extfieldsofstudy: Medicine Biology

text:
Genetic Variants Detection Based on Weighted Sparse Group Lasso

Identification of genetic variants associated with complex traits is a critical step for improving plant resistance and breeding. Although the majority of existing methods for variants detection have good predictive performance in the average case, they can not precisely identify the variants present in a small number of target genes. In this paper, we propose a weighted sparse group lasso (WSGL) method to select both common and low-frequency variants in groups. Under the biologically realistic assumption that complex traits are influenced by a few single loci in a small number of genes, our method involves a sparse group lasso approach to simultaneously select associated groups along with the loci within each group. To increase the probability of selecting out low-frequency variants, biological prior information is introduced in the model by re-weighting lasso regularization based on weights calculated from input data. Experimental results from both simulation and real data of single nucleotide polymorphisms (SNPs) associated with Arabidopsis flowering traits demonstrate the superiority of WSGL over other competitive approaches for genetic variants detection.

INTRODUCTION
Since completion of the sequencing-based structural genome project, the focus of life science research has gradually shifted from determining the composition of DNA sequences to elucidating the function of identified genes. However, the greatest challenge of functional genomics is to determine the risk genes associated with complex diseases or traits among the huge amount of DNA sequences. Approximately, 90% of all gene fragments in any two individuals of almost all organisms are identical; thus, the fragments affecting individual characteristics, diseases, or traits only appear in a small range of sequences (Tenaillon et al., 2001;Reich et al., 2002). Polygenic recombination or mutation can cause individual differences in genome sequences, resulting in genetic polymorphism. Single nucleotide polymorphisms (SNPs) are the most common form of such genetic variation. Therefore, identification and characterization of SNPs help to discover the underlying causes of various diseases or variable traits and to develop new therapeutic strategies and targets for drug development or crop improvement.
The goal of genome-wide association studies (GWAS) is to elucidate the relationship between millions of SNPs and complex traits (Klein et al., 2005). A single-locus association approach is typically used in GWAS; however, the "polygenic theory" proposes that complex traits are controlled by the action of multiple SNPs together rather than by individual genes or variants (Dudbridge, 2016). Since the number of SNPs far exceeds the number of samples in a multi-loci association study, the "curse of dimensionality" becomes the main challenge of this type of analysis (Waddell et al., 2005). Many machine-learning algorithms have been widely used to overcome this limitation and facilitate investigating the association between traits with SNPs. Based on current approaches, association studies can be divided into two main categories: one based on feature selection (FS) and the other based on statistical machine learning with regularizing penalty.
FS is the process of selecting the most effective features among a set of original features so as to reduce the dimensionality of the dataset. There are two types of FS methods: the wrapper method as a dependent classifier (Hall and Smith, 1999), and the filter method as an independent classifier (Liu and Setiono, 1996). Typically, the wrapper and filter approaches are combined as the final selected method. When applying FS methods to GWAS, the SNPs are treated as the features, phenotypes are the labels, and the candidate SNPs are then selected according to their associations with phenotypes. Numerous FS methods have been applied in genetic association studies (Evans, 2010;Batnyam et al., 2013;Anekboon et al., 2014;Alzubi et al., 2017;An et al., 2017;Setiawan et al., 2018;Tsamardinos et al., 2019). For example, Evans (2010) combined two filter FS methods with classification methods in a machine-learning approach, and obtained strong association results. To further improve the accuracy of the selected SNPs, Batnyam et al. (2013) applied four popular FS approaches (Robnik-Šikonja and Kononenko, 2003;Liang et al., 2008;Seo and Oh, 2012;Lee et al., 2013) to select novel SNPs, which were then used to generate artificial features by applying a feature fusion method. Finally, the artificial features were classified by traditional classifiers. As an alternative combinational algorithm, Anekboon et al. (2014) proposed a correlation-based FS method as a filter to first select a portion of the SNPs, followed by a wrapper phase to sequentially feed each of these SNPs into k-nearest neighbor, artificial neural network, and Ridge regression classifiers. Alzubi et al. (2017) developed a hybrid FS method by combining conditional mutual information maximization and support vector machine-recursive feature elimination (SVM-RFE). An et al. (2017) used a hierarchical feature and sample selection framework to gradually select informative features and discard ambiguous samples in multiple steps to improve the classifier learning. Setiawan et al. (2018) firstly employed random forest algorithm to reduce the search space, then selected associated SNPs by sequential forward floating selection. Tsamardinos et al. (2019) applied p-values of conditional independence tests and meta-analysis techniques to select features, and made use of parallel techonology to increase the computing speed. Current methods based on FS have sufficient ability for selecting a candidate feature set. Nevertheless, it is important to use available biological information as prior knowledge in biocomputing. Since FS methods can only reflect the dataset itself, they are not suitable to screen features based on prior biological knowledge.
Regression models with penalty can also be used for GWAS. With this approach, the SNPs correspond to the independent variables, and phenotypes are mapped to dependent variables in the regression model. Since the number of SNPs typically far exceeds the number of samples, it is necessary to regularize the sparsity of coefficients in the regression model. As a representative example, the well-established lasso method proposed by Tibshirani (1996) can learn a sparse weight vector by penalizing the weight vector with a 1-norm loss while shrinking less important coefficients to zeros. Owing to this property, lasso and its extensions have been widely applied in the detection of genetic variants (Cao et al., 2014;Arbet et al., 2017;Tamba et al., 2017;Cherlin et al., 2018;Wang et al., 2019). For example, Cao et al. (2014) incorporated prior information in lasso to further increase the selection accuracy. Arbet et al. (2017) imposed a permutation method on lasso to improve the performance of the algorithm. Tamba et al. (2017) first reduced the number of SNPs to a moderate size, then used expectation maximization Bayesian lasso to detect the quantitative trait nucleotide (QTN). Cherlin et al. (2018) used lasso to explore the association between phenotype and SNP data and achieved good prediction. Wang et al. (2019) promoted a precision lasso that utilized regularization governed by the covariance and inverse covariance matrices of explanatory variables to increase sparse variable selection. However, SNPs (features) are generally found in groups, whereas lasso does not encourage sparsity between groups. Yuan and Lin, (2006) proposed the group lasso (GL) method, which sets a regularization of the sum of the ℓ 2 norm onto groups that encourages only a few groups to be selected. The GL approach has also been successfully applied in GWAS (Li et al., 2015;Lim and Hastie, 2015;Gossmann et al., 2017;Du et al., 2018). Gossmann et al. (2017) extended sorted L1 penalized estimation (SLOPE) in the spirit of Group LASSO to handle group structures between the predictor variables. Du et al. (2018) proposed the SCCA with truncated L1 penalized and GL to improve the performance and effectiveness of discovering SNPs or QTs in imaging genetics. However, once a group is chosen, all of its comprising features are also selected, which is not compliant with the actual biological situation in which SNPs are distributed sparsely across the genome in only a few groups. Simon et al. (2013) developed sparse GL (SGL) that uses the ℓ 2 penalty to select only a subset of the groups and the ℓ 1 penalty to select only a subset of the variables within the group. Indeed, SGL has been widely applied in detecting genetic variants (Rao et al., 2015;Li et al., 2017;Samal et al., 2017;Guo et al., 2019). Samal et al. (2017) proposed a method based on SGL to identify phenotype associated extreme currents decomposed from metabolic networks data. Combined SGL with group-level graph structure, which takes advantages of gene-level priors to penalty the nucleotide-level sparsity to identify the risk SNPs. Guo et al. (2019) proposed a method that combined SGL and linear mixed model (LMM) for multivariate associations of quantitative traits, and it obtained a good power. Despite this improvement, the limitation of this method is that SGL selects sparse features within a group, but gives the same penalty for all features within the group. Consequently, this approach can easily result in swiping out low-frequency features that may play an important role in influencing phenotypes. To overcome this obstacle, it is important to assign different penalties to different features. Ideally, candidate SNPs should have a smaller penalty weight while others would have a larger penalty weight. In this way, candidate SNPs will stand out among the data more readily. To achieve this goal, we here propose a novel approach termed weighted SGL (WSGL) by introducing biological prior information for more accurate genetic variants detection. Specifically, we compute the minimum allele frequency (MAF) among a dataset of SNPs and use those values to reweight as the ℓ 1 penalty of each SNP site, which can increase the chance of retaining low-frequency variants without loss of information. To validate this approach, we compared the performance of our model with simulation and real data against the three mainstream models discussed above.

Simulation Data
We used Arabidopsis thaliana data from Atwell et al. (2010), downloaded from https://github.com/Gregor-Mendel-Institute/ atpolydb for the simulation. We used a quality control protocol on the original data. The SNPs are eliminated by the standard that Minor Allele Frequency (MAF) is < 0.01, the missing rate is > 0.05, or the allele frequencies are not in Hardy-Weinberg (P < 0.0001). After data preprocessing, we chose 200 genes on chromosome 1 covering a total of 1,993 SNPs. Twenty of these SNPs were chosen as the associated variants.

Real Data
The genotype information was the same as that obtained from the simulation data. Ten phenotypes were selected among the 107 reported. First, from chromosome 1 to 5, we chose the first 1,000 genes, which were sorted according to sequence length, including 49,962 SNPs. Second, we selected 19 genes containing 367 SNPs, which have been verified to be associated with flowering time in Arabidopsis. Thus, a total of 50,329 SNPs were analyzed in our experiments.

Statistical Model and Methods
We first give a problem statement, followed by a brief overview of lasso and its extension for application in a genetic association study. Finally, we describe our new WSGL method.
Let X = (x 1 , x 2 ,…, x n ) T denote the n × p genotype matrix, where n is the number of samples and p is the number of genotypes. Let Y = (y 1 , y 2 ,…, y n ) T represent the n × 1 phenotype vector, containing the phenotype values of the n samples. We then establish a linear model between X and y: where b = (b 1 , b 2 ,…, b n ) T is a p × 1 regression coefficients vector, and ϵ ∼N(0, 1). Tibshirani (1996) proposed the popular lasso estimator,

Lasso and Its Extension for Association Mapping
where b is the regression coefficients vector, and x, corresponding to the nonzero estimated coefficients in b, represents the candidate SNPs. ||b|| 1 is the ℓ 1 penalty item. l is a regularization parameter, and its size determines the sparsity. When l = 0, the lasso estimator is equivalent to ordinary least-squares regression. However, the lasso applies to the situation in which the variables are independent of each other. For the situation in which the variables can be divided into m groups, Yuan and Lin (2006) proposed the GL estimator, where m is the group of variables, the first part is OLS, the second part is the sum of the ℓ 2 penalty of the coefficients of each group, and l is the regularization parameter. If the size of the group is 1, it will degenerate to the standard lasso.
The GL can generate a sparse in groups; however, the variables in a group are not sparse. To solve this problem, Simon et al. (2013) proposed the SGL, where l still controls the overall penalty and a determines the ratio between ℓ 1 and ℓ 2 . When a = 1, it will be transformed into lasso, whereas when a = 0, it will be GL. SGL can either select the variables in a group-by-group manner, or screen the individual variables in the remaining groups.

Our Method
With respect to the genetic association problem, the variables in a group have different effects on the independent variable. However, the SGL uses the same penalty coefficients for all variables, regardless of the relative importance among SNPs in the screened groups.
To tackle this problem, we introduce the prior information w in the model to improve the statistical power, and propose the WSGL, The objective function in (5) is clearly convex; therefore, the optimal solution can be achieved by subgradient equations. Letb be the optimal solution of WSGL. For group k = (1, 2,…, m), the solutionb (k) satisfies where µ (k) and v (k) are subgradients of ‖b (k) ‖ 2 and ‖b (k) ‖ 1 , respectively. According to Simon et al. (2013), m (k) =b (k) = ‖b (k) ‖ 2 if b (k) ≠ 0; otherwise, || µ (k) || 2 ≤ 1. n (k) j = sign(b j (k) ) when b j (k) ≠ 0; otherwise, ‖ n (k) j ‖ 2 ≤ 0. Following the analysis in Simon et al. (2013), the condition forb (k) = 0 is whereg (−k) = y − ol≠k X lb (l) is the partial residual of y, and S is defined as (S(a, b) For each locus, MAF indicates, to some degree, its rareness. The MAF of low-frequency variants is usually small, so the associated low-frequency variants are more susceptible to sparsity regularization than other common variants. With normal sparse group lasso, the pressure of being zeroed out on each locus within the same group is equally high. In this case, those low-frequency variants are more likely to be excluded during the process. So selection of an appropriate weight can help to filter out more accurate candidate low-frequency variants.
There are several approaches for deciding the weights. For example, a small penalty can be assigned to the loci in known susceptibility genes to ensure including them into the model. Alternatively, the weights can be dependent on the MAF. For a dataset including both low-frequency and common variants, low-frequency markers are assigned smaller weights to compensate for their low frequencies. Here, we assign each locus a weight as follows: . Each weight w i is calculated in advance, which contains genotypes and biological explanations. The importance of the ith variable can be adjusted by the weight w i . Thus, to choose a locus, we can give it a relatively small penalty weight. Conversely, a larger weight can be assigned to exclude a locus. If w i = 1, our model will be transformed to the SGL. Moreover, it is important to select an optimal regularization parameter l, as a larger l will generate a sparser result. For the present model, we chose crossvalidation to select the optimal l.
A brief algorithmic description of our method is shown in Algorithm 1. Let n represent the number of samples and p be the number of genotypes. The time complexity of subgradient step in each iteration is O(np). In real data, p is usually supposed to be large, resulting in comparatively high time complexity. Therefore, in genome-wide association analysis, we suggest to analyze chromosomes individually for huge genome.

Performance Measurements
For performance evaluation of the new model, we treat the loci detection as a binary classification under class imbalance, in which associated loci are assigned the label 1, and all others are assigned the label 0. The testing frequency of each locus is then regarded as the predicted probability for label 1. The receiver operating characteristic (ROC) curve and the area under the precision-recall curve (AUPR) are typically used for performance assessments. The ROC curve is plotted based on the sensitivity and specificity, whereas AUPR is generated based on the precision and recall. In our problem, the number of variants is significantly lower than the number of all loci, resulting in an imbalanced dataset. In the ROC curve, the false positive rate cannot descend greatly when the true negative is huge. However, the AUPR is sensitive to false positive. Considering these factors, we chose the AUPR as the performance metric for this purpose.

Experiments on Simulation Data
For assessing the performance of WSGL in selecting candidate SNPs associated with a trait of interest, its performance was compared with lasso, GL, and SGL. Two parameters needed to be controlled in this experiment: a, which is the proportion of ℓ 1 and ℓ 2 loss in SGL, and l, which is the coefficient of the entire regularization term and influences the sparsity. We set a to 0.95. Based on the results of cross-validation, l was set to 0.09. Figure 1 shows the results of the four methods with the simulation data, which clearly exhibits the superior performance of WSGL. The AUPR of WSGL is 0.652, which outperformed lasso by 23.6%, GL by 50.8%, and SGL by 24.4%. Lasso uses ℓ 1 to guarantee the sparsity of selected SNPs, but does not consider the group information; therefore, the candidate SNPs may be selected from all groups equally. Although GL imposes group information on the model, it still lacks sparsity constraint within the group, which does not correspond with the biological assumption that only a small number of candidate SNPs are contained in a small number of groups. SGL considers the sparsity between and within groups, but can still easily exclude ALGORITHM 1 | Parameter estimation for weighted sparse group lasso.
Input: Genotype X, phenotype y ratio a, regularization hyperparameter l Output: Estimatedb 1: calculate w = 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MAF(1 − MAF) p ; 2: while not converge do 3: for k from 1 to number_of_groups do 4: for i from 1 to length_of_groups (k) do 5: updateb (k) i using equation (9) important SNPs with a lower MAF. By introducing biological information to adjust the penalty of SNPs in the selected groups, WGSL places less weight on the low-frequency variants and thus increases their chance of being kept out. Despite its simplicity, the simulation results demonstrated the effectiveness of this approach for screening out important SNPs.
To further compare the performance of the four algorithms, we computed their AUPR values by fixing a at 0.95 and varying l from 0.01 to 0.1 by steps of 0.01. As shown in Figure 2, with smaller l, the model shows lower sparsity. When l is 0.01 or 0.02, the model will include more SNPs, which may include more non-candidate SNPs that would cause a high false positive rate. Conversely, as l increases, the number of selected SNPs decreases, which might result in the loss of some candidate SNPs, leading to a low TP rate. However, WSGL will include more candidate low-frequency loci by introducing prior knowledge to adjust the weight. Accordingly, WSGL keeps the highest position starting from l = 0.03. When l increases from 0.02 to 0.05, the AUPR of WSGL increases significantly from 58% to 64.5%, whereas the AUPR of lasso decreases from 59.2% to 53.2%, and that of SGL decreases from 59.9% to 56.1%. Surprisingly, the AUPR of GL decreases even more sharply from 51.1% to 34.1%. When l reaches 0.05, the AUPR of WSGL tends to be stable, and the peak of 65.2% occurs at l = 0.09. The AUPR of both lasso and SGL gradually decreases, and finally drops to around 40%. When l is 0.1, the AUPR of GL drops to 13.3%. These results were consistent with our expectation that the performance of WSGL would be the best, SGL would perform better than lasso, and GL would show the worst performance overall.

Experiments on Real Data
To verify the ability of WSGL to detect candidate SNPs, we compared the performance of the four models using Arabidopsis flowering time data with known genetic associations. The dataset included 10 different phenotypes, FT10, FT16, FT22, LD, LDV, SD, SDV, LN10, LN16, and LN22, and the descriptions of the 10 phenotypes are shown in Table 1. We analyzed the associated number of genes covered by 100 SNPs with top probabilities of being target loci.
As shown in Table 2, WSGL could link more candidate genes with phenotypes FT10, FT16, FT22, LD, SD, and SDV. In particular, WSGL demonstrated excellent performance for FT10, not only by selecting less groups but also by including less SNPs within each group, and the ratio of candidate genes was 23.08%. By contrast, the ratios of candidate genes were 4.65%, 9.09%, and 5.13% for lasso, GL, and SGL, respectively. For phenotypes FT16, FT22, LD, SD, and SDV, WSGL still achieved the best detection performance. However, unexpectedly, the GL model obtained better results for the first four phenotypes. We consider that this may be due to the specific distribution of loci in the dataset. In cases for which most or all of  the candidate objects are located in only one group, GL will apparently show a good result. By contrast, all four methods could link all six genes with LDV, LN10, LN16, and LN22. This surprising result may reflect the strong association between the selected SNPs and these phenotypes, which is highly discriminable. Nevertheless, this assessment demonstrated that our new weighted method achieves the best performance overall, highlighting the importance of considering prior biological information for selection of candidate SNPs.

CONCLUSION
We proposed a method named weighted sparse group lasso (WSGL) to improve the detection of genetic variants. WSGL incorporates the ℓ 1 penalty, ℓ 2 penalty, and prior biological knowledge into a single linear regression model, and then uses SGL to either select or clear out all SNPs in a group potentially associated with a phenotype of interest. To screen candidate lowfrequency variants, we introduced the MAF as the weight to rescale each element for calculating ℓ 1 loss. In addition, WSGL can detect meaningful associations with more accuracy compared to available methods, which conforms with the general assumption that complex traits are affected by a few SNPs in a few genes. Experiments with both simulation and real data of SNPs related to the flowering time of A. thaliana demonstrated the effectiveness of our approach.


User Experience Evaluation through Automatic A/B Testing

The goal of this research is to develop an A/B testing method to automatically compare the user experience (UX) of alternative designs for a web application in a real context with a large number of users. The challenge that it poses is to find mechanisms to predict the UX with machine learning techniques. This submission outlines the motivation, research goal, current status and remaining work.

INTRODUCTION
While the UX of a web application is a crucial factor that determines its success, UX evaluation is often neglected, mainly due to its high cost. Although there are several proposals to reduce the costs of UX evaluation such as remote user testing [1] and automatic user interaction analysis [7], as Garcia et. al. [2] evidences, there is a lack of tools and methods to mediate the collaboration between designers and developers in the UX evaluation of the application being developed, which implies not only discovering the UX issues, but also the identification and the evaluation of alternative designs.
This research is focused on providing support for evaluating alternative designs of the user interface (UI) of a web application. Specifically, we are investigating a method similar to A/B testing to allow designers to compare the UX of different designs and determine the best solution for a given set of UX problems in the target application. A/B testing (also called split testing) is a wellknown technique that consists of deploying two or more versions of a UI [8]. Users are split between the versions in a persistent manner, Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). IUI '20 Companion, March 17-20, 2020, Cagliari, Italy © 2020 Copyright held by the owner/author(s). ACM ISBN 978-1-4503-7513-9/20/03. https://doi.org/ 10.1145/3379336.3381514 and key metrics are computed to determine the best version in terms of "conversion rate", which refers mostly to increased revenue.
Even though most of the commercial A/B testing tools such as Google Optimize [4] and Optimizely [9] are intended to evaluate conversion rate, there are research works that propose a variation of A/B testing to evaluate usability [10] or user's preference [11]. Both approaches require user input to compare the alternative designs, which can hinder the evaluation in a real application, lacking transparency for the subjects. The split testing proposed in [10] relies on user interaction analysis and user questionnaires to train models to predict the usability of a UI. These models are specific for the application under test. On the other side, the evaluation developed in [11] is not conducted on a working application, but on screenshots or mockups that are randomly presented to the users for them to select the preferred one. Another aspect lacking attention in the literature is the technology needed to set up the alternative designs to be tested, which is very relevant to prevent the cost of a manual implementation.
Contrarily to previous approaches, we are interested in automatically evaluating alternative designs in terms of UX, i.e., without requiring explicit users intervention, in order to easily conduct the evaluation in a real context with a large number of users. In this sense, we propose the development of Machine Learning (ML) models to assess the UX of a design. In a recent work we approached this proposal by predicting the user interaction effort with some UI elements through the analysis of user interaction metrics automatically captured [6]. Using the user interaction effort as a measure of UX, the next step is to define a method to compare alternative solutions to UX problems. Besides that, to make the A/B testing affordable, support must be provided from the beginning of the evaluation, which starts by configuring the designs to be tested.

RESEARCH GOAL
The goal of this research is to develop a method to conduct A/B testing experiments to automatically compare the UX of alternative designs of a web application. The evaluation method is aimed to support designers in the task of UX improvement once a set of UX issues have been discovered, and should work directly on production in a transparent way for the users. This research addresses two different aspects: • Set up the web application's versions to be tested. In order to allow designers to work on their own, we are investigating a method to create alternative versions of a web application without altering its source code, i.e, by directly modifying its UI in the browser. • Compare different versions of a web application. To support designers in finding the version that provides a better UX, our goal is to develop a method and a tool to automatically evaluate the UX of these versions by analyzing user interaction events. To this end, we believe that ML models are an important technique to predict users' behaviour because its multiple interaction patterns involved cannot be easily analyzed defining heuristics.

CURRENT STATUS
We have been working on both aforementioned goals. Our proposal to create alternative versions of a web application is based on the notion of Client-side web refactorings (CSWRs) [3]. Each CSWR is a predefined transformation that performs small changes in the client of a web application through a script, like adding or replacing a widget, with the aim of improving the UX while preserving functionality. Our first approach consisted in a tool that allows creating a web application version by applying CSWRs semi-automatically, once usability problems have been found by analyzing user interactions [5]. In that case, the analysis of user interaction events was based on heuristics, and the tool requires collecting a large amount of interaction events to be able to discover usability problems and suggest CSWRs that may solve them. The evaluation of this tool suggested the need of a method intended for designers to freely explore potential solutions to UX issues through CSWRs, i.e, by directly selecting an element from the UI and a CSWR to apply on it. In this sense, we are working on a visual programming tool called UX-Painter, a web-extension that allows designers to work directly on production. Currently, we are conducting experiments with UX designers to assess its validity. With respect to the evaluation of designs, since we propose the application of CSWR to create alternative UIs, our first step was to investigate a method to measure the effectiveness of the changes applied by the CSWRs in the context of a specific application. Thus, we developed a unified score called interaction effort, to assess the level of effort required by users to interact with different widgets [6]. Since alternatives CSWRs for a specific UX problem apply different changes, the proposed score aims to compare how different widgets perform for the same task. This score is predicted from micromeasures automatically captured from interaction logs, that are specific to each widget type. To be able to predict the interaction effort, we have to analyze users interacting in real web applications in two different moments: in a preliminary study to discover the micro-measures that influence the user's interaction effort for each widget type, and then with the micro-measures defined to obtain their samples to train each widget's prediction model. We already conducted the preliminary study to define the micro-measures that determine the interaction effort of two widget types, text inputs and selects, and we have validated this approach obtaining interaction samples and manually ranking them to train decision tree classifiers [6]. We decided to use decision trees as preliminary models to get feedback about the importance of each micro-measure in the resulting classification. The first results achieved allow us to go further in the interaction effort prediction, namely collect more user interaction samples and evaluate the performance of others prediction models.

NEXT STEPS
Concerning the setup of the designs to be tested, the next step is to develop a method to deploy in production the alternative versions created with UX-Painter and to split the users between them consistently. Besides that, we plan to incorporate new CSWRs to UX-Painter to offer designers different solutions for each UX issue considered.
Regarding the evaluation of alternative versions, we are creating tools to support the collection of training data, specifically, to capture the micro-measures of each widget interaction and to allow UX experts to easily rank them by looking at recorded screencasts. This will enable approaching our next challenge, i.e., to obtain a large amount of samples to implement the prediction models of all the widgets involved in the CSWRs. Also, this requires analyzing which ML model works better with the collected dataset. After being able to obtain the user's interaction effort of all widget types, we still need to define a strategy to determine which of the versions under test is better in terms of UX. The interaction effort score judges the interaction of a single user in a specific widget of a page, so first we need to define an overall interaction effort of a specific widget that considers all the user interactions gathered during the evaluation on that widget. Moreover, considering that each tested version may include different CSWRs, a method to weight the interaction effort of the different widgets is required to obtain a measure of the overall UX of each version to identify the best one. Last but not least, we may also consider other elements to judge the UX besides the interaction effort metric, by other ML models that may predict users' comfort, joy, or other aspects involved in the UX.


extfieldsofstudy:
Mathematics
Computer Science
Engineering

text:
Causal Factorization and Linear Feedback

An algebraic framework for the investigation of linear dynamic output feedback is introduced. Pivotal in the present theory is the problem of causal factorization, i.e. the problem of factoring two systems over each other through a causal factor. The basic issues are resolved with the aid of the new concept of latency kernels.

INTRODUCTION.
In recent years the system theory literature has seen a rapidly growing interest in questions associated with linear feedback. In the early 1960 's, linear control theory centered chiefly around quadratic (Gaussian) optimal problems and the resulting feedback designs. Later, interest in feedback shifted to a variety of so-called "synthesis" problems. These included the well-known problem of observer design (see Luenberger [1966] ), the pole shifting theorem and related issues (Wonham [1967] Simon and Mitter [1968], Brash and Pearson [1970], Heymann [1968]) as well as the decoupling problem (Falb and Wolovich [1967], Gilbert [1969], Wonham and Morse [1970], Morse and Wonham [1970] ). All of these feedback synthesis problems, as well as many others, were formulated and resolved within the framework of state space representations. While most of the work was done with the use of conventional state equations, the work of Wonham and Morse was distinguished by its "coordinate free" setting and initiated what later developed into the celebrated "geometric theory" of linear control (see, e.g., Wonham [1979]).
The current growing interest in linear feedback differs significantly from that of the past both in character and in its source of motivation. While previously the study of feedback was largely oriented at problem solving, the current interest is motivated by a desire of gaining insight into the general nature of linear feedback-chiefly from an algebraic point of view. Much of the motivation for the present trend can be traced back to the work of Rosenbrock [1970] , in which polynomial matrix techniques were used for the study of a variety of (linear) control theoretic questions. Particularly useful turned out to be techniques based on polynomial fraction representations of transfer functions (see, e.g., Heymann [1972], Wolovich [1974], Forney [1975], Fuhrmann [1976]). In this setting of fraction representations, feedback was first studied in Heymann [1972] (see especially Chapter 6 therein), and in a polynomial module framework the study of feedback was initiated by Eckberg [1974]. State feedback also received attention in an algebraic framework by Morse [1975]. A different approach to the study of linear feedback was taken in Hautus and Heymann [1978], where the fundamental underlying object was taken to be the input-output map of the system. There, static linear state feedback was investigated in an algebraic framework consistent with the setting of the (classical) module theory of linear realization as introduced by Kalman (see, e.g., Kalman et al. [1969, Chapter 10] ). More recently, state feedback was also examined in Fuhrmann [1979] using what he termed "polynomial models", and in Münzner and Prätzel-Wolters (1979a], [1979b], [1979c] in a module and category theoretic framework. While these various approaches to the study of feedback differ from each other substantially both in the underlying concept and in philosophy, they commonly converge on essentially the same (standard) issues that characterize state feedback. It is significant, however, that no success (and, in fact, very little effort, if any) has been reported in respect to output, as opposed to state feedback. When various fundamental questions in regard to output feedback are examined, it becomes immediately clear that difficulties arise that are completely absent in the state-feedback setting. In fact, one discovers immediately that crucial insight is missing. It turns out that the chief reason for this state of affairs is the fact that all of the presently existing algebraic theory of linear systems, and especially that of feedback, rests in one way or another on the theory of modules over the ring [ ] of polynomials and on polynomial matrices. This algebraic machinery is completely satisfactory to develop a fairly comprehensive framework for state feedback. It is not adequate, though, to deal with output-feedback where issues associated with causality become significantly more intricate. †The latex version of this paper was obtained from the original by using optical character recognition at Mathpix.com. We express our profound thanks to Mathpix. The present paper deals in a comprehensive way with the problem of causal output feedback. A related question which receives a great deal of attention in the paper and on which much of the theory hinges is the so-called causal factorization problem. This is the problem of when a given linear input-output map can be factored over another one by a causal linear map. Through the resolution of this issue, questions associated with dynamic causal output feedback are then also resolved. Attention is also given to the static factorization problem as well as the problem of static feedback where special emphasis is placed on the state-feedback case.
A crucial role in the present theory is played by the newly introduced concept of latency. In the discrete time setting, latency expresses "degree of causality" and (intuitively) refers to the intrinsic delay which inputs encounter before output responses are produced. Latency is algebraically expressed by modules over the ring −1 of power series (in −1 over a field ). These modules arise in a natural way when the concept of causality is studied algebraically and in fact are readily seen to be the natural algebraic device for the study of feedback. The paper is organized as follows. In §2 the basic concepts of Λ -linear maps causality, linear i/o maps as well as linear i/s maps, which have been investigated in detail in Hautus and Heymann [1978], are reviewed. The conceptual viewpoint, on which the present investigation of feedback rests, is discussed in §3. An important technical concept that arises in the algebraic study of linear systems both in connection with the [ ] -module theory and the −1 -module theory is that of "proper bases" and "proper independence". This is the topic of $4. Section 5 is devoted to the investigation of causal factorization, the main result being Theorem 5.2 and its corollaries. Results are also obtained on static feedback (Theorems 5.10 and 5.14 ). In §6 the problem of invariants is investigated in detail and explicit characterizations are derived and exhibited. The role of the latency kernels and latency indices is also discussed. The paper is concluded in $7 with an investigation of the interesting question of feedback (design) limitations. It is shown that the essential limitation to the possibility of causal feedback implementation of precompensators is the system's latency. In particular, precompensators can be implemented as causal feedback devices modulo a "precompensator remainder" whose dynamic order need not exceed the sum of the system's latency indices.

ΛK -LINEAR MAPS, CAUSALITY AND INPUT-OUTPUT BEHAVIOR.
We shall adopt a terminology and setup consistent with that of Hautus and Heymann [1978]. Let be a field and let be a -linear space. The class of all truncated -valued Laurent series of the form of -valued Laurent series is endowed with a field structure under the operation of convolution as multiplication and coefficientwise addition. In particular, for = ∑ ∞ = 0 − and ′ = ∑ ∞ = ′ − in Λ , the product ′ is given by and the sum + ′ is given by With Λ as the underlying field it then follows that, with convolution as the scalar multiplication and with the usual coefficientwise addition, the set Λ becomes a Λ -linear space. When is a finite dimensional -linear space, say of dimension then so is Λ as a Λ -linear space. It is readily observed that; under the same operations of convolution as multiplication and coefficientwise addition, the field Λ contains (as subobjects) also (i) the ring [ ], or in our notation Ω + , of polynomials in ; (ii) the ring −1 , or in our notation Ω − , of formal power series in −1 ; and finally, (iii) the field itself. It, thus, follows immediately that the set Λ is not only a Λ -linear space but is simultaneously also an Ω + -module, an Ω − -module and a -linear space As we shall see, these facts turn out to be of central importance in the theory. Now, we let ℤ denote the integers and for an element ∈ Λ , given by (2.1), we define the order of by If ≠ 0 and 0 = ord , we call the coefficient 0 the leading coefficient of . Let and be -linear spaces. We shall call the input value space and the output value space of an underlying linear system Σ. The Λ -linear spaces Λ and Λ are then called the extended input space and extended output space, respectively Elements = Σ − ∈ Λ and = Σ − ∈ Λ , called, respectively, (extended) inputs and (extended) outputs, are identified with time sequences and (with being identified as time marker). Let̄ ∶ Λ → Λ be a -linear map. We say that̄ is time invariant if ( ⋅ ) = ⋅̄ ( ) for all ∈ Λ , so that̄ is time invariant whenever it is a Λ -linear map (Wyman [1972].. Next, for a Λ -linear map̄ ∶ Λ → Λ we define the order of̄ by If the map̄ is the zero map then ord̄ ∶= ∞; otherwise ord̄ < ∞. While it is possible that ord̄ = −∞ we shall not concern ourselves here with this case and confine our attention to maps of finite order. This is clearly always the case when (and hence also Λ ) is finite dimensional.
We now introduce the following (see also Hautus and Heymann [1978]).
Let denote the -linear space of -linear maps → and let Λ denote the Λ -linear space of all -Laurent series. We identify this space with the space of Λ -linear maps Λ → Λ of finite order as follows. We define the -linear maps and with every Λ -linear map̄ ∶ Λ → Λ we associate the Laurent series where, for each ∈ ℤ (2.7) ∶= (̄ ) ∶=̄ ⋅̄ ⋅̄ The Laurent series (2.6) is called the impulse response or the transfer function of̄ . If = Σ − ∈ Λ is any element, then the action of̄ on is given by It is thus immediately seen that (2.9) ord̄ = min | (̄ ) ≠ 0 , whence we have the following characterization of causality in terms of the transfer function: The map̄ is causal if and only if (̄ ) = 0 for < 0 and strictly causal if and only if (̄ ) = 0 for ≦ 0. We also have the following easily verified proposition.
We associate with an extended linear i/o map̄ a restricted linear i/ o map̃ which is obtained as follows (see also Hautus and Heymann [1978]). Inputs are restricted to the subset Ω + ⊂ Λ , called the restricted input space, and consist of all inputs that terminate at = 0, i.e., elements of the form Σ ≤0 − . Outputs are observed only for ≧ 1, that is, in the subset −1 Ω − which is, of course, in bijective correspondence with the Ω + -quotient module Γ + ∶= Λ ∕Ω + which we call the restricted output space. The restricted linear i/o map̃ ∶ Ω + → Γ + associated with̄ is then defined bỹ = + ⋅̄ ⋅ + where + ∶ Ω + → Λ is the canonical injection and + ∶ Λ → Γ + is the canonical projection. Clearly, since + and + are Ω + -module homomorphisms, so is alsõ and we have the following: Definition. 2.12. A map̃ ∶ Ω + → Γ + is called a restricted linear i/o map if it is an Ω + -module homomorphism.
Next, we define the linear output response (or output value) map ∶ Ω + → associated with a given linear i/o map̄ (or̃ ) as follows: is called reachable if the associated output value map is surjective. If ∶ Ω + → is any -linear map, it can be regarded as an output value map of a linear system. In particular, the restricted and extended linear i/o maps associated with are then given by The output value map , which gives for each (restricted) input the value of the output at time = 1, is clearly a -linear map. In some special cases, there exists an Ω + -module structure on , compatible with its -vector space structure, such that the output value map is not just -linear but is also an Ω + -module homomorphism. When this is the case, then for each ∈ Ω + and for each positive integer , = ( ), whence, by (2.15), knowledge of the output value at time = 1 implies knowledge of the whole ensuing output sequence. This is therefore precisely the case when the system's output "qualifies" as state, a fact which motivates the following definition (for greater detail the reader is referred to Hautus and Heymann [1978]): Definition. 2.17. An extended linear i/o map̄ ∶ Λ → Λ is called an extended linear input-state (or i/s) map if there exists an Ω + -module structure on , compatible with its -linear structure, such that the output value map =̄ 1 ⋅̄ ⋅ + is an Ω + homomorphism. The associated restricted map̃ is called restricted linear i/s map.
If and are -linear spaces and ∶ → is a -linear map, then it induces in a natural way a Λ -linear map which we call static as follows: In a similar way induces also static Ω + and Ω − -homomorphisms. We shall need the following characterizations of linear i/s maps, from Hautus and Heymann [1978] Theorem.

FEEDBACK AND CAUSAL FACTORIZATION-GENERAL CONSIDERATIONS.
We shall be concerned with the setup described by the block diagram in Fig. 3.1. Herē ∶ Λ → Λ is an extended linear i/o map, called the open loop system,̄ ∶ Λ → Λ is a causal Λ -linear map called the (output) feedback compensator,̄ pr ∶ Λ → Λ is a Λ -linear bicausal isomorphism called (bicausal) precompensator and̄ po ∶ Λ → Λ is a Λ -linear bicausal isomorphism called (bicausal) postcompensator. In case any of the maps̄ ,̄ pr or̄ po is static we shall call it, respectively a static feedback, pre or post compensator. Now, since the map̄ is causal and̄ is strictly causal, it readily follows that the composite maps̄ ⋅̄ ∶ Λ → Λ and̄ ⋅̄ ∶ Λ → Λ are both strictly causal. Letting denote both of the corresponding identity maps, we see that both of the maps ( +̄ ̄ ) ∶ Λ → Λ and ( +̄ ̄ ) ∶ Λ → Λ are bicausal isomorphisms. It follows that the setup of Fig. 3.1 is "well-posed" in the sense that there is a strictly causal Λ -linear map Λ → Λ ∶ ↦ given by either of the following composite maps: Using again block diagrams, (3.1) and (3.2) can be described, respectively, as in Fig. 3.2a and 3.2b. In both descriptions, the dashed blocks represent bicausal mappings, so that the compensator configuration of Fig.  3.1 can always be represented equivalently Because of the obvious duality between the precompensator situation and the postcompensator situation, there is no need to discuss both of them in detail. since practical interest in postcompensators is at best limited, we shall henceforth confine our attention to precompensation, and discuss postcompensators only in connection with certain mathematical questions.
For various reasons, not to be elaborated on here, feedback compensation is preferred over external compensation whenever possible. Thus, one is interested in the following problem.
where 0 is a static Λ -linear map and h −1 is the transfer function of a strictly causal maph ∶ Λ → Λ representing the strictly causal part of̄ −1 . Hence we can always decompose the map̄ −1 as −1 = +h with static andh strictly causal. The causal feedback problem 3.3 is therefore essentially equivalent to the following.
Causal factorization problem 3.4. Let̄ ∶ Λ → Λ be a given strictly causal Λ -linear map. (a) Under what conditions can a strictly causal Λ -linear maph ∶ Λ → Λ be factored causally over̄ , i.e., when does there exist a causal map̄ ∶ Λ → Λ such thath =̄ ⋅̄ ? (b) Under what conditions can every strictly causal Λ -linear maph ∶ Λ → Λ be factored causally over̄ ? It is readily noted that the strict causality of the maps̄ andh is inessential to the causal factorization problem, and arises in problem 3.4 only because of the specific requirements of the feedback problem. Indeed, ifh factors causally over̄ , i.e., if there exists a causal̄ such thath =̄ ⋅̄ , then for each integer we also have h = ̄ ̄ =̄ ⋅ ̄ so that ℎh factors causally over ̄ , and for sufficiently large positive (unlessh or̄ are zero) the maps h and ̄ are not causal. Thus, the causal factorization problem can be stated in the following less restrictive way: Given two Λ -linear maps̄ ∶ Λ → Λ andh ∶ Λ → Λ (where , and are -linear spaces ), when does there exist a causal Λ -linear map̄ ∶ Λ → Λ such that the following diagram in Fig. 3.3 commutes FIG If the causality requirement of̄ is dropped, the factorization problem is standard (see, e.g., Greub [1967]) and ℎ factors over̄ if and only if ker̄ ⊂ kerh. Yet this condition does not say anything about the causality of̄ . To deal efficiently with the causality issue, we reintroduce the concept of causality using an approach which is algebraically more tractable.
Let̄ ∶ Λ → Λ be a Λ -linear map. We can characterize causality of̄ as follows (compare with our definitions of causality in §2 ): (3.5) The map̄ is causal if and only if ∈ Ω − implies̄ ( ) ∈ Ω − . Similarly, we have: (3.6) The map̄ is strictly causal if and only if ∈ Ω − implies̄ ( ) ∈ Ω − . Let us denote the Ω − -quotient module Λ ∕Ω − by Γ − , and let − ∶ Λ → Γ − denote the canonical projection. The following can then be easily verified by the reader. We shall use the characterizations of the above proposition extensively in the following sections.

PROPER INDEPENDENCE AND PROPER BASES.
Let be a field and let ∶= . For an element 0 ≠ ∈ Λ , denote bŷ the leading coefficient of . If = 0 we shall say that̂ = 0.
Below we derive a variety of properties of properly independent sets, of proper bases and of proper direct sum decompositions. Our objective is to develop this theory here only to the extent required in the sequel. Many further results have been omitted, and the reader can, for example, easily verify that the converses of a number of our results are also valid. A more extensive exposition of this and related topics will be published elsewhere Lemma. 4.2. If 1 , ⋯ , ∈ Λ is a properly independent set of vectors, then (i) it is Λ -linearly independent, and (ii) for every set of scalars 1 , ⋯ , ∈ Λ the following holds Proof. We shall prove the lemma by showing that if either (i) or (ii) fails to hold then the set 1 , ⋯ , is not properly independent. If 1 , ⋯ , ∈ Λ is any set of scalars then, by definition, ord ∑ =1 ≧ ∶= min ord | = 1, ⋯ , . If either (i) or (ii) fails to hold, there exist 1 , ⋯ , ∈ Λ , not all zero, such that either ∑ and consider the terms of order in Σ . This yields ∑ =1 ̂ ̂ = 0, implying that̂ 1 , ⋯ ,̂ are -linearly dependent since not all the ̂ are zero. Hence 1 ⋯ , are not properly independent, completing the proof. □ The condition of Lemma 4.2( ii ) has been called the "predictable degree property in Forney [1975], in the (analogous) setting of "minimal polynomial bases" for rational vector spaces. We shall adopt this terminology and call the property of Lemma 4.2(ii) the predictable order property.
To avoid possible confusion in the ensuing discussion where we shall deal with both -linear and Λ -linear spaces, we shall use subscripts to emphasize the field. Thus, for example, span Λ 1 , ⋯ , denotes the Λ -linear subspace spanned by 1 , ⋯ , ∈ Λ , whereas span ̂ 1 , ⋯ ,̂ denotes the -linear subspace spanned bŷ 1 , ⋯ ,̂ ∈ . Similarly, dim Λ ℛ denotes the dimension of a subspace ℛ ⊂ Λ as a ΛK -linear space (to distinguish from -linear). We next have the following theorem.
Theorem. 4.4. Every nonzero Λ -linear subspace ℛ ⊂ Λ has a proper basis. Moreover, every properly independent subset of ℛ can be extended to a proper basis.
Finally, we also have the following variant of the predictable order property.

CAUSAL FACTORIZATION.
We turn now to the causal factorization problem (3.4). As we mentioned earlier, there is no essential need, in characterizing causal factorizability, to assume strict causality, or even causality, of the maps under consideration. We shall therefore begin with the general case and turn to specific consideration of i/o maps later on. We shall assume that the spaces and are finite dimensional, in particular that = and = . For convenience of notation, we shall temporarily use the notation Λ and Λ also in connection with Λ -linear maps̄ ∶ Λ → Λ that are not necessarily i/o maps (i.e., are not necessarily strictly causal).
Let̄ ∶ Λ → Λ be a Λ -linear map and let − ∶ Λ → Γ − ∶= Λ ∕Ω − be the canonical projection. since Ω − is an Ω − -module, so is the quotient Λ ∕Ω − . Thus the map − is an Ω − -homomorphism and so is also the composite −̄ . We have Next we have the following central theorem.

Corollary. 5.4. Let̄ ∶ Λ → Λ be an extended linear i/o map. Then̄ is nonlatent if and only if for every strictly
Let̄ ∶ Λ → Λ be an extended linear i/o map and let̄ ∶ Λ → Λ be a bicausal isomorphism, i.e., a bicausal precompensator for ⃗ . Leth be the strictly causal part of̄ −1 , i.e.,̄ −1 = +h where is static. As we have seen in §3,̄ can be realized as feedback around̄ ifh factors causally over̄ . Theorem 5.2 tells us essentially that the only barrier to realizing a bicausal precompensator as feedback is the relative latency of̄ andh. Corollary 5.4 characterizes the class of i/o maps over which every bicausal precompensator can be realized as feedback. These i/o maps are, as we have seen, the nonlatent maps (a fact which motivated our choice of terminology). Now, a very special and important class of nonlatent maps is that of injective i/s maps. This fact is proved in the following theorem.
Proof. By strict causality of̄ we have that Ω − ⊂ ker −̄ , so that to prove nonlatency we need only to show that ker −1̄ ⊂ Ω − . Let ∈ ker −̄ so that̄ ( ) ∈ Ω − . Write = + + − , where + ∈ 2 Ω + and − ∈ Ω − . The proof will be completed by showing that + = 0 so that ∈ Ω − as claimed. Note that̄ ( − ) ∈ Ω − by the strict causality of̄ so that, in view of the fact that̄ ( ) =̄ + +̄ ( − ) , it follows that̄ + ∈ Ω − . By (2.16) we havē so that, in particular, that −2 + ∈ ker = ker̃ (the equality being a consequence of the i/s property (2.20) ). It follows that̄ −2 + ∈ Ω + , or alternatively, that̄ + ∈ 2 Ω + . since 2 Ω + ∩ Ω − = 0, we conclude that̄ + = 0 or that + = 0 by the injectivity of̄ . □ While Theorem 5.5 deals only with injective i/s maps, it is important to observe that this is not a serious restriction. Indeed, it is shown in Proposition 5.6 below that in the special case of i/s maps (in contrast to i/o maps in general), the kernel is "static"; i.e., if̄ is a noninjective i/s map, then ker̄ = Λ 0 where 0 ⊂ is a subspace. This means that the whole degeneracy lies in the input value space which has been chosen too large, and by restricting the input value space to a proper summand of 0 in , the injectivity is restored.
Proof. Let ∶ → Ω + ∶ ↦ be the canonical injection and define the subspace 0 ⊂ as 0 ∶= ker ⋅ , where is the output value map associated with̄ . sincē is an i/s map we have ker ⋅ = ker̃ ⋅ = ker̄ ⋅̄ with the last equality holding by the strict causality of̄ . Thus̄ 0 ⊂ ker̄ , and since ker̄ is a Λ -linear space we conclude that Λ 0 ⊂ ker̄ . To prove that ker̄ ⊂ Λ 0 , it suffices to prove that if 0 ≠ = ∑ ∞ = 0 − ∈ ker̄ then 0 ∈ 0 . By recursive application of the same argument this will then imply that ∈ 0 for all ≧ 0 . Now by formula (2.16) we have + = 0 for all ∈ ℤ, and since + 0 = 0 the results follow. □ The importance of Theorem 5.5 lies in the fact that it tells us that bicausal precompensation is equivalent, in the sense of solvability, to dynamic state feedback Let̄ ∶ Λ → Λ be an extended linear i/o map. We write (see Hautus and Heymann [1978])̄ = ⋅̄ , where is a static output map and̄ is a reachable i/s map. If̄ is injective (which is always the case when ker̄ does not contain a subspace of the form Λ , 0 ≠ ⊂ ), then every bicausal precompensator can be realized as feedback around̄ . That is, we can write every bicausal̄ ∶ Λ → Λ as̄ −1 = +̄ ̄ , wherē ∶ Λ → Λ is a causal Λ -linear map and is static.

Corollary
To see the bicausality of̄ it suffices to show that it is order preserving. Indeed, let = 1 + 2 ∈ Λ be any element with 1 ∈ Im̄ 1 and 2 ∈ ℛ 1 . Then̄ =̄ 1 1 +̄ 2 2 and using Corollary 4.8 together with the fact that Im̄ 1 and Im̄ 2 form a proper direct sum, we have that ord̄ = min ord̄ 1 1 , ord̄ 2 2 = min ord 1 , ord 2 , where the last equality follows from the order preserving property of̄ 1 and̄ 2 . Using Corollary 4.8 again, together with the fact that Im̄ 1 and ℛ 1 form a proper direct sum, gives that min ord 1 , ord 2 = ord whence ord̄ = ord as claimed and the proof is complete. □ Clearly, the bicausal Λ -linear map̄ of Corollary 5.7 can be regarded as a bicausal postcompensator for̄ 1 , and there is a kind of duality between feedback and compensation which deserves some further comments.
Let̄ ∶ Λ → Λ be an extended linear i∕o map and let̄ pr ∶ Λ → Λ be a bicausal precompensator for̄ . If ∶ Λ → Λ is the strictly causal part of̄ pr , then the causal feedback problem is that of existence of a causal Λ -linear map̄ ∶ Λ → Λ such that̄ =̄ ⋅̄ . The map̄ can be regarded essentially as a causal (but not necessarily bicausal) postcompensator for̄ . Conversely, if̄ po ∶ Λ → Λ is a bicausal postcompensator and if̄ ∶ Λ → Λ the strictly causal part of̄ po , the dual of the above causal factorization problem is that of the existence of a causal Λ -linear map̄ ∶ Λ → Λ such that̄ =̄ ⋅̄ . Herē − can be viewed as a causal, but again not necessarily bicausal, precompensator for̄ . Thus the pre-and postcompensator problems become interrelated through feedback. We can also write down the dual of Corollary 5.7 regarding the problem of bicausal precompensation.
In Hautus and Heymann [1978], the static state feedback problem was investigated. This is the following problem: Given an extended linear i/s map̄ ∶ Λ → Λ under what conditions can a bicausal precompensator̄ ∶ Λ → Λ be written as̄ −1 = + ̄ , where and are static maps. It was shown there that a necessary and sufficient condition for the static state feedback problem to have a solution is that (5.9)̄ −1 (ker̃ ) ⊂ Ω + , wherẽ ∶ Ω + → Γ + is the restricted i/s map associated with̄ . We now turn to the more general question of static output (rather than state) feedback. As we have been doing throughout this paper, we focus our attention on the static factorization problem which is characterized in the following Theorem. 5.10. Let̄ ∶ Λ → Λ andh ∶ Λ → Λ be Λ -linear maps. There exists a static Λ -linear map ∶ Λ → Λ such thath = ⋅̄ if and only if ker̄ 1 ⋅̄ ⊂ ker̄ 1 ⋅h.

FACTORIZATION INVARIANTS-EXPLICIT CALCULATION.
Throughout this section we shall assume that = and = , and we shall study properties of Λ as an Ω − -module as well as properties of submodules thereof The ring Ω − is of course a principal ideal domain, and clearly also a Euclidean domain. The units of Ω − are precisely those elements whose order is zero and each element 0 ≠ ∈ Ω − can be expressed as where 0 ∈ Ω − is a unit. It is clear, therefore, that all the ideals of Ω − are of the form − , forming a chain with −1 being the unique maximal ideal and the only prime. Thus, the ring Ω − is also a local ring and Ω − ∕ −1 is a field, isomorphic to the field 0 which consists of the units of Ω − augmented by zero. We shall make use of the special properties of the ring Ω − in the ensuing discussion.
For a fixed integer , consider the subset − Ω − ⊂ Λ . Clearly, this subset is an Ω − submodule of Λ . Moreover, while Λ itself is not a finitely generated Ω − module, the submodule − Ω − is (and hence is a free module). In fact, it is readily noted that rank Ω − − Ω − = dim Λ Λ = dim . Indeed, if 1 , ⋯ , is a basis for (as well as for Λ ), then − 1 , ⋯ , − is a basis (i.e., a free generator) for − Ω − . Let 0 ≠ Δ ⊂ Λ be an Ω − -submodule. We say that Δ is of finite order if there exists a finite integer such that Δ ⊂ − Ω − . The maximal integer for which the above holds, and which is the least order of elements in Δ, is denoted Δ and is called the order of Δ. We define the order of the zero module as infinity. We have the following: Proposition. 6.1. Let 0 ≠ Δ ⊂ Λ be an Ω − -submodule. Then Δ is finitely generated if and only if it has finite order.
Consider now the special case when = (that is, = ) and let be a nonsingular × matrix with entries in Λ . Then defines, as above, an Ω − homomorphism Ω − → Λ and also (when simply regarded as a transfer function) a Δ -linear map Λ → Λ . Denoting both maps by the same symbol , it is readily verified that the diagram in Fig. 6.1 is commutative,   FIGURE 0.5. FIG. 6.1 where − denotes the canonical injection. since the matrix is nonsingular, the Λ -linear map is invertible. We shall say that the matrix is bicausal if the associated Λ -linear map is bicausal, i.e., if the entries of are in Ω − and its determinant is a unit in this ring (that is, has order zero). In analogy we shall say that a matrix is strictly causal or causal if so is the associated Λ -linear map. Finally, an Ω − -submodule Δ = Ω − ⊂ Λ is called a full submodule if rank Δ = , i.e., if the matrix is nonsingular.
The proof of Theorem 6.2 is elementary and will be omitted. The following corollary will be useful in the sequel.

. Then Δ ⊆ Δ 1 if and only if is causal with equality if and only if is bicausal.
Let Δ ⊂ Λ be a finitely generated Ω − -submodule of rank and order Δ . Then for all integers ≦ Δ , Δ ⊂ − Ω − and for each integer ≧ Δ we define the submodule Δ ⊂ Δ by Clearly − Ω − ⊂ − Ω − for all ≧ , and it follows that (6.5) Δ = Δ Δ ⊃ Δ 2 +1 ⊃ ⋯ > Δ ⊃ Δ +1 ⋯ As an immediate consequence of the fact that if ∈ Δ then −1 ∈ Δ +1 , it is clear that rank Δ = rank Δ for all and the quotient modules (6.6) ∶= Δ ∕Δ +1 are all torsion modules with −1 as annihilators, that is, for each and for each [ ] ∈ −1 [ ] = 0. Next we shall show that the sequence of quotient modules is isomorphic to a chain of (finite dimensional) -linear subspaces of , that is, each is isomorphic to a subspace ⊂ and It is also clear that is injective, since ker = Δ +1 = [0]. Now, for each integer we define ∶= Im . Clearly is then -linearly isomorphic to and ⊂ +1 with Δ − −1 = 0 for all ≧ 0. Also, by the finite dimensionality of , there exists an integer Δ 2 ≧ Δ such that Δ −1 ≠ Δ and Δ + = Δ for all ≧ 0. We call the chain the orderchain of Δ, and the sequence of integers , ∶= dim , we call the order list of Δ In the special case when Δ = ker −̄ wherē is a linear i/o map, we refer to the order chain and the order list of Δ, respectively, also as the latency chain and latency list of̄ .
It is interesting to observe that the integer Δ is also the least integer satisfying the condition that −1 Δ = Δ +1 for all ≧ Δ . Indeed, we have seen that −1 ∕ ⊂ Δ +1 for all j. To see that −1 Δ ⊃ Δ +1 if and only if ≧ Δ , let = ∑ ∞ = +1 − ∈ Δ +1 be any element. Then we can write and clearly ∈ −1 Δ if and only if ′ ∈ Δ , This can hold for every ∈ Δ +1 only if +1 = , whence the necessity that ≧ Δ . The sufficiency of the condition is an immediate consequence of Theorem 6.11 below. Next we have the following useful result.
Proof. Let rank Δ = , let 1 , ⋯ , be a basis of Δ and define ℛ ∶= span Λ 1 , ⋯ , . It is easily seen that ℛ is the smallest Λ -linear space containing Δ and dim Λ ℛ = rank Δ. The Λ -linear space ℛ has a proper basis and (by Corollary 4.5 ) dim Λ ℛ = dim ̂ . But clearlyR = 2 and the proof is complete. □ Let and ′ be the order chains and and ′ the order lists, respectively of submodules Δ and Δ ′ of Λ . We shall say that ′ is a subchain of , denoted ′ ⊂ if, for all , ′ ⊂ . Similarly we say that the list ′ is smaller than the list , denoted ′ ≦ if ′ ≦ for all integers . As an immediate consequence of the definition we have the following, Proposition. 6.9. Let Δ, Δ ′ ⊂ Λ be Ω − -submodules with order chains and ′ and order lists and ′ , respectively. If Δ ′ ⊂ Δ then ′ ⊂ and ′ ≦ .
Let Δ ⊂ Λ be a finitely generated Ω − -submodule. A set of elements 1 , ⋯ , ∈ Δ is called properly free if the elements are properly independent as elements of Λ (regarded as a Λ -linear space), that is, if the leading coefficientŝ 1 , ⋯ ,̂ are -linearly independent. It is then clear that if 1 , ⋯ , are properly free they are also free (i.e. independent over the ring Ω − ).
Proof. (i) We shall construct an ordered proper basis for Δ which, in particular, satisfies (6.12) and (6.13). Consider the sequence of quotient modules defined by (6.6), of which Δ is the first nonzero one. Choose any equivalence class 0 ≠ 1 Δ and let 1 ∈ Δ be any representative of 1 . Then ord 1 = Δ and 1 is clearly properly free. We proceed stepwise and assume that for > 0, 1 , ⋯ , are properly free elements of Δ satisfying (6.12) and (6.13). If < , let denote the least integer such that < . Then̂ 1 , ⋯ ,̂ ∈ are -linearly independent, but they do not span , since dim = . Thus, there exists an element +1 ∈ such that for any representative +1 ∈ +1 , the set̂ 1 , ⋯ ,̂ ,̂ +1 ∈ are -linearly independent and hence the set 1 , ⋯ , +1 is properly free. Clearly (6.13) is satisfied, and since ord +1 = so is also (6.12). By Lemma 6.8, dim Δ = rank Δ = , so that we finally obtain an ordered, properly free set of elements 1 , ⋯ , ∈ Δ satisfying (6.12) and (6.13). Let Δ ′ denote the Ω − -submodule of Λ generated by 1 , ⋯ , . It remains to be shown that Δ ′ = Δ. Obviously Δ ′ ⊂ Δ and since ord ≦ Δ for all = 1, ⋯ , and since span ̂ 1 , ⋯ ,̂ = Δ , it follows also that Δ Δ ⊂ Δ ′ . Let ∈ Δ be any element and let ord = . Then̂ ∈ whence there are elements 1 , ⋯ , > . Proceeding stepwise the same way, we conclude that there are elements ∈ Δ ′ , and since ′ ∈ Δ 2 ⊂ Δ ′ , it follows also that ∈ Δ ′ and the proof of (i) is complete. To see that (ii) holds, it suffices to observe that for each integer , every ordered proper basis 1 , ⋯ , of Δ has precisely elements whose order is less than or equal to and span ̂ 1 , ⋯ ,̂ = . □ The following immediate corollary to Theorem 6.11 gives a sharp insight to the relation between ordered proper bases of Ω − -modules and their order chain Corollary. 6.14. Let Δ ⊂ Λ be an Ω − -submodule of rank with order chain and order list . Then 1 , ⋯ , is an ordered proper basis of Δ if and only if for each ,̂ 1 , ⋯ ,̂ is a basis for .
Indeed, if ∈ ker −̄ , then̄ ( ) ∈ Ω − and there are elements 1 , ⋯ , so that ord ≧ . □ In view of Proposition 6.16, it follows that the latency kernel of a given linear i/o map̄ is finitely generated if and only if̄ is injective, the case which receives, of course, most of our attention. Before proceeding further, a remark on the noninjective case is in order Remark. 6.17. It is readily noted that if̄ ∶ Λ → Λ is a Λ -linear map, then ker −̄ can (always) be written as where ℛ is a finitely generated full Ω − -submodule of Λ . However, in the above representation, ℛ is nonunique except in the special case when̄ is injective and ker̄ = 0. If̄ 1 and̄ 2 are two Λ -linear maps then ker −̄ 1 ⊂ ker −̄ 2 if and only if ker̄ 1 + ℛ 1 ⊂ ker̄ 2 + ℛ 2 . While this condition necessarily implies ker̄ 1 ⊂ ker̄ 2 , it cannot be claimed, except in the injective case, that ℛ 1 ⊂ ℛ 2 . Hence, for computational purposes it is convenient in the noninjective case to resort to the fact that ker −̄ 1 ⊂ ker −̄ 2 if and only if Δ ̄ 1 ⊂ Δ ̄ 2 for all , where Δ ̄ is as defined in Lemma 6.15 However, Δ ̄ 1 ⊂ Δ ̄ 2 for all if and only if Δ ̄ 1 ⊂ Δ ̄ 2 for any ≦ min ord ℛ 1 , ord ℛ 2 where ℛ , = 1, 2, are any submodules in the corresponding representations of ker −̄ . By Lemma 6.15 both Δ ̄ 1 and Δ ̄ 2 are full finitely generated Ω − -submodules of Λ so that the situation is thus similar to that in the injective case. □ Let̄ ∶ Λ → Λ be an injective extended linear i∕o map and let Δ = ker −̄ . Then Δ = Ω − is a full, finitely generated Ω − -submodule of Λ and the columns 1 , ⋯ , of the generating matrix form a basis of Δ. We shall next establish certain properties of possible selections of the matrix . Proposition. 6.18. Let̄ ∶ Λ → Λ be an injective extended linear i/o map. Write ker −̄ = Ω − . Then −1 exists and is strictly causal; i.e., the elements of −1 are in −1 Ω − .
Proof. The existence of −1 follows immediately from Proposition 6.16. From the strict causality of̄ it follows that Ω − ⊂ ker −̄ , whence by Theorem 6.2 there exists a causal matrix such that = . Thus −1 = −1 and −1 is clearly strictly causal. □ Let Δ ⊂ Λ be a full finitely generated Ω − -submodule and write Δ = Ω − . We call the columns 1 , ⋯ , of a polynomial be : is of Δ if the matrix is a polynomial matrix, i.e., with elements in Ω + . We call the basis a strictly polynomial basis if its elements are strict polynomials, i.e., with elements in Ω + . If in addition is a proper basis we call it a proper polynomial basis, respectively, proper strictly polynomial basis for Δ.
Theorem. 6.21. Let̄ 1 ,̄ 2 ∶ Λ → Λ be two extended linear i/o maps with and finite dimensional K-linear spaces. There exist bicausal Λ -linear compensators̄ pr ∶ Λ → Λ and̄ po ∶ Λ → Λ such that̄ 2 =̄ po ⋅̄ 1 ⋅̄ pr if and only if there exists an order preserving Ω − -isomorphism pr : ker −̄ 2 → ker −̄ 1 . We now restrict Theorem 6.21 to the injective case to obtain the following invariance characterization of the latency indices Corollary. 6.22. Let̄ 1 ,̄ 2 ∶ Λ → Λ be two injective extended linear i/o maps with and finite dimensional -linear spaces. There exist bicausal Λ -linear compensators̄ pr∶Λ → Λ and̄ po ∶ Λ → Λ such that Proof. By the injectivity of̄ 1 and̄ 2 , both Δ 1 = ker −̄ 1 and Δ 2 = ker −̄ 2 are of rank , where = dim , and in view of Theorem 6.21 it needs only to be shown that Δ 1 and Δ 2 have the same latency indices (or latency lists) if and only if there exists an order preserving Ω − -isomorphism pr ∶ Δ 2 → Δ 1 . Let 11 , ⋯ , 1 and 21 , ⋯ , 2 be ordered proper bases for Δ 1 and Δ 2 , respectively, and let 1 and 2 be the corresponding matrices. Then an order preserving isomorphism pr ∶ Δ 2 → Δ 1 exists if and only if the matrix 1 −1 2 is bicausal which is easily seen to be the case if and only if ord 1 = ord 2 for all = 1, ⋯ , . Employing Corollary 6.14 completes the proof. □ Theorem 6.21 and Corollary 6.22 could, of course, have been stated for any Λ -linear maps and not only strictly causal ones. The proofs did in no way depend on the causality properties of the maps involved. Also, Corollary 6.22 could have been obtained as an application of the existence of, so called, Smith canonical forms for matrices over Euclidean rings (see, e.g., MacDuffee [1934]).
In general, the precompensator remainder̄ is dynamic and can be represented as̄ = +̄ where is the static part of̄ and̄ ∶ Λ → Λ is strictly causal, i.e., an extended linear i/o map. We recall (see, in particular, Hautus and Heymann [1978]) that the dynamic characteristics of̄ are determined by ker +̄ ⋅ + which is an Ω + -submodule of Ω + and can be represented by where is a polynomial matrix whose columns form a basis for ker +̄ ⋅ + . The degree of the determinant of (when is nonsingular) is the dimension of the minimal state space realizinḡ . More specifically, if in (7.1) is selected to be proper, i.e., the columns of are properly free (in the sense that the leading coefficient vectors are -linearly independent just as in 84 above), then the column degrees , = 1, ⋯ , are the reachability indices of̄ and their sum is ∑ =1 = = deg ⋅ det . It is of interest in selecting a (̄ ,̄ ) pair representing a given precompensator̄ to choose the representation in such a way that the precompensator remainder̄ has least dynamic order, i.e., is realizable by a state space of least possible dimension. In this way the precompensator is realized "as much as possible" by feedback. The following theorem provides a bound on the dynamic order of the precompensator remainder̄ which need not be exceeded in the realization of any bicausal precompensator̄ , and which is dependent only on the dynamic properties (latency) of the i/o map̄ under consideration. Proof. By Hautus and Heymann [1978, Theorem 6.11] the lemma will be proved upon showing that there exists an order-preserving Ω + -isomorphism ker +̄ ⋅ + → ker +̄ −1 ⋅ + . We shall see that the map̄ itself, which is in particular also an order preserving Ω + -isomorphism, satisfies the required properties. Indeed, let ∈ ket +̄ ⋅ + be any element. Then̄ ⋅ + =̄ ∈ Ω + and since also ∈ Ω + we have =̄ −1 (̄ ) =̄ −1 + (̄ ) ∈ Ω + , whencē ∈ ker +̄ −1 ⋅ + , completing the proof. □ Let̄ ∶ Λ → Λ be an injective extended linear i/o map and let 1 , ⋯ , be a proper strictly polynomial basis for ker −̄ (see Theorem 6.19 ), and write ker −̄ = Ω − where = 1 , ⋯ , . Then −1 is also polynomial and the column degrees of −1 are (by definition) the latency indices of̄ . Below we shall not distinguish sharply between maps and their transfer functions. Let − ∶ Λ → Ω − ∶ Σ − ↦ Σ ≥0 − denote the causal truncation. Let ∶ Λ → Ω − be defined as the (unique) Λ -linear map whose transfer function is given by


Overview of the application of computer vision technology in fish farming

The issues that are currently identified in Russia during the implementation of Digital Agriculture project are considered. The main issues that need to be addressed in development of modern digital technologies in the fish farming industry using natural and artificial reservoirs are highlighted. Aqua engineering trends and scientific works of a number of teams that conduct research and use the capabilities of deep machine learning, are analyzed. Particular attention was paid to specific tasks and research results that solve applied problems in the field of aquaculture and fish farming. Conclusions are made about the prospects for implementing these objectives in Russia. The conclusions of scientific teams and new tasks set as a result of scientific experiments are considered. The main directions in the area of commercial fish farming that need active adaptation of computer vision to deal with applied problems, are identified. Questions of efficiency in introduction of neural networks of deep learning are raised, and also conclusions are drawn on introduction of the term "selectivity" to determine the relation of a data set received by a digital method, referred to quantity of the same data which would be received at their collection by means of non-digital technologies.

Introduction
With the growth of computing power, libraries for software development, popularity of digital technologies and at the same time simplification of methods of mastering programming languages, there was a call for deep digitalization of the most specific areas of national economy. For example, the team of authors of this article has been actively developing digital solutions for the problems of fish farming in the Tyumen region [1,2]. However, this article is of a review nature, since a detailed analysis of foreign practices at the "junction" of IT and fish farming is interesting, first of all, because it will make it possible to form a spread of areas that are relevant for fish farming abroad. Certainly, it is necessary to make a reservation that problems and solutions of foreign fish farming do not always correspond to the problems and solutions of Russian fish farming, however ignoring the given experience would be at least ineffective. The team of authors has analyzed a number of articles  and summed up the applicability of each solution to the identified problem in fish farming in the Tyumen region. In this regard, the logic of the presentation will follow the principle "information given in the article -summing up -applicability for fish farming in the Tyumen region".

Computer vision for fish feeding
In the first part of the article, the team of authors solves the problem of determining spatial feed distribution in the sea cage. It should be especially noted that it was the specifics of feed distribution by rotary spreaders were taken into account, which was highlighted separately in the article.
The emphasis is made to the following: 1. Experimental confirmation of the operability of the installation for measuring the degree of distribution of fish feed pellets.
2. Check of the possibility to distribute fish feed using the unmanned aerial vehicle (UAV).
3. Evaluations of the efficiency of pellets distribution using computer vision technology (by quantitative analysis of disturbances on the water surface).
Within the framework of the review, the last stated emphasis is of particular interest. The fish feeding procedure was recorded by an UAV, then the images were analyzed with the help of computer vision technology of disturbances appeared on the surface ("peak" of pixels that are brighter than pixels in the immediate neighborhood). As a result of the analysis of that activity from the UAV video records, the model of pellets distribution by rotary spreaders was received. The method adequacy for analyzing the feed pellets distribution using computer vision was confirmed experimentally, since the results obtained were confirmed by duplicating the results obtained through "manual" analysis.
As a result, the team of authors proved that with the help of computer vision, it is possible not only to make an actual assessment of the efficiency of feed distribution by various spreaders, but also to forecast distribution efficiency. Also, the provision was made to the further generation of "digital twins" for spreaders and for testing of simulation models which can take into account all kinds of spreader structures, pellet weights, etc. Unfortunately, this technology is not applicable to fish farming of the Tyumen Region, since the area of cages in the region is much smaller (due to specifics of river and lake fish farming). However, the digital part of the technology is capable of assessment of the extent of feed spreading in the area, but may also count fish by splashes. Therefore, the developed technology after some modification can be quite useful for the region as an automatic counting technology.

Computer vision for measuring fish parameters
A large list of articles analyzed by the team of authors was related to the development of technology for automatic measurement of fish parameters, based on computer vision technology. The availability of such a significant number indicates, first of all, to the demand of such application of computer vision technology in fish farming. For example, the scientific team of V. Puig-Pons et al., (2019) carried out a study related to testing 2D computer vision technique for measuring a fish parameter such as mass. For the object of study, they took exclusively "jade perch" which moves freely in a closed water reservoir. The study was divided into two stages, the first of which was to make the program accumulate the dependence of the fish shape (as I understand it also included a parameter of its size) on its weight, the second stage was to compare the actual weight of fish with the weight predicted by the computer program. The resulted determination coefficient (R) was 0.99 and a relative error of 6 ± 3 per cent compared to the value measured by the weighing scale, so computer vision technology has proven its adequacy when used in that way. By the way, only 120 images were needed to train the program.
A similar study was carried out by the team of Mohammadmehdi et al., (2018), but the difference with the previously mentioned work was in the use of infrared reflection, since the geometry of the fish back was used for learning, which allowed (unlike the previous team) to go beyond a particular fish species. The image library used 295 farmed sea basses of different species and sizes. The mathematical model was optimized due to ten-fold cross-validation of model prototypes (about ten in total). As a result of the study, a viable, maximally approximate and trained model for distinguishing the mass of fish by its shape features was obtained. The determination coefficient (R) is lower than in previous studies and is equal to 0.868, which is most likely due to the diversity of fish species. At this point, we understand that such technology will be ineffective if used in the "wilderness" without prior observation and development of a library assigned to a flock of fish being measured. But the authors state that their development is applicable to forecasting in all conditions. It would be interesting to test this technology in the conditions of river fish farming, because the degree of water transparency is obviously different and, as a consequence, it is complicates the ability of computer vision technology to analyze the parameters of fish. Such a solution can be adapted to the realities of fish farming in the Tyumen region. However, the technology needs to be improved for better analysis in conditions of complicated visibility and water contamination by large objects.
In the following article the development of computer vision library was improved by auxiliary library of acoustic vision. The result is a simultaneous convergence of technologies (acoustic locator and computer vision) enhanced by machine learning algorithms. The authors' team solved the problem of estimating the mass of tuna during transfers. The libraries were developed with side-scan sonar that operates at a frequency of 200 kHz, and stereo cameras that were positioned to the surface to register the ventral aspect of the fish. Such use undoubtedly reduced the determination coefficient, but the task was also to measure the total mass of all tunas rather than a specific specimen.
The results showed that the system provides automatic counting of tuna with an error below 10%, reaching about 1% error in the best configuration.
This study confirmed the improved efficiency of the results when combining computer vision with the analytics of an auxiliary database (in this case, the acoustic one). It would be very interesting to compare the effectiveness of a number of hybrid technologies (computer vision -acoustics, computer vision -splash analysis, etc.) and to compare which of the combinations of technologies give the most accurate result.

Computer vision for monitoring fish behavior
Researches of various types using machine learning have been conducted for a relatively long time.
The scientific team at Saberioon et al., (2016) conducted a study aimed at developing the tracking of multiple fish in three-dimensional space using sensors.
The value of image-based tracking is to provide automatic remote tracking and analysis of fish movements without human presence. In the direction of fish farming, observation and analysis of fish behavior has been carried out for a long time, for example, the scientific teams of Spampinato et al., (2012) recorded fish behavior using an underwater camera to understand how individual species behave during typhoons. Currently, the number of parameters that can be tracked using machine learning is quite large. For example, Kato et al. (2004) developed a system to quantify the behavior of individual fish, such as speed, distance that an individual can travel, and a map of individual movements. The use of machine learning for research in this area has been going on for a long time. The most common technical problem encountered by researchers in the study of fish behavior through machine learning is overlapping of several fish images during movement. Such effect occurs when several fish are moving simultaneously, or when some individuals come in close proximity to the other. This crossing can be displayed when recording in one plane, at the same time, while recording in another plane, it is visible that that the individuals are at some distance from each other.
There are several types of crossing while recording. Generally, there are two main types of such crossings: when two fish individuals swim so close that they appear as one long fish in the image, and the second type is when the trajectories intersect and the two fish are perceived as T-shaped and X-shaped images (Dolado et al., 2014). Various machine learning algorithms are used to recognize inaccurate data.
The research team considered the possibility of using special sensors permitting to receive three-dimensional image. Such sensors generate a three-dimensional geometric field to obtain depth information from objects of interest. This makes it possible to develop an affordable system for tracking and studying multiple fish in real time. Here we again encounter the use of a number of databases, which gives us an understanding of synergy of different source databases affecting the efficiency of the result. The technology is applicable to fish farming in Tyumen Region and can be used with no modifications.
The research team conducted laboratory tests using sensors for tracking and precise measurement of the three-dimensional trajectory of several fish, even in the conditions of their trajectories constantly and repeatedly crossed.
The developed and tested system in the aquarium has shown good potential, and has led to conclusions about the prospects of its application for monitoring and research of fish behavior in groups, rather than just to consider their individual behavior. Such system enhances the possibilities to study fish behavior by setting and studying more parameters, such as speed and interpersonal distances.
Other research teams faced similar difficulties.
In the next article the team of authors conducting laboratory tests of the installation using computer vision focused on processing a large number of quality images obtained from high-speed cameras. The goal set by scientists was not only to study the behavioral patterns of individuals of the fish in research, but also the duration and stability of image processing. Experimental observations made it possible to draw conclusions and recommendations regarding the necessary characteristics of the number of photos per second required for detailed analysis. The developed proprietary algorithm allowed to form further tasks connected with automatic determination of fish movement and body position, as well as spatial position. Implementation of research using computer vision has been actively used by specialists from various countries over the past decade. But unfortunately, the use of computer vision in fish farming in Russia has not become widespread, despite the fact that the industry has set applied tasks that can be solved only with the help of deep-learning neural networks.

3.3.Computer vision to study the nets integrity
For aquaculture enterprises, the integrity and safety of fishing nets is of great importance. The team of the article's authors Yun-Peng et al., (2020) is conducting a study using deep machine learning technology for research in this field.
The development of small-water aquaculture, namely coastal aquaculture, is becoming an important area of development and breeding of marine fish in China. This helps to achieve sustainable development of fish farming. A section of the allocated water resource is fenced off for use by special nets. This is done to keep the farmed fish inside a confined space, to track its population, growth, nutrition and to conduct appropriate monitoring.
However, damage to the prepared cage (Heidi et al., 2009;Hoy et al., 2012), namely holes in the cage, allows fish to enter the main water reservoir and therefore leads to losses for the aquaculture industry and has serious consequences for the ecosystem as a whole. Detection of net cage damage consists in determination of the mesh gaps between each other. Currently, there are several methods to conduct the netting integrity check, and one such method is regular netting examination by professional divers. This process is very time-consuming and not very efficient, and also presents a high safety risk. Therefore, detection of netting damage is a key problem in aquaculture development related to fish farming.
The team of authors suggests using two ways to detect holes in fencing nets. The research demonstrates the results of experiment of the netting examination with a special robot that takes high-resolution images. Further, the authors suggest two ways to process the obtained high-resolution images.
One method is based on the neural network training to recognize nodes in netting, and thus a rule for the location of node points is developed. The second method suggested by the scientists is based on recognizing holes in the netting by a special algorithm. Testing both methods to detect the netting holes has led to the conclusion that one method is mode advantageous than the other. For example, the method of hole detection in the netting can be used to detect the broken areas even when the netting is partially contaminated. This method also makes it possible to estimate how much biologically contaminated the netting is and to identify the extent of damage.

Conclusions
Summing up the analyzed articles, a number of common factors can be identified: 1. Computer vision technology when used for an applied problem in fish farming is always "backed up" by additional technology and/or a database. It can be concluded that the teams of authors, by doing so, increased not the efficiency of the technology, but the level of its selectivity. In other words, the digital technology was double-checked in more proven ways. This, in its turn, suggests that there is a lack of trust in such methods for solving applied problems in fish farming. 2. The extent of computer vision application to fish farming is increasing and shows a positive growth dynamics, but it is still a very narrowly specialized thing. Therefore, in the future it will be necessary to develop methods of adaptation of already existing computer vision solutions to the conditions of fish farms of different regions.