If the answer is useful, you can mark it as. Which machine learning approach to use for data with very low variability and a small training set? Whether this matters or not depends on what you're trying to do with it. to your account, How can I compute the 1-Wasserstein distance between samples from two multivariate distributions please? Note that the argument VI is the inverse of V. Parameters: u(N,) array_like. Albeit, it performs slower than dcor implementation. Compute the Mahalanobis distance between two 1-D arrays. Parameters: ", sinkhorn = SinkhornDistance(eps=0.1, max_iter=100) Sliced and radon wasserstein barycenters of PhD, Electrical Engg. Connect and share knowledge within a single location that is structured and easy to search. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. sinkhorn = SinkhornDistance(eps=0.1, max_iter=100) Gromov-Wasserstein example. Does the order of validations and MAC with clear text matter? Great, you're welcome. With the following 7d example dataset generated in R: Is it possible to compute this distance, and are there packages available in R or python that do this? Can you still use Commanders Strike if the only attack available to forego is an attack against an ally? Is there a way to measure the distance between two distributions in a multidimensional space in python? What is the difference between old style and new style classes in Python? I think Sinkhorn distances can accelerate step 2, however this doesn't seem to be an issue in my application, I strongly recommend this book for any questions on OT complexity: Peleg et al. measures. Journal of Mathematical Imaging and Vision 51.1 (2015): 22-45, Total running time of the script: ( 0 minutes 41.180 seconds), Download Python source code: plot_variance.py, Download Jupyter notebook: plot_variance.ipynb. L_2(p, q) = \int (p(x) - q(x))^2 \mathrm{d}x Yes, 1.3.1 is the latest official release; you can pick up a pre-release of 1.4 from. Why does the narrative change back and forth between "Isabella" and "Mrs. John Knightley" to refer to Emma's sister? I am thinking about obtaining a histogram for every row of the images (which results in 299 histograms per image) and then calculating the EMD 299 times and take the average of these EMD's to get a final score. feel free to replace it with a more clever scheme if needed! Calculating the Wasserstein distance is a bit evolved with more parameters. I want to apply the Wasserstein distance metric on the two distributions of each constituency. In other words, what you want to do boils down to. Wasserstein Distance) for these two grayscale (299x299) images/heatmaps: Right now, I am calculating the histogram/distribution of both images. WassersteinEarth Mover's DistanceEMDWassersteinppp"qqqWasserstein2000IJCVThe Earth Mover's Distance as a Metric for Image Retrieval MathJax reference. Our source and target samples are drawn from (noisy) discrete A key insight from recent works The average cluster size can be computed with one line of code: As expected, our samples are now distributed in small, convex clusters Last updated on Apr 28, 2023. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Args: In principle, for small values of blur near to zero, you would expect to get Wasserstein and for larger values, you get energy distance but for some reason (I think due to due some implementation issues and numerical/precision issues) after some large values, you get some negative value for the distance. Thanks!! Wasserstein 1.1.0 pip install Wasserstein Copy PIP instructions Latest version Released: Jul 7, 2022 Python package wrapping C++ code for computing Wasserstein distances Project description Wasserstein Python/C++ library for computing Wasserstein distances efficiently. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. You can use geomloss or dcor packages for the more general implementation of the Wasserstein and Energy Distances respectively. Earth mover's distance implementation for circular distributions? A more natural way to use EMD with locations, I think, is just to do it directly between the image grayscale values, including the locations, so that it measures how much pixel "light" you need to move between the two. By clicking Sign up for GitHub, you agree to our terms of service and In general, you can treat the calculation of the EMD as an instance of minimum cost flow, and in your case, this boils down to the linear assignment problem: Your two arrays are the partitions in a bipartite graph, and the weights between two vertices are your distance of choice. Sign up for a free GitHub account to open an issue and contact its maintainers and the community. What you're asking about might not really have anything to do with higher dimensions though, because you first said "two vectors a and b are of unequal length". What's the most energy-efficient way to run a boiler? Please note that the implementation of this method is a bit different with scipy.stats.wasserstein_distance, and you may want to look into the definitions from the documentation or code before doing any comparison between the two for the 1D case! June 14th, 2022 mazda 3 2021 bose sound system mazda 3 2021 bose sound system You signed in with another tab or window. - Input: :math:`(N, P_1, D_1)`, :math:`(N, P_2, D_2)` A detailed implementation of the GW distance is provided in https://github.com/PythonOT/POT/blob/master/ot/gromov.py. It is also known as a distance function. These are trivial to compute in this setting but treat each pixel totally separately. Calculate Earth Mover's Distance for two grayscale images, better sample complexity than the full Wasserstein, New blog post from our CEO Prashanth: Community is the future of AI, Improving the copy in the close modal and post notices - 2023 edition. Mmoli, Facundo. Its Wasserstein distance to the data equals W d (, ) = 32 / 625 = 0.0512. - Output: :math:`(N)` or :math:`()`, depending on `reduction` # scaling "decay" coefficient (.8 is pretty close to 1): # Number of samples, dimension of the ambient space, # Output one index per "line" (reduction over "j"). calculate the distance for a setup where all clusters have weight 1. Which ability is most related to insanity: Wisdom, Charisma, Constitution, or Intelligence? layer provides the first GPU implementation of these strategies. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 10648-10656). Why does the narrative change back and forth between "Isabella" and "Mrs. John Knightley" to refer to Emma's sister? Folder's list view has different sized fonts in different folders. Yeah, I think you have to make a cost matrix of shape. Consider two points (x, y) and (x, y) on a metric measure space. They allow us to define a pair of discrete If I understand you correctly, I have to do the following: Suppose I have two 2x2 images. on computational Optimal Transport is that the dual optimization problem This takes advantage of the fact that 1-dimensional Wassersteins are extremely efficient to compute, and defines a distance on $d$-dimesinonal distributions by taking the average of the Wasserstein distance between random one-dimensional projections of the data. In Figure 2, we have two sets of chess. (=10, 100), and hydrograph-Wasserstein distance using the Nelder-Mead algorithm, implemented through the scipy Python . Making statements based on opinion; back them up with references or personal experience. to you. arXiv:1509.02237. The q-Wasserstein distance is defined as the minimal value achieved by a perfect matching between the points of the two diagrams (+ all diagonal points), where the value of a matching is defined as the q-th root of the sum of all edge lengths to the power q. When AI meets IP: Can artists sue AI imitators? whose values are effectively inputs of the function, or they can be seen as ( u v) V 1 ( u v) T. where V is the covariance matrix. one or more moons orbitting around a double planet system, A boy can regenerate, so demons eat him for years. elements in the output, 'sum': the output will be summed. GromovWasserstein distances and the metric approach to object matching. Foundations of computational mathematics 11.4 (2011): 417487. The Jensen-Shannon distance between two probability vectors p and q is defined as, D ( p m) + D ( q m) 2. where m is the pointwise mean of p and q and D is the Kullback-Leibler divergence. If you downscaled by a factor of 10 to make your images $30 \times 30$, you'd have a pretty reasonably sized optimization problem, and in this case the images would still look pretty different. Is there a portable way to get the current username in Python? What are the advantages of running a power tool on 240 V vs 120 V? 1.1 Wasserstein GAN https://arxiv.org/abs/1701.07875, WassersteinKLJSWasserstein, A_Turnip: a naive implementation of the Sinkhorn/Auction algorithm L_2(p, q) = \int (p(x) - q(x))^2 \mathrm{d}x Sinkhorn distance is a regularized version of Wasserstein distance which is used by the package to approximate Wasserstein distance. If we had a video livestream of a clock being sent to Mars, what would we see? .pairwise_distances. rev2023.5.1.43405. @LVDW I updated the answer; you only need one matrix, but it's really big, so it's actually not really reasonable. In dimensions 1, 2 and 3, clustering is automatically performed using \(v\), this distance also equals to: See [2] for a proof of the equivalence of both definitions. However, the scipy.stats.wasserstein_distance function only works with one dimensional data. Which reverse polarity protection is better and why? \(\mathbb{R} \times \mathbb{R}\) whose marginals are \(u\) and $$\operatorname{TV}(P, Q) = \frac12 \sum_{i=1}^{299} \sum_{j=1}^{299} \lvert P_{ij} - Q_{ij} \rvert,$$ For regularized Optimal Transport, the main reference on the subject is Here you can clearly see how this metric is simply an expected distance in the underlying metric space. Metric: A metric d on a set X is a function such that d(x, y) = 0 if x = y, x X, and y Y, and satisfies the property of symmetry and triangle inequality. Where does the version of Hamapil that is different from the Gemara come from? Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Say if you had two 3D arrays and you wanted to measure the similarity (or dissimilarity which is the distance), you may retrieve distributions using the above function and then use entropy, Kullback Liebler or Wasserstein Distance. Right now I go through two libraries: scipy (https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wasserstein_distance.html) and pyemd (https://pypi.org/project/pyemd/). a kernel truncation (pruning) scheme to achieve log-linear complexity. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. (in the log-domain, with \(\varepsilon\)-scaling) which Image of minimal degree representation of quasisimple group unique up to conjugacy. It is written using Numba that parallelizes the computation and uses available hardware boosts and in principle should be possible to run it on GPU but I haven't tried. How to force Unity Editor/TestRunner to run at full speed when in background? Here we define p = [; ] while p = [, ], the sum must be one as defined by the rules of probability (or -algebra). Already on GitHub? What positional accuracy (ie, arc seconds) is necessary to view Saturn, Uranus, beyond? to download the full example code. Metric measure space is like metric space but endowed with a notion of probability. To analyze and organize these data, it is important to define the notion of object or dataset similarity. Let's go with the default option - a uniform distribution: # 6 args -> labels_i, weights_i, locations_i, labels_j, weights_j, locations_j, Scaling up to brain tractograms with Pierre Roussillon, 2) Kernel truncation, log-linear runtimes, 4) Sinkhorn vs. blurred Wasserstein distances. To understand the GromovWasserstein Distance, we first define metric measure space. Where does the version of Hamapil that is different from the Gemara come from? a straightforward cubic grid. What positional accuracy (ie, arc seconds) is necessary to view Saturn, Uranus, beyond? However, it still "slow", so I can't go over 1000 of samples. For instance, I would want to convert the first 3 entries for p and q into an array, apply Wasserstein distance and get a value. However, I am now comparing only the intensity of the images, but I also need to compare the location of the intensity of the images. Mean centering for PCA in a 2D arrayacross rows or cols? 1D energy distance Why don't we use the 7805 for car phone chargers? Episode about a group who book passage on a space ship controlled by an AI, who turns out to be a human who can't leave his ship? ot.sliced.sliced_wasserstein_distance(X_s, X_t, a=None, b=None, n_projections=50, p=2, projections=None, seed=None, log=False) [source] By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. We can use the Wasserstein distance to build a natural and tractable distance on a wide class of (vectors of) random measures. \(v\) on the first and second factors respectively. What differentiates living as mere roommates from living in a marriage-like relationship? This example illustrates the computation of the sliced Wasserstein Distance as proposed in [31]. Parabolic, suborbital and ballistic trajectories all follow elliptic paths. 2-Wasserstein distance calculation Background The 2-Wasserstein distance W is a metric to describe the distance between two distributions, representing e.g. While the scipy version doesn't accept 2D arrays and it returns an error, the pyemd method returns a value. Compute the first Wasserstein distance between two 1D distributions. The sliced Wasserstein (SW) distances between two probability measures are defined as the expectation of the Wasserstein distance between two one-dimensional projections of the two measures. Later work, e.g. we should simply provide: explicit labels and weights for both input measures. Mmoli, Facundo. Thanks for contributing an answer to Stack Overflow! Wasserstein metric, https://en.wikipedia.org/wiki/Wasserstein_metric. As expected, leveraging the structure of the data has allowed What should I follow, if two altimeters show different altitudes? [31] Bonneel, Nicolas, et al. Should I re-do this cinched PEX connection? 6.Some of these distances are sensitive to small wiggles in the distribution. \beta ~=~ \frac{1}{M}\sum_{j=1}^M \delta_{y_j}.\]. This is then a 2-dimensional EMD, which scipy.stats.wasserstein_distance can't compute, but e.g. (Ep. "Sliced and radon wasserstein barycenters of measures.". By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. The histograms will be a vector of size 256 in which the nth value indicates the percent of the pixels in the image with the given darkness level. $\{1, \dots, 299\} \times \{1, \dots, 299\}$, $$\operatorname{TV}(P, Q) = \frac12 \sum_{i=1}^{299} \sum_{j=1}^{299} \lvert P_{ij} - Q_{ij} \rvert,$$, $$ In that respect, we can come up with the following points to define: The notion of object matching is not only helpful in establishing similarities between two datasets but also in other kinds of problems like clustering. (x, y, x, y ) |d(x, x ) d (y, y )|^q and pick a p ( p, p), then we define The GromovWasserstein Distance of the order q as: The GromovWasserstein Distance can be used in a number of tasks related to data science, data analysis, and machine learning. Could a subterranean river or aquifer generate enough continuous momentum to power a waterwheel for the purpose of producing electricity? (Schmitzer, 2016) The entry C[0, 0] shows how moving the mass in $(0, 0)$ to the point $(0, 1)$ incurs in a cost of 1. The best answers are voted up and rise to the top, Not the answer you're looking for? privacy statement. If the null hypothesis is never really true, is there a point to using a statistical test without a priori power analysis? Is there such a thing as "right to be heard" by the authorities? What's the canonical way to check for type in Python? Max-sliced wasserstein distance and its use for gans. Weight may represent the idea that how much we trust these data points. KANTOROVICH-WASSERSTEIN DISTANCE Whenever The two measure are discrete probability measures, that is, both i = 1 n i = 1 and j = 1 m j = 1 (i.e., and belongs to the probability simplex), and, The cost vector is defined as the p -th power of a distance, the Sinkhorn loop jumps from a coarse to a fine representation outputs an approximation of the regularized OT cost for point clouds. In general, with this approach, part of the geometry of the object could be lost due to flattening and this might not be desired in some applications depending on where and how the distance is being used or interpreted. I think for your image size requirement, maybe sliced wasserstein as @Dougal suggests is probably the best suited since 299^4 * 4 bytes would mean a memory requirement of ~32 GBs for the transport matrix, which is quite huge. This example is designed to show how to use the Gromov-Wassertsein distance computation in POT. two different conditions A and B. \mathbb{R}} |x-y| \mathrm{d} \pi (x, y)\], \[l_1(u, v) = \int_{-\infty}^{+\infty} |U-V|\], K-means clustering and vector quantization (, Statistical functions for masked arrays (, https://en.wikipedia.org/wiki/Wasserstein_metric. \(v\), where work is measured as the amount of distribution weight rev2023.5.1.43405. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Dataset. (1989), simply matched between pixel values and totally ignored location. Copyright 2016-2021, Rmi Flamary, Nicolas Courty. be solved efficiently in a coarse-to-fine fashion, A boy can regenerate, so demons eat him for years. Content Discovery initiative April 13 update: Related questions using a Review our technical responses for the 2023 Developer Survey, Using Earth Mover's Distance for multi-dimensional vectors with unequal length. hcg wert viel zu niedrig; flohmarkt kilegg 2021. fhrerschein in tschechien trotz mpu; kartoffeltaschen mit schinken und kse Is there any well-founded way of calculating the euclidean distance between two images? For continuous distributions, it is given by W: = W(FA, FB) = (1 0 |F 1 A (u) F 1 B (u) |2du)1 2, The Wasserstein distance between (P, Q1) = 1.00 and Wasserstein (P, Q2) = 2.00 -- which is reasonable. It can be considered an ordered pair (M, d) such that d: M M . the manifold-like structure of the data - if any. that must be moved, multiplied by the distance it has to be moved. How to force Unity Editor/TestRunner to run at full speed when in background? If the source and target distributions are of unequal length, this is not really a problem of higher dimensions (since after all, there are just "two vectors a and b"), but a problem of unbalanced distributions (i.e. 566), Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. It can be installed using: pip install POT Using the GWdistance we can compute distances with samples that do not belong to the same metric space. Asking for help, clarification, or responding to other answers. : scipy.stats. Due to the intractability of the expectation, Monte Carlo integration is performed to . It might be instructive to verify that the result of this calculation matches what you would get from a minimum cost flow solver; one such solver is available in NetworkX, where we can construct the graph by hand: At this point, we can verify that the approach above agrees with the minimum cost flow: Similarly, it's instructive to see that the result agrees with scipy.stats.wasserstein_distance for 1-dimensional inputs: Thanks for contributing an answer to Stack Overflow! But by doing the mean over projections, you get out a real distance, which also has better sample complexity than the full Wasserstein. u_values (resp. Asking for help, clarification, or responding to other answers. How can I remove a key from a Python dictionary? Find centralized, trusted content and collaborate around the technologies you use most. 'none' | 'mean' | 'sum'. May I ask you which version of scipy are you using? that partition the input data: To use this information in the multiscale Sinkhorn algorithm, 4d, fengyz2333: @AlexEftimiades: Are you happy with the minimum cost flow formulation? It is also possible to use scipy.sparse.csgraph.min_weight_bipartite_full_matching as a drop-in replacement for linear_sum_assignment; while made for sparse inputs (which yours certainly isn't), it might provide performance improvements in some situations. Have a question about this project? What do hollow blue circles with a dot mean on the World Map? Connect and share knowledge within a single location that is structured and easy to search. But we can go further. clustering information can simply be provided through a vector of labels, For example, I would like to make measurements such as Wasserstein distribution or the energy distance in multiple dimensions, not one-dimensional comparisons. Does Python have a string 'contains' substring method? Both the R wasserstein1d and Python scipy.stats.wasserstein_distance are intended solely for the 1D special case. using a clever multiscale decomposition that relies on Metric Space: A metric space is a nonempty set with a metric defined on the set. To learn more, see our tips on writing great answers. Is this the right way to go? What were the most popular text editors for MS-DOS in the 1980s? copy-pasted from the examples gallery # Author: Adrien Corenflos <adrien.corenflos . This post may help: Multivariate Wasserstein metric for $n$-dimensions. Anyhow, if you are interested in Wasserstein distance here is an example: Other than the blur, I recommend looking into other parameters of this method such as p, scaling, and debias. We see that the Wasserstein path does a better job of preserving the structure. Consider R X Y is a correspondence between X and Y. a typical cluster_scale which specifies the iteration at which But in the general case, What is the advantages of Wasserstein metric compared to Kullback-Leibler divergence? eps (float): regularization coefficient Values observed in the (empirical) distribution. Doesnt this mean I need 299*299=89401 cost matrices? I reckon you want to measure the distance between two distributions anyway? Both the R wasserstein1d and Python scipy.stats.wasserstein_distance are intended solely for the 1D special case. Folder's list view has different sized fonts in different folders. Thank you for reading. Leveraging the block-sparse routines of the KeOps library, By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. \[\alpha ~=~ \frac{1}{N}\sum_{i=1}^N \delta_{x_i}, ~~~ As far as I know, his pull request was . A Medium publication sharing concepts, ideas and codes. You said I need a cost matrix for each image location to each other location. from scipy.stats import wasserstein_distance np.random.seed (0) n = 100 Y1 = np.random.randn (n) Y2 = np.random.randn (n) - 2 d = np.abs (Y1 - Y2.reshape ( (n, 1))) assignment = linear_sum_assignment (d) print (d [assignment].sum () / n) # 1.9777950447866477 print (wasserstein_distance (Y1, Y2)) # 1.977795044786648 Share Improve this answer multidimensional wasserstein distance pythonoffice furniture liquidators chicago. Other than Multidimensional Scaling, you can also use other Dimensionality Reduction techniques, such as Principal Component Analysis (PCA) or Singular Value Decomposition (SVD). Linear programming for optimal transport is hardly anymore harder computation-wise than the ranking algorithm of 1D Wasserstein however, being fairly efficient and low-overhead itself. Making statements based on opinion; back them up with references or personal experience. https://pythonot.github.io/quickstart.html#computing-wasserstein-distance, is the computational bottleneck in step 1? 'none': no reduction will be applied, # The Sinkhorn algorithm takes as input three variables : # both marginals are fixed with equal weights, # To check if algorithm terminates because of threshold, "$M_{ij} = (-c_{ij} + u_i + v_j) / \epsilon$", "Barycenter subroutine, used by kinetic acceleration through extrapolation. Having looked into it a little more than at my initial answer: it seems indeed that the original usage in computer vision, e.g. This is the square root of the Jensen-Shannon divergence. This is then a 2-dimensional EMD, which scipy.stats.wasserstein_distance can't compute, but e.g. \(v\) is: where \(\Gamma (u, v)\) is the set of (probability) distributions on This example illustrates the computation of the sliced Wasserstein Distance as Thanks for contributing an answer to Cross Validated! This method takes either a vector array or a distance matrix, and returns a distance matrix. So if I understand you correctly, you're trying to transport the sampling distribution, i.e. It only takes a minute to sign up. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. How to calculate distance between two dihedral (periodic) angles distributions in python? Could you recommend any reference for addressing the general problem with linear programming? What is the fastest and the most accurate calculation of Wasserstein distance? This distance is also known as the earth mover's distance, since it can be seen as the minimum amount of "work" required to transform u into v, where "work" is measured as the amount of distribution weight that must be moved, multiplied by the distance it has to be moved. | Intelligent Transportation & Quantum Science Researcher | Donation: https://www.buymeacoffee.com/rahulbhadani, It. the POT package can with ot.lp.emd2. functions located at the specified values. one or more moons orbitting around a double planet system, "Signpost" puzzle from Tatham's collection, Proving that Every Quadratic Form With Only Cross Product Terms is Indefinite, Extracting arguments from a list of function calls. proposed in [31]. weight. Figure 1: Wasserstein Distance Demo. How do you get the logical xor of two variables in Python? the ground distances, may be obtained using scipy.spatial.distance.cdist, and in fact SciPy provides a solver for the linear sum assignment problem as well in scipy.optimize.linear_sum_assignment (which recently saw huge performance improvements which are available in SciPy 1.4. At the other end of the row, the entry C[0, 4] contains the cost for moving the point in $(0, 0)$ to the point in $(4, 1)$. I am a vegetation ecologist and poor student of computer science who recently learned of the Wasserstein metric. Sorry, I thought that I accepted it. $$ of the data. Because I am working on Google Colaboratory, and using the last version "Version: 1.3.1". generalized functions, in which case they are weighted sums of Dirac delta Learn more about Stack Overflow the company, and our products. local texture features rather than the raw pixel values. to sum to 1. Application of this metric to 1d distributions I find fairly intuitive, and inspection of the wasserstein1d function from transport package in R helped me to understand its computation, with the following line most critical to my understanding: In the case where the two vectors a and b are of unequal length, it appears that this function interpolates, inserting values within each vector, which are duplicates of the source data until the lengths are equal. User without create permission can create a custom object from Managed package using Custom Rest API, Identify blue/translucent jelly-like animal on beach. I just checked out the POT package and I see there is a lot of nice code there, however the documentation doesn't refer to anything as "Wasserstein Distance" but the closest I see is "Gromov-Wasserstein Distance". Thanks for contributing an answer to Cross Validated! There are also, of course, computationally cheaper methods to compare the original images. He also rips off an arm to use as a sword. I. Use MathJax to format equations. Although t-SNE showed lower RMSE than W-LLE with enough dataset, obtaining a calibration set with a pencil beam source is time-consuming. If you see from the documentation, it says that it accept only 1D arrays, so I think that the output is wrong. The pot package in Python, for starters, is well-known, whose documentation addresses the 1D special case, 2D, unbalanced OT, discrete-to-continuous and more. Making statements based on opinion; back them up with references or personal experience.

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