing a few leading principal components is usually sufficient for visualising the structure of a pedigree.

Results

We present the open-access R package randPedPCA for rapid pedigree PCA using sparse matrices. Our rapid pedigree PCA builds on the fact that matrix-vector multiplications with the additive relationship matrix can be carried out implicitly using the extremely sparse inverse relationship factor, $\varvec{L}^{-1}$ $L^{- 1}$ , which can be directly obtained from a given pedigree. We implemented two methods. Randomised singular value decomposition tends to be faster when very few principal components are requested, and Eigen decomposition via the RSpectra library tends to be faster when more principal components are of interest. On simulated data, our package delivers a speed-up greater than 10,000 times compared to naive PCA. It further enables analyses that are impossible with naive PCA. When only two principal components are desired, the randomised PCA method can half the running time required compared to RSpectra, which we demonstrate by analysing the pedigree of the UK Kennel Club registered Labrador Retriever population of almost 1.5 million individuals.

Conclusions

The leading principal components of pedigree matrices can be efficiently obtained using randomised singular value decomposition and other methods. Scatter plots of these scores allow for intuitive visualisation of large pedigrees. For large pedigrees, this is considerably faster than rendering plots of a pedigree graph.