{"paper":{"title":"Sensitivity Analysis on the Sphere and a Spherical ANOVA Decomposition","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A function on the sphere decomposes into a sum of terms each depending on a coordinate subset and a parity vector.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Laura Weidensager","submitted_at":"2025-12-29T13:59:36Z","abstract_excerpt":"We establish sensitivity analysis on the sphere. We present formulas that allow us to decompose a function $f\\colon \\mathbb S^d\\rightarrow \\mathbb R$ into a sum of terms $f_{\\boldsymbol u,\\boldsymbol \\xi}$. The index $\\boldsymbol u$ is a subset of $\\{1,2,\\ldots,d+1\\}$, where each term $f_{\\boldsymbol u,\\boldsymbol \\xi}$ depends only on the variables with indices in $\\boldsymbol u$. In contrast to the classical analysis of variance (ANOVA) decomposition, we additionally use the decomposition of a function into functions with different parity, which adds the additional parameter $\\boldsymbol \\xi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We present formulas that allow us to decompose a function f : S^d → R into a sum of terms f_{u,ξ}. The index u is a subset of {1,2,…,d+1}, where each term f_{u,ξ} depends only on the variables with indices in u.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The natural geometry on the sphere naturally leads to the dependencies between the input variables, and suitable orthogonal basis functions exist that permit the decomposition and approximation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A parity-augmented ANOVA decomposition is established for functions on the sphere using orthogonal bases to capture geometry-induced variable dependencies.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A function on the sphere decomposes into a sum of terms each depending on a coordinate subset and a parity vector.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"43bf55e48963c9f5135caa7453382b4ca68d022590202f352dceb96d43294e87"},"source":{"id":"2512.23476","kind":"arxiv","version":2},"verdict":{"id":"58e058cc-fd5f-4475-bdb9-6fc4460718c0","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T19:17:21.819647Z","strongest_claim":"We present formulas that allow us to decompose a function f : S^d → R into a sum of terms f_{u,ξ}. The index u is a subset of {1,2,…,d+1}, where each term f_{u,ξ} depends only on the variables with indices in u.","one_line_summary":"A parity-augmented ANOVA decomposition is established for functions on the sphere using orthogonal bases to capture geometry-induced variable dependencies.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The natural geometry on the sphere naturally leads to the dependencies between the input variables, and suitable orthogonal basis functions exist that permit the decomposition and approximation.","pith_extraction_headline":"A function on the sphere decomposes into a sum of terms each depending on a coordinate subset and a parity vector."},"references":{"count":36,"sample":[{"doi":"","year":1964,"title":"M. Abramowitz and I. A. Stegun.Handbook of Mathematical Functions, With Formulas, Graphs, and Mathe- matical Tables. National Bureau of Standards, 1964","work_id":"0e36c975-0779-4a03-a3f0-cb4baa549a58","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"H.-J. Bungartz and M. Griebel. Sparse grids.Acta Numer., 13:147–269, 2004.doi:10.1017/ S0962492904000182","work_id":"c35d079f-c756-4fdf-a607-cc3f1a3b73aa","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.21314/jcf.1997.005","year":1997,"title":"R. Caflisch, W. Morokoff, and A. Owen. Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension.J. Comput. Finance, 1(1):27–46, 1997.doi:10.21314/jcf.1997.005","work_id":"d631d6b6-478d-4659-84c1-c50bf364bc44","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1137/s0895479803433295","year":2005,"title":"M. Chu, N. Del Buono, L. Lopez, and T. Politi. On the low-rank approximation of data on the unit sphere.SIAM Journal on Matrix Analysis and Applications, 27(1):46–60, Jan. 2005.doi:10.1137/s0895479803","work_id":"013c7184-367e-4404-9a51-0e94fd2be0fa","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.48550/arxiv.2101.05487","year":2021,"title":"S. da Veiga. 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