Projection Pursuit Research Articles

Visual Diagnostics for Constrained Optimisation with Application to Guided Tours

A guided tour helps to visualise high-dimensional data by showing low-dimensional projections along a projection pursuit optimisation path. Projection pursuit is a generalisation of principal component analysis, in the sense that different indexes are used to define the interestingness of the projected data. While much work has been done in developing new indexes in the literature, less has been done on understanding the optimisation. Index functions can be noisy, might have multiple local maxima as well as an optimal maximum, and are constrained to generate orthonormal projection frames, which complicates the optimization. In addition, projection pursuit is primarily used for exploratory data analysis, and finding the local maxima is also useful. The guided tour is especially useful for exploration, because it conducts geodesic interpolation connecting steps in the optimisation and shows how the projected data changes as a maxima is approached. This work provides new visual diagnostics for examining a choice of optimisation procedure, based on the provision of a new data object which collects information throughout the optimisation. It has helped to diagnose and fix several problems with projection pursuit guided tour. This work might be useful more broadly for diagnosing optimisers, and comparing their performance. The diagnostics are implemented in the R package, ferrn.

Review of Dimension Reduction Methods

Purpose: This study sought to review the characteristics, strengths, weaknesses variants, applications areas and data types applied on the various Dimension Reduction techniques. Methodology: The most commonly used databases employed to search for the papers were ScienceDirect, Scopus, Google Scholar, IEEE Xplore and Mendeley. An integrative review was used for the study where 341 papers were reviewed. Results: The linear techniques considered were Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), Singular Value Decomposition (SVD), Latent Semantic Analysis (LSA), Locality Preserving Projections (LPP), Independent Component Analysis (ICA) and Project Pursuit (PP). The non-linear techniques which were developed to work with applications that have complex non-linear structures considered were Kernel Principal Component Analysis (KPCA), Multi-dimensional Scaling (MDS), Isomap, Locally Linear Embedding (LLE), Self-Organizing Map (SOM), Latent Vector Quantization (LVQ), t-Stochastic neighbor embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP). DR techniques can further be categorized into supervised, unsupervised and more recently semi-supervised learning methods. The supervised versions are the LDA and LVQ. All the other techniques are unsupervised. Supervised variants of PCA, LPP, KPCA and MDS have been developed. Supervised and semi-supervised variants of PP and t-SNE have also been developed and a semi supervised version of the LDA has been developed. Conclusion: The various application areas, strengths, weaknesses and variants of the DR techniques were explored. The different data types that have been applied on the various DR techniques were also explored.

Research on the U.S. Economic Status under the Epidemic Based on the Comprehensive Evaluation Model of Projection Pursuit

Aiming at the impact of different candidates in the United States on the Chinese and American economies, this paper established a projection pursuit comprehensive evaluation model based on improved butterfly optimization algorithm to analyze the US economy and economy. We establish project pursuit comprehensive evaluation model based on improved butterfly optimization algorithm. First, perform data preprocessing on the 9 evaluation index data to eliminate the influence of outliers; then, establish a projection pursuit evaluation model for high-dimensional data. Secondly, use improved butterfly optimization algorithm to solve the model, and finally it is obtained that without considering the epidemic situation, Trump’s score is 8.76 and Biden is 7.89. When considering the epidemic situation, Trump’s score is 5.23, and Biden scores As 6.17. Further sensitivity analysis of the solution results proves that the model is stable.

Geodesic projection of the von Mises–Fisher distribution for projection pursuit of directional data

We investigate geodesic projections of von Mises–Fisher (vMF) distributed directional data. The vMF distribution for random directions on the (p−1)-dimensional unit hypersphere Sp−1⊂Rp plays the role of multivariate normal distribution in directional statistics. For one-dimensional circle S1, the vMF distribution is called von Mises (vM) distribution. Projections onto geodesics are one of main ingredients of modeling and exploring directional data. We show that the projection of vMF distributed random directions onto any geodesic is approximately vM-distributed, albeit not exactly the same. In particular, the distribution of the geodesic-projected score is an infinite scale mixture of vM distributions. Approximations by vM distributions are given along various asymptotic scenarios including large and small concentrations (κ→∞,κ→0), high-dimensions (p→∞), and two important cases of double-asymptotics (p,κ→∞, κ∕p→c or κ∕p→λ), to support our claim: geodesic projections of the vMF are approximately vM. As one of potential applications of the result, we contemplate a projection pursuit exploration of high-dimensional directional data. We show that in a high dimensional model almost all geodesic-projections of directional data are nearly vM, thus measures of non-vM-ness are a viable candidate for projection index.

Biased Hypothesis Formation From Projection Pursuit

Biased Hypothesis Formation From Projection Pursuit

The measurement and analysis of technological innovation diffusion in China's manufacturing industry

<abstract> <p>By constructing a technological innovation diffusion (TID) index system of the manufacturing industry with 15 indexes from four dimensions of diffusion source, diffusion path, diffusion receiver, and diffusion environment, this paper measures the TID in the manufacturing industry and analyzes its distribution characteristics from the perspective of industry and region from 2005 to 2018 by using the entropy weight method and the dynamic multi-indicator projection pursuit (PP-IPM) method. The results show that the TID in China's manufacturing industry has a good development trend in the sample period, and the diffusion resources and paths have a relatively greater impact on TID; the absorptive capacity of the receiver also has a significant impact, while the impact of the diffusion environment is relatively small. During the sample period, at the sector level, a pattern has formed in which transportation equipment manufacturing, electrical machinery and equipment manufacturing, and communication equipment, computer and electronic equipment manufacturing are the sources of diffusion, gradually spreading to other sectors according to the degree of industrial relevance. At the regional level, the diffusion pattern is that Guangdong, Jiangsu, Zhejiang, Shandong, Beijing, and Shanghai are the sources and diffuse to the surrounding and central and western regions successively.</p> </abstract>

Research on Safety Evaluation Method of Integrated Optical Storage and Charging Station

This paper studies the correlation between charging process performance indicators and charging safety of Solar-Energy storage-Charge station, analyses the influence of environmental factors, technical factors, design factors, management factors and user factors on charging process safety of energy stations. The projection pursuit algorithm is used to evaluate the influence degree of each parameter on the safety of charging process; through the establishment of charging safety evaluation system, the safety risks of battery damage and even fire caused by excessive charging current and high battery temperature are identified. The monitoring parameters corresponding to the charging safety state are determined. According to the different battery types adopted by different vehicle models, the corresponding charging mode is determined. According to the monitoring results of key data of the core equipment in the charging process, such as charging current, charging voltage, battery temperature, etc., the charging strategies of different vehicle models and different battery types are analysed, which provides reference for the safe operation of charging process.

Has ICT Contributed to Increased Carbon Productivity in Industry?

Enhancing industrial carbon productivity is the key to China's sustainable economic development. Along with the application and development of new-generation network information technologies such as big data, Internet of Things, cloud computing, etc., information communication technology (ICT) has become an essential means to improve national innovation, productivity and strengthen industrial competitiveness, and plays a crucial role in promoting industrial carbon productivity enhancement. Based on China's interprovincial panel data from 2009-2017, this study measured the ICT level through Projection Pursuit improved by an accelerated genetic algorithm and adopted a non-linear threshold regression model to investigate the non-linear impact and its Spatio-temporal heterogeneity of ICT on China's industrial carbon productivity improvement from the perspective of science and technology(S&T) human capital accumulation. The study shows that the overall level of ICT in China is high. There is a digital divide between regions; in general, ICT plays a positive role in enhancing industrial carbon productivity, but there are significant heterogeneous threshold characteristics of science and technology human capital accumulation accumulation. Once the accumulation of science and technology human resource accumulation breaks through the critical scale, the positive effect of ICT on industrial carbon productivity will gradually appear and show a strengthening trend. The temporal and spatial distribution of the threshold for accumulating science and technology human capital accumulation is different.

Influence of Trait-Patience in Goal Pursuit and Subjective Well-Being Among University Students

The present study was aimed to examine the role of trait-patience among university students in pursuit of goal and achieving subjective-wellbeing across the academic quarter. A sample of 300 university students (male = 108; female = 192) age ranged 18-35 years were selected from Rawalpindi and Islamabad. To measure study variables, Three-Factor Patience Scale (Schnitker 2012), Self-reported projects (Little, 1983), Positive and Negative Affect Scale (Watson, Clark, & Tellegen, 1988), and Satisfaction with Life Scale (Diener, Emmons, Larsen & Griffin, 1985) were used. Results revealed that individuals high in patience were more engaged in pursuit of personal projects, as compare to those with low trait- patience. It was also found that the trait-patience was significantly related with goal pursuit, achievement and well-being. In context of temporal effect, patience and goal pursuit was significant predictors of the goal achievement, while the subjective-well-being at the start of semester predicted the well-being across time. The present study would be useful for students and teachers to evaluate or manage the students to set goals and patiently strive to reach goal and well-being.

Unsupervised classification of specialty coffees in Homogeneous sensory attributes through machine learning

Brazil is the largest exporter of coffee beans, 29% world exports, 15% this volume in specialty coffees. Thereby researches are done, so that identify different segments in the market, in order to direct the end consumer to a better quality product. New technologies are explored to meet an increasing demand for high quality coffees. Therefore, in this article has an objective to propose the use of machine learning techniques combined with projection pursuit in the construction of unsupervised classification models, in a sensory acceptance experiment, applied to four groups of trained and untrained consumers, in four classes of specialty coffees in which they were evaluated sensory characteristics: aroma, body coffee, sweetness and general note. For evaluating classifier performance, in the data with reduced dimension, all instances were used, and considering four groupings, the models were adjusted. The results obtained from the groupings formed were compared with pre-established classes to confirm the model. Success and error rates were obtained, considering the rate of false positives and false negatives, sensitivity and classification methods accuracy. It was concluded that, machine learning use in data with reduced dimensions is feasible, as it allows unsupervised classification of specialty coffees, produced at different altitudes and processes, considering the heterogeneity among consumers involved in sensory analysis, and the high homogeneity of sensory attributes among the analyzed classes, obtaining good hit rates in some classifiers. Key words: Classification models; Data dimension reduction; Groupings identification; Projection pursuit.

Proposition of a new index for projection pursuit in the multiple factor analysis

Proposition of a new index for projection pursuit in the multiple factor analysis

Spatial Spillover Effect of Rural Finance on Ecological Environment Construction: Based on Genetic Algorithm Projection Pursuit Model and Spatial Durbin Model

Spatial Spillover Effect of Rural Finance on Ecological Environment Construction: Based on Genetic Algorithm Projection Pursuit Model and Spatial Durbin Model

Novel method for measuring regional precipitation complexity characteristics based on multiscale permutation entropy combined with CMFO-PPTTE model

The influence of entropy values obtained under different multiscale permutation entropy scales on the complexity and unconstrained extreme value problem of traditional bionics algorithms was assessed. The multiscale permutation entropy (MSPE) eigenvalue calculation method based on the chaotic moth-flame optimization-based projection pursuit threat target evaluation (CMFO-PPTTE) model is proposed (CMFO-PPTTE-MSPE). The CMFO-PPTTE-MSPE not only solves the problems of rough calculation and information loss in traditional eigenvalue calculation but also improves the shortcomings of slow convergence and local optimality in the intelligent bionic algorithm. For this purpose, the monthly precipitation of 13 administrative regions in Heilongjiang Province under the Global Precipitation Climatology Centre dataset (GPCC) from 1967 to 2017 was evaluated to improve the precipitation complexity entropy accuracy. The main influencing factors were altitude (p < 0.05), water area (p < 0.05), urban construction area (p < 0.05) and forestland area (p < 0.01). Radial basis function (RBF) neural networks were used to forecast the precipitation in Heilongjiang Province over 36 months, and a better forecast was obtained. To verify the rationality of CMFO-PPTTE-MSPE under GPCC data, compared with the situ data, it is found that GPCC data can more accurately classify the complexity grade of Daxing’anling region. At the same time, the administrative discrimination of GPCC data (1.107) is significantly higher than the situ data (1.023). To verify the rationality of the CMFO-PPTTE-MSPE, the partial mean of MSPE (MSPE-PM), whale optimization algorithm PPTTE (WOA-PPTTE) and ω-particle swarm optimization PPTTE (ω-PSO-PPTTE) models were also used to calculate the MSPE eigenvalues under GPCC data. Comparisons and evaluations were performed after dividing the grade based on geographical discrimination (CMFO-PPTTE-MSPE (1.107) > WOA-PPTTE-MSPE (1.094) > W-PSO-PPTTE-MSPE (1.090) > MSPE-PM (1.063)) and administrative discrimination (CMFO-PPTTE-MSPE (1.146) > W-PSO-PPTTE-MSPE (1.002) > MSPE-PM (1.012) > WOA-PPTTE-MSPE (1.002)). The CMFO-PPTTE-MSPE had a significantly higher distinguishing capabilities than the other algorithms and thus could better distinguish the complexity of precipitation in different regions and geographic locations. In summary, the CMFO-PPTTE-MSPE is beneficial for analyzing precipitation complexity and represents a novel approach for mining the fine structural features of regional hydrological series. However, determining how to apply artificial intelligence algorithms to reasonably calibrate key parameters of the MSPE algorithm to further improve the accuracy of complexity diagnosis for hydrological series will be an important avenue of research.

Direct Multivariate Simulation - A stepwise conditional transformation for multivariate geostatistical simulation

Several applications in geoscience require the generation of multiple realizations of random fields of physical properties to mimic their spatial distribution and quantify the model uncertainty. Some modeling problems present complex multivariate distributions with heteroscedasticity and non-linear relations among the variables. We propose a new algorithm, namely Direct Multivariate Simulation, for the simulation of random fields of non-parametric multivariate joint distributions. The methodology is based on a generalization of the normal score and back transformation for multivariate distributions, also called Stepwise Conditional transformation. The sequential sampling of each variable is performed by decomposing the target joint distribution into a product of univariate marginal and conditional probability density functions. We propose numerical solutions to improve the algorithm efficiency for real case applications with a large number of variables. The method is demonstrated through an application where we sample a 6-variate joint distributions with strong nonlinear dependence among the variables. The results are validated by comparing the results to the Projection Pursuit Multivariate Transform and through the computation of the experimental semi-variograms, the marginal distributions, and the bi-histograms of the simulated variables.

Projection pursuit emulation for many-input computer experiments

This paper studies projection pursuit emulation for computer experiments with many input variables. This method aims at capturing the most influential directions of the inputs to the response, and thus the active dimensionality is reduced. Its interpolation property is proved under certain conditions. We also propose a two-stage method to handle the case where the projection pursuit method does not converge. Simulation studies show that the proposed methods are more efficient than the traditional Kriging methods.

Identifying urban vitality in metropolitan areas of developing countries from a comparative perspective: Ho Chi Minh City versus Shanghai

Urban vitality is widely recognized as an authentic philosophy that reflects chaotic urbanization and orthodox planning in the developed world. However, comparative studies on the urban vitality of developing countries are scarce. This study developed a framework for analyzing urban vitality in developing countries. Using Ho Chi Minh City and Shanghai as cases, we measured urban vitality and analyzed its spatial pattern by using the projection pursuit model based on three dimensions of human activity, built environment, and their linkage. Both cities show a declining gradient of urban vitality from the urban cores to suburbs. Shanghai also fosters several peaks of urban vitality in its subcenters. The different spatial patterns of urban vitality are largely determined by the monocentric or polycentric urban form. A similar pattern of high urban vitality in both urban cores may be associated with the European-style block planning in the former concession areas. Recently, these two cities launched large-scale transport projects and replicated the modern style of broad and grid roads from the US, thereby reducing their urban vitality. This comparative study can improve our understanding of urban vitality patterns in developing countries and provide planning implications that can help nurture urban vitality.

Identify and Assess Hydropower Project’s Multidimensional Social Impacts with Rough Set and Projection Pursuit Model

To realize the coordinated and sustainable development of hydropower projects and regional society, comprehensively evaluating hydropower projects’ influence is critical. Usually, hydropower project development has an impact on environmental geology and social and regional cultural development. Based on comprehensive consideration of complicated geological conditions, fragile ecological environment, resettlement of reservoir area, and other factors of future hydropower development in each country, we have constructed a comprehensive evaluation index system of hydropower projects, including 4 first-level indicators of social economy, environment, safety, and fairness, which contain 26 second-level indicators. To solve the problem that existing models cannot evaluate dynamic nonlinear optimization, a projection pursuit model is constructed by using rough set reduction theory to simplify the index. Then, an accelerated genetic algorithm based on real number coding is used to solve the model and empirical study is carried out with the Y hydropower station as a sample. The evaluation results show that the evaluation index system and assessment model constructed in our paper effectively reduce the subjectivity of index weight. Applying our model to the social impact assessment (SIA) of related international hydropower projects can not only comprehensively analyze the social impact of hydropower projects but also identify important social influencing factors and effectively analyze the social impact level of each dimension. Furthermore, SIA assessment can be conducive to project decision-making, avoiding social risks and social stability.

Tempo-spatial changes of ecological vulnerability in resource-based urban based on genetic projection pursuit model

Due to the intensification of large-scale mining and of human activities, the ecological integrity of resource-based cities has been severely damaged. Studying its ecological vulnerability (EV) will help improve the environmental degradation and ecosystem imbalance in the region and provide a decision-making basis for its sustainable development. In this paper, genetic projection pursuit and ecological vulnerability index model were used to quantitatively evaluate the EV of Datong from 2000 to 2017. This study shows that it is feasible to apply genetic projection pursuit on ecological vulnerability evaluation. This method avoids artificial assumptions, and thus provides objective results that successfully execute multi indicator evaluation process and analysis under non-linear systems in ecological environment. From 2000 to 2017, under the dual influence of national development plans and coal mining activities, Datong City's EV showed a downward trend, and the highest and lowest EV occurred in 2010 and 2017 respectively; the differences in natural conditions in different regions made the spatial distribution of EV high in the west and low in the east, the spatial variation was relatively stable, and the regions with large changes were concentrated in the Urban District, Zuoyun County and Lingqiu County; due to rapid industrialization and urbanization, the centroid of EV shifted to the southwest. The maximum offset was 3.81 km, and the EV distribution was uneven. Local governments should take corresponding measures in different ecologically vulnerable areas. In the buffer and transition zone, the idea of attaching importance to ecology should continue to be maintained and the ecological asset management models and mechanisms based on ecological protection should be actively explored. In the key ecological restoration area, efforts should be made to return farmland to forests (grass), accelerate the process of vegetation restoration, and rationally develop coal resources. In the absolute ecological protection zone, establish a mine reclamation monitoring mechanism and promote the development of the tertiary industry represented by tourism.

ACCURACY OF NONPARAMETRIC DENSITY ESTIMATION FOR UNIVARIATE GAUSSIAN MIXTURE MODELS: A COMPARATIVE STUDY

Flexible and reliable probability density estimation is fundamental in unsupervised learning and classification. Finite Gaussian mixture models are commonly used for this purpose. However, the parametric form of the distribution is not always known. In this case, non-parametric density estimation methods are used. Usually, these methods become computationally demanding as the number of components increases. In this paper, a comparative study of accuracy of some nonparametric density estimators is made by means of simulation. The following approaches have been considered: an adaptive bandwidth kernel estimator, a projection pursuit estimator, a logspline estimator, and a k-nearest neighbor estimator. It was concluded that data clustering as a pre-processing step improves the estimation of mixture densities. However, in case data does not have clearly defined clusters, the pre-preprocessing step does not give that much of advantage. The application of density estimators is illustrated using municipal solid waste data collected in Kaunas (Lithuania). The data distribution is similar (i.e., with kurtotic unimodal density) to the benchmark distribution introduced by Marron and Wand. Based on the homogeneity tests it can be concluded that distributions of the municipal solid waste fractions in Kutaisi (Georgia), Saint-Petersburg (Russia), and Boryspil (Ukraine) are statistically indifferent compared to the distribution of waste fractions in Kaunas. The distribution of waste data collected in Kaunas (Lithuania) follows the general observations introduced by Marron and Wand (i.e., has one mode and certain kurtosis).

A Comprehensive Evaluation Model of Regional Water Resource Carrying Capacity: Model Development and a Case Study in Baoding, China

Scientific water resource carrying capacity (WRCC) evaluations are necessary for providing guidance for the sustainable utilization of water resources. Based on the driving-pressure-state-impact-response feedback loop, this paper selects 21 indicators under five dimensions to construct a regional WRCC comprehensive evaluation framework. The projection pursuit clustering (PPC) method is implemented with the matter-element extension (MEE) model to overcome the limitations of subjective deviation and indicator attribute incompatibility in traditional comprehensive assessment methods affecting the accuracy of evaluations. The application of the integrated evaluation model is demonstrated in Baoding city in the Jing-Jin-Ji area from 2010 to 2017. The results indicate that the economic water consumption intensity is the most influential factor that impacts the WRCC change in Baoding, and the pressure subsystem and response subsystem are dominant in the entire system. The WRCC in Baoding significantly improved between 2010 and 2017 from a grade V extremely unsafe state to a grade III critical state. Natural water shortages and large population scales are the main negative factors during this period; however, the existing measures are still insufficient to achieve an optimal WRCC status. Considering the future population and industry inflow, additional actions must be proposed to maintain and promote harmonious conditions.