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 <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · MonotoneSplines.jl</title><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>MonotoneSplines.jl</a></span></div><form class="docs-search" action="search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Home</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="examples/monofit/">Monotone Fitting</a></li><li><a class="tocitem" href="examples/conditions/">Conditions</a></li><li><a class="tocitem" href="examples/ph/">Application: Polarization-hole</a></li><li><a class="tocitem" href="examples/monofit_mlp/">MLP Generator (fitting curve)</a></li><li><a class="tocitem" href="examples/monoci_mlp/">MLP Generator (confidence band)</a></li><li><a class="tocitem" href="examples/diff_sort/">Differentiable Sort</a></li></ul></li><li><a class="tocitem" href="api/">API</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/szcf-weiya/MonotoneSplines.jl/blob/master/docs/src/index.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="MonotoneSplines.jl-Documentation"><a class="docs-heading-anchor" href="#MonotoneSplines.jl-Documentation">MonotoneSplines.jl Documentation</a><a id="MonotoneSplines.jl-Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#MonotoneSplines.jl-Documentation" title="Permalink"></a></h1><blockquote><p>Wang, L., Fan, X., Li, H., &amp; Liu, J. S. (2023). Monotone Cubic B-Splines (arXiv:2307.01748). arXiv. https://doi.org/10.48550/arXiv.2307.01748</p></blockquote><p><em>MonotoneSplines.jl</em> is a Julia package for monotone splines, which impose a monotonicity constraint on the smoothing splines. </p><p class="math-container">\[\underset{\color{red}{f\textbf{ is monotonic}}}{\arg\min} \sum_{i=1}^n\left\{y_i-f(x_i)\right\}^2 + \lambda \int \left\{f&#39;&#39;(t)\right\}^2dt\,,\]</p><p>where <span>$f$</span> is formed with B-spline basis <span>$f(x) = \sum_{j=1}^J\gamma_j B_j(x)$</span>. A sufficient condition for <span>$f$</span> to be monotonic is <span>$\gamma_1,\ldots,\gamma_J$</span> is monotonic. With matrix notation <span>${\mathbf y} = [y_1,\ldots, y_n], {\mathbf B}_{ij} = B_j(x_i), {\boldsymbol\Omega}_{ij} = \int B_i&#39;&#39;(s)B_j&#39;&#39;(s)ds$</span>, the problem can be rewritten as</p><p class="math-container">\[\begin{aligned}
 \underset{\gamma}{\arg\min} &amp; \Vert {\mathbf y} - {\mathbf B} \gamma\Vert_2^2 + \lambda \gamma^T\boldsymbol\Omega\gamma\\
 \text{subject to } &amp; \alpha \gamma_1 \le \alpha \gamma_2\le \cdots \le \alpha\gamma_J\,,
-\end{aligned}\]</p><p>where <span>$\alpha=1$</span> implies non-decreasing and <span>$\alpha=-1$</span> indicates non-increasing.</p><p>The package provides two algorithms (frameworks) for fitting the monotone splines.</p><ul><li>Convert the problem into a classical convex second-order cone optimization problem. There are many mature existing optimization toolboxes can be used, such as <a href="https://github.com/jump-dev/ECOS.jl">ECOS.jl</a>.</li><li>Approximate the solution with an Multi-Layer Perceptrons (MLP) generator, using the powerful representation ability of neural network.</li></ul><p>Particularly, the second approach can achieve good approximations and it can save much time by avoiding repeating to run the optimization problems of the first approach when we conduct bootstrap to estimate the confidence band. </p><p>We do not <em>reinvent the wheel</em>. Instead, we fully take advantage of the existing widely-used implementations in other programming languages with the help of the flexible integration feature of Julia. For example, the package adopts the calculation of B-splines from R&#39;s <code>splines</code> package via <a href="https://github.com/JuliaInterop/RCall.jl">RCall.jl</a>, and provides the PyTorch deep learning backend via <a href="https://github.com/JuliaPy/PyCall.jl">PyCall.jl</a> as an alternative to the pure-Julia deep learning framework <a href="https://github.com/FluxML/Flux.jl">Flux.jl</a>.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="examples/monofit/">Monotone Fitting »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 20 November 2023 21:32">Monday 20 November 2023</span>. Using Julia version 1.8.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{aligned}\]</p><p>where <span>$\alpha=1$</span> implies non-decreasing and <span>$\alpha=-1$</span> indicates non-increasing.</p><p>The package provides two algorithms (frameworks) for fitting the monotone splines.</p><ul><li>Convert the problem into a classical convex second-order cone optimization problem. There are many mature existing optimization toolboxes can be used, such as <a href="https://github.com/jump-dev/ECOS.jl">ECOS.jl</a>.</li><li>Approximate the solution with an Multi-Layer Perceptrons (MLP) generator, using the powerful representation ability of neural network.</li></ul><p>Particularly, the second approach can achieve good approximations and it can save much time by avoiding repeating to run the optimization problems of the first approach when we conduct bootstrap to estimate the confidence band. </p><p>We do not <em>reinvent the wheel</em>. Instead, we fully take advantage of the existing widely-used implementations in other programming languages with the help of the flexible integration feature of Julia. For example, the package adopts the calculation of B-splines from R&#39;s <code>splines</code> package via <a href="https://github.com/JuliaInterop/RCall.jl">RCall.jl</a>, and provides the PyTorch deep learning backend via <a href="https://github.com/JuliaPy/PyCall.jl">PyCall.jl</a> as an alternative to the pure-Julia deep learning framework <a href="https://github.com/FluxML/Flux.jl">Flux.jl</a>.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="examples/monofit/">Monotone Fitting »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Sunday 18 February 2024 04:58">Sunday 18 February 2024</span>. Using Julia version 1.8.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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