Mathematics for Machine Learning

Deisenroth, Marc Peter; Faisal, A. Aldo; Ong, Cheng Soon

doi:10.1017/9781108679930

Cited by 271 publications

(140 citation statements)

References 132 publications

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“…The training and test set were forwarded respectively to the training process, where the process of parameter tuning was done and the model was assessed in a process of cross-validation, and to the model selection step, which is detailed in the next paragraphs. It is important to keep the test split completely out of this process to avoid bias in the trained model [35]. Train-test cross-validation split methodology used in this paper.…”

Section: Machine Learningmentioning

confidence: 99%

A Method for Chlorophyll-a and Suspended Solids Prediction through Remote Sensing and Machine Learning

Kupssinskü

Guimarães

Souza

et al. 2020

Sensors

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† This paper is an extended version of our paper published in Kupssinskü, L.; Guimarães, T.; Freitas, R.; de Souza, E.; Rossa, P.; Ademir Marques, J.; Veronez, M.; Junior, L.G.; Mauad, F.; Cazarin, C. Prediction of chlorophyll-a and suspended solids through remote sensing and artificial neural networks.Abstract: Total Suspended Solids (TSS) and chlorophyll-a concentration are two critical parameters to monitor water quality. Since directly collecting samples for laboratory analysis can be expensive, this paper presents a methodology to estimate this information through remote sensing and Machine Learning (ML) techniques. TSS and chlorophyll-a are optically active components, therefore enabling measurement by remote sensing. Two study cases in distinct water bodies are performed, and those cases use different spatial resolution data from Sentinel-2 spectral images and unmanned aerial vehicles together with laboratory analysis data. In consonance with the methodology, supervised ML algorithms are trained to predict the concentration of TSS and chlorophyll-a. The predictions are evaluated separately in both study areas, where both TSS and chlorophyll-a models achieved R-squared values above 0.8.Sensors 2020, 20, 2125 2 of 18 bodies [3]. Moreover, the suspended solids may indicate erosive processes within the watershed and additional water pollution since it can carry or store pollutants [4].Generally, once the samples get to a laboratory, their concentrations are analyzed, and manual measurements are taken to extract these parameters. The involved processes are costly, time-consuming, and require skilled personnel [5]. Although there are accurate evaluations, they do not indicate the individual spatio-temporal variation of water quality [6].In this sense, applications involving remote sensing can overcome the drawbacks of conventional monitoring since they allow real-time, spatial, continuous, and long-term monitoring in large areas [7,8]. The presence of components alters the spectral characteristics of the pure water, called Optically Active Components (OAC). The main OACs in water bodies are chlorophyll, suspended solids, and organic material [4]. The possibility of monitoring non-accessible areas and composing time series of water quality from historical data of remote sensing [9] are also factors that make this tool essential and its use promising.Among the most common satellites in OAC remote sensing studies are Landsat 8 and Sentinel-2. Although Landsat is more widely used, the European Sentinel-2 satellite has promising characteristics: it has higher spatial resolution than Landsat 8 and also spectral bands in the region called the red edge, which is of great interest for water and vegetation studies.Remote sensing serves as a powerful technique for monitoring environmental and seasonal changes, and its ability to remotely monitor water resources has increased in recent decades because of the quality and availability of satellite imagery data [10]. Even so, the analysis of small water bodies may not be adequ...

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Section: Machine Learningmentioning

confidence: 99%

A Method for Chlorophyll-a and Suspended Solids Prediction through Remote Sensing and Machine Learning

Kupssinskü

Guimarães

Souza

et al. 2020

Sensors

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“…The hyperplane f w,b is learned during the "training" window and then evaluated in the development window to evaluate performance. Finally, Platt calibration, which fits a logistic regression model to the SVM scores, was used to transform the outputs of the SVM model into a probabilistic quantity, which we will call the state signal:ρ(f w,b = 1|Z PLS ) = 1 1+exp (Afw,b(ZPLS)+B) ,where the parameters A and B are optimized using gradient descent to minimize the cross-entropy error [41].…”

Section: The State Signalsmentioning

confidence: 99%

A computational paradigm for real-time MEG neurofeedback for dynamic allocation of spatial attention

et al. 2020

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Background: Neurofeedback aids volitional control of one's own brain activity using non-invasive recordings of brain activity. The applications of neurofeedback include improvement of cognitive performance and treatment of various psychiatric and neurological disorders. During real-time magnetoencephalography (rt-MEG), sensor-level or source-localized brain activity is measured and transformed into a visual feedback cue to the subject. Recent real-time fMRI (rt-fMRI) neurofeedback studies have used pattern recognition techniques to decode and train a brain state to link brain activities and cognitive behaviors. Here, we utilize the real-time decoding technique similar to ones employed in rt-fMRI to analyze time-varying rt-MEG signals. Results: We developed a novel rt-MEG method, state-based neurofeedback (sb-NFB), to decode a time-varying brain state, a state signal, from which timings are extracted for neurofeedback training. The approach is entirely data-driven: it uses sensor-level oscillatory activity to find relevant features that best separate the targeted brain states. In a psychophysical task of spatial attention switching, we trained five young, healthy subjects using the sb-NFB method to decrease the time necessary for switch spatial attention from one visual hemifield to the other (referred to as switch time). Training resulted in a decrease in switch time with training. We saw that the activity targeted by the training involved proportional changes in alpha and beta-band oscillations, in sensors at the occipital and parietal regions. We also found that the state signal that encodes whether subjects attend to the left or right visual field effectively switches consistently with the task. Conclusion: We demonstrated the use of the sb-NFB method when the subject learns to increase the speed of shifting covert spatial attention from one visual field to the other. The sb-NFB method can target timing features that would otherwise also include extraneous features such as visual detection and motor response in a simple reaction time task.

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“…From Bolzano-Weierstrass theorem [39], there must be at least one accumulation point of the sequence {P k } +∞ k=1 . We denote one of the points P * = {L * , E * , M * }.…”

Section: Tensor Robust Principal Component Analysis With Non-convex Rmentioning

confidence: 99%

“…(see the proof of Proposition 3 of [30] and Remark 2.3. of [45]) and using the chain rule [39], we can deduce that…”

Section: Tensor Robust Principal Component Analysis With Non-convex Rmentioning

confidence: 99%

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Tensor Robust Principal Component Analysis via Non-Convex Low Rank Approximation

et al. 2019

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Tensor Robust Principal Component Analysis (TRPCA) plays a critical role in handling high multi-dimensional data sets, aiming to recover the low-rank and sparse components both accurately and efficiently. In this paper, different from current approach, we developed a new t-Gamma tensor quasi-norm as a non-convex regularization to approximate the low-rank component. Compared to various convex regularization, this new configuration not only can better capture the tensor rank but also provides a simplified approach. An optimization process is conducted via tensor singular decomposition and an efficient augmented Lagrange multiplier algorithm is established. Extensive experimental results demonstrate that our new approach outperforms current state-of-the-art algorithms in terms of accuracy and efficiency.

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Mathematics for Machine Learning

Cited by 271 publications

References 132 publications

A Method for Chlorophyll-a and Suspended Solids Prediction through Remote Sensing and Machine Learning

A Method for Chlorophyll-a and Suspended Solids Prediction through Remote Sensing and Machine Learning

A computational paradigm for real-time MEG neurofeedback for dynamic allocation of spatial attention

Tensor Robust Principal Component Analysis via Non-Convex Low Rank Approximation

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