Local Linear Approximation Research Articles

Environmental Modelling through Chaotic Approach for Malaysian West Coast Sea Level.

Sea level forecasting is an essential task for coastal engineering, geodetic application, navigation, and recreational activities. Predicting the behavior of future sea level is important for monitoring and forecasting of changes in fishery and marine ecosystems as well as for protection of coastal. This research is focused on the analysis and prediction of hourly sea level time series data at the benchmark station located in Penang by using chaotic approach. The purpose of this research was to identify the presence of chaotic behavior by the phase space reconstruction and Cao methods, and also a local linear approximation method is applied for prediction purposes. The results notified that the value of correlation coefficient between the observed and predicted time series is 0.8838 which is near to one. This reveals that the local linear approximation method can be used to predict the sea level time series in Malaysia. Certainly, the result of this research is expected to help stakeholder such as Department of Survey and Mapping Malaysia (JUPEM) in having a better sea level management.

A cubic spline penalty for sparse approximation under tight frame balanced model

The study of non-convex penalties has recently received considerable attentions in sparse approximation. The existing non-convex penalties are proposed on the principle of seeking for a continuous alternative to the l0-norm penalty. In this paper, we come up with a cubic spline penalty (CSP) which is also continuous but closer to l0-norm penalty compared to the existing ones. As a result, it produces the weakest bias among them. Wavelet tight frames are efficient for sparse approximation due to its redundancy and fast implementation algorithm. We adopt a tight frame balanced model with our proposed cubic spline penalty since the balanced model takes the advantages of both analysis and synthesis model. To solve the non-convex CSP penalized problem, we employ a proximal local linear approximation (PLLA) algorithm and prove the generated sequence converges to a stationary point of the model if it is bounded. Under additional conditions, we find that the limit point behaves as well as the oracle solution, which is obtained by using the exact support of the ground truth signal. The efficiency of our cubic spline penalty are further demonstrated in applications of variable selection and image deblurring.

LOCAL COMPOSITE QUANTILE REGRESSION SMOOTHING: A FLEXIBLE DATA STRUCTURE AND CROSS-VALIDATION

In this paper, we study the local composite quantile regression estimator for mixed categorical and continuous data. The local composite quantile estimator is an efficient and safe alternative to the local polynomial method and has been well-studied for continuous covariates. Generalization of the local composite quantile regression estimator to a flexible data structure is appealing to practitioners as empirical studies often encounter categorical data. Furthermore, we study the theoretical properties of the cross-validated bandwidth selection for the local composite quantile estimator. Under mild conditions, we derive the rates of convergence of the cross-validated smoothing parameters to their optimal benchmark values for both categorical and continuous covariates. Monte Carlo experiments show that the proposed estimator may have large efficiency gains compared with the local linear estimator. Furthermore, we illustrate the robustness of the local composite quantile estimator using the Boston housing dataset.

Nonparametric Quantile Regression Estimation With Mixed Discrete and Continuous Data

In this article, we investigate the problem of nonparametrically estimating a conditional quantile function with mixed discrete and continuous covariates. A local linear smoothing technique combining both continuous and discrete kernel functions is introduced to estimate the conditional quantile function. We propose using a fully data-driven cross-validation approach to choose the bandwidths, and further derive the asymptotic optimality theory. In addition, we also establish the asymptotic distribution and uniform consistency (with convergence rates) for the local linear conditional quantile estimators with the data-dependent optimal bandwidths. Simulations show that the proposed approach compares well with some existing methods. Finally, an empirical application with the data taken from the IMDb website is presented to analyze the relationship between box office revenues and online rating scores. Supplementary materials for this article are available online.

Efficient estimation for the volatility of stochastic interest rate models

The joint analysis of non-stationary and high frequency financial data poses theoretical challenges due to that such massive data varies with time and possesses no fixed density function. This paper proposes the local linear smoothing to estimate the unknown volatility function in scalar diffusion models based on Gamma asymmetric kernels for high frequency financial big data. Under the mild conditions, we obtain the asymptotic normality for the estimator at both interior and boundary design points. Besides the standard properties of the local linear estimator such as simple bias representation and boundary bias correction, the local linear smoothing using Gamma asymmetric kernels possesses some extra advantages such as variance reduction and resistance to sparse design, which is validated through finite sample simulation study and empirical analysis on 6-month Shanghai Interbank Offered Rate (abbreviated as Shibor) in China.

FDA: theoretical and practical efficiency of the local linear estimation based on the kNN smoothing of the conditional distribution when there are missing data

ABSTRACTWe aim to estimate effectively the conditional distribution function (CDF) of a scalar response variable, with missing data at random, given a functional co-variable. For this aim, we combine the local linear approach with the kernel nearest neighbours procedure to construct a new estimator of the CDF. A fundamental issue of interest is to study the impact of the missing observations on the performances of estimators. We establish, under less restrictive conditions, the strong consistency of the constructed estimator. Then, we test first its effectiveness on simulated and real datasets, and then we conclude by a comparison study with classical estimators of the CDF.

Bias correction for local linear regression estimation using asymmetric kernels via the skewing method

The skewing method, which has been originally proposed as a bias correction device for local linear regression estimation using standard symmetric kernels, is extended to the cases of asymmetric kernels. The method is defined as a convex combination of three local linear estimators. It is demonstrated that the skewed estimator using asymmetric kernels with properly chosen weights can accelerate the bias convergence from O(b) to O(b2) as b → 0 under sufficient smoothness of the unknown regression curve while not inflating the variance in an order of magnitude, where b is the smoothing parameter and the regressor is assumed to have at least one boundary. As a consequence, the estimator has optimal pointwise convergence of n−4/9 when best implemented, where n is the sample size. It is noteworthy that these properties are the same as those for a local cubic regression estimator. Finite-sample properties of the skewed estimator are assessed in comparison with local linear and local cubic estimators. An application of the skewed estimation to real data is also considered.

A Method of Local Linear Approximation for the Nonlinear Discrete Equations

We obtain new conditions for the existence of bounded solutions of nonlinear discrete equations with the use of local linear approximation of these equations.

Time-varying income and price elasticities for energy demand: Evidence from a middle-income panel

Abstract We estimate time-varying income and price elasticities for energy demand for a 26-country, middle-income balanced panel that spans 1996–2014. To do so, we employ a recently developed local linear dummy estimation method to estimate the trend and coefficient functions. We find that the price elasticity for energy demand is either insignificant or positive and small. While the income elasticity for energy demand behaves in a non-linear fashion over-time, it is always less than unity and is generally within 0.6–0.8. A GDP elasticity of less than one suggests that these middle-income countries are on the right-hand-side of an inverted-U energy intensity-GDP path that is consistent with the dematerialization process. Also, this finding suggests that energy intensity — but not energy consumption — in these countries will fall with economic growth. Hence, intensity-based targets may be met in a business-as-usual setting, but aggregate or per capita-based carbon emissions targets would likely require policy interventions.

Detection and reconstruction of catastrophic breaks of high-frequency financial data with local linear scaling approximation

Catastrophic breaks of high-frequency financial data usually contain important information, and it is crucial for investment decision making to accurately detect and analyze these catastrophic breaks. In this paper, we use a robust algorithm, the local linear scale approximation (LLSA), which enriches nonlinear filtering methods in data mining to detect and reconstruct catastrophic breaks. Guided by this methodology, we detect the catastrophic breaks in the high-frequency financial data of the Shanghai composite index from 2015 to 2018, and ten catastrophic breaks are found. Our results confirm that the LLSA is able to preserve smooth trends and accurately display any catastrophic breaks. In addition, we find that all these breaks are consistent with three types of economic events — the adjustment of monetary policies, the introduction of macroeconomic policies or government guidance, and the revolutionary events of the securities market. Explanations are given for the emergence of these events.

Local Polynomial Order in Regression Discontinuity Designs

The local linear estimator has become the standard in the regression discontinuity design literature, but we argue that it should not always dominate other local polynomial estimators in empirical studies. We show that the local linear estimator in the data generating processes (DGP’s) based on two well- known empirical examples does not always have the lowest (asymptotic) mean squared error (MSE). Therefore, we advocate for a more flexible view towards the choice of the polynomial order, p, and suggest two complementary approaches for picking p: comparing the MSE of alternative estimators from Monte Carlo simulations based on an approximating DGP, and comparing the estimated asymptotic MSE using actual data.Length: 47 pages

AMiBI: Application of Flood Mitigation in Indonesia based on the Results of Statistical Analyses of Causal Factors using Local Linear Estimators in Nonparametric Regression Model

This article describes a new idea called AMiBI. It is a mitigation platform based on the fact that flood still becomes an annual problem in Indonesia. According to the National Disaster Management Agency or Badan Nasional Penanggulangan Bencana (BNPB), 649 flood incidents occurred from 2011 until 2019 in Indonesia. Natural factors are certainly not the only main factors causing it. Environmental damage plays a more crucial role. One of the causes of this damage is the existence of settlements along the riverbanks. This factor exactly should be controlled by humans. Still, economic needs have become the main reason driving people to survive in big cities by establishing illegal settlements along riverbanks. Regarding these facts, AMiBi was also built under statistical analysis by modeling the flood incidents based on the number of settlements along riverbanks using the local linear nonparametric regression. Its result shows that the model has R2 value of 51.48% and a Mean Square Error (MSE) of 24.26. It also performs a linear relationship between those variables, which means that the existence of settlements along riverbanks significantly affects the number of flood incidents. Regarding those analyses as the basis of development, this digital platform performs several services for reducing loss potency caused and supporting the awareness to build a sustainable environment in riverbanks. Considering AMiBI as the only platform that uses statistical modeling as the basis of services and implementation, it has a significant role in supporting Indonesia as a smart country for mitigation.

Fast extremum seeking using multisine dither and online complex curve fitting

Fast online optimization of uncertain Wiener systems using extremum seeking control (ESC) is investigated. Derivative estimation in extremum seeking is herefore described as an online parametric system identification problem. Multisine dithering is applied with frequencies around the first resonance frequency of the system to remove the time scale separation between dither and plant dynamics which is commonly required in ESC. Recursive use of the Fourier transform, over a moving window of historic data, provides a frequency response function estimate of the system’s local best linear approximation. Continuous online complex curve fitting is then applied to extrapolate to an estimate of the steady-state response which coincides with the local gradient of the steady-state objective function. An analysis of the closed-loop dynamics is provided. Transient improvements and robustness of the approach against plant variation are demonstrated with a simulation example.

Correcting an estimator of a multivariate monotone function with isotonic regression.

In many problems, a sensible estimator of a possibly multivariate monotone function may fail to be monotone. We study the correction of such an estimator obtained via projection onto the space of functions monotone over a finite grid in the domain. We demonstrate that this corrected estimator has no worse supremal estimation error than the initial estimator, and that analogously corrected confidence bands contain the true function whenever the initial bands do, at no loss to band width. Additionally, we demonstrate that the corrected estimator is asymptotically equivalent to the initial estimator if the initial estimator satisfies a stochastic equicontinuity condition and the true function is Lipschitz and strictly monotone. We provide simple sufficient conditions in the special case that the initial estimator is asymptotically linear, and illustrate the use of these results for estimation of a G-computed distribution function. Our stochastic equicontinuity condition is weaker than standard uniform stochastic equicontinuity, which has been required for alternative correction procedures. This allows us to apply our results to the bivariate correction of the local linear estimator of a conditional distribution function known to be monotone in its conditioning argument. Our experiments suggest that the projection step can yield significant practical improvements.

The Shanghai- Hong Kong Stock Connect: An Application of the Semi-CGARCH and Semi-EGARCH

This paper examines the impact on volatility on the Shanghai Stock Exchange and the Hong Kong Stock Exchange before and after the connection on November 17, 2014. We test whether this event led to a structural break. For this purpose, the volatility series are shown and analysed in more detail using two new models. We test both semiparametric GARCH extensions, the Semi-EGARCH based on the EGARCH model (Nelson, 1991) and the Semi-CGARCH model, based on the CGARCH (Engle & Lee, 1999). Both univariate models work with a constrained local linear estimator for the scale function and a fully data-driven algorithm developed under weak moment conditions. The proposed method is applied to the two major Asian financial indices and to stocks from the banking sector. Furthermore, our focus is to improve the quantitative risk management analysis. When a parametric GARCH model provides satisfactory results for the calculation of e.g the Value at Risk (VaR), semiparametric can also be applied to improve the quality of measurement. This article compares the results of various parametric and semiparametric approaches regarding the VaR to show how the extensions increase the performance.

System Identification of Ships Using Machine Learning and Database

Ships are more difficult to automate than other vehicles. This is because they are exposed to disturbances such as tidal currents, waves, winds, etc., and their dynamic characteristics fluctuate greatly during the voyage. To date, autopilots have become popular for assisting voyages. However, there are various issues in its complete automation. For example, in a questionnaire regarding the disadvantages, there are answers that "they are susceptible to the tide" or "the displacement due to disturbance cannot be corrected". In this study, an online system identification method is designed. Specifically, the system parameters of the hull motion model are updated only when the error of the turning angular velocity between the ship and the estimated hull motion model exceeds a reference value. At that time, after a rough local linear approximation model is designed using a database, the finish is adjusted using machine learning by particle swarm optimization. Finally, the effectiveness of the method is quantitatively verified by simulation using the input and output data of the ship.

Conceptual Challenges for Interpretable Machine Learning

As machine learning has gradually entered into ever more sectors of public and private life, there has been a growing demand for algorithmic explainability. How can we make the predictions of complex statistical models more intelligible to end users? A sub-discipline of computer science known as interpretable machine learning (iML) has emerged to address this urgent question. Numerous influential methods have been proposed, from local linear approximations to rule lists and counterfactuals. In this article, I highlight three conceptual challenges that are largely overlooked by authors in this area. I argue that the vast majority of iML algorithms are plagued by: (1) ambiguity with respect to their true target; (2) a disregard for error rates and severe testing; and (3) an emphasis on product over process. Each point is developed at length, drawing on relevant debates in epistemology and philosophy of science. Examples and counterexamples from iML are considered, demonstrating how failure to acknowledge these problems can result in counter-intuitive and potentially misleading explanations. Without greater care for the conceptual foundations of iML, future work in this area is doomed to repeat the same mistakes.

A Spatiotemporal Deformation Modelling Method Based on Geographically and Temporally Weighted Regression

The geographically and temporally weighted regression (GTWR) model is a dynamic model which considers the spatiotemporal correlation and the spatiotemporal nonstationarity. Taking into account these advantages, we proposed a spatiotemporal deformation modelling method based on GTWR. In order to further improve the modelling accuracy and efficiency and considering the application characteristics of deformation modelling, the inverse window transformation method is used to search the optimal fitting window width and furthermore the local linear estimation method is used in the fitting coefficient function. Moreover, a comprehensive model for the statistical tests method is proposed in GTWR. The results of a dam deformation modelling application show that the GTWR model can establish a unified spatiotemporal model which can represent the whole deformation trend of the dam and furthermore can predict the deformation of any point in time and space, with stronger flexibility and applicability. Finally, the GTWR model improves the overall temporal prediction accuracy by 43.6% compared to the single-point time-weighted regression (TWR) model.

The Horseshoe-Like Regularization for Feature Subset Selection

Feature subset selection arises in many high-dimensional applications of statistics, such as compressed sensing and genomics. The l0 penalty is ideal for this task, the caveat being it requires the NP-hard combinatorial evaluation of all models. A recent area of considerable interest is to develop efficient algorithms to fit models with a non-convex lγ penalty for γ ∈ (0,1), which results in sparser models than the convex l1 or lasso penalty, but is harder to fit. We propose an alternative, termed the horseshoe regularization penalty for feature subset selection, and demonstrate its theoretical and computational advantages. The distinguishing feature from existing non-convex optimization approaches is a full probabilistic representation of the penalty as the negative of the logarithm of a suitable prior, which in turn enables efficient expectation-maximization and local linear approximation algorithms for optimization and MCMC for uncertainty quantification. In synthetic and real data, the resulting algorithms provide better statistical performance, and the computation requires a fraction of time of state-of-the-art non-convex solvers.

Modeling of hypertension risk factors using local linear of additive nonparametric logistic regression

Hypertension has become a serious health problem in Indonesia because of its prevalence, however, the causative factors could not be ascertained for about ninety percent of the patients. Various studies have found several risk factors causing hypertension to be obesity, family history, stress levels, heart rate, and an unhealthy lifestyle. In this case, the variables are considered influential on hypertension through a regression curve without a specific pattern. Also, we need to describe the functional relationships between several predictor variables with binary or dichotomous response variables and need to describe locally effect of predictor variables to the response variable. Therefore, in this study, to model the case of hypertension by age, body mass index, heart rate, stress levels we use the additive nonparametric logistic regression approach based on local linear estimators. The results of the study showed that hypertension was most prevalent among respondents over 65 years of age with BMI between 25-30 kg/m2 (obesity) and normal heart rate (60-100) bpm and most of them were found to be experiencing mild stress conditions. The model obtained a classification accuracy of 95 percent (in-sample) and 89.47 percent (out-sample) with a cut off probability value of 0.4.