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This problem is an instance of root cause discovery. Specifically, we aim to identify the intervened variable in one interventional sample using a set of observational samples as reference. We consider a linear structural equation model where the causal ordering is unknown. We begin by examining a simple method that uses squared z-scores and characterize the conditions under which this method succeeds and fails, showing it generally cannot identify the root cause. We then prove, without additional assumptions, that the root cause is identifiable even if the causal ordering is not. Two key ingredients of this identifiability result are the use of permutations and the Cholesky decomposition, which allow us to exploit an invariant property across different permutations to discover the root cause. Furthermore, we characterize permutations that yield the correct root cause and, based on this, propose a valid method for root cause discovery. We also adapt this approach to high-dimensional settings. Finally, we evaluate our methods through simulations and apply the high-dimensional method to discover disease-causing genes in the gene expression dataset that motivates this work.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkaf066","type":"journal-article","created":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T12:13:58Z","timestamp":1758716038000},"page":"779-798","source":"Crossref","is-referenced-by-count":1,"title":["Root cause discovery via permutations and Cholesky decomposition"],"prefix":"10.1093","volume":"88","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4054-4104","authenticated-orcid":false,"given":"Jinzhou","family":"Li","sequence":"first","affiliation":[{"name":"Department of Statistics and Data Science, National University of Singapore , Singapore ,","place":["Singapore"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8342-2361","authenticated-orcid":false,"given":"Benjamin 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