Geographically Weighted Regression as a Predictive Tool ...1351572/FULLTEXT01.pdf · This thesis studies a new regression method, Geographically Weighted Regression (GWR) to predict

IN THE FIELD OF TECHNOLOGYDEGREE PROJECT CIVIL ENGINEERING AND URBAN MANAGEMENTAND THE MAIN FIELD OF STUDYTHE BUILT ENVIRONMENT,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2019

Geographically Weighted Regression as a Predictive Tool for Station-Level RidershipThe Case of Stockholm

KARIM OUNSI

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT

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Abstract

English/ Engelska/ Anglais

This thesis studies a new regression method, Geographically Weighted Regression (GWR) to predict ridership at the station level for future stations. The case study of Stockholm’s blue line is used as new stations will be built by 2030. This paper is written in English.

Historically, linear regression methods, independent of the geographical location of the

observations, was and is still used as the Ordinary Least Square regression method. With the rise of GIS-softwares these last decades, geographically dependent regression can be used and previous preliminary studies have shown a dependency between ridership and location of the station within the network.

GWR equations for new stations are determined and used to predict their respective

ridership. GIS-data was collected using Geodata and Traffikverket (Traffic Authority) and ridership as well as socio-economic related material for the base year of 2016 was used in order to determine, first, significant variables from a group of candidate ones (Workers, number of bus lines and type of change were chosen) and, second the OLS and GWR equations. Significances of both models were compared and the OLS equation was used in order to determine the hypothetical ridership of the new stations if they were present in 2016. GWR equations were then determined using these calculated ridership of these new stations. Having all GWR equations (each station having its own equation), ridership was thus predicted for the new stations for 2030 using assumptions and planned, programmed development around the stations (population, apartment to be built…) and compared with the official predictions.

The results show that the GWR method, generally, overpredicts ridership when compared

to the official predictions. Many reasons can explain this overprediction like the assumptions made with regards to the number of buses as well as the method followed to calculate the number of workers around each station.

Three main conclusions were drawn for this case study. One main conclusion, specific for

this study and two other, more general, conclusions were deduced from this study. First, GWR is a good predicting tool for future stations that are close to most currently available stations. Second, GWR is a good predicting method for stations where limited changes in the future environment will occur.

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Sammanfattning

Swedish/ Svenska/ Suédois

Denna avhandling studerar en ny regressionsmetod, Geografically Weighted Regression (GWR) för att förutsäga antal resenärer på stationsnivå för framtida stationer. Fallstudien av Stockholms blå linje används eftersom nya stationer kommer att byggas år 2030. Denna rapport skrivs på engelska.

Historiskt används linjära regressionsmetoder oberoende av observationens geografiska

placering som den ordinarie Least Square-regressionsmetoden. Med ökningen av GIS-programvaror de senaste decennierna kan geografiskt beroende regression användas och tidigare preliminära studier har visat ett beroende mellan antal resenärer och plats för stationen i nätverket.

GWR-ekvationer för nya stationer bestäms och används för att förutsäga deras respektive

antal resenärer. GIS-data samlades in med hjälp av Geodata och Traffikverket och antal resenärer samt socioekonomiskt relaterat material för basåret 2016 användes för att först fastställa betydande variabler från en grupp kandidater (Arbetare, antal busslinjer och typ av förändring valdes) och för det andra OLS- och GWR-ekvationerna. Betydelsen av båda modellerna jämfördes och OLS-ekvationen användes för att bestämma det hypotetiska antal resenärer för de nya stationerna om de var närvarande 2016. GWR-ekvationerna bestämdes sedan med hjälp av dessa beräknade antal resenärer för dessa nya stationer. Med alla GWR-ekvationer (varje station har sin egen ekvation) förutsades således antal resenärer för de nya stationerna för 2030 med antaganden och planerad, programmerad utveckling runt stationerna (befolkning, lägenhet som ska byggas ...) och jämförs med de officiella förutsägelserna.

Resultaten visar att GWR-metoden generellt sett förutsäger antalet resenärer jämfört med

de officiella antalet resenärer. Många orsaker kan förklara denna överförutsägelse som antaganden om antalet bussar och metoden som följdes för att beräkna antalet arbetare runt varje station.

Tre huvudsakliga slutsatser drogs för denna fallstudie. En huvudsaklig slutsats, specifik för

denna studie och två andra, mer generella, slutsatser härleddes från denna studie. För det första är GWR ett bra förutsägningsverktyg för framtida stationer som ligger nära de flesta tillgängliga stationer. För det andra är GWR en bra förutsägningsmetod för stationer där begränsade förändringar i den framtida miljön kommer att inträffa.

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Résumé

French/ Franska/ Français

Cette thèse étudie une nouvelle méthode de régression, la régression géographiquement pondérée (GWR), pour prédire le nombre de voyageurs au niveau des stations pour de futures stations. L’étude de cas de la ligne bleue de Stockholm est prise vu que de nouvelles stations seront construites d’ici 2030. Cette thèse est rédigée en anglais.

Historiquement, les méthodes de régression linéaire, indépendantes de la localisation géographique de des observations, étaient et sont toujours utilisées comme méthode de régression des moindres carrés ordinaires (OLS). Avec le développement des logiciels SIG au cours des dernières décennies, l’utilisation de régression géographiquement dépendante devient plus accessible et des études préliminaires antérieures ont montré une dépendance entre le nombre de voyageurs et l'emplacement de la station dans le réseau.

Les équations GWR pour les nouvelles stations sont déterminées et utilisées pour prédire leurs nombres de voyageurs respectives. Les données SIG ont été collectées à l’aide de Geodata et de Traffikverket (Autorité des transports). Le nombre de passagers ainsi que les données socio-économiques pour l’année de référence de 2016 ont été utilisés afin de déterminer, en premier lieu, les variables significatives d’un groupe de candidats (travailleurs, nombre de lignes de bus type de changement ont été choisis) et, deuxièmement, les équations de OLS et de GWR. Les valeurs significatives des deux modèles ont été comparées et l'équation OLS a été utilisée afin de déterminer le nombre de voyageurs hypothétique des nouvelles stations si elles étaient présentes en 2016. Les équations GWR ont ensuite été déterminées à l'aide de ce nombre de voyageurs calculé de ces nouvelles stations. Disposant de toutes les équations GWR (chaque station ayant sa propre équation), le nombre de voyageurs des nouvelles stations pour 2030 a donc été prédite à l'aide d'hypothèses et de développements planifiés et programmés autour des stations (population, appartement à construire…) et comparés aux prévisions officielles.

Les résultats montrent que la méthode GWR surestime d’une façon générale le nombre de voyageurs par rapport aux prévisions officielles. Plusieurs raisons peuvent expliquer cette surestimation, telles que les hypothèses émises concernant le nombre d'autobus et la méthode suivie pour calculer le nombre de travailleurs autour de chaque station.

Trois principales conclusions ont été tirées pour cette étude de cas. Une conclusion principale, spécifique à cette étude et deux autres conclusions, plus générales, ont été déduites de cette étude. Premièrement, le GWR est un bon outil de prévision pour les futures stations proches de la plupart des stations actuellement présentes. Deuxièmement, le GWR est une bonne méthode de prévision pour les stations où des changements limités dans l’environnement futur auront lieu.

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Introduction

Background

According to the World Bank in 2018, more and more people are moving to cities leaving behind rural areas. In fact, since 2007, more people live in these cities than in rural areas for the first time in history. Even if the rate of urbanization is decreasing, it has constantly been positive with a value always superior than 1,9% since 1960. More people in cities means a higher number of people moving around during the day leading to an increase to the number of passengers within the public transportation network. SL, the region of Stockholm authority, quantifies this increase to more than 2 million passengers per day in 2013 from 1,6 million in 2003, a 20% increase in merely 10 years.

Figure 1: Diagram illustrates travelers per day by cars and public transportation in thousands (Trafikförvaltningen, n.d.)

Investing for a more efficient and demand-satisfying public transportation network is thus more important than ever. According to the American Public Transportation Association, “every $1 invested in public transportation generates $4 in economic returns” (American Public Transportation Association, 2019). Moreover, funding for public transportation is increasing from a bit over $40 billion to over $70 billion in 20 years in the United States of America.

Figure 2: Total Funding in Public Transportation in the USA (in billions of 2017 dollars)

This continuous increase is met with a growing concern within the transportation authorities, policymakers and even private firms demanding a more precise, detailed and accurate prediction of ridership in order to explain this never-reaching level of investment. Today’s models, mainly lineal regression models, are used in order to predicted ridership or the transit share for a specific region. However, the assumptions previously made in order to ease their use and limit their cost are leading to errors in predictions, creating uncertainties and

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Stockholm is growing – and so is public transport Stockholm County is growing rapidly, in recent years by about 40,000 inhabitants every year. By 2030 the (county’s) population is expected to have increased to about 2.6 million (from just under 2.1 million in 2010). This will increase pressure on public transport services. Roads and railways are already congested, particularly in the central parts of the city and during peak traffic. In-commuting from other counties will also increase and accessibility to public transport will need to be adapted to the changing needs.

PUBLIC TRANSPORT should be perceived as the most attractive form of travel for every-one, including the elderly and travellers with disabilities. It is therefore crucial to the Stockholm region of the future that public transport develop at the same pace, at least, as the population increases and that the entire transport system be planned so as to facilitate public transport’s long-term expansion.

THE COUNTY COUNCIL invests billions in public transport every year. Over the coming years, the county council will be investing more than ever to meet the needs of a growing population. The biggest investments will be made in upgrading the infra-structure but to an increasing degree also new construction and expansion.

MAJOR INVESTMENTS over the next ten-year period include:

• extension of the metro

• the metro’s Red Line

• the Commuter Train programme

• extension of the Roslagen Line.

Public transport’s positive development from 2003 to 2013

The diagram illustrates travel per day by car ( ) and public transport ( ) in thousands.

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1500

2000

2500

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1,900

1,800

1,700

1,600

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2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Year

Thousands

Public transport

Car 1,742

2,017

1,785

1,8481,907

1,663

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risks that are increasingly higher to bear for investors. One of these assumptions is space independency between the observations. In 2000, Fotheringham et al. proved that global regression models estimate a limited number of parameters between the dependent and independent variables with estimated parameters independently determined with regards to spatial characteristics. This lead to a huge disadvantage of this models with regards to observations that are geographically dependent. In fact, multiple studies like the one by Cordazo et al. in 2012, have proven a high correlation between transit use or ridership and geographic locations of the observation, with high spatial autocorrelation with closer observation having a higher influence than farther ones. This means that when using global, traditional regression models in order to predicted ridership, errors are generated due to the assumptions that observation are geographically independent while they are actually not. As explained above, taking into consideration geographic location of the observation at the time was both time and cost consuming. Today, with the development of GIS, Geographical Information System models and softwares during the last decade, makes these excuses obsolete. These new softwares can acknowledge the geographic locations of the observations while developing a regression model. The Geographically Weighted Regression is one of these new methods that can be used in order to explain and (maybe) predicted ridership.

Objectives and Goals

The objective of this thesis is to determine if Geographically Weighted Regression method can be used in order to predict the station-level ridership at future metro stations.

Three kind of stations are to be compared, stations with different geographic locations compared to the rest of the network and built in different changing environment: Stations close to the existing metro stations of the network that will not experience major changes in their surrounding environment, stations built close to the existing metro station that will, however, experience considerable change to their surrounding environment and, finally, stations that will be built far away from the actual network that will experience big change to their surrounding environment.

The comparison between these three types of station will determine if and/or when is are

predictions possible following the method use in this study. The case of Stockholm is studied here.

Thesis Structure and Flow

This paper is divided into seven parts. First, the literature review talks the way predictions were done until today, while introducing GWR, both theoretically and with regards to previous studies using it. Second, a methodology is presented in which detailed steps performed in this study are described in order to reach the results that are needed. Third, Stockholm as a city and its metro network is presented describing the extension of the metro network. This part also lists the candidate variables as well as the assumptions laid for this study. Fourth, results are presented (the models used, parameters, significance of the models and finally the ridership predictions). Fifth,

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the analysis of these results is done extensively dividing the analysis into general, regional and station-specific. Sixth, the limitations of the studies are described mainly in order to present what should be avoided for future similar studies that are to be performed. Finally, the conclusions are presented both case study specific and generally.

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ABSTRACT 1

English/ Engelska/ Anglais 1

SAMMANFATTNING 2

Swedish/ Svenska/ Suédois 2

RESUME 3

French/ Franska/ Français 3

INTRODUCTION 4

Background 4

Objectives and Goals 5

Thesis Structure and Flow 5

LITERATURE REVIEW 10

Old Forecasting Methods 10

Geographically Weighted Regression 11 Description 11 Difference between Spatial Autocorrelation and Spatial Non-Stationality: Accuracy of the model 14 Spatial Autocorrelation 15 Spatial Non-Stationality 16

Geographically Weighted Regression as Forecasting Method 17 Previous works and fields in use 17 Comparison between OLS and GWR 18 Predictions using GWR compared to other methods 18

METHODOLOGY 21

STOCKHOLM’S CASE STUDY AND DATA COLLECTION 24

Stockholm’s Metro System: Present Situation, Forecasting and Future Development 24 Stockholm today with ridership and population 24 Preliminary studies with Sampers for Stockholm’s new station 25 General idea with map of the planned extension 26 Transit-oriented development: Stockholm Case Study 28

Data and Candidate Variables 29 Line chosen 29 Candidate Variable 29 Prediction assumptions for the candidate variables 32

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RESULTS 32

GWR Equations with Existing Conditions 32

Predictions Using the Determined GWR Equations 41

ANALYSIS 45

Division between the North and the South 45

The Model in Numbers 45

General Reasons 48

Bus Assumptions and Effect on Predictions 48

Special Case of Sofia 48

LIMITATIONS OF THE STUDY 49

CONCLUSION 51

REFERENCES 52

APPENDIX 56

Appendix I: Method and Tools in Determining Data in ArcGIS (ArcMap) 56 Income, Workers, Population and Age 56 Road density (m/m2) 56 Number of bus lines at a 200-meter buffer around the entrances of the metro 56 Terminal Station 56 Type of change 57 Commuting distance 57

Appendix II: Table of GWR Equations and Predictions for the 2016 Situation 58

Appendix III: Table of GWR Equations and Predictions for the 2016 Situation with New Stations 73

Appendix IV: GWR Prediction by ArcGIS (ArcMap) 89

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FIGURE 1: DIAGRAM ILLUSTRATES TRAVELERS PER DAY BY CARS AND PUBLIC TRANSPORTATION IN THOUSANDS (TRAFIKFÖRVALTNINGEN, N.D.) 4

FIGURE 2: TOTAL FUNDING IN PUBLIC TRANSPORTATION IN THE USA (IN BILLIONS OF 2017 DOLLARS) 4 FIGURE 3:TRADITIONAL FOUR-STEP TRANSPORT MODEL (ADAPTED FROM WHITEHEAD & BUTTON, 1977, P.117) 10 FIGURE 4: DIFFERENT TYPES OF THE KERNEL FUNCTION (INSTITUT NATIONAL DE LA STATISTIQUE ET DES ETUDES

ECONOMIQUES, 2018) 13 FIGURE 5:100-METER SQUARES IN RENNES, SAMPLED IN RED (FLOCH, 2015) 20 FIGURE 6: BOX-PLOT OF THE RCEQMR FOR THE HORWITZ-THOMPSON (1), REPRESSION (2) AND GWR (3)

ESTIMATORS (FLOCH, 2016) 21 FIGURE 7: CATCHMENT AREA (SERVICE AREAS) FOR EACH STATION (OLD STATIONS IN BEIGE, NEW STATIONS IN

PURPLE). 22 FIGURE 8: SERVICE AREA FOR THE BLUE LINE STATION TO BE PREDICTED FOR 2030 24 FIGURE 9: METRO NETWORK WITH THE ADDITIONAL STATIONS IN DASHED LINE. 26 FIGURE 10: PROJECTED RIDERSHIP ON THE BLUE LINE DURING THE MORNING PEAK HOUR WITH A FOUR-MINUTE

HEADWAY (NYLÉN, 2017; HARDERS AND BJÖRKMAN, 2016) 27 FIGURE 11: PREDICTED RIDERSHIP ON THE YELLOW LINE DURING THE MORNING PEAK HOUR BY 2030. 28 FIGURE 12: RIDERSHIP VS. NUMBER OF WORKERS 36 FIGURE 13: OLS (UP) AND GWR (DOWN) STANDARD DEVIATIONS 37 FIGURE 14: GWR STANDARD DEVIATION WITH NEW STATIONS 40 FIGURE 15: THE DISTRIBUTION OF THE INTERCEPT OVER THE STUDIED AREA (FROM THE LOWEST IN BLUE TO THE

HIGHEST IN RED; THIS APPLIES TO ALL DISTRIBUTIONS TO FOLLOW) 45 FIGURE 16: THE DISTRIBUTION OF THE WORKER’S COEFFICIENTS OVER THE STUDIED AREA 46 FIGURE 17: THE DISTRIBUTION OF THE BUS’S COEFFICIENTS OVER THE STUDIED AREA 47 FIGURE 18: THE DISTRIBUTION OF THE CHANGE’S COEFFICIENTS OVER THE STUDIED AREA 47 TABLE 1: EQUIVALENT PRESENT STATIONS FOR FUTURE STATIONS (WSP ANALYS & STRATEGI, 2013) ..................... 26 TABLE 2: CANDIDATE VARIABLES EVALUATION AND SELECTION ................................................................................ 32 TABLE 3: SUMMARY OF MULTICOLLINEARITY ............................................................................................................. 33 TABLE 4: PERCENTAGE OF SEARCH CRITERIA PASSED ................................................................................................. 34 TABLE 5: MORAN’S I TESTS ON THE DEPENDENT AND CANDIDATE VARIABLES ......................................................... 34 TABLE 6: OLS EQUATION ............................................................................................................................................. 38 TABLE 7: GWR EQUATIONS ......................................................................................................................................... 38 TABLE 8: PREDICTED RIDERSHIP USING OLS EQUATION FOR NEW STATION IN 2016 ................................................ 39 TABLE 9: GWR EQUATIONS WITH THE NEW STATIONS ............................................................................................... 40 TABLE 10: NUMBER OF ADDITIONAL POPULATION AND WORKERS BY 2030 ............................................................. 41 TABLE 11: COEFFICIENTS AND GWR ESTIMATIONS FOR NEW STATIONS ................................................................... 43 TABLE 12: NUMBER OF ADDITIONAL POPULATION AND WORKERS BY 2030 FOR BARKARBYSTADEN AND BARKARBY

STATION ............................................................................................................................................................. 44 TABLE 13: COEFFICIENTS AND GWR ESTIMATIONS FOR NEW STATIONS AFTER THE CHANGE IN WORKERS ............. 44

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Literature Review

Having efficient and reliable ridership estimation is important for all stakeholders. Passengers can plan their trips by choosing confidently the time and route of their choice will be sure of their time of arrival to their destination. It can also create a routine when it comes to regular trips, mainly work-bound trips in the morning, increasing adequate planning and thus efficiency and productivity. Transit operators can plan efficiently for the needed capacities and frequencies by securing funds early on while spending them in the required areas and departments. Public authorities and operators can forecast resourcefully the funds needed for the future, the dispatching and evolution of jobs and population on the interested region as well as implementing policies and strategies in order to lead the region towards a more sustainable future.

Old Forecasting Methods

Transit ridership are usually estimated using comparison methods to equivalent situations, professional and elasticity analysis and travel demand models (Litman, 2004; Boyle, 2006). The first models are typically employed for route evaluation while the latter is used in assessing new amenities providing transit ridership only as a part of the prediction with no particular focus on transit ridership and public transportation travel that is treated as another mode (Zhang & Wang, 2014).

Transport forecasting and modelling took a serious turn in the fifties’ when the four-step

model, a method that predicts traffic patterns at an aggregate level, was created (Horowitz, 1984). It has since been the dominant model for transport modelling and was adopted for transit ridership as well over the years (McNally, 2007). The four-step model is characteristic by its four step process by first generating the demand for travel in specific region, second distributing this demand by creating Origin-Destination region pairs, third assigning a mode of transport the travelers are going to use (public transport, car, walking, biking, etc.), and finally by assigning a specific route to each trip.

Figure 3:Traditional four-step transport model (adapted from Whitehead & Button, 1977, p.117)

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Activity-based models are used for forecasting and prediction. This method focuses on predicting individual travel behavior at a disaggregated level (Hildebrand, 2003).

However, these overall travel demand forecasting methods require a huge number of

surveys, data collection and processing. These are only a couple of reasons why this method is costly to implement and maintain (Marshall & Grady, 2006). They also fail to capture subtle land-use characteristics in specific areas that might influence ridership more or less than in the other region (Cervero, 2006). For these reasons, other methods, such as regression models, were developed in order to have efficient and reliable forecasts and predictions of transit ridership. They are also faster and cheaper to develop. Creating a straight relationship between a couple of predefined factors (independent variables) with transit ridership (dependent variable), regression models are easy to use with less trouble in defining them providing a rapid alternative. According to consulted papers, these predefined factors are usually grouped in 4 categories: Demographic features, socio-economic indexes, land-use arrangement, geographic information.

Nevertheless, most current regression models accept spatial-independence in ridership estimation. The problem with this assumption is the fact that many (if not all) factors are spatially correlated (Zhang & Wang, 2014). New methods must then be utilized to acknowledge this spatial dependency between the different observations.

An Ordinary Least Squared regression (OLS) is a regression method that determines the

parameters of the linear regression model by minimizing the square of the errors. The following equation summarizes this regression.

Geographically Weighted Regression

In the following section, the technical information of the geographically weighted regression is described as well as the different ways to evaluate its significance. Description

Global calibrating models, using all the observations provided for a concerned region, predict global estimates for the whole interested region, while local models, using a handful of observation like GWR predict local models that for each interested observation. The main difference between the two methods is that the first emphasizes on the spatial similarities while the latter emphasizes on the spatial differences in the interested region.

As a reminder, the OLS equation is presented below:

! = # + #1'( + #2'*+. . . +#,'- +./

Where y is the dependent observed variable, xj is the j-th independent observations,

variables or predictors (j = 1, ..., p), βj is the j-th model parameters to be estimated (j = 0, 1, ..., p) and epsilon y is the error at for observation y. An Ordinary Least Squared regression (OLS) is a

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regression method that determines the parameters of the linear regression model by minimizing the square of the errors. The following equation summarizes this regression. It is important to note that OLS the βj are the same for all the studied region and do not change with space or dependent observation.

In fact, traditional global regression models, like OLS, assume that the whole studied

region can be explained and predicted using one common equation with common parameters on the whole region. This rational is easily discredited by local regression models, like GWR (Fotheringham, Brunsdon & Charlton, 2002). This has been explained in section SOMETHING with case study examples from previous studies.

An advantage of GWR over other spatial methods, like multi-level modelling is that each

calibration yields equations for each observation where each one of them is treated independently from the others, capturing geographic heterogeneity (Zhang et Wang, 2014).

The main idea behind the GWR method is the fact that each point i to be predicted is

surrounded by an area of influence that decreases the farther the sampled observations are from point i, thus creating as many regression equations as there are observations to predicted. This is done by incorporating the geographical coordinates of each observation in its equation.

As stated before, GWR is inspired by OLS. OLS can be, actually, seen as an exception of

GWR where all function is constant over space. Indeed, GWR uses a weighted least squares method to predict the parameters (Fotheringham and Charlton, 1998). The following function summarizes the GWR model:

!0 = #1(30, 50) +78

#8(30, 50)'08 + .0

Where !0 is the dependent observed variable at location (ui,vi), #1(30, 50) is the intercept parameter at location (ui,vi), #8(30, 50) are the independent parameters for observation (ui,vi), '08 are the observation k at (ui,vi) and .0 is the error term for observation (ui,vi). Beta best is thus equivalent to this equation:

#9(:) = (;<=(30, 50);)>(;<=(30, 50)? Where Wi (uivi) is the weighting function. This weighting function is spatially dependent providing a weight for the observation depending on its location from (and thus distance to) the observation that is to be predicted. It can be represented in a diagonal matrix where the primary diagonal line represents the weight function at location i (Fotheringham et al, 1998). Here is the weight matrix used:

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=(:) = @

A0( 0 … 00 A0* … 00 0 … 00 0 … A0D

E

Where A0F(G = 1, 2, … , H) is the weight given at location j

Coming back to the fact that OLS is an exception of GWR, it is clearer now that the weight function is introduced. In fact, one can assume that the weight function is equal to 1 for all points in the studied area.

These functions vary depending on the predicted information. There are several options from Gaussian, exponential, bi-square and kernel just to name the most used ones.

One can differentiate between continuous weight functions where a weighting value is given to each observation in the studied area from weight function with compact support where the latter tends to a zero-value reached at the determined bandwidth value and assigned to observations having a distance greater than this bandwidth (Institut national de la statistique et des études économiques, 2018). However, according to Brunsdon, Fotheringham, and Charlton in 1998, there are the choice weight function has no significant effect on the results.

Figure 4: Different types of the kernel function (Institut national de la statistique et des études économiques, 2018)

The following curves are written explicitly in the following equations, respectively uniform kernel, Gaussian kernel and Exponential kernel.

AIJ0FK = 1

AIJ0FK = L>(*(MNOP )Q

AIJ0FK = L>(*(RMNORP )

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Another way to differentiate them is by classifying them in either a fixed or adaptive weight function. The main difference between the two is observation density and sample size and whether the bandwidth is constant or variable. The first kind determines the spread of the weight function according to a fixed distance (bandwidth) identical in the whole studied area to be used when one has high density of observation and sample points. The second determined the spread of the weight function according to the number of neighboring observations a point of interest has led to a greater spread when the density is low and varying the distance (bandwidth) (Fotheringham et al., 2002). The changing function can also determine the optimal bandwidth for highly dense observations. In fact, this optimal value of bandwidth, that is a variable in the weight function equations with compact support, can be determined in other ways for a stated weighting function. The bandwidth value has the biggest influence on the results (Institut national de la statistique et des études économiques, 2018). Examples of these data-driven criteria are cross-validation (CV), generalized cross-validation (GCV), Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The most in use are CV and AIC and are, thus, presented here (Fotheringham et al., 2002).

ST =7

D

0U(

[!0 − !XY0(ℎ)]*

Where !XY0(ℎ) is the value of y at i predicted where developing the model with all observation except !0. The optimum value of the bandwidth would be 0 if all the observations are used to estimate the model, meaning that the only available point in the model is !0, leading to !0 = !X. Generally, the bandwidth that minimizes CV is the one that maximizes the predictive capacity of the model.

\]S(ℎ) = 2H ^H ^H(_X) + H ^H ^H(2`) + H aH + (b)

H − 2 − (b)c

Where n is the sample size, _X the estimate of the standard deviation of the error term, (b) the trace of the projection matrix of the observed variable y on the estimated variable !X.

When using one of these two statistical criteria, the bandwidth is determined when minimizing their values. The main difference between the two is that CV maximizes the predictive power of the model while AIC compromises between this predictive power and the model’s complexity. The weaker the bandwidth, the more the global model is complex. In general, AIC determines larger bandwidth than CV. Difference between Spatial Autocorrelation and Spatial Non-Stationality: Accuracy of the model

The estimated GWR model needs to be evaluated and diagnosed passing by the same process of global models.

The estimated GWR can be evaluated thanks to coefficient of determination (R-Square), t-values and p-values.

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d* = 1 −∑0 .*

∑0 (!0 − !)*

Where . is the residual between the observed value and the predicted one by the model, !0 is the observed value i (or at location i if dealing with GWR) and ! is the average of the observed values. As stated before, the Akaike Information Criterion (AIC) can determine the optimal bandwidth for a given weighting function in the studied area. However, AIC can also evaluate the goodness-of-fit of the model. In general, an AIC value greater than 3 suggest a good fit of the model (Fotheringham et al, 2002).

In addition to these estimations and because local statistics are considered spatially disaggregated compared to global models, new evaluation statistics must also be performed to evaluate the model for local characteristics.

These characteristics can be classified into two categories: spatial autocorrelation and spatial

non-stationality (Anselin, 1999). They have been challenging issues to deal with according to Fotheringham in 2002 and GWR permits to consider while considering the coordinates of the observation for spatial autocorrelation when calculating the intercept and spatial non-stationality when estimating the parameters. On one hand, spatial autocorrelation refers to an interaction in space, in other words the value of a certain variable compared to the value of its neighbours. On the other hand, spatial non-stationality refers to the structure in space (Anselin, 1999).

Spatial Autocorrelation

Spatial autocorrelation can be tested using the error of the GWR model created. In fact, GWR

hypothesis that the error terms are identically distributed. Thus, a test for validating or not this independence of the distribution is the establishment of a hypothesis test (Leung, Mei and Zhang, 2000).

Ho: No spatial autocorrelation among the disturbance.

H1: There exists either positive or negative spatial autocorrelation among the disturbances with respect to a specific spatial weight matrix W.

In order to accept or reject the null hypothesis, the test statistics Moran’s I is used. The values

range between -1 and 1, where 0, theoretically, represent no spatial correlation (Rosenberg, 2010). However, for a defined sample size, the value representing no spatial correlation is -1/(N-1) where N is the number of spatial observations: it is the expected value. This value (or 0) is the expected value of the null hypothesis. Moran’s I is given as follows:

]1 =.̂<g<=g.̂.̂<g<g.̂

16

Where W = a specific symmetric spatial weight matrix of order n; and N

g = ] − h = ] −

⎣⎢⎢⎡;(

<[;<=(1);]>(;<=(1);*<[;<=(2);]>(;<=(2)

…;D<[;<=(H);]>(;<=(H)⎦

⎥⎥⎤

A probability must be thus determined as well in order to accept or reject the null

hypothesis. The equation given below calculates theoretically this probability when the Moran’s I value is less than a given value p (Leung, Mei and Zhang, 2000).

o(]1 ≤ q) =12−1`rs

1

t:H t:H[u(v)]vw(v)

Jv

Where u(v) = (

*∑D0U( xqyvxH xqyvxH(z0v), w(v) = ∏D

0U( (1 + z0*v*)1,*| and z0 =

g<(= − q])g.

This equation can be simplified after multiple assumption to the following expression (Leung, Mei and Zhang, 2000):

r}

1

t:H t:H[u(v)]vw(v)

Jv

Spatial Non-Stationality

For GWR, the dependent and a given independent variable are linked geographically. In other

words, the hypothesis for GWR is that the variables are stationary in a given geographic area. In order to evaluate the efficiency of the model, it might be interesting to test the spatial non-stationality of the variables (Institut national de la statistique et des études économiques, 2018). Technically, the calculation of the variance of the variables for a given variable should be able to give a satisfying answer (Fotheringham et al, 2002):

Txq~#9(:)� = [(;<=(:);)>(;<=(:)][(;<=(:);)>(;<=(:)]<_*

Where _* = ∑0 (!0 − !X0)/(H − 25( + 5*)

However, the theoretical distribution of each variable is unknown leading to a difficulty in

using the above method. Thus, another method, the Monte Carlo Simulation, is used to help reject or accept the null hypothesis of the following hypothesis test:

H0 : ∀k, βk(u1,v1) = βk(u2,v2) = ... = βk(un,vn)

H1 : ∃k, all βk(ui,vi) are not equal.

17

In fact, if there is spatial non-stationality in the studied area, the locations (coordinates) of the observations are irrelevant and changing them will yield the same value of the variance. When dealing with the Monte Carlo Simulation, the geographical coordinates of the observations are permuted n times, finding n spatial variance estimations of the observations. The p-value of the spatial variability of the coefficients is then estimated. This p-value can determine if the null hypothesis should be accepted or rejected (Institut national de la statistique et des études économiques, 2018). Finally, the bandwidth can give an indication of the efficiency and reliability of the model. Its value is very important and when compared with the extent of the studied area can give, even if not precise, important information about the model nay if GWR should be even used in the first place.

On one hand, if the bandwidth yields toward the maximum value possible (over the whole studied area), local autocorrelation and spatial stationality are weak and GWR should not be used. On the other hand, if the bandwidth is really small, it is important to check for randomness in the process (Gollini et al, 2015).

Geographically Weighted Regression as Forecasting Method

Geographically Weighted Regression or GWR, developed by Fotheringham and Charlton in 1998, is a new regression model inspired by the usual Ordinary Least Square (OLS) method regression that, however, takes into consideration spatial dependency when forecasting equations and its parameters. The authors refer to a family of “spatial adjusted” regression. Previous works and fields in use

Since its introduction as a spatial data analysis in the late 1990’s, GWR has been used in a large number of areas. From health and healthcare (Zhang, Wong, So & Lin, 2012) and forestry (Pineda, Bosque-Sendra, Gómez-Delgado & Franco, 2010) to real estate (Dimopoulos & Moulas, 2016; Institut national de la statistique et des études économiques, 2018), passing by land and urban space use (Luo & Wei, 2009; Tu & Guo, 2008) and poverty rates (Floch, 2016), GWR has been more and more present in transport science, mainly as an explanatory method.

The field of transportation was no exception. Determining the explanatory variables in order to identify potential causes and relations with transportation related issue is the main use today of this relatively new technique. Traffic accidents, average commute distances, transport-land use interaction and influence and public transport share (Chow et al, 2006) are only a couple of fields that were tackled by GWR over the years. In 2015, Qian and Ukkusuri evaluated after a comparison of the performances of the Ordinary Least Square (OLS) and GWR, the causes behind the taxi ridership in New-York city. Liu, Ji, Shi and Gao presented a research on the effect of the built environment on student’s metro commuting to their schools and back home in Nanjing, China.

However, ridership forecasting on a station-based level is not developed enough when it

comes to utilising the GWR method. A couple of preliminary studies in this field were, nevertheless, presented having mainly as the core subject the comparison of the level of efficiency and reliability between OLS and GWR. In 2015, Chiou, Jou and Yang determined the predictors

18

for ridership data for the state of Taiwan with the help of both OLS and GWR. Another study conducted by Blainey and Mulley in 2013 also determines the predictors for ridership data for railway stations in the Sydney region of New South Wales by also comparing both OLS and GWR methods. Studies in Madrid, Spain, Sydney, Australia and Adelaide, Australia conduct similar studies by examining the local characteristics of their respectable studied areas. They also predict some ridership data with the models that they created (Cardozo, García-Palomares, Javier Gutiérrez, 2012; Somenahalli, 2011; Blainey, Mulley, 2013). Comparison between OLS and GWR

As stated above, global regression models, like OLS, assumes that the relationship between the dependent and independent variables are uniform over the study area, ignoring spatial characteristics such as distances to stations. OLS does not consider variations due to spatial autocorrelation (Fotheringham, Brunsdon and Charlton, 2000; Lloyd and Shuttleworth, 2005; Cardozo, 2012). In 2010, Harris et al. compared multiple methods, mainly variations of Kriging, multiple regression methods and GWR and concluded that GWR-based models out-performed MLR models.

Previous studies that have focused on comparing OLS and GWR methods have frequently

discovered that GWR provides more predictability than GWR due to this concern for spatial correlation.

In fact, according to Hadayeghi, Shalaby and Persaud, estimation errors are smaller in a

majority of cases when using GWR compared to OLS. They explain this result by claiming that the problem of spatial autocorrelation is reduced nay eliminated. This is the case for Cardozo, García-Palomares and Javier Gutiérrez when, in 2012 they compare station-level ridership forecasting between the OLS and GWR methods. Their analysis of the residuals proved better results with GWR than with OLS. In addition, when comparing Moran’s I values that were calculated, the one generated for GWR was closer to the expected value than the OLS one, concluding that spatial autocorrelation was reduced. In this same study, GWR performed better with p-values and z-values, describing less variance and greater likelihood for random distribution.

Another study, already presented above, compares a global regression model with GWR

in Taiwan and found an adjusted R-Squared for GWR more than 0,2 units higher than the traditional MLR model in addition to a better performance with regards to spatial autocorrelation for the GWR model (Chiou, Jou and Yang, 2015). Zhao et al. also, in 2005, found a better prediction performance with GWR with regards to OLS for the county of Broward, Florida.

Predictions using GWR compared to other methods

With the establishment of the formulas and equations for each observation, one can use the

calculated coefficients to analyze how relationships vary across the studied area and study any possible patterns they might create. These coefficients can provide local understanding of the observed dependent variable (Fotheringham et al, 2002). In fact, these coefficients, seeing that they are localized, can provide information on the influence of changes in the station’s direct

19

environment will have on the dependent variable, in this case station-level ridership. Any evolution of population, jobs or even land use can provide detailed and specific information for the station.

This leads to a more accurate and detailed forecast, compared to a generalized forecast with

standardized coefficients for the whole studied region when dealing with global regression models (Lloyd, 2010). The model(s) can be evaluated over the whole studied area, determining the areas with better fits, variations of estimated coefficient and significance.

Any future development, such as new residential buildings or workplaces can be thus

evaluated independently for each station environment and area. In fact, for example, an increase in the number of jobs in one area can lead to an increase in ridership while in another it can have the opposite effect. This is where localized policies and plans, land-use for example, come in action and can be implemented, with more insurance and less risks with these more realistic predictions, in the area in order to improve ridership and/or the level of service of the station and/or public transport network. De Smith et al. (2009) call in “place-based” techniques.

GWR can also be used in predicting models in areas where dependent variable observations

are not available. The use of approaches based on the use of “Best Linear Unbiased Predictors (BLUP) estimators is more and more common nowadays (Chambers and Clark, 2012). This method is based on the replacement of non-observed dependent variables by predicted values thanks to a model where the parameters are estimated from observed dependent variables. Recent literature seems to prefer the use of GWR in estimation methods rather than methods originated from other methods. The fact that GWR considers spatial heterogeneity is regarded as theoretically improve the precision of the estimators.

Floch presented in 2016 at the JMS (Journée de Méthodologie Statistique) a study of

Rennes’ Iris zones where they wanted to predict the number of households with low incomes in areas of the city that was not observed. For that reason, and having the number of households beneficiation from free health care due to low incomes in all the studied area, they calculated first the GWR models for all the observed Iris zones determining the equations.

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Figure 5:100-meter squares in Rennes, sampled in red (Floch, 2015)

Each Iris zone is divided in 100-meter squares. Presenting the notation used, y represents the number of people with low income, x is the number of people having free health care. In U, all squares (N=2141) are assigned their coordinates, the number of low-income people, the number of people with free health care and the Iris it belongs to. A sample s of n/N=40% is selected. r is the complement of s in U where all yi’s are known in i ∈ s and xi are known for all squares in i ∈ U.

In order to determine the predicted number of households with low incomes, three different estimation methods (Horvitz-Thompson, the classic regression one and the GWR one respectively) were calculated.

Determining K=1000 estimators by Iris, the relative quadric average error of Monte Carlo

estimation is calculated before calculating its square root (RCEQMR).

ÑÖÜ áv/à (G)â = ä>(7ã

8U(

(v/à (G)8 − v/(G))*

dSÑÖÜd áv/à (G)â = åÑÖÜáv/à (G)â

v/(G)

Where v/à (G)8 is the estimator for the total of the variable y of the Iris j and the simulation k.

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Figure 6: Box-Plot of the RCEQMR for the Horwitz-Thompson (1), repression (2) and GWR (3) estimators (Floch, 2016)

One can realize that the best outcome was the GWR estimation. In fact, with a RCEQMR of 0.4, the Horwitz-Thompson estimator was the least performing of the three. The RCEQMR of both the “classic” regression and the GWR estimators are quite similar with 0,178 and 0,156 respectively. This is where the box-plot gives more information about these two RCEQMR showing a smaller spread regarding GWR.

It can be thus safely said that the results showed the GWR estimator as the best compared to the other methods.

Methodology A precise and extensive methodology was developed in order to determine the GWR

equations for each station, for present as well as future conditions. The software used is ArcGIS (ArcMap).

The first step is to choose a line to investigate thoroughly. In fact, even if evaluating the

whole network would also yield interesting findings, limited time and resources constrained to selecting one specific line.

The next step is to gather the data needed to produce the analysis, required by the candidate

variable such as population, income, age, type of station and distance to the center of the network. Having gathered all the data needed to create both OLS and GWR equations, catchment

areas (or service areas) around each metro station were defined. In fact, a catchment area is considered as the area around the station where the walking distance to the station is 800 meters or less, representing 10 minutes or less of walking time. This number was determined after previous literature and research review. It was evaluated that a station’s “neighborhood” or the

22

willingness distance to walk to and from a rail (and thus a metro) station is 800 meters (O’Neill et al., 1992; Hsiao et al., 1997; Murray, 2001; Zhao et al., 2003; Kuby et al., 2004; Sallis, 2008; Gutiérrez et al., 2011). In 2008, Gutiérrez et al. proved that network distance provides better estimates than Euclidean distances. Following this reasoning, determining the 800-meter catchment area following the road network around the respective stations was executed. It was also proven that riders are more willing to walk a larger distance at end stations.

Figure 7: Catchment Area (Service areas) for each station (old stations in beige, new stations in purple).

New station names are also presented on this map.

T-Centralen is present here and on all following map by a black point for orientation.

At some locations, mainly in the central area of the urbanized region and of the network,

the dense metro network lead to stations being situated at less than 1600 meters, meaning that an overlap in catchment areas is unavoidable. Following the reasoning described above, possible riders have more than one station to choose from within walking distance in order to travel. In order to avoid double counting when determining the equations and forecasting the ridership and following the steps of various previous studies, Thiessen polygons as catchment areas were generated for these special cases, meaning that possible riders always choose the closest stations at walking distance (Cardozo et al., 2012; Zhang and Wang, 2014). This method is also reinforced by Wardman’s 2004 study in Manhattan that suggest that riders are more likely to choose the closest station due to a higher out-of-vehicle value a time compared with in-vehicle value of time.

The next step is to determine the significant variables from the previously presented

candidate variables as well as the best OLS equation. In fact, ArcGIS permits the evaluation candidate variables by creating OLS regression equation. The most significant equation is chosen in order to proceed with the analysis. The choice is made after comparing the significant indicators such as p-value, R-Squared, the variance inflation factor and the global Moran’s I p-value. Moran’s

23

I is also performed individually on each candidate variable determining the most significant variables with regards to spatial correlation.

Using ArcGIS, the now determined OLS equation is used to forecast ridership for the whole

network, at the future stations that are being built, as if they were existent today. Even if these ridership computations are completely hypothetical, they are necessary in determining GWR equations for these stations (and all station more generally).

In order to formulate GWR equations, the weight function must be determined. Multiple

functions are adequate for the job, however, according to Zhao et al. in 2005, adaptive kernel does not have a limited number of observations meaning an advantage for observation on the limits of the study area.

Having the knowledge of all dependent and independent variables on the whole system for 2016 including the hypothetical ridership for the new stations on the blue line as well as the most adequate weighted function for the presented situation, GWR equations can be computed by ArcGIS for all stations on the blue line for current conditions. These different equations are then evaluated and compared with regards to significance and goodness-of-fit using R2, AIC, p-value and t-test methods with regards to the previously computed OLS equation.

Having the GWR equations for the whole network, the chosen line with its new stations is

isolated and updated with the predicted data for the time of prediction in question. Depending on the variables that were chosen, the data needed would be different. Some data might be lacking. Some assumptions can thus be made in order to remedy this. Population data, when detailed estimations are lacking, can be determined by multiplying the official number of apartments planned to be built by the average number of persons per household. When this is also lacking, the projected evolution of the population in the region can be used for the service areas. Regarding workers, they can be obtained by multiplying the population by a ratio Population to Workers determined by the stations’ equivalent stations (stations that are present today in the network with similar characteristics to a future stations). Regarding the median income, a trend line is determined from past statistics and interpolated into the time of prediction. The change in land use incorporated and updated to the state of the time of prediction should also be considered. Other information is assumed to remain unchanged from today.

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Figure 8: Service Area for the Blue Line Station to be predicted for 2030

Having determined the ridership according to GWR models, ridership is compared to official forecasts and analyzed.

Finally, each new station is evaluated with regards to its GWR model and the ridership it

forecasted with regards to the official forecast. The determined parameters can help determine policies that can be proposed. For instance, the choice of building new apartments in specific areas can also be analyzed and other areas, not considered, can be proposed.

Stockholm’s Case Study and Data Collection The following section presents the present metro situation in Stockholm as well as the plans for a future extension of this current network. It also presents the data collected through candidate variable that can explain the ridership for current and future states.

Stockholm’s Metro System: Present Situation, Forecasting and Future Development

Stockholm’s metro network developed over the years starting in the 1950’s to become a complex start network with T-Centralen in its core. Forecasting for the ridership has been done using a special model, Sampers, developed by Trafikverket, the transportation authority in Sweden. This model has been used in preliminary studies to forecast the ridership on stations and lines to be constructed by 2030.

Stockholm today with ridership and population

In the 1940s, Sweden decided to build a metro network in its capital, even if, at the time, the number of inhabitants of Stockholm did not technically require, this new form of public

25

transportation. The green line in the 50s, the red one in the 60s and the blue one in the 70s were constructed with gradual extensions over time.

Today the system welcomes almost a million passengers every day (MTR, 2018). This

number is expected to increase by 170 000 in 2030 (Stockholms läns landsting, 2016). In fact, Stockholm is expected to welcome between 30 000 to 35 000 new residents every year until 2030, making it the fastest-growing European capital. The last seven years saw a growth by 250 000 people in Stockholm county. The current capacity of the network would not sustain this influx of new residents.

In addition, Stockholm is facing, with the rest of Sweden, a housing scarcity problem. The

official rent-controlled queue has around 500 000 people waiting in line for an accommodation. The average waiting time is nine years with some neighborhoods having an average time up to 20 years. For this reason, Stockholm’s city council is backing the construction of 40 000 new permanent homes by 2020 and 100 000 more by 2030. Movable modular homes and a big co-living space for global entrepreneurs are also considered to be adopted to ease the housing problem (Savage, 2016).

For these reasons, Stockholm has decided to expand and enlarge its public transportation

network, namely its metro network, constructing new housing next to these new stations. Preliminary studies with Sampers for Stockholm’s new station

Preliminary studies were done starting from 2007 and were updated continuously with the progression and direction the metro expansion project took.

The forecasting part and projection of ridership for the new completed network by 2030,

including these new stations were determined thanks to Sweden’s national model system, Sampers. According to Trafikverket’s website, updated in 2018, it uses a cross-sectional analysis for determining future passengers and traffic volumes for different scenarios. Its main variables are GDP, fuel prices, employment and population growth. In addition to forecasting new ridership or/traffic flows, Sampers provides impact assessments and investment calculations for land-use or transport changes such as new residential project or infrastructure project for possible transport policy measures.

According to Prognos över resandeutveckling for both the extension of the blue line and

the construction of the new yellow line, the traffic analysis for 2030 was executed using both PTV Visum and Sampers. Ridership was thus found using these methods. However, the report presenting the results for the yellow line presents limits for PTV Visum. In fact, “VISUM is a generalizing model that includes the whole Stockholm län and whose strength is, first, providing a general analysis”. It continues in warning that detailed analysis of the results should be done in caution.

Equivalent present stations to potential stations during the preliminary analysis has been

determined by WSP in its Effekter på värdet på handelsfastigheter vid etablering av nya

26

tunnelbanelinjer i Stockholmsregionen report. The following table presents the equivalent station of the selected stations of the blue line:

Table 1: Equivalent Present Stations for Future Stations (WSP Analys & Strategi, 2013)

Station Equivalent Station

Station Equivalent Station

Kungträdgården (New)

Hötorget Järla Västra Skogen

Sofia Mariatorget Nacka Solna Centrum

Hammerby Kannal

Alvik Barkerbystaden Farsta strand

Sickla Järla Barkerby Station

Farsta Strand

General idea with map of the planned extension

As explained above, the new stations will be built on two different lines: the blue line and

a new yellow line.

Figure 9: Metro network with the additional stations in dashed line.

Notice that the current arm of the green line towards Hagsätra will be integrated to the blue line by 2030 (Stockholms läns landsting, 2016)

On one hand, the blue line, already existent, is facing an extension from both sides. In the

north, a small extension will see the line grow by two stations after Akalla, Barkarbystaden and Barkarby. It is in the south that the extension is going to be significant with the addition of five new stations: Sofia, Hammarby kanal, Sickla, Järla and Nacka. In addition, at Sofia, the blue line

27

is splitting in two different branches, one that goes until Nacka passing by all the named stations and another that is connecting to the present green branch to Hagsätra at Gullmarsplan, turning it blue. The new station, Slackhusetområdet, is replacing the present Globen and Enskede gård stations (Nylén, 2017). The below figure XX presents the forecasted ridership on the whole line, with an obvious emphasis on the new parts of the line.

Figure 10: Projected ridership on the blue line during the morning peak hour with a four-minute headway (Nylén, 2017; Harders

and Björkman, 2016)

On the other hand, the yellow line is going to be built from scratch from Odenplan to Arenastaden, with two stations between them, Hagastaden and Södra Hagalund. After Odenplan the line is joining the green line continuing to either Farsta Strand or Snarpnäck. The figure below (Figure XX) present the ridership between Odenplan and Arenastaden. It is important to note however that the ridership presented here was determined with the assumption that Odenplan is the terminal station and with a line having only three stations instead of the current four (Harders and Björkman, 2016). In fact, the decision to change the initial assumptions was made in 2017.

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Figure 11: Predicted ridership on the yellow line during the morning peak hour by 2030.

The upper graph represents the ridership heading towards Odenplan and the lower one heading towards Arenastaden. Legend: Dark red: Boarding with a five-minute headway, Pink: Boarding with a 10-minute headway, Dark blue: Alighting with a five

minute headway, Light blue: Alighting with a 10 minute headway, Bold line: Load for a 5 minute headway, Light line: Load for a 10 minute headway (Harders and Björkman, 2016)

Transit-oriented development: Stockholm Case Study There have been multiple studies drawing a connection between transit use and Transit-Oriented Development (TOD). In Stockholm, the planned construction of multiple residences as well as workplaces around not only the new stations but current stations show the interest of Stockholm’s policy makers in TOD. By definition, TOS is the development of urbanized neighborhood where transit is easily accessible. Mixed land use as well as pedestrian oriented mobility in such regions is a core aspect in addition to a densely designed roads and buildings (Zhao et al., 2005).

Even if no clear conclusion can be drawn, some studies, like the one done by Parker et al in 2002, show transit ridership can be increased by up to 40% at individual stations after TOD was implemented around them. A case study of Portland asserts this finding after a survey in 1994 concluded that transit share is higher and car ownership lower in TOD in comparison to traditionally developed neighborhoods (Lawton, 1997).

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With regards to Stockholm län a goal of 140 000 new residences by 2030 is shared among the different municipalities, mainly Nacka, Järfalla and Solna as the latter will have the new stations built within their municipal boundaries. A total of 78 000 of these new residences will be built around these stations (Stockholms läns landsting, 2016). For example, around the future Hammerby Kannal station, around 2140 residences and a total of 73 000 square meters of new locals and offices will be built (Stockholm växer, 2018).

However, in order to strengthen and increase the share of transit in the region, existing

stations like Kista, Rinkeby and Kristineberg will also see a densification of their neighborhoods with additional residences and workplaces. As an example, the area around Kista will see the construction of a 1600 new residential building and a couple of new office buildings.

These projects are intended to densify and diversify, if not already present TOD’s, potential

one in order to increase the share of transit in the whole Stockholm region in general and more specifically, lead to relatively high transit use around future stations.

This is where GWR comes in. In fact, according to Somenahalli, in 2011, GWR were better

in developing the relationship between transit use and TOD’s.

Data and Candidate Variables

Line chosen

The blue line was selected to be investigated in this case. There are mainly two reasons behind this bias. On one hand, the fact that the blue line is indeed expanding greatly, will thus allow the careful analysis and discussion over both old and new stations, discarding both green and red lines. On the other hand, even if one can consider the newly constructed yellow line as part of the green line, the preliminary studies were conducted as if the line would end at Odenplan with a missing station leaving behind problematic results for this study. The complexity and uncertainty of the newly constructed green line regarding, for example, headway, the number of stations and the number of branches were also additional reasons that favored the use of the blue line for this study.

All stations, as explained in the methodology were assigned an 800-meter service area

each, except for terminal stations, in this case, Nacka Barkarby Station and Hjulsta where they were assigned a 1000-meter service area each.

Candidate Variable

The choice of candidate variables was established thanks to studies while insuring the logic

with the case in hand, i.e. Stockholm. The dependent variable will be the number of boarding passengers at each station during

the morning peak hour, meaning between 7:30 and 8:30. This criterion was selected as a simple goal to be able to compare predicted data from the model developed here with official predictions. The present (2016) data was available in the yearly published report by SL, AB Storstockholms

30

Localtrafik: SL och ländet 2016. This data is the base for the OLS and GWR equations developed for forecasting and evaluation for 2030 situation.

Multiple independent variables were chosen in order to explain the ridership starting with

the socio-economic factors and ending with accessibility ones. Population, income, age distribution and workers for 2016 were all acquired from the

Läntmateriet and GeoData Portal website, respectively the official Swedish authority in gathering statistical and infrastructural information and the website where this information is published. Both these websites make this data public for research purposes. This data sets were already divided into small area, called SAMS in the data. The road network was found on NVDB website. It provided the road network for the whole Stockholm Län. Finally, the metro network for 2030 was provided by Torbjörn Ekerot and Henrik Sarri, respectively an IT-manager from SLL and Metria. The shape file had also the network of the other mode of public transport for the county, such as the bus network and stops as well as the commuter rail (pendeltåg) network and stops. It also assigned each station, present and future, if it is to be considered a significant changing point, a regional one, or not at all. The latter information was taken as a base to determine the type of change that takes place at each station. Unfortunately, detailed land use was only available for current state and only for the municipality of Stockholm. It was thus disregarded as a candidate variable.

The first explanatory variable is the density and size of the population living around the stations. According to Messenger and Ewing in 2007, the relationship between the public transport ridership and population density, even if not direct, exists and is regularly used a factor to justify transportation station expansion or upgrades. Multiple other authors have also asserted the existence of this relationship (Javier Gutiérrez et al, 2011; Sekhar Somenahalli, 2011). In 1996, Seskin et al. presented sufficient evidence to establish a positive relationship between the two.

The second potential explanatory variable is the number of workers around a station. In

fact, even if one can easily hypothesis and deduce it from what was explained with the population explanatory variable, Murray et al. (1998) discovered that the more workers live around a transit service the greater the probability of the latter will be used.

The third potential significant variable is the income of the population respectively to where they live. In fact, it was used in Chow et al., in 2006 and proven to be a significant variable. In addition, an increase of income in specific areas leads to a decrease in transit use for the benefit of the car (Gómez-Ibáñez, 1996; Wachs, 1989; Kitamura, 1989).

The fourth but last socio-economic variable considered is age. In fact, multiple previous

studies (Cristaldi, 2005) consider it and even incorporate it in their respective models like in Bernetti et al, in 2008. Age groups can be created, with the first one between 0 and 19 years old, representing mainly the minor, non-active and unlicenced population, the second between 20 and 64, representing the active population and possibly licenced population and the last third group from 64 onwards representing the retired population. These groups were defined accordingly given the available data for Stockholm and the way the age groups are defined in previous literature.

31

The fifth candidate but first accessibility variable is road density. It was used in multiple previous studies such as Cardozo et al in 2012 and Zhao et al in 2005. In fact, road density can determine to a certain extent the accessibility of the metro station and consequently the number of alternatives to reach this station leading to a shorter and easier reach with high road density.

The sixth one would define the number of bus lines in a 200-meter radius around each

station. A study in 2004 by Kuby et al discovered a connection between feeder modes, for instance bus stops or bus lines accommodating a specific station and ridership at the station in question.

The seventh candidate variable is the type of station itself one is dealing with, mainly with

regards to terminal stations and change and transfer stations. The first type of stations tends to attract residents for larger areas than intermediate stations due to the fact that this station is the closest one to the network inclining riders to walk more than for other stations (O’Sullivan and Morral, 1996). The second type of stations attracts more riders than normal stations (Gutiérrez, Cardozo and García-Palomares, 2011). In fact, be it an interchange station or an intermodal one, they both usually have higher boarding than non-interchange non-intermodal stations. Dummy variables for both these kinds of stations can be used as it was done in Kuby et al, in 2004.

The last accessibility and eighth candidate variable is the commuting distance to the central

business district (CBD) or the central region of the network. In Stockholm case, T-centralen was assumed to be the central point of the network, being the start of the star network system of Stockholm and where all line meet. This was chosen after studying both the papers of Pushkarev and Zupan (1982) and Kuby et al (2004) where it was defined that passengers usually commute to the central part of the network, especially during the morning’s peak hour to reach their places of work.

Finally, the last candidate variable is the land use around the station. According to multiple

studies, such as the ones by Parsons Brinckerhoff in 1996, land use plays a role in transit ridership. In a study by Bhat and Gossen in 2004, an equation was developed in order to quantify the type of land use present in the area of interest, categorising land use in three different groups: residential, commercial/industrial/office and other types.

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The value ranges between 0 and 1 with 0 being no land use diversity and 1 being perfect land use diversity. This equation was used in the study as multinomial logit model variable for the San Francisco area. Prediction assumptions for the candidate variables

When the predictions are made, these assumptions are taken for the following independent variables. Regarding the population data, lacking detailed and precise number of inhabitants around the stations, except for Barkarbystaden and Barkarby Station where the exact number of inhabitants is known, the number of apartments planned to be built by 2030 are multiplied by the average number of persons per household. An increase in the population of 5% and this according to RUFS (Regional utvecklingsplan för Stockholmsregionen, 2010) is done on remaining stations where there is a lack of information with regards to specific population evolution and the number of apartments. Workers also lack detailed predictions. They are thus multiplied by a ratio Population to Workers determined by the stations’ equivalent stations, presented in an earlier part. Regarding the median income, a trend line is determined from past statistics and interpolated into 2030. Other information is assumed to remain unchanged from today.

Results The main results are split into two parts, the first being for present conditions where the GWR equations were determined for the blue line and the second presenting the results and ridership by station for 2030.

GWR Equations with Existing Conditions

The presented candidate variables were analyzed on ArcMap using a tool called Explanatory variables.

Table 2: Candidate variables evaluation and selection

(AdjR2 is Adjusted R-Squared, AICc the Akaike's Information Criterion, p- value the Koenker Statistic p-value, VIF the Max Variance Inflation Factor and the variable’s significance at 0,01 is in yellow)

Number of Variables

Highest AdjR2

AICc p-value

VIF Model

1 of 11 0,33 1552,13 0,01 1,00 Bus

0,23 1565,55 0,10 1,00 Workers

0,22 1566,38 0,09 1,00 Age 2

2 of 11 0,47 1530,64 0,01 1,04 Workers Bus

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0,46 1531,71 0,01 1,04 Age 2 Bus

0,45 1534,35 0,01 1,04 Pop Bus

3 of 11 0,52 1522,11 0,00 1,33 Workers Bus Change

0,51 1524,02 0,00 150,46 Pop Age 2 Bus

0,51 1524,07 0,00 1,32 Age 2 Bus Change

4 of 11 0,56 1515,00 0,00 150,49 Pop Age 2 Bus Change

0,54 1518,24 0,00 7,65 Age 1 Age 2 Bus Change

0,54 1518,46 0,00 38,62 Pop Workers Bus Change

5 of 11 0,56 1514,84 0,00 152,03 Pop Income Age 2 Bus Change

0,56 1514,89 0,00 162,54 Pop Age 2 Buses Change Dist

0,56 1516,48 0,00 308,66 Pop Workers Age 2 Bus Change

Table 2 presents ArcGIS’s explanatory variable analysis in which the software analyses all

giving variables, in this case all candidate variables. The analysis presents each possible equation for a specific number of variables in these equations in function of the three highest adjusted R-Squared. It also provides a multicollinearity table in which it presents each candidate variable’s covariates. The table goes even forward in explicitly stating that a combination of variables was not possible due to perfect multicollinearity.

Table 3: Summary of Multicollinearity

Variable VIF Violations

Covariates

Pop 577,62 321 Workers (98,47), Age 1 (93,13), Age 2 (93,13), Age 3(93,13)

Worker 222,84 309 Age 2 (98,47), Age 3(98,47), Pop (98,47), Age 1 (54,96)

Inc 3,44 0

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Age 1 46,83 201 Pop (93,13), Age 2 (77,10), Workers (54,96), Age 3 (38,93)

Age 2 770,17 312 Workers (98,47), Age 3 (93,13), Pop (93,13), Age 1 (77,10)

Age 3 31,80 279 Workers (98,47), Age 2 (93,13), Pop (93,13), Age 1 (38,93)

Road density

1,09 0

Buses 1,55 0

Terminal 1,14 0

Change 1,40 0

Distance 3,35 0

The following table presents the number and percentage of equations generated and passed according the presented criteria.

Table 4: Percentage of Search Criteria Passed

Search Criterion

Cutoff Trials # Passed % Passed

Min Adjusted R-Squared

> 0,50 1015 115 11,33

Max Coefficient p-value

< 0,05 1015 69 6,80

Max VIF Value < 7,50 1015 417 41,08

Moran’s I test was also done in order to evaluate autocorrelation with regards to ridership. The following table presents the outcomes as well as the estimated Moran’s I value.

Table 5: Moran’s I tests on the Dependent and Candidate Variables

Variable Moran's Index

Expected Index

Variance

z-score p-value Pattern

Ridership 0,019397 -0,010204

0,00308 0,533356 0,593787 Clustered

Population 0,513214 -0,010204

0,00463 7,69199 0 Clustered

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Age 1 0,375009 -0,010204

0,005029 5,435004 0 Clustered

Age 2 0,548009 -0,010204

0,004924 7,957756 0 Clustered

Age 3 0,675734 -0,010204

0,003933 10,929235 0 Clustered

Workers 0,551187 -0,010204

0,004596 8,28109 0 Clustered

Med Inc 0,659357 -0,010204

0,004869 9,5952 0 Clustered

Road Density

0,268342 -0,010204

0,004724 4,052722 0,000051 Clustered

Buses 0,149382 -0,010204

0,004242 2,450214 0,014277 Clustered

Change -0,08105 -0,010204

0,004801 -1,022436 0,306575 Random

Terminal 0,009537 -0,010204

0,004629 0,290162 0,771692 Mixed

Distance 0,872311 -0,010204

0,005166 12,281491 0 Clustered

The best fit was determined to include the number of workers in the station’s proximity,

the type of station when it comes to changes and the number of bus lines in a 200-meter radius. However, the initial equation presented both the p-value and the t-value of the number of

workers insignificant. Plotting the ridership as a function of the number of workers, it was shown that T-Centralen, Slussen and Gullmarsplan were in all of them huge outliers and when out of the data, the worker variable is significant with an R-Squared of 0,02166 before removing them and 0,23722 after removing them. It was thus decided not to include these three stations in future studies both for OLS and GWR.

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Figure 12: Ridership vs. Number of Workers

(R-Squared values show a better fit when T-centralen, Slussen and Gullmarsplan are taken out of the expression)

After taking these outliers out, both OLS and GWR analysis were executed again leading to the following residual maps.

y = 0,1758x + 1006,7R² = 0,0283

y = 0,2083x + 619,99R² = 0,236

0

2000

4000

6000

8000

10000

12000

14000

16000

0 2000 4000 6000 8000 10000 12000

Ride

rshi

p

WorkersRidership with Outliers Ridership without outliers

Linear (Ridership with Outliers) Linear (Ridership without outliers)

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Figure 13: OLS (up) and GWR (down) Standard Deviations

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The significance of the OLS parameters as well as for both equations are presented in the following tables.

Table 6: OLS Equation

Variables Coefficients Standard Error

t-Statistic Probability VIF

Intercept -552,050799 351,437275 -1,570837 0,119624 -

Workers 0,311537 0,054621 5,703612 0,000000 1,044079

Buses 88,258312 18,137573 4,866049 0,000005 1,327416

Change 1038,326730 283,930296 3,656978 0,000429 1,277995

Number of Observation

97

Number of Variables

4

Adj R-Squared

0,546342

Residual Square

33224590,808728

AICc 1631,798893

Table 7: GWR Equations

Variables Minimum Maximum Mean Standard Deviation

Intercept -2446,301797 657,6131415 -563,7553011 716,045689

Workers -0,004097741 0,258132412 0,149459243 0,050484

Buses -2,954762007 88,47400346 39,9531998 23,725989

Change -130,9206048 2793,393959 883,5952445 738,663661

Number of Variables

4

Number of Neighbors

42

39

Adj R-Squared

0,683548

Residual Square

16766876,5866

AICc 1500,09803 Looking at the maps, one can see different cluster in the OLS residual map that were reduced nay sometimes corrected in the GWR residual map. Looking closely at the value, one can safely say that the GWR has a higher predictability with an overall adjusted R-Squared of 0,68 compared to 0,55 for the normal OLS model. In addition, the residual square is almost half for the GWR model compared with the OLS one. It is important to specify that when the OLS equation was determined, ArcGIS launched a warning stating that the model should be checked for spatial autocorrelation, hinting that a spatial independent model might not be the best fit. This was proven with both Moran’s I test as well as the comparison between the OLS equation and GWR ones. The OLS equation was thus used in order to predict the ridership at the new stations for current situation. The following table presents the predicted ridership.

Table 8: Predicted Ridership using OLS Equation for New Station in 2016

Station Predicted Ridership

Nacka 2940

Järla 1053

Sickla 1118

Hammarby Kannal 1854

Sofia 2026

Barkarbystaden 265

Barkarby Station 608 The following map show the overall standard deviation of residual when GWR equations are calibrated for the new stations with this OLS estimated ridership.

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Figure 14: GWR Standard Deviation with New Stations

The coefficients of the GWR equations as well as their significances are presented bellow.

Table 9: GWR Equations with the New Stations

Variables Minimum Maximum Mean Standard Deviation

Intercept -2367,47257 625,462321 -570,298174 670,481597

Workers 0,009195 0,24977 0,151743 0,043329

Buses 7,585861 86,22674 42,060432 19,308832

Change -120,334075 2667,703535 873,7549 688,076229

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Number of Variables

4

Number of Neighbors

46

Adj R-Squared

0,69915

Residual Square

17569889,2664

AICc 1600,959011

Predictions Using the Determined GWR Equations

Having determined the GWR equations, future ridership can be thus determined. The future number of workers in the respective station’s service area should be thus determined. For the presented stations, the number of planned residences has been multiplied by the average number of persons per household. The average number of persons per household depends on the municipality and is respectively 2,12 and 2,47 for Stockholm and Nacka in 2018 (Statistiska centralbyrån (SCB), n.d.). Each new station’s equivalent station is also presented as well as the ratio Population to Workers. Finally, this ratio is multiplied to the number of people living in the area, leading to the additional number of workers living in the area. Adding these numbers to the current number of workers in the area gives the information needed to input into the equation.

Table 10: Number of Additional Population and Workers by 2030

(*Note: For Barkarbystaden and Barkarby Station, the total population and the number of workers is presented)

Station Number of residences

Household

Additional Population

Equivalent Station

Ratio Pop/Wor

Additional Workers

Nacka 6500 2,47 16055 Solna Centrum

0,540875 8684

Järla 950 2,47 2347 Västra Skogen 0,462653 1086

Sickla 2000 2,47 4940 Liljeholmen 0,555743 2746

Hammerby Kannal

2140 2,12 4537 Alvik 0,577865 2622

Sofia 155 2,12 329 Mariatorget 0,580132 191

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Kungträd- gården

0 2,12 0 Hötorget 0,563254 0

Fridhems- plan

134 2,12 285 - 0,612255 174

Stadshagen

1975 2,12 4187 - 0,630363 2640

Rinkeby 1100 2,12 2332 - 0,294039 686

Tensta 1030 2,12 2184 - 0,347523 759

Kista 1600 2,12 3392 - 0,467844 1587

Akalla 1000 2,12 2120 - 0,443841 941

Barkarby- Staden*

- - 4946 Farsta Strand 0,462653 2288

Barkarby Station*

- - 1755 Farsta Strand 0,462653 811

Regarding Barkarbystaden and Barkarby Station, the future number of residences is already known by SAM. Adding the population to each service area by proceeding in the same manner done for the current situation, one can proceed in multiplying this population data by the ratio Population to Workers and getting the number of workers in the service area. Remaining stations on the blue line where no significant information has been found saw an increase of 5% of its worker’s population and this according to RUFS’ estimated increase of the population by 2030. T-Centralen’s service area remained untouched, however. Using the GWR equations, the ridership of the chosen blue line stations is predicted. The following table presents the GWR equation for each station as well as the final determined ridership. Having all the needed data and statistics, a deep analysis and discussion of these results is discussed in the following section.

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Table 11: Coefficients and GWR Estimations for New Stations

Name Intercept Worker Coeff.

Bus Coeff.

Change Coeff.

Predicted by GWR

Predicted by SLL

Nacka -587,71361

32 0,13881581 36,517037

07 949,625133

7 4200

3000

Järla -514,98852

84 0,14376295

6 38,812961

65 844,623781

4 1169

600

Sickla -628,53274

11 0,15307304

8 36,370452

15 942,764456

8 1444

800

Hammarby Kanal

-1667,6785

14 0,17223955

6 21,962985

43 2007,36610

6 2298

1200

Sofia -1487,5451

22 0,16621622

7 18,791536

61 1860,82295

4 1632

1600

Barkarbystaden

-6,7022540

07 0,12151317

6 52,664246

18 282,178624

6 554

2300

Barkarby station

-28,110621

92 0,11704238

7 54,708347

43 306,278404

3 701

1200

After looking at the results, one can see that the ridership predicted for Barkarbystaden and Barkarby Station were very much under predicted, less than a quarter of the official prediction for Barkarbystaden. For this reason, the way the workers were determined for these two stations was reevaluated and calculated the same way it was done for other stations with information about the number of planned residences to be built by 2030. The following table shows the newly calculated additional workers.

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Table 12: Number of Additional Population and Workers by 2030 for Barkarbystaden and Barkarby Station

Station Number of residences

Household

Additional Population

Equivalent Station

Ratio Pop/Wor

Additional Workers

Barkarbystaden

16593 2,42 40156 Farsta Strand 0,46265 18578

Barkarby Station

5078 2,42 12289 Farsta Strand 0,46265 5686

Recalculating the ridership for these two stations, the final ridership predictions for all stations new stations are as presented below.

Table 13: Coefficients and GWR Estimations for New Stations after the change in workers

Name Intercept Worker Coeff.

Bus Coeff.

Change Coeff.

Predicted by GWR

Predicted by City

Nacka -587,71361

32 0,13881581 36,517037

07 949,62513

37 4200

3000

Järla -514,98852

84 0,14376295

6 38,812961

65 844,62378

14 1169

600

Sickla -628,53274

11 0,15307304

8 36,370452

15 942,76445

68 1444

800

Hammarby Kanal

-1667,6785

14 0,17223955

6 21,962985

43 2007,3661

06 2298

1200

Sofia -1487,5451

22 0,16621622

7 18,791536

61 1860,8229

54 1632

1600

Barkarbystaden

-6,7022540

07 0,12151317

6 52,664246

18 282,17862

46 2566

2300

Barkarby station

-28,110621

92 0,11704238

7 54,708347

43 306,27840

43 1342

1200

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Analysis Analyzing the predicted ridership for the new stations, one can see that they are, generally overestimated compared to official results, thus with regards to the Sampers model used in Vissum. This can be explained by multiple reasons both general, regional and station specific and the assumptions behind them.

Division between the North and the South

One can see a clear schism between the north stations (Barkarbystaden and Barkarby Station) and the south stations (Sofia, Hammerby Kannal, Sickla, Järla and Nacka). The first mentioned have ridership estimates that are close to the estimations made by SLL, even if slightly over estimated.

However, it is important to stress that the models of the northern stations has been developed with only one ridership that was predicted by the OLS model while in the southern station’s models, this number is definitely higher, especially when moving further away from the center of the study area towards Nacka.

The Model in Numbers

The model’s coefficients are very broad and different. Looking into the intercepts, stations like Sofia and Hammerby Kannal have values well

below the average of -570 with -1487,54 and -1667,67 while the stations in the north (Barkarbystaden and Barkarby Station).

Figure 15: The distribution of the intercept over the studied area (from the lowest in blue to the highest in red; this applies to all

distributions to follow)

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Looking into the workers coefficients, these values seem to decrease the further one moves away from the city center towards the end of the lines. This means that, at least for the blue line, the further one goes from the city center, the less workers use the metro to get to their respective places of work.

Figure 16: The distribution of the worker’s coefficients over the studied area

Looking into the bus coefficient, south stations have values below the average of 42,06 with respectively from Sofia to Nacka, 18,79, 21,96, 36,37, 38,81 and 36,52. With regards to northern stations, they are slightly above average with respectively 52,66 and 54,70 for Barkarbystaden and Barkarby Station. The same pattern can be observed than the one observed for workers, yet inversely. In fact, the farther the station is compared to the central area the greater the bus coefficient influences the estimated ridership. This is quite common for metro network, especially star network like the one in Stockholm: in central area the dense metro stations presents does not present the need to use buses to get to a specific station while the further one goes from the center, the lower the density of metro stations, leading to the use of buses to reach these stations.

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Figure 17: The distribution of the bus’s coefficients over the studied area

Looking into the change coefficient, the station in the north have coefficients that are below average while the south stations have coefficient above the 873-mean value. The same pattern can also be deduced. The change coefficient in stations in the central part of Stockholm increase the number of passengers in each specific station compared to stations outside of the city center.

Figure 18: The distribution of the change’s coefficients over the studied area

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General Reasons

In general, this might be due to the use of the average number of persons per household. In fact, even if these new station with the exception of Sofia and Hammerby Kannal are outside of municipal Stockholm, they are relatively close to it. This means that the average number of persons per household in this area might be closer and thus lower than their average municipal one, i.e. Nacka and Järfalla.

The major change of the areas between 2016 and 2030 might affect the way the GWR

equations are predicted leading to a change in the calibration of the variables that are geographically dependent. The GWR equations were estimated for a spatial area that is different in 2030 than it is today, like Sickla, Järla and Nacka. Sofia are predicted relatively accurately due to the fact that the environment has not changed much.

Bus Assumptions and Effect on Predictions

Looking closer into the difference between the north and south stations in the first place might explain the difference seen between GWR values and the one by SLL: this is mainly due to the assumption taken with regards to the number of bus lines. In fact, the number of buses in the areas of interests were left as is. Although, in 2030, with new development and metro stations built, the current state will definitely evolve. On one hand, the stations in the north of the blue line see a slight overprediction of respectively 11,6% and 11,8% for Bakarbystaden and Barkarby Station compared to SLL estimations. This is mainly due to the fact that a limited number of bus lines were found around the future stations, none for Barkarbystaden and six for Barkarby Station. An increase in the number of residence and two new metro stations will definitely lead to new lines stopping in the area, especially in Barkarbystaden. On the other hand, the stations in the south and in municipal Nacka are overwhelmingly overestimated. In fact, the predicted ridership at Järla by the GWR model is almost twice as the ridership predicted by SLL. Here, a huge number of bus lines that pass in the region that lead to Nacka and beyond east will be removed by 2030. Today, these lines mainly start at Slussen. In 2030, it is planned that the lines that lead to Nacka will drastically decrease and the ones that leads farther east moved to Nacka’s bus terminal. This means that a lot of the buses accounted for when doing the predictions will be gone in the future and thus should not be accounted for in order to get a closer projected ridership to the SLL one.

Unfortunately, no data is available today on the number of bus lines that are planned to cover both areas.

Special Case of Sofia

Sofia is the only station that has an estimation that is almost identical to the one predicted by SLL.

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A simple explanation might be the fact that the region barely changes with regards to the socio-economic dynamic as well as the transit characteristic with regards to 2016. In fact, hardly any residence will be built in the area, only 155, a thin increase of less than 200 workers. The area has already a high-service when it comes with bus stops and bus lines that pass through it. A change in the number of lines will thus be limited in the area.

Limitations of the Study Multiple limitations can be drawn in this study dipping a toe into the multiple defects of the model and the study itself.

First, land use variables are, according to preliminary studies an important component that can explain a part of ridership. However, accurate land use data was not available for the whole study area. Only municipal Stockholm had land use data with the required accuracy. For this reason, land use was disregarded when calculation both OLS and GWR equations. Lack of land use data and variable automatically puts limits for this study as it is a crucial information to incorporate in the study, especially when debating on a subject that has the change of land use for the future service areas of the new stations in its core.

Second, when service areas were drawn on ArcGIS, three main critics can be said. First, when Thiessen polygons were used for close stations, the latter assumes, at least in the central part of Stockholm that people always choose the closest station independently of the type of station. Behavioral studies show that this is not always the case. Second, problems with generating the service area on ArcGIS led to having some stations with either service areas way bigger or way smaller than 800 meters road length wise and the fix was thus done manually and might not be precis. Third, the network layer used in order to create the service areas had missing pedestrian tracks for some regions, meaning that some service areas might have been created with flawed road and pedestrian network especially at the outskirts of the metro network where the problem was considerable. Manual fixes were thus undertaken in order to resolve the problem.

Third, assumptions and hypotheses taken in order to be able to pursue the study can definitely be questioned.

The way the GWR equations for the new stations were determined is completely problematic. Determining ridership for stations that do not yet exist in a year they definitely will never do can never be observed. This hypothetical situation has led to error being included and dragged in the study that can never be identified.

The assumption about the population and workers being homogeneously distributed over the multiple SAMS is far from the situation on the ground and can lead to inaccuracy and misrepresentation, especially in relatively large zones (which are mainly clustered in the outskirts of the metro network) where irregularities can be more pronounced. As already discussed above, the unchanged bus network between 2016 and 2030 is a huge drawback for accurate prediction at the new stations.

Finally, the choice of the number of persons per household and of the equivalent stations while calculating the number of workers in service areas is unrealistic. First, the exact number of persons per household cannot be known and the value that was used was for the current state of 2018. Second, to assume that the ratio of workers to population in new stations is identical to their

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chosen current equivalent station cannot be proven yet and is totally hypothetical done for the purpose of easing the calculation of the potential number of workers in the service areas.

The ridership predicted by SLL was assumed to be correct and unfaulty, However , this

predictions cannot be observed yet. Comparing the GWR predictions with other predictions cannot prove for sure how GWR performs with regards to the situation on the ground.

In future studies, these limits should be considered and tackled in order to improve the

potential predictiveness of the model.

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Conclusion

After going through the analysis of these results, one case-specific conclusion and two

main conclusions were deduced from this study.

In this case, GWR equations overpredicted the ridership for future stations with regards to the official predictions. However, a generalization of this results cannot be drawn due, first, the many case-specific assumptions taken here and, second, the lack of other studies available today using the same process.

The general conclusion are as follows. First, GWR is a good predicting tool for future

stations that are closely located from most present stations. Second, GWR is a good predicting method for stations where limited changes in the future environment will occur.

These conclusions help in realizing when to use GWR. In fact, according to these results

the conclusion determined, Geographically Weighted Regression can be used as both an explanatory and predictive tool but in limited cases. With regards to its explanatory power, the model can help explaining in details and area specific the current situation or the possibility of building one or more stations for a given area which will not change drastically. This can also be taken further in determining the ridership using the GWR equation.

GWR is thus performs good for a limited number of stations built next to an existing, stable

environment. For example, densely populated cities with an existing but limited public transportation or metro network are ideal cases for the use of this method. In addition, developed countries with limited funds are good situation where an area/station-specific equation explains and predicts the ridership reassuring policymakers and investors. The determined parameters can also help determine area-specific and thus more accurate and precise policies that can be proposed. For instance, the choice of building new apartments in specific areas can also be analyzed and other areas, not considered, can be proposed.

However, this study is still preliminary and experimental in the field of the

predictive power of GWR with regards to new metro stations. More studies should be performed for different contexts, tackling the limitations in order to get more accurate and significant results.

52

References American Public Transportation Association. (2019). Public Transportation Facts - American Public Transportation

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56

Appendix

Appendix I: Method and Tools in Determining Data in ArcGIS (ArcMap)

Income, Workers, Population and Age

All of the above was done by a method of intersect and dissolve like in NYC. The service area was divided on the given SAM’s areas. The proportional population (with the respective age groups) and workers were then added to each division that fell within each respective SAM’s area. It was assumed that the population and the number of workers were homogeneously distributed on the SAM’s area and thus on the division of the service area that fell in each SAM area.

A step further was taken in calculating the median income. A weighted average in function of the number of workers was done. The following equation explains how the median income was determined for each service area.

∑=çqéLqt_ê!_ë:5:t:çH ∗ ÜLJ]Hy∑=çqéLqt_ê!_ë:5:t:çH

Where the Workers_by_Division was calculated as explained above.

Road density (m/m2) The layer was intersected using the tool “Intersect” then the summation of the length of roads was done in the area, the all divided by the area of the service area.

Number of bus lines at a 200-meter buffer around the entrances of the metro

The number of buses located at a 200-meter radius was considered. The buffer tool was used. Some adjustments were done manually in order to correct any mistakes. The number of bus lines were finally divided by 2 in order to account for only the feeder type of the buses in the morning leading to the station.

Terminal Station

Terminal station were added regarding their respective years. In 2016, Akalla, Hjulsta and Kungsträdgården on the blue line, Hässelby Strand, Hagsätra, Farsta Strand and Skarpnäck on the green line and Mörby Centrum, Ropsten, Norsborg and Fruängen on the red line were assigned the dummy value of 2 while all other stations were assigned 1. In 2030, Kungsträdgården is replaced by Nacka and Akalla by Barkarby. All other stations remain the same.

57

Type of change

The type of station was just translated to a dummy variable using the “bytespunkt” column from SLL of future stations.

Commuting distance

The point to distance tool was used to determine the distance from a station to T-centralen. Euclidian distances were used as the Stockholm metro network has a star shape and limited curves in its network.

58

Appendix II: Table of GWR Equations and Predictions for the 2016 Situation

Stat

ion

Obs

erve

d

Loca

l R2

Pred

icte

d

Inte

rcep

t

Wor

kers

Bus

Chan

ge

Resid

ual

Std

Erro

r

Std

Err_

Int

Std

Err

Wor

kers

St

d Er

r Bus

Std

Err

Chan

ge

StdR

esid

Äng

bypl

an

400,

0000

0000

0000

000

0,74

5808

5427

3200

0 59

5,87

4423

8949

9995

0 - 57

,098

7578

9659

9999

- 0,

0040

9774

1460

700

71,1

2596

8780

4000

01

514,

8025

9472

6000

050

- 195,

8744

2389

5000

010

440,

9814

5246

6000

010

381,

0086

0604

3999

980

0,08

2431

8699

9290

0 27

,907

9925

6609

9999

31

9,18

3645

8380

0002

0 - 0,

4441

7837

2581

000

Aka

lla

800,

0000

0000

0000

000

0,67

1086

8392

7900

0 93

3,80

5454

0240

0003

0 44

,963

6597

0650

0003

0,

1150

8926

5256

000

51,5

2738

5300

7999

99

249,

9766

1361

1999

990

- 133,

8054

5402

4000

000

390,

0152

8203

1000

030

302,

6912

2643

2000

010

0,05

9364

5491

7470

0 11

,278

7089

5919

9999

22

9,47

4690

8640

0000

0 - 0,

3430

7746

4369

000

Fittj

a

600,

0000

0000

0000

000

0,06

7274

1394

5750

0 91

3,86

1968

7460

0000

0 59

5,27

1498

8180

0005

0 0,

1173

6388

3033

000

21,4

9569

4744

4000

01

- 105,

8248

0043

0000

000

- 313,

8619

6874

6000

000

426,

6336

1665

2000

000

415,

9698

4168

0999

990

0,13

9072

9690

1400

0 36

,149

3607

5380

0003

30

4,26

7640

4229

9999

0 - 0,

7356

7097

5037

000

Brom

map

lan

3000

,000

0000

0000

0000

0,

6896

8980

3382

000

2594

,867

6502

7999

9900

- 48

6,80

5166

8219

9999

0 0,

0969

5239

9292

300

54,6

5231

8977

1000

02

831,

2969

5161

8999

970

405,

1323

4971

8000

000

166,

4111

4393

5000

010

382,

8379

8985

4999

990

0,06

1248

8680

1600

0 28

,419

4697

4320

0001

35

0,35

5978

8910

0001

0 2,

4345

2655

9560

000

Lilje

holm

en

4200

,000

0000

0000

0000

0,

8292

7186

9873

000

3755

,908

7095

8000

0000

- 22

79,9

4394

0149

9999

00

0,15

5161

9448

8700

0 37

,011

1633

7999

9999

26

42,3

2660

0120

0000

00

444,

0912

9042

4000

020

203,

5263

5604

2000

000

363,

3476

5862

9000

020

0,05

0474

9233

0690

0 27

,175

9831

9360

0000

34

2,91

8497

3170

0000

0 2,

1819

8418

6520

000

59

Örn

sber

g

900,

0000

0000

0000

000

0,75

0006

1087

9600

0 80

4,33

3990

0229

9996

0 - 14

35,5

2524

8859

9999

00

0,17

2199

0080

1700

0 73

,274

2496

7349

9995

16

73,4

8241

4460

0000

00

95,6

6600

9977

2000

05

438,

8380

2864

2999

970

384,

6743

9868

3999

980

0,05

8211

1461

4880

0 30

,089

8171

4140

0001

31

9,91

4168

8880

0001

0 0,

2179

9844

9845

000

Kär

rtorp

900,

0000

0000

0000

000

0,49

6910

9286

1900

0 65

2,07

0832

7229

9997

0 - 78

,980

2940

9580

0005

0,

1510

3419

2061

000

34,4

9912

6787

5999

98

396,

4958

0479

9999

970

247,

9291

6727

7000

000

450,

8317

2477

8000

020

367,

2393

1636

8000

000

0,06

1071

0451

8290

0 26

,322

7332

7100

0001

28

7,48

6143

4050

0001

0 0,

5499

3726

8499

000

Stur

eby

500,

0000

0000

0000

000

0,43

8176

8446

7900

0 66

1,16

4172

2509

9997

0 - 24

1,95

6260

9360

0001

0 0,

1445

6254

8257

000

59,0

2243

5228

5000

01

503,

3231

5617

6000

000

- 161,

1641

7225

1000

000

449,

2946

1506

4000

030

346,

2971

2979

2000

020

0,06

5277

5087

6440

0 28

,539

7469

6699

9999

27

5,96

5378

8439

9999

0 - 0,

3587

0488

2826

000

Glo

ben

300,

0000

0000

0000

000

0,60

2723

7730

7900

0 55

0,35

2684

1719

9995

0 - 10

93,4

9460

7679

9999

00

0,15

6560

1973

7600

0 40

,755

0528

0210

0003

13

45,9

2159

1990

0000

00

- 250,

3526

8417

2000

010

458,

2904

1066

5000

020

359,

4516

8974

0000

010

0,06

2474

3132

5000

0 28

,705

3812

9879

9999

28

5,98

7991

0729

9999

0 - 0,

5462

7519

6569

000

Huv

udsta

600,

0000

0000

0000

000

0,63

1011

1965

7000

0 92

6,23

8926

6249

9998

0 - 77

0,26

1623

8099

9995

0 0,

1936

6421

3989

000

30,6

8981

6188

1999

98

928,

3731

7775

0999

960

- 326,

2389

2662

4999

980

448,

4876

0379

9000

000

328,

1205

7522

7000

020

0,03

9174

7078

1510

0 20

,981

5975

9949

9999

29

6,87

3984

7019

9997

0 - 0,

7274

2016

4706

000

Uni

vers

itete

t

500,

0000

0000

0000

000

0,59

7904

8678

4400

0 82

2,45

7341

4369

9997

0 - 75

4,89

3479

6590

0004

0 0,

1456

0841

2684

000

33,0

7125

7853

2000

02

1061

,770

4260

0000

0000

- 32

2,45

7341

4370

0003

0 42

9,22

3630

4269

9999

0 28

2,02

0067

7340

0001

0 0,

0342

2284

9464

000

11,9

6863

9394

2000

00

257,

3876

2027

0000

010

- 0,75

1257

1968

9700

0

Ham

mar

byhö

jde

n 90

0,00

0000

0000

0000

0 0,

5605

2280

1223

000

988,

6445

0089

1000

010

- 558,

0317

1047

1000

000

0,15

8867

3267

6700

0 32

,349

9297

5929

9997

84

9,76

1543

5879

9999

0 - 88

,644

5008

9049

9998

46

0,33

7759

2919

9999

0 36

9,98

9969

1730

0002

0 0,

0579

7470

5077

500

26,7

3637

8512

0000

02

292,

3979

0117

4000

030

- 0,19

2564

0447

7200

0

60

Joha

nnel

und

200,

0000

0000

0000

000

0,64

0398

6013

4900

0 49

9,50

3251

4120

0000

0 67

,535

0184

4340

0006

0,

0621

5205

2741

000

71,5

7583

5008

9000

00

254,

3221

7367

9999

990

- 299,

5032

5141

2000

000

431,

6275

9706

0000

030

372,

7506

0285

0000

010

0,08

5770

0649

7600

0 25

,259

0797

9370

0000

30

2,57

1291

7180

0000

0 - 0,

6938

9272

9410

000

Hag

sätra

1200

,000

0000

0000

0000

0,

3862

2809

6510

000

676,

4634

4108

4000

010

- 112,

0135

4510

6000

000

0,24

5129

7689

3300

0 66

,796

5999

2480

0006

24

7,35

6807

5990

0001

0 52

3,53

6558

9159

9999

0 44

0,89

9094

6920

0001

0 37

7,71

3177

6389

9998

0 0,

0879

7411

8596

600

33,1

7154

7694

2000

01

301,

2679

3064

8999

990

1,18

7429

4259

6000

0

Mäl

arhö

jden

700,

0000

0000

0000

000

0,59

1631

5791

4000

0 84

2,25

4058

6970

0000

0 - 75

6,55

7067

2400

0004

0 0,

2223

8022

7498

000

61,5

0530

5715

6999

97

975,

1882

7678

2999

990

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3969

4779

0000

010

369,

1189

5173

9000

010

310,

0142

1823

5999

980

0,03

4220

3902

5360

0 16

,724

3808

5690

0002

29

3,93

2742

6070

0002

0 - 2,

1277

6110

2730

000

Sund

bybe

rg

1500

,000

0000

0000

0000

0,

5935

5462

1426

000

2113

,684

1935

6000

0000

- 34

9,83

6604

2000

0001

0 0,

1540

6081

6299

000

48,8

0956

1265

7999

99

520,

0809

6552

2000

040

- 613,

6841

9356

4000

000

400,

8963

7748

0000

010

311,

9165

7746

9000

000

0,04

0106

4207

8610

0 13

,199

0997

6410

0000

24

7,36

8569

9670

0001

0 - 1,

5307

8009

1910

000

69

Åke

shov

500,

0000

0000

0000

000

0,71

0611

1012

3600

0 58

1,86

0181

3620

0005

0 - 17

5,30

1210

1180

0000

0 0,

0287

9733

6472

600

64,8

4085

7528

3000

06

600,

8421

4018

6000

050

- 81,8

6018

1362

0000

06

436,

0845

9174

2999

980

377,

2296

2683

9999

980

0,07

2810

0863

3230

0 27

,776

7244

1820

0001

32

1,78

9331

1620

0000

0 - 0,

1877

1628

9252

000

Sved

myr

a

400,

0000

0000

0000

000

0,45

1790

6354

8900

0 64

1,67

2465

7390

0002

0 - 20

9,28

1569

5220

0001

0 0,

1408

8003

0524

000

54,1

5733

7540

2999

99

479,

2972

4570

5000

020

- 241,

6724

6573

8999

990

438,

7935

0462

0000

020

349,

8867

1455

8999

980

0,06

4577

2159

1700

0 28

,063

3116

7830

0000

27

6,26

5738

1760

0001

0 - 0,

5507

6582

3092

000

Fars

ta st

rand

700,

0000

0000

0000

000

0,36

9912

5191

5600

0 10

20,8

7316

1870

0000

00

155,

1699

4445

3999

990

0,16

9378

8682

8900

0 37

,454

0626

5679

9998

17

0,02

7653

7860

0000

0 - 32

0,87

3161

8709

9998

0 34

5,17

0703

3710

0000

0 34

6,27

6223

0089

9998

0 0,

0706

1912

2909

800

32,0

232 7

0303

4999

99

275,

9345

3968

5000

000

- 0,92

9607

1733

1400

0

Sock

enpl

an

500,

0000

0000

0000

000

0,49

1468

8287

9200

0 58

1,14

3183

8610

0001

0 - 37

4,94

9291

1510

0001

0 0,

1335

6366

2847

000

55,8

0957

8995

8000

03

636,

8018

5336

8000

020

- 81,1

4318

3860

7000

06

461,

9641

4607

8000

000

357,

0885

9251

8999

970

0,06

5957

0629

1710

0 29

,079

5765

3260

0001

28

2,03

0503

4819

9997

0 - 0,

1756

4822

8439

000

Bred

äng

1100

,000

0000

0000

0000

0,

4419

9738

9828

000

1076

,979

6052

6000

0000

- 22

3,70

4196

2479

9999

0 0,

2357

9020

6964

000

67,4

5337

4499

6000

02

432,

6531

7313

8000

000

23,0

2039

4736

8000

00

445,

4306

1250

0999

980

429,

6112

6843

5000

000

0,08

6332

4378

2030

0 27

,336

0487

8479

9999

34

1,91

2517

3560

0002

0 0,

0516

8121

3842

700

Skog

skyr

kogå

rde

n 40

0,00

0000

0000

0000

0 0,

4665

5515

7685

000

741,

6258

0585

9000

020

- 101,

9519

9617

1000

000

0,14

7651

9894

1700

0 41

,736

1158

1899

9998

40

0,47

4721

3570

0002

0 - 34

1,62

5805

8590

0002

0 46

0,52

1819

8499

9999

0 35

1,17

6656

7970

0002

0 0,

0618

4739

1114

100

26,5

8259

4784

2000

01

273,

8246

1389

5000

030

- 0,74

1823

2777

1200

0

Mid

som

mar

kran

sen

900,

0000

0000

0000

000

0,76

3772

7983

7500

0 88

6,38

5013

8999

9999

0 - 17

95,8

9534

8380

0001

00

0,15

2855

6614

7000

0 65

,085

9725

1320

0005

20

95,0

3945

0190

0000

00

13,6

1498

6100

1999

99

374,

5766

0104

1999

990

382,

6848

4910

7999

980

0,06

6216

4951

0140

0 36

,614

4113

6599

9999

33

1,39

2967

7159

9999

0 0,

0363

4766

8440

500

70

Tens

ta

900,

0000

0000

0000

000

0,63

0302

7641

0500

0 70

8,47

1445

9450

0003

0 13

,064

5065

5670

0000

0,

1057

1943

1990

000

53,1

4901

1678

9999

97

292,

4380

5827

0999

990

191,

5285

5405

5000

000

455,

0566

5826

6999

990

304,

2419

1955

9000

000

0,06

2207

4865

1610

0 11

,953

7810

5330

0000

22

9,54

9677

5180

0001

0 0,

4208

8946

6345

000

Stor

a m

osse

n

500,

0000

0000

0000

000

0,68

9576

4162

7800

0 59

9,83

4146

0399

9995

0 - 10

11,6

6904

5660

0001

00

0,18

1706

0148

7300

0 37

,727

4445

6299

9999

12

42,0

0107

7550

0000

00

- 99,8

3414

6039

4999

98

442,

6444

7279

6000

000

356,

9096

7013

5000

000

0,04

6169

8214

7990

0 26

,446

9982

6890

0000

34

6,24

5833

5200

0002

0 - 0,

2255

4025

2223

000

Stad

ion

700,

0000

0000

0000

000

0,63

5028

6707

6000

0 11

10,8

2656

2120

0001

00

- 1261

,170

1128

8999

9900

0,

1396

8471

3089

000

0,83

8112

5463

8100

0 18

24,2

9288

0840

0000

00

- 410,

8265

6211

8000

030

462,

3062

3559

6000

020

317,

9646

2515

7999

990

0,03

4812

7206

1840

0 18

,876

1427

0199

9999

31

2,78

4702

5739

9999

0 - 0,

8886

4594

6098

000

Skär

holm

en

800,

0000

0000

0000

000

0,22

2348

2304

9300

0 12

55,3

8527

3680

0000

00

169,

4783

0307

9999

990

0,22

6824

5455

8500

0 45

,939

0767

9969

9997

96

,406

7294

0189

9995

- 45

5,38

5273

6760

0000

0 34

9,02

9954

1320

0000

0 39

8,75

6201

6410

0002

0 0,

1105

3916

3734

000

27,5

8571

1070

3999

99

288,

2830

0686

7999

970

- 1,30

4716

8825

6000

0

Vår

berg

1100

,000

0000

0000

0000

0,

1686

0012

4614

000

526,

0413

0552

5999

970

326,

6084

8999

9000

030

0,17

6835

6165

4600

0 44

,198

5566

0780

0001

1,

7512

9128

6740

000

573,

9586

9447

4000

030

398,

1110

4286

6999

980

395,

0872

5776

4000

010

0,11

7854

8832

8800

0 28

,577

2395

0540

0001

29

0,43

6883

5889

9999

0 1,

4417

0503

3700

000

Häs

selb

y str

and

700,

0000

0000

0000

000

0,67

4047

4395

1500

0 71

6,33

3303

7260

0005

0 46

,333

7686

9290

0001

0,

0672

5061

5545

600

76,8

5473

5540

9999

97

262,

1754

2385

8999

980

- 16,3

3330

3725

4999

99

451,

9986

0165

5000

020

388,

9909

5735

6000

020

0,09

4088

9903

5150

0 26

,098

0883

6630

0001

31

5,10

6225

2460

0001

0 - 0,

0361

3573

9503

900

Häg

erste

nsås

en 10

00,0

0000

0000

0000

00

0,59

7229

9785

7800

0 79

6,12

1720

1040

0002

0 - 89

7,01

5075

5670

0005

0 0,

1782

1895

6362

000

80,1

8026

6424

3000

01

1135

,526

2090

9000

0100

20

3,87

8279

8960

0001

0 44

0,01

3351

7169

9998

0 41

0,36

4521

1269

9999

0 0,

0686

7531

4011

400

35,5

2756

6082

6999

98

347,

6052

1554

7999

990

0,46

3345

6668

9900

0

71

Väs

tra sk

ogen

1300

,000

0000

0000

0000

0,

6459

0222

8646

000

600,

7215

1355

9999

950

- 897,

5524

7891

0999

980

0,21

3675

1157

9900

0 16

,091

1945

0620

0001

10

79,2

0852

2220

0001

00

699,

2784

8644

0000

050

443,

0670

1547

7999

970

315,

3539

7270

9000

000

0,03

7702

5212

1500

0 19

,955

9913

5450

0000

29

3,37

9526

3289

9999

0 1,

5782

6798

6580

000

Mar

iato

rget

1900

,000

0000

0000

0000

0,

7393

5191

5749

000

1756

,509

7653

6000

0100

- 19

24,5

9825

5950

0001

00

0,17

8761

5085

1500

0 2,

1592

2020

9290

000

2334

,935

2874

4000

0100

14

3,49

0234

6410

0000

0 40

2,72

2681

5240

0000

0 33

5,93

9103

6719

9999

0 0,

0410

2789

3464

900

21,9

8474

2328

5000

01

349,

7227

7308

8999

990

0,35

6300

3556

1500

0

Blac

kebe

rg

500,

0000

0000

0000

000

0,72

3517

9132

0600

0 56

5,56

3354

0159

9995

0 9,

5658

4797

3550

000

0,01

7669

9300

7300

0 76

,242

3953

8999

9999

38

2,98

0256

5179

9998

0 - 65

,563

3540

1619

9995

44

9,98

4071

5870

0002

0 38

0,32

5816

8699

9998

0 0,

0864

1032

4258

000

27,0

3241

8951

8999

99

320,

1894

6008

5000

010

- 0,14

5701

4995

7800

0

Råck

sta

1100

,000

0000

0000

0000

0,

6749

2348

0757

000

629,

9899

6982

6999

980

42,1

8093

4125

5999

97

0,03

4374

8337

6040

0 73

,023

1463

6229

9997

32

5,56

4805

4120

0000

0 47

0,01

0030

1730

0002

0 45

6,74

6585

5560

0001

0 36

4,04

1598

1040

0000

0 0,

0814

5529

5604

100

25,6

0122

3515

8999

99

302,

5591

8535

0000

010

1,02

9038

9573

5000

0

Asp

udde

n

1000

,000

0000

0000

0000

0,

7768

7771

2488

000

1053

,380

7790

5000

0000

- 16

78,9

0187

6680

0000

00

0,14

8994

2297

8100

0 73

,704

3187

0439

9995

19

50,4

5106

8800

0000

00

- 53,3

8077

9053

6000

01

453,

0881

5845

2000

020

374,

2404

7 716

7999

980

0,06

3776

9112

8500

0 35

,170

1655

3999

9999

31

2,97

2036

3910

0002

0 - 0,

1178

1543

6263

000

Rops

ten

3800

,000

0000

0000

0000

0,

6216

4459

2548

000

2664

,041

7735

8000

0200

- 11

06,7

0397

4340

0001

00

0,11

2785

5568

8600

0 10

,618

8913

6289

9999

16

97,7

6131

2760

0000

00

1135

,958

2264

2000

0100

34

2,73

9677

7110

0001

0 32

0,10

4193

3560

0000

0 0,

0352

8835

7934

900

14,1

5582

1723

7000

01

319,

3052

1740

0000

000

3,31

4347

0111

5000

0

Nor

sbor

g

400,

0000

0000

0000

000

0,06

5499

4193

8290

0 77

5,04

5439

2310

0005

0 62

3,28

7192

7899

9995

0 0,

0634

9577

1962

700

31,2

5174

0709

6000

00

- 129,

7298

1167

3000

000

- 375,

0454

3923

0999

990

259,

9591

5393

0000

010

414,

7859

1884

5000

030

0,14

6985

2282

5200

0 34

,235

9850

7919

9999

31

0,83

1266

1769

9997

0 - 1,

4427

0910

8570

000

72

Hal

lund

a

300,

0000

0000

0000

000

0,06

1613

7368

9130

0 89

2,98

0589

6530

0004

0 62

5,46

5650

4700

0001

0 0,

0790

2522

6269

800

27,5

6382

2097

3000

01

- 127,

9415

0668

6000

000

- 592,

9805

8965

3000

040

418,

0844

7536

5000

000

417,

0960

7203

7000

000

0,14

5125

9733

4300

0 35

,000

0691

4490

0003

30

9,27

4601

4269

9999

0 - 1,

4183

2721

5180

000

73

Appendix III: Table of GWR Equations and Predictions for the 2016 Situation with New Stations

Stat

ion

Obs

erve

d

Loca

lR2

Pred

icte

d

Inte

rcep

t

Wor

kers

Bus

Chan

ge

Resid

ual

Std

Erro

r

Std

Err I

nt

Std

Err

Wor

kers

St

d Er

r Bus

Std

Err

Chan

ge

Std

Resid

Äng

bypl

an

400,

0000

0000

0000

000

0,72

7369

2341

9800

0 58

7,38

8321

5719

9999

5 - 81

,939

3552

3659

9994

0,

0139

8781

2387

000

70,0

6781

6737

8000

01

515,

2601

8219

5999

960

- 187,

3883

2157

1999

995

433,

1980

8421

5999

984

361,

2608

6556

0000

013

0,07

6271

1518

7810

0 26

,936

7417

9989

9998

30

6,74

6430

0500

0001

5 - 0,

4325

6959

8990

000

Aka

lla

800,

0000

0000

0000

000

0,69

9401

7182

6100

0 92

7,32

9708

6750

0000

6 0,

2248

9695

2387

000

0,12

6336

3590

1500

0 51

,782

3458

1449

9998

26

6,83

7991

5310

0000

0 - 12

7,32

9708

6750

0000

6 38

8,83

9882

6839

9997

4 28

2,96

1593

7410

0000

2 0,

0536

5378

3224

400

10,8

8269

1585

9000

00

221,

9512

7054

4000

010

- 0,32

7460

5161

2800

0

Fittj

a

600,

0000

0000

0000

000

0,07

8572

8103

4390

0 93

8,01

1866

1190

0004

7 53

7,23

9307

3380

0000

3 0,

1244

7410

4369

000

27,4

0168

2787

1999

99

- 86,1

8470

2012

9000

04

- 338,

0118

6611

8999

990

419,

2909

0940

6000

026

393,

9943

7839

0000

008

0,12

8862

2512

6800

0 33

,485

1190

1740

0002

29

2,55

9789

4320

0000

2 - 0,

8061

5119

1301

000

Brom

map

lan

3000

,000

0000

0000

0000

0,

6674

9263

7530

000

2576

,626

5592

2000

0217

- 53

6,56

5773

9329

9994

6 0,

1334

9341

4738

000

55,5

1024

5585

4000

00

816,

5178

6778

8999

979

423,

3734

4077

6999

985

182,

5143

2626

1000

008

338,

2088

0207

2000

026

0,05

1643

3753

7140

0 26

,135

7611

2559

9998

30

8,45

7002

0169

9997

9 2,

3196

7237

5590

000

Lilje

holm

en

4200

,000

0000

0000

0000

0,

8032

6067

1494

000

3606

,655

3792

9999

9822

- 21

39,2

6651

4139

9997

99

0,15

4922

8248

7900

0 37

,578

3648

1310

0002

24

94,1

9462

2619

9997

91

593,

3446

2070

4000

022

228,

2158

0217

1999

997

332,

3145

5162

4999

979

0,04

2765

4370

2670

0 21

,870

0962

8750

0001

30

8,87

4828

1809

9999

8 2,

5999

2785

3620

000

74

Örn

sber

g

900,

0000

0000

0000

000

0,74

9512

1592

5600

0 79

2,47

9968

4939

9999

1 - 14

49,2

4409

4910

0000

58

0,17

4508

9244

5300

0 66

,788

3121

9740

0003

16

97,6

2515

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Appendix IV: GWR Prediction by ArcGIS (ArcMap)

ArcMap proposes an option while calculating the GWR model to predict dependent variables at given location as well as their respective GWR equations. The tool is very useful and the results for 2016 are presented in the table below with each respective prediction done with the OLS equation.

The way these predictions were executed and calculated by ArcMap are unclear. It was for this reason that the GWR equations for the new stations were finally determined using the prediction from the OLS equation.

TRITA TRITA-ABE-MBT-19647

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Documents

Geographically Weighted Regression as a Predictive Tool ...1351572/FULLTEXT01.pdf · This thesis studies a new regression method, Geographically Weighted Regression (GWR) to predict