scholarly journals Inference in multiscale geographically weighted regression

2018 ◽  
Author(s):  
Hanchen Yu ◽  
Stewart Fotheringham ◽  
Ziqi Li ◽  
Taylor Oshan ◽  
Wei Kang ◽  
...  
Keyword(s):  
Geographically Weighted Regression ◽  
Additive Model ◽  
Simulated Data ◽  
Weighted Regression ◽  
Parameter Estimates ◽  
Real World Data ◽  
Data Set ◽  
Smoothing Factor ◽  
Hat Matrix ◽  
Response Vector

A recent paper (Fotheringham et al. 2017) expands the well-known Geographically Weighted Regression (GWR) framework significantly by allowing the bandwidth or smoothing factor in GWR to be derived separately for each covariate in the model – a framework referred to as Multiscale GWR (MGWR). However, one limitation of the MGWR framework is that, until now, no inference about the local parameter estimates was possible. Formally, the so-called “hat matrix,” which projects the observed response vector into the predicted response vector, was available in GWR but not in MGWR. This paper addresses this limitation by reframing GWR as a Generalized Additive Model (GAM), extending this framework to MGWR and then deriving standard errors for the local parameters in MGWR. In addition, we also demonstrate how the effective number of parameters (ENP) can be obtained for the overall fit of an MGWR model and for each of the covariates within the model. This statistic is essential for comparing model fit between MGWR, GWR, and traditional global models, as well as adjusting for multiple hypothesis tests. We demonstrate these advances to the MGWR framework with both a simulated data set and a real-world data set.

scholarly journals On the Measurement of Bias in Geographically Weighted Regression Models

2019 ◽  
Author(s):  
Hanchen Yu ◽  
Alexander Stewart Fotheringham ◽  
Ziqi Li ◽  
Taylor M. Oshan ◽  
Levi John Wolf
Keyword(s):  
Geographically Weighted Regression ◽  
Regression Models ◽  
Simulated Data ◽  
Information Criterion ◽  
Weighted Regression ◽  
Parameter Estimates ◽  
Local Parameter ◽  
Data Set ◽  
Trade Off ◽  
Parameter Values

Under the realization that Geographically Weighted Regression (GWR) is a data-borrowing technique, this paper derives expressions for the amount of bias introduced to local parameter estimates by borrowing data from locations where the processes might be different from those at the regression location. This is done for both GWR and Multiscale GWR (MGWR). We demonstrate the accuracy of our expressions for bias through a comparison with empirically derived estimates based on a simulated data set with known local parameter values. By being able to compute the bias in both models we are able to demonstrate the superiority of MGWR. We then demonstrate the utility of a corrected Akaike Information Criterion statistic in finding optimal bandwidths in both GWR and MGWR as a trade-off between minimizing both bias and uncertainty. We further show how bias in one set of local parameter estimates can affect the bias in another set of local estimates. The bias derived from borrowing data from other locations appears to be very small.

scholarly journals Exploring the spatio-temporal dynamics of geographical processes with geographically weighted regression and geovisual analytics

2008 ◽  
Vol 7 (3-4) ◽  
pp. 181-197 ◽  
Author(s):  
Urška Demšar ◽  
A. Stewart Fotheringham ◽  
Martin Charlton
Keyword(s):  
Geographically Weighted Regression ◽  
Temporal Dynamics ◽  
Simulated Data ◽  
Weighted Regression ◽  
Parameter Estimates ◽  
Data Set ◽  
Complex Processes ◽  
Regression Parameters ◽  
Geovisual Analytics ◽  
Spatio Temporal

The paper examines the potential for combining a spatial statistical methodology – Geographically Weighted Regression (GWR) – with geovisual analytical exploration to help understand complex spatio-temporal processes. This is done by applying the combined statistical – exploratory methodology to a simulated data set in which the behaviour of regression parameters was controlled across space and time. A variety of complex spatio-temporal processes was captured through space-time (i.e. as spatio-temporal) varying parameters whose values were known. The task was to see if the proposed methodology could uncover these complex processes from the data alone. The results of the experiment confirm that the combined methodology can successfully identify spatio-temporal patterns in the local GWR parameter estimates that correspond to the controlled behaviour of the original parameters.

Fast Geographically Weighted Regression (FastGWR): a scalable algorithm to investigate spatial process heterogeneity in millions of observations

2019 ◽  
Vol 33 (1) ◽  
pp. 155-175 ◽  
Author(s):  
Li ◽  
Fotheringham ◽  
Li ◽  
Oshan
Keyword(s):  
Los Angeles ◽  
Open Source ◽  
Geographically Weighted Regression ◽  
Message Passing Interface ◽  
House Price ◽  
Weighted Regression ◽  
Spatial Process ◽  
Parameter Estimates ◽  
Single Family ◽  
Scalable Algorithm

Geographically Weighted Regression (GWR) is a widely used tool for exploring spatial heterogeneity of processes over geographic space. GWR computes location-specific parameter estimates, which makes its calibration process computationally intensive. The maximum number of data points that can be handled by current open-source GWR software is approximately 15,000 observations on a standard desktop. In the era of big data, this places a severe limitation on the use of GWR. To overcome this limitation, we propose a highly scalable, open-source FastGWR implementation based on Python and the Message Passing Interface (MPI) that scales to the order of millions of observations. FastGWR optimizes memory usage along with parallelization to boost performance significantly. To illustrate the performance of FastGWR, a hedonic house price model is calibrated on approximately 1.3 million single-family residential properties from a Zillow dataset for the city of Los Angeles, which is the first effort to apply GWR to a dataset of this size. The results show that FastGWR scales linearly as the number of cores within the High-Performance Computing (HPC) environment increases. It also outperforms currently available open-sourced GWR software packages with drastic speed reductions – up to thousands of times faster – on a standard desktop.

scholarly journals Computational improvements to multiscale geographically weighted regression

2019 ◽  
Author(s):  
Ziqi Li ◽  
Alexander Stewart Fotheringham
Keyword(s):  
Geographically Weighted Regression ◽  
Parallel Implementation ◽  
Additive Model ◽  
Large Datasets ◽  
Computational Method ◽  
Weighted Regression ◽  
Multi Scale ◽  
Single Scale ◽  
Gwr Model ◽  
Python Package

Geographically Weighted Regression (GWR) has been broadly used in various fields to model spatially non-stationary relationships. Multi-scale Geographically Weighted Regression (MGWR) is a recent advancement to the classic GWR model. MGWR is superior in capturing multi-scale processes over the traditional single-scale GWR model by using different bandwidths for each covariate. However, the multiscale property of MGWR brings additional computation costs. The calibration process of MGWR involves iterative back-fitting under the additive model (AM) framework. Currently, MGWR can only be applied on small datasets within a tolerable time and is prohibitive on moderately large datasets (greater than 5,000 observations). In this paper, we propose a parallel implementation that has crucial computational improvements to MGWR calibration. This improved computational method reduces both memory footprint and runtime to allow MGWR modelling to be applied to moderate-to-large datasets (up to 100,000 observations). These improvements are integrated into the mgwr python package and MGWR 2.0 software, both of which are freely available to download.

scholarly journals Measuring Bandwidth Uncertainty in Multiscale Geographically Weighted Regression Using Akaike Weights

2019 ◽  
Author(s):  
Ziqi Li ◽  
Alexander Stewart Fotheringham ◽  
Taylor M. Oshan ◽  
Levi John Wolf
Keyword(s):  
Geographically Weighted Regression ◽  
Regression Models ◽  
Bandwidth Selection ◽  
Information Criterion ◽  
Weighted Regression ◽  
Parameter Estimates ◽  
Multi Scale ◽  
Single Scale ◽  
Heterogeneous Processes ◽  
Key Parameter

Bandwidth, a key parameter in geographically weighted regression models, is closely related to the spatial scale at which the underlying spatially heterogeneous processes being examined take place. Generally, a single optimal bandwidth (geographically weighted regression) or a set of covariate-specific optimal bandwidths (multiscale geographically weighted regression) is chosen based on some criterion such as the Akaike Information Criterion (AIC) and then parameter estimation and inference are conditional on the choice of this bandwidth. In this paper, we find that bandwidth selection is subject to uncertainty in both single-scale and multi-scale geographically weighted regression models and demonstrate that this uncertainty can be measured and accounted for. Based on simulation studies and an empirical example of obesity rates in Phoenix, we show that bandwidth uncertainties can be quantitatively measured by Akaike weights, and confidence intervals for bandwidths can be obtained. Understanding bandwidth uncertainty offers important insights about the scales over which different processes operate, especially when comparing covariate-specific bandwidths. Additionally, unconditional parameter estimates can be computed based on Akaike weights accounts for bandwidth selection uncertainty.

scholarly journals Forecasting for smart energy: an accurate and effificient negative binomial additive model

2020 ◽  
Vol 20 (2) ◽  
pp. 1000
Author(s):  
Yousef-Awwad Daraghmi ◽  
Eman Yaser Daraghmi ◽  
Motaz Daadoo ◽  
Samer Alsaadi
Keyword(s):  
Negative Binomial ◽  
Load Forecasting ◽  
Additive Model ◽  
Additive Models ◽  
Electric Load ◽  
Short Term ◽  
Real World Data ◽  
Smooth Functions ◽  
Data Set ◽  
Electric Load Forecasting

<div>Smart energy requires accurate and effificient short-term electric load forecasting to enable effificient</div><div>energy management and active real-time power control. Forecasting accuracy is inflfluenced by the char</div><div>acteristics of electrical load particularly overdispersion, nonlinearity, autocorrelation and seasonal patterns.</div><div>Although several fundamental forecasting methods have been proposed, accurate and effificient forecasting</div><div>methods that can consider all electric load characteristics are still needed. Therefore, we propose a novel</div><div>model for short-term electric load forecasting. The model adopts the negative binomial additive models</div><div>(NBAM) for handling overdispersion and capturing the nonlinearity of electric load. To address the season</div><div>ality, the daily load pattern is classifified into high, moderate, and low seasons, and the autocorrelation of</div><div>load is modeled separately in each season. We also consider the effificiency of forecasting since the NBAM</div><div>captures the behavior of predictors by smooth functions that are estimated via a scoring algorithm which has</div><div>low computational demand. The proposed NBAM is applied to real-world data set from Jericho city, and its</div><div>accuracy and effificiency outperform those of the other models used in this context.</div>

scholarly journals mgwr: A Python Implementation of Multiscale Geographically Weighted Regression for Investigating Process Spatial Heterogeneity and Scale

2019 ◽  
Vol 8 (6) ◽  
pp. 269 ◽  
Author(s):  
Taylor Oshan ◽  
Ziqi Li ◽  
Wei Kang ◽  
Levi Wolf ◽  
A. Fotheringham
Keyword(s):  
Spatial Heterogeneity ◽  
Geographically Weighted Regression ◽  
Linear Models ◽  
Statistical Technique ◽  
Weighted Regression ◽  
Parameter Estimates ◽  
Local Models ◽  
Spatial Processes ◽  
Geographic Scale ◽  
Global Regression

Geographically weighted regression (GWR) is a spatial statistical technique that recognizes that traditional ‘global’ regression models may be limited when spatial processes vary with spatial context. GWR captures process spatial heterogeneity by allowing effects to vary over space. To do this, GWR calibrates an ensemble of local linear models at any number of locations using ‘borrowed’ nearby data. This provides a surface of location-specific parameter estimates for each relationship in the model that is allowed to vary spatially, as well as a single bandwidth parameter that provides intuition about the geographic scale of the processes. A recent extension to this framework allows each relationship to vary according to a distinct spatial scale parameter, and is therefore known as multiscale (M)GWR. This paper introduces mgwr, a Python-based implementation of MGWR that explicitly focuses on the multiscale analysis of spatial heterogeneity. It provides novel functionality for inference and exploratory analysis of local spatial processes, new diagnostics unique to multi-scale local models, and drastic improvements to efficiency in estimation routines. We provide two case studies using mgwr, in addition to reviewing core concepts of local models. We present this in a literate programming style, providing an overview of the primary software functionality and demonstrations of suggested usage alongside the discussion of primary concepts and demonstration of the improvements made in mgwr.

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