From 1727083c7ec5c657f68d94037a5a9e644ef04877 Mon Sep 17 00:00:00 2001 From: Niharika Karnik Date: Thu, 14 Aug 2025 10:38:37 -0700 Subject: [PATCH 1/4] Adding the JOSS 2 paper --- docs/JOSS1/paper.bib | 820 +++++++++++++++++++++++++++++++++++++++++++ docs/JOSS1/paper.md | 98 ++++++ docs/JOSS2/Fig1.jpeg | Bin 0 -> 643488 bytes docs/JOSS2/Fig2.jpeg | Bin 0 -> 895154 bytes docs/JOSS2/paper.bib | 229 ++++++++++++ docs/JOSS2/paper.md | 140 ++++++++ 6 files changed, 1287 insertions(+) create mode 100644 docs/JOSS1/paper.bib create mode 100644 docs/JOSS1/paper.md create mode 100644 docs/JOSS2/Fig1.jpeg create mode 100644 docs/JOSS2/Fig2.jpeg create mode 100644 docs/JOSS2/paper.bib create mode 100644 docs/JOSS2/paper.md diff --git a/docs/JOSS1/paper.bib b/docs/JOSS1/paper.bib new file mode 100644 index 0000000..16fc61f --- /dev/null +++ b/docs/JOSS1/paper.bib @@ -0,0 +1,820 @@ +@article{ko1995exact, + title={An exact algorithm for maximum entropy sampling}, + author={Ko, Chun-Wa and Lee, Jon and Queyranne, Maurice}, + journal={Operations Research}, + volume={43}, + number={4}, + pages={684--691}, + year={1995}, + publisher={INFORMS}, + doi={10.1287/opre.43.4.684}, +} + +@inproceedings{narayanan2020toolkit, + title={A toolkit for spatial interpolation and sensor placement}, + author={Narayanan, S Deepak and Patel, Zeel B and Agnihotri, Apoorv and Batra, Nipun}, + booktitle={Proceedings of the 18th Conference on Embedded Networked Sensor Systems}, + pages={653--654}, + year={2020}, + doi={10.1145/3384419.3430407}, +} + +@article{manohar2019optimized, + title={Optimized sampling for multiscale dynamics}, + author={Manohar, Krithika and Kaiser, Eurika and Brunton, Steven L and Kutz, J Nathan}, + journal={Multiscale Modeling \& Simulation}, + volume={17}, + number={1}, + pages={117--136}, + year={2019}, + publisher={SIAM}, + doi={10.1137/17M1162366}, +} + +@article{clark2018greedy, + title={Greedy sensor placement with cost constraints}, + author={Clark, Emily and Askham, Travis and Brunton, Steven L and Kutz, J Nathan}, + journal={IEEE Sensors Journal}, + volume={19}, + number={7}, + pages={2642--2656}, + year={2018}, + publisher={IEEE}, + doi={10.1109/JSEN.2018.2887044}, +} + +@article{manohar2018data, + title={Data-driven sparse sensor placement for reconstruction: Demonstrating the benefits of exploiting known patterns}, + author={Manohar, Krithika and Brunton, Bingni W and Kutz, J Nathan and Brunton, Steven L}, + journal={IEEE Control Systems Magazine}, + volume={38}, + number={3}, + pages={63--86}, + year={2018}, + publisher={IEEE}, + doi={10.1109/mcs.2018.2810460} +} + +@article{brunton2016sparse, + title={Sparse sensor placement optimization for classification}, + author={Brunton, Bingni W and Brunton, Steven L and Proctor, Joshua L and Kutz, J Nathan}, + journal={SIAM Journal on Applied Mathematics}, + volume={76}, + number={5}, + pages={2099--2122}, + year={2016}, + publisher={SIAM}, + doi={10.1137/15M1036713} +} + +@article{summers2015submodularity, + title={On submodularity and controllability in complex dynamical networks}, + author={Summers, Tyler H and Cortesi, Fabrizio L and Lygeros, John}, + journal={IEEE Transactions on Control of Network Systems}, + volume={3}, + number={1}, + pages={91--101}, + year={2015}, + publisher={IEEE} +} + +@article{chen2011h, + title={H 2 optimal actuator and sensor placement in the linearised complex Ginzburg-Landau system}, + author={Chen, Kevin K and Rowley, Clarence W}, + journal={Journal of Fluid Mechanics}, + volume={681}, + number={241-260}, + year={2011}, + publisher={Cambridge University Press, The Edinburgh Building, Cambridge CB 2 2 RU~…}, + doi={10.1017/jfm.2011.195} +} + +@article{krause2008near, + title={Near-optimal sensor placements in Gaussian processes: Theory, efficient algorithms and empirical studies}, + author={Krause, Andreas and Singh, Ajit and Guestrin, Carlos}, + journal={Journal of Machine Learning Research}, + volume={9}, + number={Feb}, + pages={235--284}, + year={2008} +} + +@article{joshi2008sensor, + title={Sensor selection via convex optimization}, + author={Joshi, Siddharth and Boyd, Stephen}, + journal={IEEE Transactions on Signal Processing}, + volume={57}, + number={2}, + pages={451--462}, + year={2009}, + publisher={IEEE} +} + +@article{willcox2006unsteady, + title={Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition}, + author={Willcox, Karen}, + journal={Computers \& fluids}, + volume={35}, + number={2}, + pages={208--226}, + year={2006}, + publisher={Elsevier}, + doi={10.2514/6.2004-2415} +} + +@article{brunton2016pnas, + title = {Discovering governing equations from data by sparse identification of nonlinear dynamical systems}, + author = {Brunton, Steven L. and Proctor, Joshua L. and Kutz, J. 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Albuquerque, NM: Sandia National Laboratories}, + year={2017}, + doi={10.2172/1405271}, +} + +@article{scikit-learn, + title={Scikit-learn: Machine Learning in {P}ython}, + author={Pedregosa, F. and Varoquaux, G. and Gramfort, A. and Michel, V. + and Thirion, B. and Grisel, O. and Blondel, M. and Prettenhofer, P. + and Weiss, R. and Dubourg, V. and Vanderplas, J. and Passos, A. and + Cournapeau, D. and Brucher, M. and Perrot, M. and Duchesnay, E.}, + journal={Journal of Machine Learning Research}, + volume={12}, + pages={2825--2830}, + year={2011}, +} diff --git a/docs/JOSS1/paper.md b/docs/JOSS1/paper.md new file mode 100644 index 0000000..a082f87 --- /dev/null +++ b/docs/JOSS1/paper.md @@ -0,0 +1,98 @@ +--- +title: 'PySensors: A Python package for sparse sensor placement' +tags: + - Python + - machine learning +authors: + - name: Brian M. de Silva + affiliation: 1 + orcid: 0000-0003-0944-900X + - name: Krithika Manohar + affiliation: 2 + orcid: 0000-0002-1582-6767 + - name: Emily Clark + affiliation: 3 + - name: Bingni W. Brunton + affiliation: 4 + orcid: 0000-0002-4831-3466 + - name: Steven L. Brunton + affiliation: 2 + orcid: 0000-0002-6565-5118 + - name: J. Nathan Kutz + affiliation: 1 + orcid: 0000-0002-6004-2275 +affiliations: + - name: Department of Applied Mathematics, University of Washington + index: 1 + - name: Department of Mechanical Engineering, University of Washington + index: 2 + - name: Department of Physics, University of Washington + index: 3 + - name: Department of Biology, University of Washington + index: 4 +date: 13 February 2021 +bibliography: paper.bib +--- + + +# Summary + +Successful predictive modeling and control of engineering and natural processes is often entirely determined by *in situ* measurements and feedback from sensors [@Brunton2019book], which provide measurements of the state of these processes at specific points in space and time. +However, deploying sensors into complex environments, including in application areas such as manufacturing [@Manohar2018jms], geophysical environments [@Yildirim:2009], and biological processes [@colvert2017local;@Mohren2018pnas], is often expensive and challenging. +Furthermore, modeling outcomes are extremely sensitive to the location and number of these sensors, motivating optimization strategies for the principled placement of sensors for different decision-making tasks. +In general, choosing the globally optimal placement within the search space of a large-scale complex system is an intractable computation, in which the number of possible placements grows combinatorially with the number of candidates [@ko1995exact]. +While sensor placement has traditionally been guided by expert knowledge and first principles models, increases in system complexity, emerging sensor technologies, and innovations in data-driven modeling strategies motivates automated algorithms for optimizing sensor placements. + +`PySensors` is a Python package for the scalable optimization of sensor placement from data. In particular, `PySensors` provides tools for sparse sensor placement optimization approaches that employ data-driven dimensionality reduction [@brunton2016sparse;@manohar2018data]. This approach results in near-optimal placements for various decision-making tasks and can be readily customized using different optimization algorithms and objective functions. + +The `PySensors` package can be used by both researchers looking to advance state-of-the-art methods and practitioners seeking simple sparse sensor selection methods for their applications of interest. +Straightforward methods and abundant examples help new users to quickly and efficiently leverage existing methods to their advantage. +At the same time, modular classes leave flexibility for users to experiment with and plug in new sensor selection algorithms or dimensionality reduction techniques. +Users of `scikit-learn` will find `PySensors` objects familiar, intuitive, and compatible with existing `scikit-learn` routines such as cross-validation [@scikit-learn]. + +# Statement of need +Maximizing the impact of sensor placement algorithms requires tools to make them accessible to scientists and engineers across various domains and at various levels of mathematical expertise and sophistication. `PySensors` unifies the algorithms developed in the papers [@manohar2018data;@clark2018greedy;@brunton2016sparse] and their accompanying codes `SSPOR_pub` and `SSPOC_pub` into one software package. The only other packages in this domain of which we are aware are `Chama` [@klise2017sensor] and `Polire` [@narayanan2020toolkit]. While these packages and `PySensors` all enable sparse sensor placement optimization, `Chama` and `Polire` are geared towards event detection and Gaussian processes respectively, whereas `PySensors` is aimed at signal reconstruction and classification tasks. +As such, there are marked differences in the objective functions optimized by `PySensors` and its precursors. +In addition to these two packages, researchers and practitioners have made available various custom scripts for sensor placement. +Currently, researchers seeking to employ modern sensor placement methods must choose between implementing them from scratch or manually augmenting existing unpolished codes. + +Reconstruction and classification tasks often arise in the modeling, prediction, and control of complex processes in geophysics, fluid dynamics, biology, and manufacturing. +The goal of _reconstruction_ is to recover a high-dimensional signal $\mathbf{x}\in\mathbb{R}^N$ from a limited number of $p$ measurements $y_ i = \mathbf{c}_ i^\top \mathbf{x}$, where each $\mathbf{c}_ i \in \mathbb{R}^N$ represents the action of a sensor. For example, $\mathbf{c}_ i^\top = [1, 0, 0, \dots, 0]$ represents a sensor which takes a point measurement of the first dimension of the signal $\mathbf{x}$. `PySensors` selects a set of $p$ sensors out of $N$ candidates $\mathbf{c}_ i^\top$ (rows of a measurement matrix $\mathbf{C}:\mathbf{y} = \mathbf{Cx}$) that minimize reconstruction error in a data-dependent basis $\mathbf{\Phi}\in\mathbb{R}^{N\times r}$ +$$ \mathbf{C}^ \star= \underset{\mathbf{C}\in\mathbb{R}^{p\times N}}{\arg\min} \|\mathbf{x} - \mathbf{\Phi}(\mathbf{C\Phi})^{\dagger} \mathbf{y}\|_ 2^2, $$ +where $\dagger$ denotes the Moore-Penrose pseudoinverse. The key innovation is to recover the low-dimensional representation $\mathbf{x}_ r \in \mathbb{R}^r$ satisfying $\mathbf{x} = \mathbf{\Phi x}_ r$ via the reconstruction map $\mathbf{\Phi}(\mathbf{C\Phi})^{\dagger}$, ultimately reducing sensor placement to a highly efficient matrix pivoting operation [@manohar2018data]. Similarly, sensor placement for _classification_ [@brunton2016sparse] optimizes the sparsest vector $\mathbf{s}^ \star$ that reconstructs $\mathbf{w}: \mathbf{\Phi}^\dagger\mathbf{s} = \mathbf{w}$ in the low-dimensional feature space, where $\mathbf{w}$ is the the set of weights learned by a linear classifier fit to $\mathbf{x}_ r$. +In this case, the optimal sensor locations are determined by the nonzero components of $\mathbf{s}^ \star$. + +The basis $\mathbf{\Phi}$ is explicitly computed from the data using powerful dimensionality reduction techniques such as principal components analysis (PCA) and random projections, which enable significant compression of most signals to $r\ll N$ dimensions. PCA extracts the dominant spatial correlations or _principal components_, the leading eigenvectors of the data covariance matrix. It is computed using the matrix singular value decomposition (SVD) and is closely related to proper orthogonal decomposition (POD); POD modes and principal components are equivalent. +Other basis choices are possible, such as dynamic mode decomposition for extracting temporally correlated features [@manohar2019optimized]. + + + + +# Features + +`PySensors` enables the sparse placement of sensors for two classes of problems: reconstruction and classification. +For reconstruction problems the package implements a unified `SensorSelector` class, with methods for efficiently analyzing the effects that data or sensor quantity have on reconstruction performance [@manohar2018data]. +Sensor selection is based on the computationally efficient QR algorithm. +Often different sensor locations impose variable costs, e.g. if measuring sea-surface temperature, it may be more expensive to place buoys/sensors in the middle of the ocean than close to shore. +These costs can be taken into account during sensor selection via a built-in cost-sensitive optimization routine [@clark2018greedy]. +For classification tasks, the package implements the Sparse Sensor Placement Optimization for Classification (SSPOC) algorithm [@brunton2016sparse], allowing one to optimize sensor placement for classification accuracy. +The algorithm is related to compressed sensing optimization [@Candes2006cpam;@Donoho2006ieeetit;@Baraniuk2007ieeespm], but identifies the sparsest set of sensors that reconstructs a discriminating plane in a feature subspace. +This SSPOC implementation is fully general in the sense that it can be used in conjunction with any linear classifier. +Additionally, `PySensors` provides methods to enable straightforward exploration of the impacts of primary hyperparameters like the number of sensors or basis modes. + +It is well known [@manohar2018data] that the basis in which one represents measurement data can have a pronounced effect on the sensors that are selected and the quality of the reconstruction. +Users can readily switch between different bases typically employed for sparse sensor selection, including POD modes and random projections. +Because `PySensors` was built with `scikit-learn` compatibility in mind, it is easy to use cross-validation to select among possible choices of bases, basis modes, and other hyperparameters. + +Finally, included with `PySensors` is a large suite of examples, implemented as Jupyter notebooks. +Some of the examples are written in a tutorial format and introduce new users to the objects, methods, and syntax of the package. +Other examples demonstrate intermediate-level concepts such as how to visualize model parameters and performance, how to combine `scikit-learn` and `PySensors` objects, selecting appropriate parameter values via cross-validation, and other best-practices. +Further notebooks use `PySensors` to solve challenging real-world problems. +The notebooks reproduce many of the examples from the papers upon which the package is based [@manohar2018data;@clark2018greedy;@brunton2016sparse]. +To help users begin applying `PySensors` to their own datasets even faster, interactive versions of every notebook are available on Binder. +Together with comprehensive documentation, the examples will compress the learning curve of learning a new software package. + +# Acknowledgments +The authors acknowledge support from the Air Force Office of Scientific Research (AFOSR FA9550-19-1-0386) and The Boeing Corporation. The work of KM is supported by the National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship (award 1803663). JNK acknowledges support from the Air Force Office of Scientific Research (AFOSR FA9550-19-1-0011) + +# References \ No newline at end of file diff --git a/docs/JOSS2/Fig1.jpeg b/docs/JOSS2/Fig1.jpeg new file mode 100644 index 0000000000000000000000000000000000000000..88f60860c2890ae26786580664012088bae83db2 GIT binary patch literal 643488 zcmeFa30zZ4moK^z6&WN5GRPDa1(}sW6p%QAfPl=C5D^gp5t$@FNR)XHIWj6!RAd%V znPiO22T_n&f&vL5$RG(r*g&%Lwx?hB`M$nizt?^5>-+ljW%El??9{GQ)vj9gU;kQb zZPsVj1i)vYtFH^Nu>rtE_zz%_fzuZ-ZZ`nH$Ow=F0ALru7Uc$Phf{DCKooufz_zE` z{`m~Yy{GK|oZf!o*O#mX;Kb#dUVdJ_H@)ucS5-O+oH(m*wEb5J@bO3b@E=LO0f`4A z3Ba4}Kkjipo(Iq3Sd?u?^v|6`Up6z*)xUJ%k4kM5ak_Ko)~|~_Z~OU}>7CtgV{5mc z<2ziNU-f|xCuiR~XD(m9^eg?(pTE+79`vDq%GMg z{lgU3br)Y}_!|cNI_iAK#}DoZ1vq^G2zu78vEyygGvcYhtfTFT>c+tM8V+zY?Nf%AYH z;0xRVZUIg}Ag~{}45j>0dMU;PYU3m^2s|GWPb{yY~p4gjZ!EEcQo&vOPu0C=qd09-YHo;w1Mi9Or^ z@O#Z2Cm*Ljierc0x7~n8*it?K?6w5}flu%?_P#e%RsQ&%jpr-?>=7ySAU;xW~@62iUfUjeQRrs}(@PJ+*`FkK>Pm;5W8y 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Soo and Cogliati, Joshua J and Skifton, Richard S and Calderoni, Pattrick and Brunton, Steven L and Manohar, Krithika}, + journal={IEEE Sensors Journal}, + year={2024}, + publisher={IEEE} +} + +@article{klishin2025doubledescent, + title={Origins and mitigation of double descent in reduced order modeling}, + author={Klishin, Andrei A and Otto, Samuel E and Kutz, J Nathan and Manohar, Krithika}, + journal={In preparation}, + year={2025} +} + +@article{erichson2020shallow, + title={Shallow neural networks for fluid flow reconstruction with limited sensors}, + author={Erichson, N Benjamin and Mathelin, Lionel and Yao, Zhewei and Brunton, Steven L and Mahoney, Michael W and Kutz, J Nathan}, + journal={Proceedings of the Royal Society A}, + volume={476}, + number={2238}, + pages={20200097}, + year={2020}, + publisher={The Royal Society Publishing} +} + +@article{manohar2018predicting, + title={Predicting shim gaps in aircraft assembly with machine learning and sparse sensing}, + author={Manohar, Krithika and Hogan, Thomas and Buttrick, Jim and Banerjee, Ashis G and Kutz, J Nathan and Brunton, Steven L}, + journal={Journal of manufacturing systems}, + volume={48}, + pages={87--95}, + year={2018}, + publisher={Elsevier} +} + +@article{alonso2010novel, + title={A novel strategy for radar imaging based on compressive sensing}, + author={Alonso, Mariv{\'\i} Tello and L{\'o}pez-Dekker, Paco and Mallorqu{\'\i}, Jordi J}, + journal={IEEE Transactions on Geoscience and Remote Sensing}, + volume={48}, + number={12}, + pages={4285--4295}, + year={2010}, + publisher={IEEE} +} + +@article{krause2008near, + title={Near-optimal sensor placements in Gaussian processes: Theory, efficient algorithms and empirical studies.}, + author={Krause, Andreas and Singh, Ajit and Guestrin, Carlos}, + journal={Journal of Machine Learning Research}, + volume={9}, + number={2}, + year={2008} +} + +@article{alexanderian2021optimal, + title={Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: A review}, + author={Alexanderian, Alen}, + journal={Inverse Problems}, + volume={37}, + number={4}, + pages={043001}, + year={2021}, + publisher={IOP Publishing} +} + +@article{donoho2006compressed, + title={Compressed sensing}, + author={Donoho, David L}, + journal={IEEE Transactions on information theory}, + volume={52}, + number={4}, + pages={1289--1306}, + year={2006}, + publisher={IEEE} +} + +@article{summers2015submodularity, + title={On submodularity and controllability in complex dynamical networks}, + author={Summers, Tyler H and Cortesi, Fabrizio L and Lygeros, John}, + journal={IEEE Transactions on Control of Network Systems}, + volume={3}, + number={1}, + pages={91--101}, + year={2015}, + publisher={IEEE} +} + +@article{joshi2008sensor, + title={Sensor selection via convex optimization}, + author={Joshi, Siddharth and Boyd, Stephen}, + journal={IEEE Transactions on Signal Processing}, + volume={57}, + number={2}, + pages={451--462}, + year={2008}, + publisher={IEEE} +} + +@article{de2021pysensors, + title={PySensors: A Python package for sparse sensor placement}, + author={de Silva, Brian M and Manohar, Krithika and Clark, Emily and Brunton, Bingni W and Brunton, Steven L and Kutz, J Nathan}, + journal={arXiv preprint arXiv:2102.13476}, + year={2021} +} + +@article{brunton2016sparse, + title={Sparse sensor placement optimization for classification}, + author={Brunton, Bingni W and Brunton, Steven L and Proctor, Joshua L and Kutz, J Nathan}, + journal={SIAM Journal on Applied Mathematics}, + volume={76}, + number={5}, + pages={2099--2122}, + year={2016}, + publisher={SIAM} +} + +@article{clark2020multi, + title={Multi-fidelity sensor selection: Greedy algorithms to place cheap and expensive sensors with cost constraints}, + author={Clark, Emily and Brunton, Steven L and Kutz, J Nathan}, + journal={IEEE Sensors Journal}, + volume={21}, + number={1}, + pages={600--611}, + year={2020}, + publisher={IEEE} +} + +@article{klishin2023data, + title={Data-induced interactions of sparse sensors}, + author={Klishin, Andrei A and Kutz, J Nathan and Manohar, Krithika}, + journal={arXiv preprint arXiv:2307.11838}, + year={2023} +} + +@article{baraniuk2010model, + title={Model-based compressive sensing}, + author={Baraniuk, Richard G and Cevher, Volkan and Duarte, Marco F and Hegde, Chinmay}, + journal={IEEE Transactions on information theory}, + volume={56}, + number={4}, + pages={1982--2001}, + year={2010}, + publisher={IEEE} +} + +@article{candes2006stable, + title={Stable signal recovery from incomplete and inaccurate measurements}, + author={Candes, Emmanuel J and Romberg, Justin K and Tao, Terence}, + journal={Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences}, + volume={59}, + number={8}, + pages={1207--1223}, + year={2006}, + publisher={Wiley Online Library} +} + +@article{schmid2010dynamic, + title={Dynamic mode decomposition of numerical and experimental data}, + author={Schmid, Peter J}, + journal={Journal of fluid mechanics}, + volume={656}, + pages={5--28}, + year={2010}, + publisher={Cambridge University Press} +} + +@article{barrault2004empirical, + title={An ‘empirical interpolation’method: application to efficient reduced-basis discretization of partial differential equations}, + author={Barrault, Maxime and Maday, Yvon and Nguyen, Ngoc Cuong and Patera, Anthony T}, + journal={Comptes Rendus Mathematique}, + volume={339}, + number={9}, + pages={667--672}, + year={2004}, + publisher={Elsevier} +} + +@article{chaturantabut2010nonlinear, + title={Nonlinear model reduction via discrete empirical interpolation}, + author={Chaturantabut, Saifon and Sorensen, Danny C}, + journal={SIAM Journal on Scientific Computing}, + volume={32}, + number={5}, + pages={2737--2764}, + year={2010}, + publisher={SIAM} +} + +@article{drmac2016new, + title={A new selection operator for the discrete empirical interpolation method---improved a priori error bound and extensions}, + author={Drmac, Zlatko and Gugercin, Serkan}, + journal={SIAM Journal on Scientific Computing}, + volume={38}, + number={2}, + pages={A631--A648}, + year={2016}, + publisher={SIAM} +} + +@article{manohar2018data, + title={Data-driven sparse sensor placement for reconstruction: Demonstrating the benefits of exploiting known patterns}, + author={Manohar, Krithika and Brunton, Bingni W and Kutz, J Nathan and Brunton, Steven L}, + journal={IEEE Control Systems Magazine}, + volume={38}, + number={3}, + pages={63--86}, + year={2018}, + publisher={IEEE} +} + +@article{manohar2021optimal, + title={Optimal sensor and actuator selection using balanced model reduction}, + author={Manohar, Krithika and Kutz, J Nathan and Brunton, Steven L}, + journal={IEEE Transactions on Automatic Control}, + volume={67}, + number={4}, + pages={2108--2115}, + year={2021}, + publisher={IEEE} +} diff --git a/docs/JOSS2/paper.md b/docs/JOSS2/paper.md new file mode 100644 index 0000000..aafa1c4 --- /dev/null +++ b/docs/JOSS2/paper.md @@ -0,0 +1,140 @@ +--- +title: 'PySensors 2.0: A Python Package for Sparse Sensor Placement' +tags: + - Python + - machine learning +authors: + - name: Niharika Karnik + affiliation: 1 + orcid: 0000-0002-4259-0294 + - name: Yash Bhangale + affiliation: 1 + orcid: 0009-0008-4163-6538 + - name: Mohammad G. Abdo + affiliation: 3 + orcid: 0000-0001-9845-6978 + - name: Andrei A. Klishin + affiliation: 4 + orcid: 0000-0002-5740-8520 + - name: Joshua J. Cogliati + affiliation: 3 + orcid: 0000-0003-2471-8095 + - name: Bingni W. Brunton + affiliation: 5 + orcid: 0000-0002-4831-3466 + - name: J. Nathan Kutz + affiliation: 2 + orcid: 0000-0002-6004-2275 + - name: Steven L. Brunton + affiliation: 1 + orcid: 0000-0002-6565-5118 + - name: Krithika Manohar + affiliation: 1 + orcid: 0000-0002-1582-6767 +affiliations: + - name: Department of Mechanical Engineering, University of Washington + index: 1 + - name: Department of Applied Mathematics, University of Washington + index: 2 + - name: Idaho National Laboratory + index: 3 + - name: Department of Mechanical Engineering, University of Hawai'i at Mānoa + index: 4 + - name: Department of Biology, University of Washington + index: 5 +date: 12 August 2025 +bibliography: paper.bib +--- + + +# Summary + +`PySensors` is a Python package for selecting and placing a sparse set of sensors for reconstruction and classification tasks. In this major update to `PySensors`, we introduce spatially constrained sensor placement capabilities, allowing users to enforce constraints such as maximum or exact sensor counts in specific regions, incorporate predetermined sensor locations, and maintain minimum distances between sensors. We extend functionality to support custom basis inputs, enabling integration of any data-driven or spectral basis. We also propose a thermodynamic approach that goes beyond a single ``optimal'' sensor configuration and maps the complete landscape of sensor interactions induced by the training data. This comprehensive view facilitates integration with external selection criteria and enables assessment of sensor replacement impacts. The new optimization technique also accounts for over- and under-sampling of sensors, utilizing a regularized least squares approach for robust reconstruction. Additionally, we incorporate noise-induced uncertainty quantification of the estimation error and provide visual uncertainty heat maps to guide deployment decisions. To highlight these additions, we provide a brief description of the mathematical algorithms and theory underlying these new capabilities. We demonstrate the usage of new features with illustrative code examples and include practical advice for implementation across various application domains. Finally, we outline a roadmap of potential extensions to further enhance the package's functionality and applicability to emerging sensing challenges. + +# Statement of need +Sensor placement is critical for efficient monitoring, control, and decision-making in modern engineering systems. Sensors play a crucial role in characterizing spatio-temporal dynamics in high-dimensional, non-linear systems such as fluid flows [erichson2020shallow], manufacturing [manohar2018predicting], geophysical [alonso2010novel] and nuclear systems [karnik2024constrained]. Optimal sensor placement ensures accurate, real-time tracking of key system variables with minimal hardware and enables cost-effective, real-time system analysis and control. In general, sensor placement optimization is NP-hard and cannot be solved in polynomial time. There are ${n \choose p} = n!/((n-p)!p!)$ possible combinations of choosing $p$ sensors from an $n$-dimensional state. +Common approaches to optimizing sensor placement include maximizing the information criteria [krause2008near], Bayesian Optimal Experimental Design [alexanderian2021optimal], compressed sensing [donoho2006compressed], and heuristic methods. Many sensor placement methods have submodular objective form, which sets guarantees on how close an efficient greedy placement can be to the unknown true optimum [summers2015submodularity]. Sub-modular objectives can be efficiently optimized for hundreds or thousands of candidate locations using convex [joshi2008sensor] or greedy optimization approaches [summers2015submodularity] . + +![An overview image of capabilities of Pysensors](docs/JOSS2/Fig1.jpeg "PySensors 2.0 expands its capabilities by introducing custom basis functions, optimizers, constraints, solvers, and uncertainty quantification, enabling constrained sensing, over- and under-sampling, and uncertainty quantification in the presence of noisy sensor measurements.") + + +`PySensors` is a Python package [de2021pysensors] dedicated to solving the complex challenge of optimal sensor placement in data-driven systems. It implements advanced sparse optimization algorithms that use dimensionality reduction techniques to identify the most informative measurement locations with remarkable efficiency [manohar2018data;brunton2016sparse;clark2020multi]. It helps users identify the best locations for sensors when working with complex high dimensional data, focusing on both reconstruction [manohar2018data] and classification [brunton2016sparse] tasks. The package follows `scikit-learn` conventions for user-friendly access while offering advanced customization options for experienced users. Designed with researchers and practitioners in mind, `PySensors` provides open-source, accessible tools that support model discovery across various scientific applications. + +This new version of `Pysensors` focuses specifically on practical engineering applications where measurement data is inherently noisy and spatial deployment constraints are unavoidable. Key improvements include constraint-aware optimization methods that handle spatial restrictions and sensor density limitations. In addition, the framework introduces uncertainty quantification metrics that track how measurement noise propagates through reconstruction algorithms, enabling robust error estimation in sensor outputs. This version of `Pysensors` implements methodologies introduced by Klishin et al. [klishin2023data] and Karnik et al. [karnik2024constrained] to make them accessible to scientists and engineers in all domains. These enhancements transform `PySensors` from a purely academic tool into a practical platform for solving real-world sensing challenges while maintaining mathematical rigor. + +Pysensors implements `Sparse Sensor Placement Optimization for Reconstruction (SSPOR)`[manohar2018data] of full fields $\mathbf{x}\in\reals^n$ from $p$ noise-corrupted sensor measurements $\mathbf{y}\in\reals^p$ +$$ + \mathbf{y} = \mathbb{S} \mathbf{x} + \boldsymbol\eta, +$$ +where $\boldsymbol{\eta}$ consists of zero-mean, Gaussian independent and identically distributed (i.i.d.) components, and $\mathbb{S} \in \reals^{p\times n}$ is the desired sensor (measurement) selection operator. +This measurement selection operator $\mathbb{S}$ encodes point measurements with unit entries in a sparse matrix $\mathbb{S} = \begin{bmatrix} \mathbf{e}_{\gamma_1} & \mathbf{e}_{\gamma_2} & \hdots & \mathbf{e}_{\gamma_ p}\end{bmatrix}^T$ where $\mathbf{e}_j$ are canonical basis vectors for $\mathbb{R}^n$, with a unit entry in component $j$ (where a sensor should be placed) and zeroes elsewhere. Here, $\gamma = \{\gamma_1, \gamma_2, \hdots, \gamma_p\} \subset \{1, 2, \hdots, n\}$ denotes the index set of sensor locations with cardinality $p$. +Sensor selection then corresponds to the components of $\mathbf{x}$ that were chosen to be measured: $\mathbf{\mathbb{S} x} = \begin{bmatrix} x_{\gamma_1} & x_{\gamma_2} & \hdots & x_{\gamma_p}\end{bmatrix}^T$. +The `SSPOR` class selects these sensors through a cohesive computational framework for strategically minimizing sensor deployment while maintaining reconstruction fidelity in high-dimensional systems. When initialized with measurement data $\mathbf{x}\in\reals^n$, `SSPOR` first applies dimensional reduction through basis identification, then employs the computationally efficient QR factorization algorithm to determine optimal sensor locations. This approach, which has demonstrated efficacy in both reduced-order modeling and sparse sensing applications, strategically leverages the inherent structure of the identified basis to prioritize the most informative measurement points. + +The `Sparse Sensor Placement Optimization for Classification (SSPOC)` framework identifies minimal sensor configurations that can classify high dimensional signals $\mathbf{x}\in\reals^n$ as one of $c$ classes. Unlike traditional compressed sensing [baraniuk2010model;donoho2006compressed;candes2006stable] approaches that focus on signal reconstruction, `SSPOC` specifically targets the preservation of classification boundaries in feature space $\boldsymbol{\Psi}_r$ by identifying the sparsest sensor set capable of reconstructing discriminant hyperplanes. +The `SSPOC` architecture accepts any combination of basis representation and linear classifier, defaulting to Linear Discriminant Analysis (LDA) and Identity basis when not otherwise specified. The optimization pipeline proceeds through multiple stages: dimensional reduction via basis fitting, classifier training within the reduced space, sparse optimization incorporating both classifier weights and basis structures, sensor selection based on optimization outputs, and optional classifier retraining using only measurements from the selected locations. + +High dimensional field dynamics $\mathbf{x}\in\reals^n$ can be represented as a linear combination of spatial basis modes $\boldsymbol{\Psi}$ weighted by time-varying coefficients $\mathbf{a}$ +$$\mathbf{x} = \boldsymbol{\Psi}_r\mathbf{a}.$$ +This basis, which can be built from spectral or data-driven decomposition methods, is typically chosen so that the embedding dimension is as small as possible, i.e., $r\ll n$. Different basis functions can significantly impact sensor selection effectiveness and reconstruction quality [manohar2018data]. PySensors offers several interchangeable `basis` options for sparse sensor selection: +- `Identity`: Processes raw measurement data directly without transformation. +- `SVD`: Utilizes truncated singular value decomposition's $\mathbf{X} = \mathbf{U}_r \mathbf{\Sigma}_r \mathbf{V}^*_r$ left singular vectors $\mathbf{U}_r$, computing only the specified number of modes to reduce computational demands. A randomized SVD option further enhances efficiency. +- `RandomProjection`: Projects measurements into a new space by multiplying them with random Gaussian vectors, connecting to compressed sensing methodologies from established literature [baraniuk2010model;donoho2006compressed;candes2006stable]. +- `Custom`: This is a new option included in `Pysensors 2.0` that enables users to leverage custom basis functions beyond `PySensors` built-in options. Researchers can transform their data into an alternative basis such as dynamic mode decomposition modes [schmid2010dynamic], before passing it to a PySensors instance configured with the Custom basis. + +When a user specifies $p$ sensors, SSPOR first fits a basis $\bTi_r$ to the data and optimizes sensor placement by minimizing the following objective function: +$$ +\gamma_* = \argmax_{\gamma, |\gamma|= p} \; \log \det((\mathbb{S} \boldsymbol{\Psi}_r)^T(\mathbb{S} \boldsymbol{\Psi}_r)).\label{eq:detobj} \tag{1} +$$ +where $\gamma_*$ denotes the index set of optimized sensor locations with cardinality $p$. When $p = r$, is equivalent to the maximizer of $\log |\det(\mathbb{S} \boldsymbol{\Psi}_r)|$. +Direct optimization of this criterion leads to a brute force combinatorial search. This sensor placement approach builds upon the empirical interpolation method (EIM) [barrault2004empirical] and discrete empirical interpolation method (DEIM) [chaturantabut2010nonlinear] to develop a greedy strategy for optimizing sensor selection built upon the pivoted QR factorization[drmac2016new;manohar2018data;manohar2018predicting;manohar2021optimal]. + +PySensors implements the `QR` optimizer for optimal sensor selection in unconstrained scenarios where the number of sensors $p$ equals the number of modes $r$. The framework incorporates heterogeneous cost functions into the optimization process to accommodate practical deployment constraints. For example, when monitoring oceanographic parameters, the system can account for the substantially higher costs of deep-sea sensors relative to coastal installations[clark2020multi]. This capability is implemented through the `Cost-Constrained QR (CCQR)` algorithm in the optimizers submodule, allowing users to balance information capture against resource limitations when designing sensor networks for complex physical systems. + +Traditional QR factorization presents challenges in under-sampling $p < r$ and over-sampling $p > r$ scenarios, as well as when spatial constraints must be considered. `PySensors 2.0` addresses these limitations through two new optimization algorithms: `Generalized QR (GQR)` and `Two Point GReedy (TPGR)`. + +# New Features + +`PySensors` has been enhanced to address critical challenges in sensor placement by incorporating spatial constraints, noise-induced uncertainties, over/under sampling and sensor interactions. These advancements optimize reconstruction performance for complex processes across nuclear energy, fluid dynamics, biological systems, and manufacturing applications, enabling more accurate modeling, prediction, and control capabilities. +The previous version of `PySensors` implements two sensor placement approaches: (1) an unconstrained optimization formulated as `QR` that allows sensors to be placed anywhere in the domain, and (2) a cost-constrained optimization formulated as `CCQR` that incorporates variable placement costs, making certain regions more expensive for sensor deployment. + +Many engineering applications have extreme operating conditions, high costs, limited accessibility and safety regulations that impose significant constraints on spatial locations of sensors. Implementing spatially constrained sensor placement requires a deeper intervention in the underlying QR optimization framework. To address this requirement, we have developed a new optimization functionality called General QR `GQR` based on Karnik et. al. [@karnik2024constrained], which provides the architectural flexibility needed to handle complex spatial constraints. In `Pysensors 2.0` we enhance the algorithm's capabilities by incorporating diverse spatial constraints defined by users through `_norm_calc.py` in `utils`. The three types of spatial constraints handled by the algorithm are: + +- *Region constrained:* This type of constraint arises when we can place either a _maximum_ of or _exactly_ $s$ sensors in a certain region, while the remaining $r-s$ sensors must be placed outside the constraint region. + - *Maximum:* This case deals with applications in which the number of sensors in the constraint region should be less than or equal to $s$. This functionality has been implemented through `max_n` in `_norm_calc.py`. + - *Exact:* This case deals with applications in which the number of sensors in the constraint region should equal $s$. This functionality has been implemented through `exact_n` in `_norm_calc.py`. +- *Predetermined:* This type of constraint occurs when a certain number of sensor locations $s$ are already specified, and optimized locations for the remaining sensors are desired. This functionality has been implemented through `predetermined` in `_norm_calc.py`. +- *Distance constrained:* This constraint enforces a minimum distance $d$ between selected sensors. This functionality has been implemented through `distance` in `_norm_calc.py`. + +To implement spatial constraints in our sensor placement methodology, we must first define the shape of our constraint region and then identify which grid points fall within the designated constraint regions. We accomplish this using the specialized classes and functions provided in the `_constraints.py` module in `utils`. The classes define constraint regions in various geometric shapes including `Circle`, `Ellipse`, `Polygon`, `Parabola`, `Line`, and `Cylinder`. Additionally, we provide `UserDefinedConstraints` options that allow users to supply their own constraints either as a Python`.py` file containing their constraint shape definition or as an equation string. These classes work with two distinct data formats: image data defined by pixel coordinates and tabular data stored in dataframes with explicit x, y, and z values. Upon instantiation of the class, users must provide the input `data` in one of two formats: a matrix representation for images or a structured dataframe for tabular data. In the latter case, the class requires additional keyword arguments `kwargs` specifying the columns corresponding to `X_axis`, `Y_axis`, `Z_axis`, and `Field` parameters, thereby enabling the extraction of relevant data from their respective columns for subsequent analysis. + +The D-optimal objective in \ref{detobj} suffers from two limitations: it is not defined for the under-sampling case $p r)scenarios , select TPGR optimizer. In noisy environments enable uncertainty quantification for robust results.") + + +# Acknowledgments +The authors acknowledge support from the Boeing Company, NSF AI Institute in Dynamic Systems under grant 2112085 and through the Idaho National Laboratory (INL) Laboratory Directed Research \& Development (LDRD) Program under DOE Idaho Operations Office Contract DE-AC07-05ID14517 for LDRD-22A1059-091FP. + +# References \ No newline at end of file From a7791b28bc5911967e9cf4f1bb0952fa117c7a8a Mon Sep 17 00:00:00 2001 From: Niharika Karnik Date: Thu, 14 Aug 2025 10:48:30 -0700 Subject: [PATCH 2/4] Fixing figure path, references and some of the equations --- docs/JOSS2/paper.md | 40 ++++++++++++++++++++-------------------- 1 file changed, 20 insertions(+), 20 deletions(-) diff --git a/docs/JOSS2/paper.md b/docs/JOSS2/paper.md index aafa1c4..1829c8b 100644 --- a/docs/JOSS2/paper.md +++ b/docs/JOSS2/paper.md @@ -52,44 +52,44 @@ bibliography: paper.bib `PySensors` is a Python package for selecting and placing a sparse set of sensors for reconstruction and classification tasks. In this major update to `PySensors`, we introduce spatially constrained sensor placement capabilities, allowing users to enforce constraints such as maximum or exact sensor counts in specific regions, incorporate predetermined sensor locations, and maintain minimum distances between sensors. We extend functionality to support custom basis inputs, enabling integration of any data-driven or spectral basis. We also propose a thermodynamic approach that goes beyond a single ``optimal'' sensor configuration and maps the complete landscape of sensor interactions induced by the training data. This comprehensive view facilitates integration with external selection criteria and enables assessment of sensor replacement impacts. The new optimization technique also accounts for over- and under-sampling of sensors, utilizing a regularized least squares approach for robust reconstruction. Additionally, we incorporate noise-induced uncertainty quantification of the estimation error and provide visual uncertainty heat maps to guide deployment decisions. To highlight these additions, we provide a brief description of the mathematical algorithms and theory underlying these new capabilities. We demonstrate the usage of new features with illustrative code examples and include practical advice for implementation across various application domains. Finally, we outline a roadmap of potential extensions to further enhance the package's functionality and applicability to emerging sensing challenges. # Statement of need -Sensor placement is critical for efficient monitoring, control, and decision-making in modern engineering systems. Sensors play a crucial role in characterizing spatio-temporal dynamics in high-dimensional, non-linear systems such as fluid flows [erichson2020shallow], manufacturing [manohar2018predicting], geophysical [alonso2010novel] and nuclear systems [karnik2024constrained]. Optimal sensor placement ensures accurate, real-time tracking of key system variables with minimal hardware and enables cost-effective, real-time system analysis and control. In general, sensor placement optimization is NP-hard and cannot be solved in polynomial time. There are ${n \choose p} = n!/((n-p)!p!)$ possible combinations of choosing $p$ sensors from an $n$-dimensional state. -Common approaches to optimizing sensor placement include maximizing the information criteria [krause2008near], Bayesian Optimal Experimental Design [alexanderian2021optimal], compressed sensing [donoho2006compressed], and heuristic methods. Many sensor placement methods have submodular objective form, which sets guarantees on how close an efficient greedy placement can be to the unknown true optimum [summers2015submodularity]. Sub-modular objectives can be efficiently optimized for hundreds or thousands of candidate locations using convex [joshi2008sensor] or greedy optimization approaches [summers2015submodularity] . +Sensor placement is critical for efficient monitoring, control, and decision-making in modern engineering systems. Sensors play a crucial role in characterizing spatio-temporal dynamics in high-dimensional, non-linear systems such as fluid flows [@erichson2020shallow], manufacturing [@manohar2018predicting], geophysical [@alonso2010novel] and nuclear systems [@karnik2024constrained]. Optimal sensor placement ensures accurate, real-time tracking of key system variables with minimal hardware and enables cost-effective, real-time system analysis and control. In general, sensor placement optimization is NP-hard and cannot be solved in polynomial time. There are ${n \choose p} = n!/((n-p)!p!)$ possible combinations of choosing $p$ sensors from an $n$-dimensional state. +Common approaches to optimizing sensor placement include maximizing the information criteria [@krause2008near], Bayesian Optimal Experimental Design [@alexanderian2021optimal], compressed sensing [@donoho2006compressed], and heuristic methods. Many sensor placement methods have submodular objective form, which sets guarantees on how close an efficient greedy placement can be to the unknown true optimum [@summers2015submodularity]. Sub-modular objectives can be efficiently optimized for hundreds or thousands of candidate locations using convex [joshi2008sensor] or greedy optimization approaches [@summers2015submodularity] . -![An overview image of capabilities of Pysensors](docs/JOSS2/Fig1.jpeg "PySensors 2.0 expands its capabilities by introducing custom basis functions, optimizers, constraints, solvers, and uncertainty quantification, enabling constrained sensing, over- and under-sampling, and uncertainty quantification in the presence of noisy sensor measurements.") +![An overview image of capabilities of Pysensors](../Fig1.jpeg "PySensors 2.0 expands its capabilities by introducing custom basis functions, optimizers, constraints, solvers, and uncertainty quantification, enabling constrained sensing, over- and under-sampling, and uncertainty quantification in the presence of noisy sensor measurements.") -`PySensors` is a Python package [de2021pysensors] dedicated to solving the complex challenge of optimal sensor placement in data-driven systems. It implements advanced sparse optimization algorithms that use dimensionality reduction techniques to identify the most informative measurement locations with remarkable efficiency [manohar2018data;brunton2016sparse;clark2020multi]. It helps users identify the best locations for sensors when working with complex high dimensional data, focusing on both reconstruction [manohar2018data] and classification [brunton2016sparse] tasks. The package follows `scikit-learn` conventions for user-friendly access while offering advanced customization options for experienced users. Designed with researchers and practitioners in mind, `PySensors` provides open-source, accessible tools that support model discovery across various scientific applications. +`PySensors` is a Python package [@de2021pysensors] dedicated to solving the complex challenge of optimal sensor placement in data-driven systems. It implements advanced sparse optimization algorithms that use dimensionality reduction techniques to identify the most informative measurement locations with remarkable efficiency [@manohar2018data;@brunton2016sparse;@clark2020multi]. It helps users identify the best locations for sensors when working with complex high dimensional data, focusing on both reconstruction [@manohar2018data] and classification [@brunton2016sparse] tasks. The package follows `scikit-learn` conventions for user-friendly access while offering advanced customization options for experienced users. Designed with researchers and practitioners in mind, `PySensors` provides open-source, accessible tools that support model discovery across various scientific applications. -This new version of `Pysensors` focuses specifically on practical engineering applications where measurement data is inherently noisy and spatial deployment constraints are unavoidable. Key improvements include constraint-aware optimization methods that handle spatial restrictions and sensor density limitations. In addition, the framework introduces uncertainty quantification metrics that track how measurement noise propagates through reconstruction algorithms, enabling robust error estimation in sensor outputs. This version of `Pysensors` implements methodologies introduced by Klishin et al. [klishin2023data] and Karnik et al. [karnik2024constrained] to make them accessible to scientists and engineers in all domains. These enhancements transform `PySensors` from a purely academic tool into a practical platform for solving real-world sensing challenges while maintaining mathematical rigor. +This new version of `Pysensors` focuses specifically on practical engineering applications where measurement data is inherently noisy and spatial deployment constraints are unavoidable. Key improvements include constraint-aware optimization methods that handle spatial restrictions and sensor density limitations. In addition, the framework introduces uncertainty quantification metrics that track how measurement noise propagates through reconstruction algorithms, enabling robust error estimation in sensor outputs. This version of `Pysensors` implements methodologies introduced by Klishin et al. [@klishin2023data] and Karnik et al. [@karnik2024constrained] to make them accessible to scientists and engineers in all domains. These enhancements transform `PySensors` from a purely academic tool into a practical platform for solving real-world sensing challenges while maintaining mathematical rigor. -Pysensors implements `Sparse Sensor Placement Optimization for Reconstruction (SSPOR)`[manohar2018data] of full fields $\mathbf{x}\in\reals^n$ from $p$ noise-corrupted sensor measurements $\mathbf{y}\in\reals^p$ +Pysensors implements `Sparse Sensor Placement Optimization for Reconstruction (SSPOR)`[@manohar2018data] of full fields $\mathbf{x}\in\mathbb{R}^n$ from $p$ noise-corrupted sensor measurements $\mathbf{y}\in\mathbb{R}^p$ $$ - \mathbf{y} = \mathbb{S} \mathbf{x} + \boldsymbol\eta, +\mathbf{y} = \mathbb{S} \mathbf{x} + \boldsymbol\eta $$ -where $\boldsymbol{\eta}$ consists of zero-mean, Gaussian independent and identically distributed (i.i.d.) components, and $\mathbb{S} \in \reals^{p\times n}$ is the desired sensor (measurement) selection operator. +where $\boldsymbol{\eta}$ consists of zero-mean, Gaussian independent and identically distributed (i.i.d.) components, and $\mathbb{S} \in \mathbb{R}^{p\times n}$ is the desired sensor (measurement) selection operator. This measurement selection operator $\mathbb{S}$ encodes point measurements with unit entries in a sparse matrix $\mathbb{S} = \begin{bmatrix} \mathbf{e}_{\gamma_1} & \mathbf{e}_{\gamma_2} & \hdots & \mathbf{e}_{\gamma_ p}\end{bmatrix}^T$ where $\mathbf{e}_j$ are canonical basis vectors for $\mathbb{R}^n$, with a unit entry in component $j$ (where a sensor should be placed) and zeroes elsewhere. Here, $\gamma = \{\gamma_1, \gamma_2, \hdots, \gamma_p\} \subset \{1, 2, \hdots, n\}$ denotes the index set of sensor locations with cardinality $p$. Sensor selection then corresponds to the components of $\mathbf{x}$ that were chosen to be measured: $\mathbf{\mathbb{S} x} = \begin{bmatrix} x_{\gamma_1} & x_{\gamma_2} & \hdots & x_{\gamma_p}\end{bmatrix}^T$. -The `SSPOR` class selects these sensors through a cohesive computational framework for strategically minimizing sensor deployment while maintaining reconstruction fidelity in high-dimensional systems. When initialized with measurement data $\mathbf{x}\in\reals^n$, `SSPOR` first applies dimensional reduction through basis identification, then employs the computationally efficient QR factorization algorithm to determine optimal sensor locations. This approach, which has demonstrated efficacy in both reduced-order modeling and sparse sensing applications, strategically leverages the inherent structure of the identified basis to prioritize the most informative measurement points. +The `SSPOR` class selects these sensors through a cohesive computational framework for strategically minimizing sensor deployment while maintaining reconstruction fidelity in high-dimensional systems. When initialized with measurement data $\mathbf{x}\in\mathbb{R}^n$, `SSPOR` first applies dimensional reduction through basis identification, then employs the computationally efficient QR factorization algorithm to determine optimal sensor locations. This approach, which has demonstrated efficacy in both reduced-order modeling and sparse sensing applications, strategically leverages the inherent structure of the identified basis to prioritize the most informative measurement points. -The `Sparse Sensor Placement Optimization for Classification (SSPOC)` framework identifies minimal sensor configurations that can classify high dimensional signals $\mathbf{x}\in\reals^n$ as one of $c$ classes. Unlike traditional compressed sensing [baraniuk2010model;donoho2006compressed;candes2006stable] approaches that focus on signal reconstruction, `SSPOC` specifically targets the preservation of classification boundaries in feature space $\boldsymbol{\Psi}_r$ by identifying the sparsest sensor set capable of reconstructing discriminant hyperplanes. +The `Sparse Sensor Placement Optimization for Classification (SSPOC)` framework identifies minimal sensor configurations that can classify high dimensional signals $\mathbf{x}\in\mathbb{R}^n$ as one of $c$ classes. Unlike traditional compressed sensing [@baraniuk2010model;@donoho2006compressed;@candes2006stable] approaches that focus on signal reconstruction, `SSPOC` specifically targets the preservation of classification boundaries in feature space $\boldsymbol{\Psi}_r$ by identifying the sparsest sensor set capable of reconstructing discriminant hyperplanes. The `SSPOC` architecture accepts any combination of basis representation and linear classifier, defaulting to Linear Discriminant Analysis (LDA) and Identity basis when not otherwise specified. The optimization pipeline proceeds through multiple stages: dimensional reduction via basis fitting, classifier training within the reduced space, sparse optimization incorporating both classifier weights and basis structures, sensor selection based on optimization outputs, and optional classifier retraining using only measurements from the selected locations. -High dimensional field dynamics $\mathbf{x}\in\reals^n$ can be represented as a linear combination of spatial basis modes $\boldsymbol{\Psi}$ weighted by time-varying coefficients $\mathbf{a}$ +High dimensional field dynamics $\mathbf{x}\in\mathbb{R}^n$ can be represented as a linear combination of spatial basis modes $\boldsymbol{\Psi}$ weighted by time-varying coefficients $\mathbf{a}$ $$\mathbf{x} = \boldsymbol{\Psi}_r\mathbf{a}.$$ -This basis, which can be built from spectral or data-driven decomposition methods, is typically chosen so that the embedding dimension is as small as possible, i.e., $r\ll n$. Different basis functions can significantly impact sensor selection effectiveness and reconstruction quality [manohar2018data]. PySensors offers several interchangeable `basis` options for sparse sensor selection: +This basis, which can be built from spectral or data-driven decomposition methods, is typically chosen so that the embedding dimension is as small as possible, i.e., $r\ll n$. Different basis functions can significantly impact sensor selection effectiveness and reconstruction quality [@manohar2018data]. PySensors offers several interchangeable `basis` options for sparse sensor selection: - `Identity`: Processes raw measurement data directly without transformation. - `SVD`: Utilizes truncated singular value decomposition's $\mathbf{X} = \mathbf{U}_r \mathbf{\Sigma}_r \mathbf{V}^*_r$ left singular vectors $\mathbf{U}_r$, computing only the specified number of modes to reduce computational demands. A randomized SVD option further enhances efficiency. -- `RandomProjection`: Projects measurements into a new space by multiplying them with random Gaussian vectors, connecting to compressed sensing methodologies from established literature [baraniuk2010model;donoho2006compressed;candes2006stable]. -- `Custom`: This is a new option included in `Pysensors 2.0` that enables users to leverage custom basis functions beyond `PySensors` built-in options. Researchers can transform their data into an alternative basis such as dynamic mode decomposition modes [schmid2010dynamic], before passing it to a PySensors instance configured with the Custom basis. +- `RandomProjection`: Projects measurements into a new space by multiplying them with random Gaussian vectors, connecting to compressed sensing methodologies from established literature [@baraniuk2010model;@donoho2006compressed;@candes2006stable]. +- `Custom`: This is a new option included in `Pysensors 2.0` that enables users to leverage custom basis functions beyond `PySensors` built-in options. Researchers can transform their data into an alternative basis such as dynamic mode decomposition modes [@schmid2010dynamic], before passing it to a PySensors instance configured with the Custom basis. -When a user specifies $p$ sensors, SSPOR first fits a basis $\bTi_r$ to the data and optimizes sensor placement by minimizing the following objective function: +When a user specifies $p$ sensors, SSPOR first fits a basis $\boldsymbol{\Psi}_r$ to the data and optimizes sensor placement by minimizing the following objective function: $$ \gamma_* = \argmax_{\gamma, |\gamma|= p} \; \log \det((\mathbb{S} \boldsymbol{\Psi}_r)^T(\mathbb{S} \boldsymbol{\Psi}_r)).\label{eq:detobj} \tag{1} $$ where $\gamma_*$ denotes the index set of optimized sensor locations with cardinality $p$. When $p = r$, is equivalent to the maximizer of $\log |\det(\mathbb{S} \boldsymbol{\Psi}_r)|$. -Direct optimization of this criterion leads to a brute force combinatorial search. This sensor placement approach builds upon the empirical interpolation method (EIM) [barrault2004empirical] and discrete empirical interpolation method (DEIM) [chaturantabut2010nonlinear] to develop a greedy strategy for optimizing sensor selection built upon the pivoted QR factorization[drmac2016new;manohar2018data;manohar2018predicting;manohar2021optimal]. +Direct optimization of this criterion leads to a brute force combinatorial search. This sensor placement approach builds upon the empirical interpolation method (EIM) [@barrault2004empirical] and discrete empirical interpolation method (DEIM) [@chaturantabut2010nonlinear] to develop a greedy strategy for optimizing sensor selection built upon the pivoted QR factorization[@drmac2016new;@manohar2018data;@manohar2018predicting;@manohar2021optimal]. -PySensors implements the `QR` optimizer for optimal sensor selection in unconstrained scenarios where the number of sensors $p$ equals the number of modes $r$. The framework incorporates heterogeneous cost functions into the optimization process to accommodate practical deployment constraints. For example, when monitoring oceanographic parameters, the system can account for the substantially higher costs of deep-sea sensors relative to coastal installations[clark2020multi]. This capability is implemented through the `Cost-Constrained QR (CCQR)` algorithm in the optimizers submodule, allowing users to balance information capture against resource limitations when designing sensor networks for complex physical systems. +PySensors implements the `QR` optimizer for optimal sensor selection in unconstrained scenarios where the number of sensors $p$ equals the number of modes $r$. The framework incorporates heterogeneous cost functions into the optimization process to accommodate practical deployment constraints. For example, when monitoring oceanographic parameters, the system can account for the substantially higher costs of deep-sea sensors relative to coastal installations[@clark2020multi]. This capability is implemented through the `Cost-Constrained QR (CCQR)` algorithm in the optimizers submodule, allowing users to balance information capture against resource limitations when designing sensor networks for complex physical systems. Traditional QR factorization presents challenges in under-sampling $p < r$ and over-sampling $p > r$ scenarios, as well as when spatial constraints must be considered. `PySensors 2.0` addresses these limitations through two new optimization algorithms: `Generalized QR (GQR)` and `Two Point GReedy (TPGR)`. @@ -108,7 +108,7 @@ Many engineering applications have extreme operating conditions, high costs, lim To implement spatial constraints in our sensor placement methodology, we must first define the shape of our constraint region and then identify which grid points fall within the designated constraint regions. We accomplish this using the specialized classes and functions provided in the `_constraints.py` module in `utils`. The classes define constraint regions in various geometric shapes including `Circle`, `Ellipse`, `Polygon`, `Parabola`, `Line`, and `Cylinder`. Additionally, we provide `UserDefinedConstraints` options that allow users to supply their own constraints either as a Python`.py` file containing their constraint shape definition or as an equation string. These classes work with two distinct data formats: image data defined by pixel coordinates and tabular data stored in dataframes with explicit x, y, and z values. Upon instantiation of the class, users must provide the input `data` in one of two formats: a matrix representation for images or a structured dataframe for tabular data. In the latter case, the class requires additional keyword arguments `kwargs` specifying the columns corresponding to `X_axis`, `Y_axis`, `Z_axis`, and `Field` parameters, thereby enabling the extraction of relevant data from their respective columns for subsequent analysis. -The D-optimal objective in \ref{detobj} suffers from two limitations: it is not defined for the under-sampling case $p r)scenarios , select TPGR optimizer. In noisy environments enable uncertainty quantification for robust results.") +![A flowchart to suggest which method to use](../Fig2.jpeg "When selecting a sensing method in PySensors, consider your primary objective: For field reconstruction in standard settings, use QR with Identity or SVD basis. For classification tasks, leverage SVD basis with SSPOC optimizer. When facing spatial constraints, choose GQR optimizer. For under-sampling (p < r) and over-sampling cases (p > r)scenarios , select TPGR optimizer. In noisy environments enable uncertainty quantification for robust results.") # Acknowledgments From 7a02ec1920b7dd48542818257b1df7022118d8c2 Mon Sep 17 00:00:00 2001 From: Niharika Karnik Date: Thu, 14 Aug 2025 10:50:20 -0700 Subject: [PATCH 3/4] Fixing path of images --- docs/JOSS2/paper.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/JOSS2/paper.md b/docs/JOSS2/paper.md index 1829c8b..9554a97 100644 --- a/docs/JOSS2/paper.md +++ b/docs/JOSS2/paper.md @@ -55,7 +55,7 @@ bibliography: paper.bib Sensor placement is critical for efficient monitoring, control, and decision-making in modern engineering systems. Sensors play a crucial role in characterizing spatio-temporal dynamics in high-dimensional, non-linear systems such as fluid flows [@erichson2020shallow], manufacturing [@manohar2018predicting], geophysical [@alonso2010novel] and nuclear systems [@karnik2024constrained]. Optimal sensor placement ensures accurate, real-time tracking of key system variables with minimal hardware and enables cost-effective, real-time system analysis and control. In general, sensor placement optimization is NP-hard and cannot be solved in polynomial time. There are ${n \choose p} = n!/((n-p)!p!)$ possible combinations of choosing $p$ sensors from an $n$-dimensional state. Common approaches to optimizing sensor placement include maximizing the information criteria [@krause2008near], Bayesian Optimal Experimental Design [@alexanderian2021optimal], compressed sensing [@donoho2006compressed], and heuristic methods. Many sensor placement methods have submodular objective form, which sets guarantees on how close an efficient greedy placement can be to the unknown true optimum [@summers2015submodularity]. Sub-modular objectives can be efficiently optimized for hundreds or thousands of candidate locations using convex [joshi2008sensor] or greedy optimization approaches [@summers2015submodularity] . -![An overview image of capabilities of Pysensors](../Fig1.jpeg "PySensors 2.0 expands its capabilities by introducing custom basis functions, optimizers, constraints, solvers, and uncertainty quantification, enabling constrained sensing, over- and under-sampling, and uncertainty quantification in the presence of noisy sensor measurements.") +![An overview image of capabilities of Pysensors](/Fig1.jpeg "PySensors 2.0 expands its capabilities by introducing custom basis functions, optimizers, constraints, solvers, and uncertainty quantification, enabling constrained sensing, over- and under-sampling, and uncertainty quantification in the presence of noisy sensor measurements.") `PySensors` is a Python package [@de2021pysensors] dedicated to solving the complex challenge of optimal sensor placement in data-driven systems. It implements advanced sparse optimization algorithms that use dimensionality reduction techniques to identify the most informative measurement locations with remarkable efficiency [@manohar2018data;@brunton2016sparse;@clark2020multi]. It helps users identify the best locations for sensors when working with complex high dimensional data, focusing on both reconstruction [@manohar2018data] and classification [@brunton2016sparse] tasks. The package follows `scikit-learn` conventions for user-friendly access while offering advanced customization options for experienced users. Designed with researchers and practitioners in mind, `PySensors` provides open-source, accessible tools that support model discovery across various scientific applications. @@ -131,7 +131,7 @@ For any choice of the reconstruction matrix $\mathbf{A}$, the reconstruction $\h \mathbf{K}=\boldsymbol{\Psi}_r \mathbf{B} \mathbf{B}^T \boldsymbol{\Psi}_r^T;\quad \mathbf{B}=\eta \mathbf{A}. \label{eqn:K} \end{align} -![A flowchart to suggest which method to use](../Fig2.jpeg "When selecting a sensing method in PySensors, consider your primary objective: For field reconstruction in standard settings, use QR with Identity or SVD basis. For classification tasks, leverage SVD basis with SSPOC optimizer. When facing spatial constraints, choose GQR optimizer. For under-sampling (p < r) and over-sampling cases (p > r)scenarios , select TPGR optimizer. In noisy environments enable uncertainty quantification for robust results.") +![A flowchart to suggest which method to use](/Fig2.jpeg "When selecting a sensing method in PySensors, consider your primary objective: For field reconstruction in standard settings, use QR with Identity or SVD basis. For classification tasks, leverage SVD basis with SSPOC optimizer. When facing spatial constraints, choose GQR optimizer. For under-sampling (p < r) and over-sampling cases (p > r)scenarios , select TPGR optimizer. In noisy environments enable uncertainty quantification for robust results.") # Acknowledgments From 6880d77b786bc9a4ff1931e0c25ca1444e62f70e Mon Sep 17 00:00:00 2001 From: Niharika Karnik Date: Thu, 14 Aug 2025 11:12:59 -0700 Subject: [PATCH 4/4] Correction in the JOSS2 paper --- docs/JOSS/paper.bib | 820 -------------------------------------------- docs/JOSS/paper.md | 98 ------ docs/JOSS2/paper.md | 29 +- 3 files changed, 15 insertions(+), 932 deletions(-) delete mode 100644 docs/JOSS/paper.bib delete mode 100644 docs/JOSS/paper.md diff --git a/docs/JOSS/paper.bib b/docs/JOSS/paper.bib deleted file mode 100644 index 16fc61f..0000000 --- a/docs/JOSS/paper.bib +++ /dev/null @@ -1,820 +0,0 @@ -@article{ko1995exact, - title={An exact algorithm for maximum entropy sampling}, - author={Ko, Chun-Wa and Lee, Jon and Queyranne, Maurice}, - journal={Operations Research}, - volume={43}, - number={4}, - pages={684--691}, - year={1995}, - publisher={INFORMS}, - doi={10.1287/opre.43.4.684}, -} - -@inproceedings{narayanan2020toolkit, - title={A toolkit for spatial interpolation and sensor placement}, - author={Narayanan, S Deepak and Patel, Zeel B and Agnihotri, Apoorv and Batra, Nipun}, - booktitle={Proceedings of the 18th Conference on Embedded Networked Sensor Systems}, - pages={653--654}, - year={2020}, - doi={10.1145/3384419.3430407}, -} - -@article{manohar2019optimized, - title={Optimized sampling for multiscale dynamics}, - author={Manohar, Krithika and Kaiser, Eurika and Brunton, Steven L and Kutz, J Nathan}, - journal={Multiscale Modeling \& Simulation}, - volume={17}, - number={1}, - pages={117--136}, - year={2019}, - publisher={SIAM}, - doi={10.1137/17M1162366}, -} - -@article{clark2018greedy, - title={Greedy sensor placement with cost constraints}, - author={Clark, Emily and Askham, Travis and Brunton, Steven L and Kutz, J Nathan}, - journal={IEEE Sensors Journal}, - volume={19}, - number={7}, - pages={2642--2656}, - year={2018}, - publisher={IEEE}, - doi={10.1109/JSEN.2018.2887044}, -} - -@article{manohar2018data, - title={Data-driven sparse sensor placement for reconstruction: Demonstrating the benefits of exploiting known patterns}, - author={Manohar, Krithika and Brunton, Bingni W and Kutz, J Nathan and Brunton, Steven L}, - journal={IEEE Control Systems Magazine}, - volume={38}, - number={3}, - pages={63--86}, - year={2018}, - publisher={IEEE}, - doi={10.1109/mcs.2018.2810460} -} - -@article{brunton2016sparse, - title={Sparse sensor placement optimization for classification}, - author={Brunton, Bingni W and Brunton, Steven L and Proctor, Joshua L and Kutz, J Nathan}, - journal={SIAM Journal on Applied Mathematics}, - volume={76}, - number={5}, - pages={2099--2122}, - year={2016}, - publisher={SIAM}, - doi={10.1137/15M1036713} -} - -@article{summers2015submodularity, - title={On submodularity and controllability in complex dynamical networks}, - author={Summers, Tyler H and Cortesi, Fabrizio L and Lygeros, John}, - journal={IEEE Transactions on Control of Network Systems}, - volume={3}, - number={1}, - pages={91--101}, - year={2015}, - publisher={IEEE} -} - -@article{chen2011h, - title={H 2 optimal actuator and sensor placement in the linearised complex Ginzburg-Landau system}, - author={Chen, Kevin K and Rowley, Clarence W}, - journal={Journal of Fluid Mechanics}, - volume={681}, - number={241-260}, - year={2011}, - publisher={Cambridge University Press, The Edinburgh Building, Cambridge CB 2 2 RU~…}, - doi={10.1017/jfm.2011.195} -} - -@article{krause2008near, - title={Near-optimal sensor placements in Gaussian processes: Theory, efficient algorithms and empirical studies}, - author={Krause, Andreas and Singh, Ajit and Guestrin, Carlos}, - journal={Journal of Machine Learning Research}, - volume={9}, - number={Feb}, - pages={235--284}, - year={2008} -} - -@article{joshi2008sensor, - title={Sensor selection via convex optimization}, - author={Joshi, Siddharth and Boyd, Stephen}, - journal={IEEE Transactions on Signal Processing}, - volume={57}, - number={2}, - pages={451--462}, - year={2009}, - publisher={IEEE} -} - -@article{willcox2006unsteady, - title={Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition}, - author={Willcox, Karen}, - journal={Computers \& fluids}, - volume={35}, - number={2}, - pages={208--226}, - year={2006}, - publisher={Elsevier}, - doi={10.2514/6.2004-2415} -} - -@article{brunton2016pnas, - title = {Discovering governing equations from data by sparse identification of nonlinear dynamical systems}, - author = {Brunton, Steven L. and Proctor, Joshua L. and Kutz, J. 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Brunton - affiliation: 4 - orcid: 0000-0002-4831-3466 - - name: Steven L. Brunton - affiliation: 2 - orcid: 0000-0002-6565-5118 - - name: J. Nathan Kutz - affiliation: 1 - orcid: 0000-0002-6004-2275 -affiliations: - - name: Department of Applied Mathematics, University of Washington - index: 1 - - name: Department of Mechanical Engineering, University of Washington - index: 2 - - name: Department of Physics, University of Washington - index: 3 - - name: Department of Biology, University of Washington - index: 4 -date: 13 February 2021 -bibliography: paper.bib ---- - - -# Summary - -Successful predictive modeling and control of engineering and natural processes is often entirely determined by *in situ* measurements and feedback from sensors [@Brunton2019book], which provide measurements of the state of these processes at specific points in space and time. -However, deploying sensors into complex environments, including in application areas such as manufacturing [@Manohar2018jms], geophysical environments [@Yildirim:2009], and biological processes [@colvert2017local;@Mohren2018pnas], is often expensive and challenging. -Furthermore, modeling outcomes are extremely sensitive to the location and number of these sensors, motivating optimization strategies for the principled placement of sensors for different decision-making tasks. -In general, choosing the globally optimal placement within the search space of a large-scale complex system is an intractable computation, in which the number of possible placements grows combinatorially with the number of candidates [@ko1995exact]. -While sensor placement has traditionally been guided by expert knowledge and first principles models, increases in system complexity, emerging sensor technologies, and innovations in data-driven modeling strategies motivates automated algorithms for optimizing sensor placements. - -`PySensors` is a Python package for the scalable optimization of sensor placement from data. In particular, `PySensors` provides tools for sparse sensor placement optimization approaches that employ data-driven dimensionality reduction [@brunton2016sparse;@manohar2018data]. This approach results in near-optimal placements for various decision-making tasks and can be readily customized using different optimization algorithms and objective functions. - -The `PySensors` package can be used by both researchers looking to advance state-of-the-art methods and practitioners seeking simple sparse sensor selection methods for their applications of interest. -Straightforward methods and abundant examples help new users to quickly and efficiently leverage existing methods to their advantage. -At the same time, modular classes leave flexibility for users to experiment with and plug in new sensor selection algorithms or dimensionality reduction techniques. -Users of `scikit-learn` will find `PySensors` objects familiar, intuitive, and compatible with existing `scikit-learn` routines such as cross-validation [@scikit-learn]. - -# Statement of need -Maximizing the impact of sensor placement algorithms requires tools to make them accessible to scientists and engineers across various domains and at various levels of mathematical expertise and sophistication. `PySensors` unifies the algorithms developed in the papers [@manohar2018data;@clark2018greedy;@brunton2016sparse] and their accompanying codes `SSPOR_pub` and `SSPOC_pub` into one software package. The only other packages in this domain of which we are aware are `Chama` [@klise2017sensor] and `Polire` [@narayanan2020toolkit]. While these packages and `PySensors` all enable sparse sensor placement optimization, `Chama` and `Polire` are geared towards event detection and Gaussian processes respectively, whereas `PySensors` is aimed at signal reconstruction and classification tasks. -As such, there are marked differences in the objective functions optimized by `PySensors` and its precursors. -In addition to these two packages, researchers and practitioners have made available various custom scripts for sensor placement. -Currently, researchers seeking to employ modern sensor placement methods must choose between implementing them from scratch or manually augmenting existing unpolished codes. - -Reconstruction and classification tasks often arise in the modeling, prediction, and control of complex processes in geophysics, fluid dynamics, biology, and manufacturing. -The goal of _reconstruction_ is to recover a high-dimensional signal $\mathbf{x}\in\mathbb{R}^N$ from a limited number of $p$ measurements $y_ i = \mathbf{c}_ i^\top \mathbf{x}$, where each $\mathbf{c}_ i \in \mathbb{R}^N$ represents the action of a sensor. For example, $\mathbf{c}_ i^\top = [1, 0, 0, \dots, 0]$ represents a sensor which takes a point measurement of the first dimension of the signal $\mathbf{x}$. `PySensors` selects a set of $p$ sensors out of $N$ candidates $\mathbf{c}_ i^\top$ (rows of a measurement matrix $\mathbf{C}:\mathbf{y} = \mathbf{Cx}$) that minimize reconstruction error in a data-dependent basis $\mathbf{\Phi}\in\mathbb{R}^{N\times r}$ -$$ \mathbf{C}^ \star= \underset{\mathbf{C}\in\mathbb{R}^{p\times N}}{\arg\min} \|\mathbf{x} - \mathbf{\Phi}(\mathbf{C\Phi})^{\dagger} \mathbf{y}\|_ 2^2, $$ -where $\dagger$ denotes the Moore-Penrose pseudoinverse. The key innovation is to recover the low-dimensional representation $\mathbf{x}_ r \in \mathbb{R}^r$ satisfying $\mathbf{x} = \mathbf{\Phi x}_ r$ via the reconstruction map $\mathbf{\Phi}(\mathbf{C\Phi})^{\dagger}$, ultimately reducing sensor placement to a highly efficient matrix pivoting operation [@manohar2018data]. Similarly, sensor placement for _classification_ [@brunton2016sparse] optimizes the sparsest vector $\mathbf{s}^ \star$ that reconstructs $\mathbf{w}: \mathbf{\Phi}^\dagger\mathbf{s} = \mathbf{w}$ in the low-dimensional feature space, where $\mathbf{w}$ is the the set of weights learned by a linear classifier fit to $\mathbf{x}_ r$. -In this case, the optimal sensor locations are determined by the nonzero components of $\mathbf{s}^ \star$. - -The basis $\mathbf{\Phi}$ is explicitly computed from the data using powerful dimensionality reduction techniques such as principal components analysis (PCA) and random projections, which enable significant compression of most signals to $r\ll N$ dimensions. PCA extracts the dominant spatial correlations or _principal components_, the leading eigenvectors of the data covariance matrix. It is computed using the matrix singular value decomposition (SVD) and is closely related to proper orthogonal decomposition (POD); POD modes and principal components are equivalent. -Other basis choices are possible, such as dynamic mode decomposition for extracting temporally correlated features [@manohar2019optimized]. - - - - -# Features - -`PySensors` enables the sparse placement of sensors for two classes of problems: reconstruction and classification. -For reconstruction problems the package implements a unified `SensorSelector` class, with methods for efficiently analyzing the effects that data or sensor quantity have on reconstruction performance [@manohar2018data]. -Sensor selection is based on the computationally efficient QR algorithm. -Often different sensor locations impose variable costs, e.g. if measuring sea-surface temperature, it may be more expensive to place buoys/sensors in the middle of the ocean than close to shore. -These costs can be taken into account during sensor selection via a built-in cost-sensitive optimization routine [@clark2018greedy]. -For classification tasks, the package implements the Sparse Sensor Placement Optimization for Classification (SSPOC) algorithm [@brunton2016sparse], allowing one to optimize sensor placement for classification accuracy. -The algorithm is related to compressed sensing optimization [@Candes2006cpam;@Donoho2006ieeetit;@Baraniuk2007ieeespm], but identifies the sparsest set of sensors that reconstructs a discriminating plane in a feature subspace. -This SSPOC implementation is fully general in the sense that it can be used in conjunction with any linear classifier. -Additionally, `PySensors` provides methods to enable straightforward exploration of the impacts of primary hyperparameters like the number of sensors or basis modes. - -It is well known [@manohar2018data] that the basis in which one represents measurement data can have a pronounced effect on the sensors that are selected and the quality of the reconstruction. -Users can readily switch between different bases typically employed for sparse sensor selection, including POD modes and random projections. -Because `PySensors` was built with `scikit-learn` compatibility in mind, it is easy to use cross-validation to select among possible choices of bases, basis modes, and other hyperparameters. - -Finally, included with `PySensors` is a large suite of examples, implemented as Jupyter notebooks. -Some of the examples are written in a tutorial format and introduce new users to the objects, methods, and syntax of the package. -Other examples demonstrate intermediate-level concepts such as how to visualize model parameters and performance, how to combine `scikit-learn` and `PySensors` objects, selecting appropriate parameter values via cross-validation, and other best-practices. -Further notebooks use `PySensors` to solve challenging real-world problems. -The notebooks reproduce many of the examples from the papers upon which the package is based [@manohar2018data;@clark2018greedy;@brunton2016sparse]. -To help users begin applying `PySensors` to their own datasets even faster, interactive versions of every notebook are available on Binder. -Together with comprehensive documentation, the examples will compress the learning curve of learning a new software package. - -# Acknowledgments -The authors acknowledge support from the Air Force Office of Scientific Research (AFOSR FA9550-19-1-0386) and The Boeing Corporation. The work of KM is supported by the National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship (award 1803663). JNK acknowledges support from the Air Force Office of Scientific Research (AFOSR FA9550-19-1-0011) - -# References \ No newline at end of file diff --git a/docs/JOSS2/paper.md b/docs/JOSS2/paper.md index 9554a97..322c77a 100644 --- a/docs/JOSS2/paper.md +++ b/docs/JOSS2/paper.md @@ -49,13 +49,13 @@ bibliography: paper.bib # Summary -`PySensors` is a Python package for selecting and placing a sparse set of sensors for reconstruction and classification tasks. In this major update to `PySensors`, we introduce spatially constrained sensor placement capabilities, allowing users to enforce constraints such as maximum or exact sensor counts in specific regions, incorporate predetermined sensor locations, and maintain minimum distances between sensors. We extend functionality to support custom basis inputs, enabling integration of any data-driven or spectral basis. We also propose a thermodynamic approach that goes beyond a single ``optimal'' sensor configuration and maps the complete landscape of sensor interactions induced by the training data. This comprehensive view facilitates integration with external selection criteria and enables assessment of sensor replacement impacts. The new optimization technique also accounts for over- and under-sampling of sensors, utilizing a regularized least squares approach for robust reconstruction. Additionally, we incorporate noise-induced uncertainty quantification of the estimation error and provide visual uncertainty heat maps to guide deployment decisions. To highlight these additions, we provide a brief description of the mathematical algorithms and theory underlying these new capabilities. We demonstrate the usage of new features with illustrative code examples and include practical advice for implementation across various application domains. Finally, we outline a roadmap of potential extensions to further enhance the package's functionality and applicability to emerging sensing challenges. +`PySensors` is a Python package for selecting and placing a sparse set of sensors for reconstruction and classification tasks. In this major update to `PySensors`, we introduce spatially constrained sensor placement capabilities, allowing users to enforce constraints such as maximum or exact sensor counts in specific regions, incorporate predetermined sensor locations, and maintain minimum distances between sensors. We extend functionality to support custom basis inputs, enabling integration of any data-driven or spectral basis. We also propose a thermodynamic approach that goes beyond a single "optimal" sensor configuration and maps the complete landscape of sensor interactions induced by the training data. This comprehensive view facilitates integration with external selection criteria and enables assessment of sensor replacement impacts. The new optimization technique also accounts for over- and under-sampling of sensors, utilizing a regularized least squares approach for robust reconstruction. Additionally, we incorporate noise-induced uncertainty quantification of the estimation error and provide visual uncertainty heat maps to guide deployment decisions. To highlight these additions, we provide a brief description of the mathematical algorithms and theory underlying these new capabilities. We demonstrate the usage of new features with illustrative code examples and include practical advice for implementation across various application domains. Finally, we outline a roadmap of potential extensions to further enhance the package's functionality and applicability to emerging sensing challenges. # Statement of need Sensor placement is critical for efficient monitoring, control, and decision-making in modern engineering systems. Sensors play a crucial role in characterizing spatio-temporal dynamics in high-dimensional, non-linear systems such as fluid flows [@erichson2020shallow], manufacturing [@manohar2018predicting], geophysical [@alonso2010novel] and nuclear systems [@karnik2024constrained]. Optimal sensor placement ensures accurate, real-time tracking of key system variables with minimal hardware and enables cost-effective, real-time system analysis and control. In general, sensor placement optimization is NP-hard and cannot be solved in polynomial time. There are ${n \choose p} = n!/((n-p)!p!)$ possible combinations of choosing $p$ sensors from an $n$-dimensional state. Common approaches to optimizing sensor placement include maximizing the information criteria [@krause2008near], Bayesian Optimal Experimental Design [@alexanderian2021optimal], compressed sensing [@donoho2006compressed], and heuristic methods. Many sensor placement methods have submodular objective form, which sets guarantees on how close an efficient greedy placement can be to the unknown true optimum [@summers2015submodularity]. Sub-modular objectives can be efficiently optimized for hundreds or thousands of candidate locations using convex [joshi2008sensor] or greedy optimization approaches [@summers2015submodularity] . -![An overview image of capabilities of Pysensors](/Fig1.jpeg "PySensors 2.0 expands its capabilities by introducing custom basis functions, optimizers, constraints, solvers, and uncertainty quantification, enabling constrained sensing, over- and under-sampling, and uncertainty quantification in the presence of noisy sensor measurements.") +![An overview image of capabilities of Pysensors](./Fig1.jpeg "PySensors 2.0 expands its capabilities by introducing custom basis functions, optimizers, constraints, solvers, and uncertainty quantification, enabling constrained sensing, over- and under-sampling, and uncertainty quantification in the presence of noisy sensor measurements.") `PySensors` is a Python package [@de2021pysensors] dedicated to solving the complex challenge of optimal sensor placement in data-driven systems. It implements advanced sparse optimization algorithms that use dimensionality reduction techniques to identify the most informative measurement locations with remarkable efficiency [@manohar2018data;@brunton2016sparse;@clark2020multi]. It helps users identify the best locations for sensors when working with complex high dimensional data, focusing on both reconstruction [@manohar2018data] and classification [@brunton2016sparse] tasks. The package follows `scikit-learn` conventions for user-friendly access while offering advanced customization options for experienced users. Designed with researchers and practitioners in mind, `PySensors` provides open-source, accessible tools that support model discovery across various scientific applications. @@ -67,8 +67,8 @@ $$ \mathbf{y} = \mathbb{S} \mathbf{x} + \boldsymbol\eta $$ where $\boldsymbol{\eta}$ consists of zero-mean, Gaussian independent and identically distributed (i.i.d.) components, and $\mathbb{S} \in \mathbb{R}^{p\times n}$ is the desired sensor (measurement) selection operator. -This measurement selection operator $\mathbb{S}$ encodes point measurements with unit entries in a sparse matrix $\mathbb{S} = \begin{bmatrix} \mathbf{e}_{\gamma_1} & \mathbf{e}_{\gamma_2} & \hdots & \mathbf{e}_{\gamma_ p}\end{bmatrix}^T$ where $\mathbf{e}_j$ are canonical basis vectors for $\mathbb{R}^n$, with a unit entry in component $j$ (where a sensor should be placed) and zeroes elsewhere. Here, $\gamma = \{\gamma_1, \gamma_2, \hdots, \gamma_p\} \subset \{1, 2, \hdots, n\}$ denotes the index set of sensor locations with cardinality $p$. -Sensor selection then corresponds to the components of $\mathbf{x}$ that were chosen to be measured: $\mathbf{\mathbb{S} x} = \begin{bmatrix} x_{\gamma_1} & x_{\gamma_2} & \hdots & x_{\gamma_p}\end{bmatrix}^T$. +This measurement selection operator $\mathbb{S}$ encodes point measurements with unit entries in a sparse matrix $\mathbb{S} = \begin{bmatrix} \mathbf{e}_{\gamma_1} & \mathbf{e}_{\gamma_2} & \dots & \mathbf{e}_{\gamma_p}\end{bmatrix}^T$ where $\mathbf{e}_j$ are canonical basis vectors for $\mathbb{R}^n$, with a unit entry in component $j$ (where a sensor should be placed) and zeroes elsewhere. Here, $\gamma = \{\gamma_1, \gamma_2, \dots, \gamma_p\} \subset \{1, 2, \dots , n\}$ denotes the index set of sensor locations with cardinality $p$. +Sensor selection then corresponds to the components of $\mathbf{x}$ that were chosen to be measured: $\mathbf{\mathbb{S} x} = \begin{bmatrix} x_{\gamma_1} & x_{\gamma_2} & \dots & x_{\gamma_p}\end{bmatrix}^T$. The `SSPOR` class selects these sensors through a cohesive computational framework for strategically minimizing sensor deployment while maintaining reconstruction fidelity in high-dimensional systems. When initialized with measurement data $\mathbf{x}\in\mathbb{R}^n$, `SSPOR` first applies dimensional reduction through basis identification, then employs the computationally efficient QR factorization algorithm to determine optimal sensor locations. This approach, which has demonstrated efficacy in both reduced-order modeling and sparse sensing applications, strategically leverages the inherent structure of the identified basis to prioritize the most informative measurement points. The `Sparse Sensor Placement Optimization for Classification (SSPOC)` framework identifies minimal sensor configurations that can classify high dimensional signals $\mathbf{x}\in\mathbb{R}^n$ as one of $c$ classes. Unlike traditional compressed sensing [@baraniuk2010model;@donoho2006compressed;@candes2006stable] approaches that focus on signal reconstruction, `SSPOC` specifically targets the preservation of classification boundaries in feature space $\boldsymbol{\Psi}_r$ by identifying the sparsest sensor set capable of reconstructing discriminant hyperplanes. @@ -84,7 +84,7 @@ This basis, which can be built from spectral or data-driven decomposition method When a user specifies $p$ sensors, SSPOR first fits a basis $\boldsymbol{\Psi}_r$ to the data and optimizes sensor placement by minimizing the following objective function: $$ -\gamma_* = \argmax_{\gamma, |\gamma|= p} \; \log \det((\mathbb{S} \boldsymbol{\Psi}_r)^T(\mathbb{S} \boldsymbol{\Psi}_r)).\label{eq:detobj} \tag{1} +\gamma_* = \argmax_{\gamma, |\gamma|= p} \; \log \det((\mathbb{S} \boldsymbol{\Psi}_r)^T(\mathbb{S} \boldsymbol{\Psi}_r)) $$ where $\gamma_*$ denotes the index set of optimized sensor locations with cardinality $p$. When $p = r$, is equivalent to the maximizer of $\log |\det(\mathbb{S} \boldsymbol{\Psi}_r)|$. Direct optimization of this criterion leads to a brute force combinatorial search. This sensor placement approach builds upon the empirical interpolation method (EIM) [@barrault2004empirical] and discrete empirical interpolation method (DEIM) [@chaturantabut2010nonlinear] to develop a greedy strategy for optimizing sensor selection built upon the pivoted QR factorization[@drmac2016new;@manohar2018data;@manohar2018predicting;@manohar2021optimal]. @@ -108,11 +108,11 @@ Many engineering applications have extreme operating conditions, high costs, lim To implement spatial constraints in our sensor placement methodology, we must first define the shape of our constraint region and then identify which grid points fall within the designated constraint regions. We accomplish this using the specialized classes and functions provided in the `_constraints.py` module in `utils`. The classes define constraint regions in various geometric shapes including `Circle`, `Ellipse`, `Polygon`, `Parabola`, `Line`, and `Cylinder`. Additionally, we provide `UserDefinedConstraints` options that allow users to supply their own constraints either as a Python`.py` file containing their constraint shape definition or as an equation string. These classes work with two distinct data formats: image data defined by pixel coordinates and tabular data stored in dataframes with explicit x, y, and z values. Upon instantiation of the class, users must provide the input `data` in one of two formats: a matrix representation for images or a structured dataframe for tabular data. In the latter case, the class requires additional keyword arguments `kwargs` specifying the columns corresponding to `X_axis`, `Y_axis`, `Z_axis`, and `Field` parameters, thereby enabling the extraction of relevant data from their respective columns for subsequent analysis. -The D-optimal objective in \ref{detobj} suffers from two limitations: it is not defined for the under-sampling case $p r)scenarios , select TPGR optimizer. In noisy environments enable uncertainty quantification for robust results.") +$$ + \mathbf{K}=\boldsymbol{\Psi}_r \mathbf{B} \mathbf{B}^T \boldsymbol{\Psi}_r^T;\quad \mathbf{B}=\eta \mathbf{A}. +$$ +In practice, the whole covariance matrix is both hard to store as it takes $n\times n$ space in computer memory, and hard to interpret as it has a low rank. Instead, PySensors 2.0 computes a reduced metric: the \emph{uncertainty heatmap} $\sigma=\sqrt{diag(\mathbf{K})}$ or the marginal standard deviation of each pixel of the reconstruction. +Note that for linear reconstructions, the error metrics do not depend on the system state, and thus can be computed with fairly fast numerical linear algebra in time that does not scale with the number of train or test snapshots $N$. +![A flowchart to suggest which method to use](./Fig2.jpeg "When selecting a sensing method in PySensors, consider your primary objective: For field reconstruction in standard settings, use QR with Identity or SVD basis. For classification tasks, leverage SVD basis with SSPOC optimizer. When facing spatial constraints, choose GQR optimizer. For under-sampling (p < r) and over-sampling cases (p > r)scenarios , select TPGR optimizer. In noisy environments enable uncertainty quantification for robust results.") # Acknowledgments