Non-negative Matrix Factorization Method Research Articles

Dual local learning regularized nonnegative matrix factorization and its semi-supervised extension for clustering

Nonnegative matrix factorization (NMF) has received considerable attention in data representation due to its strong interpretability. However, traditional NMF methods neglect the discriminative information and geometric structure of both the data space and the feature space, simultaneously. In this paper, we propose a dual local learning regularized nonnegative matrix factorization (DLLNMF) method, which not only considers the geometric structure of both the data manifold and the feature manifold, simultaneously, but also takes advantage of the discriminative information of both the data space and the feature space. To make full use of the partial label information among samples, we further propose its semi-supervised extension, called dual local learning regularized nonnegative matrix factorization with label constraint (DLLNMF-LC), which imposes the label information as a hard constraint without additional parameters. Experimental results on some benchmark datasets have demonstrated the effectiveness of our proposed methods.

A new non-negative matrix factorization method to build a recommender system

The main aim of this paper is to apply non-negative matrix factorization to build a recommender system. In a recommender system there are a group of users that rate to a set of items. These ratings can be represented by a rating matrix. The main problem is to estimate the unknown ratings and then predict the interests of the users to the items which haven’t rated. The main innovation of this paper is to propose a new algorithm to compute matrix factorization in a way that the factorized matrixes would be a good approximation for the initial rating matrix and moreover would be a good source to predict the unknown ratings of the items precisely. The results show that the proposed matrix factorization improves the estimated ratings considerably.

A New Non-Negative Matrix Factorization Method to Build a Recommender System

The main aim of this paper is to apply non-negative matrix factorization to build a recommender system. In a recommender system there are a group of users that rate to a set of items. These ratings can be represented by a rating matrix. The main problem is to estimate the unknown ratings and then predict the interests of the users to the items which haven’t rated. The main innovation of this paper is to propose a new algorithm to compute matrix factorization in a way that the factorized matrixes would be a good approximation for the initial rating matrix and moreover would be a good source to predict the unknown ratings of the items precisely. The results show that the proposed matrix factorization improves the estimated ratings considerably.

Robust semi-supervised nonnegative matrix factorization for image clustering

Nonnegative matrix factorization (NMF) is a powerful dimension reduction method, and has received increasing attention in various practical applications. However, most traditional NMF based algorithms are sensitive to noisy data, or fail to fully utilize the limited supervised information. In this paper, a novel robust semi-supervised NMF method, namely correntropy based semi-supervised NMF (CSNMF), is proposed to solve these issues. Specifically, CSNMF adopts a correntropy based loss function instead of the squared Euclidean distance (SED) in constrained NMF to suppress the influence of non-Gaussian noise or outliers contaminated in real world data, and simultaneously uses two types of supervised information, i.e., the pointwise and pairwise constraints, to obtain the discriminative data representation. The proposed method is analyzed in terms of convergence, robustness and computational complexity. The relationships between CSNMF and several previous NMF based methods are also discussed. Extensive experimental results show the effectiveness and robustness of CSNMF in image clustering tasks, compared with several state-of-the-art methods.

Unsupervised feature selection and NMF de-noising for robust Speech Emotion Recognition

Speech feature fusion is the most commonly used phenomenon for improving the accuracy in Speech Emotion Recognition (SER). But in this, there is a disadvantage of increasing the complexity in SER system in terms of processing time. Besides this, some of the features could be redundant and do not contribute for SER and lead to incorrect emotion prediction and reduction in SER accuracy. To overcome this problem, in this paper, unsupervised feature selection is applied to the feature set with the combination of INTERSPEECH 2010 paralinguistic features, Gammatone Cepstral Coefficients (GTCC) and Power Normalized Cepstral Coefficients (PNCC). The Feature Selection with Adaptive Structure Learning (FSASL), Unsupervised Feature Selection with Ordinal Locality (UFSOL) and a novel Subset Feature Selection (SuFS) algorithm are the feature dimension reduction techniques used to acquire better SER performance in this work. The proposed SER system is analyzed in both clean and noisy environments. The EMO-DB and IEMOCAP emotion databases are considered for evaluating the proposed SER performance. For noise analysis, the clean speech is corrupted with different noises of Aurora noise database and white Gaussian noise at different Signal to Noise Ratio (SNR) levels from −5dB to 20 dB. Support Vector Machine (SVM) classifier with linear and Radial Basis Function (RBF) kernels using 10-fold cross-validation and hold-out validation is used in this analysis with classification accuracy and computation time as the performance metrics. The results show that the proposed SER system outperforms the baseline SER system as well as many of the existing literature works both in clean and noisy conditions. For SNR levels >15 dB, the proposed SER system in presence of different noises performs same as the SER in clean environments. Whereas, for lower SNRs <15 dB the performance is likely to be reduced. Therefore, to overcome this drawback and improve the SER performance in noisy conditions, a dense Non-Negative Matrix Factorization (denseNMF) method is adopted for de-noising the noisy speech signal prior to SER achieving noise robustness.

Background image-assisted divide-and-conquer reconstruction method for ECT

Electrical capacitance tomography (ECT) is a potential image-based measurement technology for monitoring time-varying industrial processes, but its applicability is challenged by low-quality images. To address this conundrum, a two-stage reconstruction (TSR) method with more effective priors and optimizer is presented in this work. In the first stage, the random vector functional link network (RVFLN) is developed to calculate a data-dependent background image, and a new distributed computing method is developed to achieve efficient training. To decrease the expensive computation load in the RVFLN training, the regularized projective nonnegative matrix factorization (RPNMF) method is developed to diminish the size of the sample data. A new optimization problem (OP) is proposed to model the RPNMF problem, which is solved by a new optimizer efficiently. In the second stage, a new OP that achieves the confluence of the domain knowledge-based prior related to imaging objects and the measurement physics is built, and a new divide-and-conquer optimizer is devised to solve the OP. To reduce the difficulty of parameter adjustment, the background image is used to initialize the proposed computing algorithm, and such treatment not only achieves the simultaneous fusion of the domain knowledge-based prior and the data-dependent prior but also decreases the computational difficulty. Numerical results show that the TSR algorithm is not only robust but also can achieve high precision reconstruction in comparison with popular imaging techniques.

Machinery signal separation using non-negative matrix factorization with real mixing

A big challenge in detecting damage occurs when the sound of a machine mixes with the sound of another machine. This paper proposes the separation of mixed acoustic signals using Non-negative Matrix Factorization (NMF) method for fault diagnosis. The NMF method is an effective solution for finding hidden parameters when the number of observations obtained by the sensor is less than the number of sources. The real mixing process is done by placing two microphones in front of the machine. Two microphones will be used as sensors to capture a mixture of four machinery signals. Performance testing of signal separation is done by comparing baseline signals with estimated signals through the mean log spectral distance (LSD) and the mean square error (MSE). The smallest spectral distance between the estimated signal and the baseline signal is found in Ŝ2 with an average LSD of 1.26. The estimated signal Ŝ2 is the closest to the baseline signal with MSE of 1.15 x 10-2. The pattern of bearing damage in the male screw compressor can be identified from the spectrum of estimated signal through harmonic frequencies as in the estimated signal Ŝ3 which is seen at 11x fundamental frequency, 12x fundamental frequency, 15x fundamental frequency, and 16x fundamental frequency.

Community detection method using improved density peak clustering and nonnegative matrix factorization

Community detection in networks is valuable in analyzing, designing, and optimizing complex network. Recently, the nonnegative matrix factorization (NMF) method has successfully uncovered the community structure in the complex networks. However, most of community detection methods based on NMF require the number of the community as a prior information. To address this problem, in this paper, we use the improved density peak clustering (DPC) to obtain the number of centers as the pre-assigned parameter for nonnegative matrix factorization. The proposed algorithm first calculates the modified PageRank of nodes as the density indexes and then draws a decision graph to obtain the hubs of the network. By means of Markov chain model of a random walk, we execute NMF on one expansion feature matrix. Finally, we compare and analyze the performance of different algorithms on artificial networks and real-world networks. Experimental results indicate that the proposed method is superior to the state-of-the-art methods.

An Improved Non-negative Matrix Factorization Method for Dynamic Industrial Fault Diagnosis

Most classical fault diagnosis methods such as principal component analysis (PCA), are extracting comprehensive information to represent data features in fault diagnosis. In comparison, Non-Negative Matrix Factorization (NMF) is a method for dimension reduction and feature extraction, and because its characteristic matrix has sparsity, this method is superior in the ability of extracting the local feature and suppressing noises; however, the NMF method is not applicable for dynamic industrial processes. In this paper, we introduce the past information of industrial processes for fault diagnosis, proposing Canonical Variate Analysis - Non-Negative Matrix Factorization (CVA-NMF) methods to improve the dynamic performance of NMF. The experimental results via TE process indicate that the proposed approach could handle a dynamic production process such as Fault 2 and Fault 5 and retain the superior performance of NMF.

Spatial gradients of geomagnetic temporal variations causing the instability of inter-station transfer functions

Spatial gradients in the primary geomagnetic fields directly contribute to both the amplitudes and phases of inter-station transfer functions (IS-TFs). This suggests that, for the analysis of subsurface resistivity structures, IS-TFs should be carefully treated by checking the establishment of the plane-wave assumption. Geomagnetic time-series data include various and complicated characteristics and accordingly, time–frequency domain analysis is suitable for the discussion of spatial gradients of time-varying geomagnetic fields. However, such evaluations are complicated by the huge amount of information contained in the spectrograms from several stations. Therefore, we propose a Multi-Channel Nonnegative Matrix Factorization (MC-NMF) method that can decompose raw spectrograms into several components, allowing the spatial gradient of each geomagnetic temporal variation to be identified. We confirm that such components actually affect the estimation of IS-TFs using data acquired at the Kakioka and Memambetsu magnetic observatories in Japan. We derive the year-to-year changes in IS-TFs from each set of paired stations among Kakioka, Kanoya, and Memambetsu observatories. Although the IS-TFs should exhibit opposite polarities (a negative correlation) when the input and output observatories are swapped; surprisingly, some of them have “identical” polarities. The application of MC-NMF shows that the analyzed geomagnetic data include several components that have various spatial gradients. Although IS-TFs sometimes fail to give the expected implication regarding the spatial gradients of geomagnetic temporal variations, MC-NMF can verify whether the IS-TFs exhibit any spatial gradients. Thus, the use of IS-TFs with MC-NMF can yield better implications regarding subsurface resistivity information.

A variable neighborhood search heuristic for nonnegative matrix factorization with application to microarray data

Nonnegative matrix factorization (NMF) has become a popular method for establishing a low-dimensional approximation of a two-mode (nonnegative) data matrix and, in some instances, to also establish partitions for the objects associated with the two modes of the matrix. Although similar to singular-value-decomposition, as its name implies, NMF requires nonnegative elements for the factors and this assures a ‘sum of the parts’ fit to the data. There are a variety of alternative objective functions and heuristic methods for NMF. Using both simulated and real-world two-mode data, we demonstrate that a multiple restart (multistart) heuristic for NMF commonly fails to produce optimal objective function values. A new variable neighborhood search (VNS) heuristic is shown to outperform the multistart approach with respect to both solution quality and computation time. Although the objective function value improvements associated with VNS are often small (on a percentage basis), such improvements can sometimes lead to differences in the partitions obtained for the row and column objects. Application to two well-known microarray datasets is used to support the merits of the proposed VNS heuristic.

FER based on the improved convex nonnegative matrix factorization feature

Facial expression recognition is an important research issue in the pattern recognition field. In this paper, we intend to present an accurate facial expression recognition (FER) system, which employs an improved convex non-negative matrix factorization (ICNMF) method based on a novel objective function and smaller iterative step sizes for feature extraction. Since negative values appearing in the facial expression feature will weaken the features and reduce the recognition rate, the nonnegative matrix factorization (NMF) methods are adopted to guarantee the non-negativity of the extracted feature value to improve the recognition rate. To enhance the performance of NMF method for FER, the ICNMF approach based on a novel convergent objective function and smaller iterative step sizes is proposed, and the FER rate can be improved effectively. In the FER system, the face region is detected firstly, and is enhanced by histogram specification, secondly the ICNMF approach is adopted to extract features and then the feature coefficient matrix is achieved. Finally, the SVM classifier is applied to recognize the extracted features. To validate the effectiveness of FER system, four public available datasets of MultiPIE, CK+, FER2013 and SFEW are tested and then high recognition rates can be achieved based on ICNMF method. In addition, the proposed ICNMF approach is compared with the methods of multi-layer NMF, sparse non-negative matrix factorization (SNMF), the traditional convex non-negative matrix factorization (CNMF), deep belief networks (DBN) and stacked auto-encoder (SAE), and results of experiments show that the proposed ICNMF approach is significantly effective contrasting to the other five expression extraction methods.

Using Topic Modeling Methods for Short-Text Data: A Comparative Analysis.

With the growth of online social network platforms and applications, large amounts of textual user-generated content are created daily in the form of comments, reviews, and short-text messages. As a result, users often find it challenging to discover useful information or more on the topic being discussed from such content. Machine learning and natural language processing algorithms are used to analyze the massive amount of textual social media data available online, including topic modeling techniques that have gained popularity in recent years. This paper investigates the topic modeling subject and its common application areas, methods, and tools. Also, we examine and compare five frequently used topic modeling methods, as applied to short textual social data, to show their benefits practically in detecting important topics. These methods are latent semantic analysis, latent Dirichlet allocation, non-negative matrix factorization, random projection, and principal component analysis. Two textual datasets were selected to evaluate the performance of included topic modeling methods based on the topic quality and some standard statistical evaluation metrics, like recall, precision, F-score, and topic coherence. As a result, latent Dirichlet allocation and non-negative matrix factorization methods delivered more meaningful extracted topics and obtained good results. The paper sheds light on some common topic modeling methods in a short-text context and provides direction for researchers who seek to apply these methods.

Correntropy-Based Hypergraph Regularized NMF for Clustering and Feature Selection on Multi-Cancer Integrated Data.

Non-negative matrix factorization (NMF) has become one of the most powerful methods for clustering and feature selection. However, the performance of the traditional NMF method severely degrades when the data contain noises and outliers or the manifold structure of the data is not taken into account. In this article, a novel method called correntropy-based hypergraph regularized NMF (CHNMF) is proposed to solve the above problem. Specifically, we use the correntropy instead of the Euclidean norm in the loss term of CHNMF, which will improve the robustness of the algorithm. And the hypergraph regularization term is also applied to the objective function, which can explore the high-order geometric information in more sample points. Then, the half-quadratic (HQ) optimization technique is adopted to solve the complex optimization problem of CHNMF. Finally, extensive experimental results on multi-cancer integrated data indicate that the proposed CHNMF method is superior to other state-of-the-art methods for clustering and feature selection.

An Integrative Framework of Heterogeneous Genomic Data for Cancer Dynamic Modules Based on Matrix Decomposition.

Cancer progression is dynamic, and tracking dynamic modules is promising for cancer diagnosis and therapy. Accumulated genomic data provide us an opportunity to investigate the underlying mechanisms of cancers. However, as far as we know, no algorithm has been designed for dynamic modules by integrating heterogeneous omics data. To address this issue, we propose an integrative framework for dynamic module detection based on regularized nonnegative matrix factorization method (DrNMF) by integrating the gene expression and protein interaction network. To remove the heterogeneity of genomic data, we divide the samples of expression profiles into groups to construct gene co-expression networks. To characterize the dynamics of modules, the temporal smoothness framework is adopted, in which the gene co-expression network at the previous stage and protein interaction network are incorporated into the objective function of DrNMF via regularization. The experimental results demonstrate that DrNMF is superior to state-of-the-art methods in terms of accuracy. For breast cancer data, the obtained dynamic modules are more enriched by the known pathways, and can be used to predict the stages of cancers and survival time of patients. The proposed model and algorithm provide an effective integrative analysis of heterogeneous genomic data for cancer progression.

Application of a Deep Matrix Factorization Model on Integrated Gene Expression Data

Background: Non-negative Matrix Factorization (NMF) has been extensively used in gene expression data. However, most NMF-based methods have single-layer structures, which may achieve poor performance for complex data. Deep learning, with its carefully designed hierarchical structure, has shown significant advantages in learning data features. Objective: In bioinformatics, on the one hand, to discover differentially expressed genes in gene expression data; on the other hand, to obtain higher sample clustering results. It can provide the reference value for the prevention and treatment of cancer. Method: In this paper, we apply a deep NMF method called Deep Semi-NMF on the integrated gene expression data. In each layer, the coefficient matrix is directly decomposed into the basic and coefficient matrix of the next layer. We apply this factorization model on The Cancer Genome Atlas (TCGA) genomic data. Results: The experimental results demonstrate the superiority of Deep Semi-NMF method in identifying differentially expressed genes and clustering samples. Conclusion: The Deep Semi-NMF model decomposes a matrix into multiple matrices and multiplies them to form a matrix. It can also improve the clustering performance of samples while digging out more accurate key genes for disease treatment.

Self-Paced Nonnegative Matrix Factorization for Hyperspectral Unmixing

The presence of mixed pixels in the hyperspectral data makes unmixing to be a key step for many applications. Unsupervised unmixing needs to estimate the number of endmembers, their spectral signatures, and their abundances at each pixel. Since both endmember and abundance matrices are unknown, unsupervised unmixing can be considered as a blind source separation problem and can be solved by nonnegative matrix factorization (NMF). However, most of the existing NMF unmixing methods use a least-squares objective function that is sensitive to the noise and outliers. To deal with different types of noises in hyperspectral data, such as the noise in different bands (band noise), the noise in different pixels (pixel noise), and the noise in different elements of hyperspectral data matrix (element noise), we propose three self-paced learning based NMF (SpNMF) unmixing models in this article. The SpNMF models replace the least-squares loss in the standard NMF model with weighted least-squares losses and adopt a self-paced learning (SPL) strategy to learn the weights adaptively. In each iteration of SPL, atoms (bands or pixels or elements) with weight zero are considered as complex atoms and are excluded, while atoms with nonzero weights are considered as easy atoms and are included in the current unmixing model. By gradually enlarging the size of the current model set, SpNMF can select atoms from easy to complex. Usually, noisy or outlying atoms are complex atoms that are excluded from the unmixing model. Thus, SpNMF models are robust to noise and outliers. Experimental results on the simulated and two real hyperspectral data sets demonstrate that our proposed SpNMF methods are more accurate and robust than the existing NMF methods, especially in the case of heavy noise.

Organising the knowledge from stack overflow about location‐sensing of Android applications

The number of Android applications using location information has increased significantly in recent years. Over time, there have been many improvements made to the location application programme interfaces (APIs), providing newer challenges and difficulties to the developers. Therefore, there is a need to summarise the existing knowledge and to highlight the unsolved issues to bring them to the attention of expert developers. The authors used the non-negative matrix factorisation (NMF) method to identify the topics discussed by the developers on stack overflow. They found the following ten topics: fundamental, background service, global positioning system (GPS) provider, application error, location updates, programming aspects, GPS alternatives, location settings, NULL location, and location testing. In addition, they performed a manual analysis to add more qualitative insights into the results. They applied the NMF method on 3165 question posts and produced ten related topics. This study aims at organising the knowledge about location-sensing strategies by answering three relevant research questions. They also analysed the most popular and unanswered topics in recent years. An important finding of this study is that the changes that occurred in the Google Location APIs have had a significant impact on the location-sensing strategies followed by the developers.

0016 Genetic Basis of Daytime Napping and Consequence on Cardiometabolic Health

Abstract Introduction Although daytime napping is a common, evolutionarily conserved behavior, its genetic basis is unknown. Elucidating its genetic basis may clarify relevant underlying biological pathways and determine causal links with cardiometabolic health. Methods We performed a genome-wide association study of self-reported daytime napping using linear regression in adults of European ancestry in the UK Biobank (n=452,633) and assessed robustness of signals with accelerometer-derived daytime inactivity duration (n=84,671). Next, we conducted a phenome-wide association study in a hospital-based clinical biobank (n=30,683) using napping genome-wide polygenic score (GPS), and Mendelian randomization (MR) with cardiometabolic traits. To deconstruct the napping genetic variants, we applied a novel “soft clustering” Bayesian nonnegative matrix factorization method and generated partitioned cluster-specific polygenic risk scores (PRS). Results We identified 121 distinct genome-wide significant loci for daytime napping, with lead signals at or near genes KSR2 (kinase-suppressor of ras 2), HCRTR1/HCRTR2 (hypocretin-receptor 1/2), SKOR2 (SKI family transcriptional-corepressor 2), and MAPT (microtubule-associated protein tau), among others. The loci associated with accelerometer-derived daytime inactivity duration. Gene enrichment analyses pointed to pathways involved in neurogenesis and others including nervous system development and opioid signaling. Genetic overlaps were evident in a clinical biobank where highest, compared to lowest, decile of napping GPS associated with 30%, 40%, and 50% higher odds for essential hypertension, obesity, and nonalcoholic liver disease, respectively (P&amp;lt;0.0001). In MR, potential causal links were identified with higher diastolic blood pressure (2.67 mmHg per napping category-increase, 95% CI 1.62–3.23, P=6.80e-07), systolic blood pressure (3.65mmHg, 1.86–5.44, P=6.40e-05), and waist circumference (0.28 SD-units, 0.11–0.45, P=0.0015). The clustering of variants identified 3 robust clusters (cluster-1: “higher sleep propensity”; cluster-2: “more fragmented/inefficient night sleep”; cluster-3: “early sleep timing”). Only clusters 2 and 3 PRSs were associated with worse cardiometabolic health outcomes, including higher BMI, waist circumference, CRP, and triglycerides (all P&amp;lt;0.05). Conclusion These findings expand our understanding of the genetic architecture of napping implicating multiple biological pathways, indicating possible genetic overlap and causal links to cardiometabolic traits, and suggesting distinct nap-promoting mechanisms with differential associations with health outcomes. Support This work is supported by grants NIH-F32DK102323, NIH-4T32HL007901, NIH-R01DK107859, NIH-R35HL135818, and MGH Research Scholar Fund.

Sensory Evaluation of Odor Approximation Using Nmf with Kullback-Leibler Divergence and Itakura-Saito Divergence in Mass Spectrum Space

Introduction The odor reproduction is achieved by approximating mass spectra of various odors by blending a set of odor components. These reproduced odors should be as close to the target odor as possible. The fidelity of odor reproduction should be as in the same manner as reproducing colors. Since no primary odors have been found, thus finding an appropriate set of odor components to perform odor reproduction is essential. The odor components should be kept as small as possible whereas it should cover a wide range of odors. In the present study, we performed a sensory evaluation of odors approximated with Kullback-Leibler and Itakura-Saito divergences on mass spectrum space. Method In our previous study, the non-negative matrix factorization (NMF) method was used to extract basis vector corresponding to odor components [1], [2]. NMF was performed to analyze the mass spectrum data space of essential oils as shown in Figure 1. Non-negativity property of NMF is useful in exploring odor components. We created a mass spectrum database of 185 essential oils originating from plants. Mass spectrum data were gathered at mass to charge (m/z) region from 50 to 250. Region below 50 was discarded because we diluted the samples with ethanol (molecular weight of 46). An example of essential oils' mass spectra that we gathered is shown in Figure 2.NMF with two different cost functions were performed in odor component exploration. Kullback-Leibler divergence (NMF-KL) and Itakura-Saito divergence (NMF-IS) were performed for the odor component exploration. IS divergence can extract small peaks in the high m/z region, with much contribution to human perception [3]. Those small peaks are ignored or degraded when using KL divergences [4]. Thus NMF-IS can enhance the approximation accuracy of the small peaks in high m/z area which might have a contribution to human olfaction.Non-negative least squares methods were performed to approximate the recipe for odor approximation. Non-negative least squares methods were applied twice. First one was used to obtain the recipe to approximate odor components explored by NMF and the second one to obtain the recipe to approximate the target odor based on odor components. Non-negative least squares with IS divergence based on optimization algorithm was applied to the result of NMF-IS, whereas normal non-negative least squares were applied to the result of NMF-KL. Results and Discussions The previous study revealed that 30 odor components worked well in approximating odor. We evaluated NMF-IS and NMF-KL only using 10 odor components. Although 10 odor components were insufficient for obtaining good approximation accuracy, the difference between the two methods became large. We extracted odor components from essential oils based on the recipe approximated by first non-negative least squares method.We approximated 6 target odors (clove bud, ylang-ylang, orange, origanum, mint) by using both NMF-IS and NMF-KL methods. Duo trio sensory test was performed to evaluate the approximated odors. A comparison of the original target odor and approximated odor mass spectrum are shown in Figure 3. The peaks locations were actually reproduced in spite of certain errors in magnitude even if we used only 10 odor components. A subject was asked to pick up either of the two approximated odor (one based on NMF-IS and another one based on NMF-KL) closer to the target one. We evaluated the Z score to the results of sensory tests (Table 1, P<0.05). The Z score of “+2” means the approximated odor by NMF-IS is significantly closer to the target odor whereas the Z score of “-2” means the approximated odor by NMF-KL is significantly closer to the target odor. The Z score of between +2 and -2 means none of approximated odor from both NMFs is closer to the target odor [5]. Three of the approximated odors (orange, mint, and ylang-ylang) by NMF-IS were closer to the target odor than that by NMF-KL. One approximated odor (cypress) by NMF-KL was closer to the target odor than that by NMF-IS. There was no significant closeness for the rest of two odors (clove bud and origanum). Overall, the sensory test reveals that approximated odors by NMF-IS were closer to the target odor than the odor approximated by NMF-KL. The Approximation quality based upon IS divergence was better than that based upon KL divergence.