{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:16:07Z","timestamp":1760148967807,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T00:00:00Z","timestamp":1687305600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We present a novel nonparametric adaptive partitioning and stitching (NAPS) algorithm to estimate a probability density function (PDF) of a single variable. Sampled data is partitioned into blocks using a branching tree algorithm that minimizes deviations from a uniform density within blocks of various sample sizes arranged in a staggered format. The block sizes are constructed to balance the load in parallel computing as the PDF for each block is independently estimated using the nonparametric maximum entropy method (NMEM) previously developed for automated high throughput analysis. Once all block PDFs are calculated, they are stitched together to provide a smooth estimate throughout the sample range. Each stitch is an averaging process over weight factors based on the estimated cumulative distribution function (CDF) and a complementary CDF that characterize how data from flanking blocks overlap. Benchmarks on synthetic data show that our PDF estimates are fast and accurate for sample sizes ranging from 29 to 227, across a diverse set of distributions that account for single and multi-modal distributions with heavy tails or singularities. We also generate estimates by replacing NMEM with kernel density estimation (KDE) within blocks. Our results indicate that NAPS(NMEM) is the best-performing method overall, while NAPS(KDE) improves estimates near boundaries compared to standard KDE.<\/jats:p>","DOI":"10.3390\/a16070310","type":"journal-article","created":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T03:18:38Z","timestamp":1687317518000},"page":"310","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Probability Density Estimation through Nonparametric Adaptive Partitioning and Stitching"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6224-800X","authenticated-orcid":false,"given":"Zach D.","family":"Merino","sequence":"first","affiliation":[{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada"},{"name":"Department of Physics and Optical Science, University of North Carolina Charlotte, Charlotte, NC 28213, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7953-1044","authenticated-orcid":false,"given":"Jenny","family":"Farmer","sequence":"additional","affiliation":[{"name":"Department of Bioinformatics and Genomics, University of North Carolina Charlotte, Charlotte, NC 28213, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7711-1639","authenticated-orcid":false,"given":"Donald J.","family":"Jacobs","sequence":"additional","affiliation":[{"name":"Department of Physics and Optical Science, University of North Carolina Charlotte, Charlotte, NC 28213, USA"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1214\/aoms\/1177728190","article-title":"Remarks on Some Nonparametric Estimates of a Density Function","volume":"27","author":"Rosenblatt","year":"1956","journal-title":"Ann. 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