How Does the Effect Fade over Distance? An Inquiry into the Decay Pattern of Distance Effect on Property Values in the Case of Taipei, Taiwan

land
Article
How Does the Effect Fade over Distance? An Inquiry into the
Decay Pattern of Distance Effect on Property Values in the Case
of Taipei, Taiwan
Lin-Han Chiang Hsieh

 Department of Environmental Engineering, Chung Yuan Christian University, Taoyuan 320, Taiwan;
 chianghsieh@cycu.edu.tw

 Abstract: It is generally accepted that the perception of homeowners towards certain potential risks
 or amenities fades as distance from the risk or amenity increases. This study aims to illustrate the
 distance–decay pattern with an appropriate mathematical function. Distance–decay functions and
 parameters that yield the minimum residual sum of squares (RSS) for a given regression model are
 considered to be the optimal approximation for the pattern of decay. The effect of flood risk and
 mass rapid transit (MRT) accessibility on residential housing prices in Taipei, Taiwan, are used as
 examples to test the optimization process. The results indicate that the type of distance function
 affects both the significance and the magnitude of the regression coefficients. In the case of Taipei,
 concave functions provide better fits for both the flood risk and MRT accessibility. RSS reduction
 is up to 10% compared to the blank. Surprisingly, the impact range for the flood risk is found to
 



 
 be larger than that for MRT accessibility, which suggested that the impact range of perception for
Citation: Chiang Hsieh, L.-H. How uncertain risks is larger than expected.
Does the Effect Fade over Distance?
An Inquiry into the Decay Pattern of Keywords: distance–decay function; hedonic price analysis; flood risk perception; public transportation
Distance Effect on Property Values in accessibility
the Case of Taipei, Taiwan. Land 2021,
10, 1238. https://doi.org/10.3390/
land10111238
 1. Introduction
Academic Editors: Tsung-Yu Lee,
 When considering the spatial variation of the perception of residents towards certain
Chia-Jeng Chen, Yin-Phan Tsang and
 potential risk or amenities, it is widely accepted that perceptions of risk decay as distance
Shao-Yiu Hsu
 increases [1–3]. Yet, the theoretical background underlying this consensus remains un-
 clear [4]. The pattern of how the perception decays with distance, i.e., the shape of the
Received: 13 October 2021
Accepted: 10 November 2021
 distance–decay function, is rarely the focus of discussion [5]. This study describes how per-
Published: 12 November 2021
 ception changes using an optimal distance–decay function. The attitudes of homeowners
 towards potential flood risk and public transportation accessibility in Taipei, Taiwan, are
Publisher’s Note: MDPI stays neutral
 used as case studies to demonstrate the process of determining the optimal distance–decay
with regard to jurisdictional claims in
 function and its parameters.
published maps and institutional affil- This study addresses the problem through the identification of two distance effects
iations. on residential housing prices: the effect of the distance to the nearest flood risk area and
 the distance to the nearest mass rapid transit (MRT) station. The distance to flood risk
 area is selected because it represents attitudes of home buyers towards a negative risk
 factor, while the distance to an MRT station represents perceptions towards favorable
Copyright: © 2021 by the author.
 amenities. For both distance variables, a negative association between distance and the
Licensee MDPI, Basel, Switzerland.
 magnitude of the effect has been shown to be statistically significant in previous work [6].
This article is an open access article
 However, the exact distance–decay function, i.e., the shape of the decay curve, has yet to
distributed under the terms and be thoroughly investigated.
conditions of the Creative Commons The decay curves for flood risk and MRT accessibility are presumed to be different.
Attribution (CC BY) license (https:// An illustration of different types of distance–decay function is shown in Figure 1. Generally,
creativecommons.org/licenses/by/ these functions can be categorized as concave and convex functions. Concave functions
4.0/). capture a distance effect that decays slowly over a short distance and then falls quickly

Land 2021, 10, 1238. https://doi.org/10.3390/land10111238 https://www.mdpi.com/journal/land

Land 2021, 10, x FOR PEER REVIEW 2 of 11

 The decay curves for flood risk and MRT accessibility are presumed to be different.
 Land 2021, 10, 1238 An illustration of different types of distance–decay function is shown in Figure 1. Gener-
 2 of 10
 ally, these functions can be categorized as concave and convex functions. Concave func-
 tions capture a distance effect that decays slowly over a short distance and then falls
 quickly after a certain point. The distance effect of MRT accessibility is presumed to be a
 after a certain point. The distance effect of MRT accessibility is presumed to be a concave
 concave function, since the premium for a property located close to an MRT station should
 function, since the premium for a property located close to an MRT station should remain
 remain significant only within walking distance. Convex functions, on the other hand,
 significant only within walking distance. Convex functions, on the other hand, describe
 describe a distance effect that decays dramatically within relatively short distances. The
 a distance effect that decays dramatically within relatively short distances. The effect of
 effect of flood risk on property value is presumed to be a convex function because the
 flood risk on property value is presumed to be a convex function because the impact of
 impact of flooding falls immediately beyond the flood potential area.
 flooding falls immediately beyond the flood potential area.

 Figure 1. A schematic illustration of the shape of distance–decay functions.
 Figure 1. A schematic illustration of the shape of distance–decay functions.
 Different patterns of distance–decay functions for different distance effects have
 Different
 seldom beenpatterns
 addressed of distance–decay
 in previous studies. functions for different
 In this study, the distance
 optimaleffects have sel-
 distance–decay
 dom been addressed in previous studies. In this study, the
 functions for both the flood risk and MRT accessibility effect are defined as curves optimal distance–decay func-
 with
 tions for both the flood risk and MRT accessibility effect are defined
 minimum residual sum of squares (RSS). For each distance–decay function, a corresponding as curves with mini-
 mum residual
 distance sumisof
 variable squares (RSS).
 generated. For each
 By applying eachdistance–decay
 distance variable function, a corresponding
 to an identical regression
 distance variable
 model, the is generated.
 RSS measures By applying
 the goodness of fiteach
 of thedistance variable to
 distance–decay an identical
 function. regres-
 The distance–
 sion model,
 decay the RSS
 function withmeasures
 the minimum the goodness
 correspondingof fit ofRSS
 the is
 distance–decay
 identified as the function.
 optimalThe dis-
 function
 tance–decay
 to describe howfunction with thechange
 perceptions minimum corresponding
 as distance increases. RSS is identified as the optimal
 functionThetoresults
 describe how perceptions
 indicate that the selectionchange of aas distance increases.
 distance–decay function is an important issue
 The results indicate that the selection of
 for distance variables. The RSS reduction ranges from 0.1% to 10% a distance–decay function is an important
 for different distance–
 issue
 decayforfunctions,
 distance variables.
 showing that The the
 RSSselection
 reduction ofranges
 function from 0.1% affects
 directly to 10%the for goodness
 different dis-
 of fit
 tance–decay
 of a model.functions,
 In addition, showing that the
 the results selection
 in Taipei show of function directlywith
 that compared affects the goodness
 convex functions,
 ofconcave
 fit of a model.
 functions In addition,
 are generally the results in Taipei show that
 better approximations compared
 to the decay ofwith effectsconvex func-
 for both the
 tions,
 floodconcave
 risk andfunctions are generally
 MRT accessibility betterThe
 variables. approximations
 main difference to the decayvariables
 between of effectslies
 forin
 the the
 both range of the
 flood riskeffect,
 and MRT rather than the pattern
 accessibility of decay.
 variables. The main difference between variables
 The results of this study have several
 lies in the range of the effect, rather than the pattern of decay. theoretical and practical implications. Theo-
 retically, the methodology
 The results of this study used have in this study
 several is shown
 theoretical to be an implications.
 and practical appropriate method Theoret-to
 determine
 ically, the optimal distance–decay
 the methodology used in this study function
 is shownfor any distance-related
 to be an appropriatevariable.
 methodPractically,
 to deter-
 in the
 mine thecase of Taipei,
 optimal the results demonstrate
 distance–decay function for any the distance-related
 effectiveness of avariable.
 concave Practically,
 function and in a
 carefully
 the case of determined
 Taipei, the results impact demonstrate
 range for each thedistance-related
 effectiveness ofvariable.
 a concave It isfunction
 recommended
 and a
 to first determine
 carefully determinedthe function
 impact range typeforand
 eachimpact range for any
 distance-related distance
 variable. It is variables
 recommendedbefore
 applying them in a regression model. This article is structured
 to first determine the function type and impact range for any distance variables before in five sections: introduction,
 previousthem
 applying literatures, material model.
 in a regression and methods, results
 This article is and discussion,
 structured in fiveand conclusion.
 sections: introduc-
 tion, previous literatures, material and methods, results and discussion, and conclusion.
 2. Previous Literatures
 Most of the literature concerning distance–decay focuses on the determination of
 distance–decay parameters, rather than the formulation of distance–decay functions or
 their theoretical background. For example, in the field of geography, the distance–decay
 parameter used in gravity models has attracted attention since the 1970s [7,8]. Gravity
 models, which borrow ideas from the theory of gravitational attraction, describe the

Land 2021, 10, 1238 3 of 10

 interaction patterns between separated locations. The discussion thus focuses on the
 determination and calibration of the distance–decay parameter in gravity models [9–11].
 Some authors have further investigated heterogeneity within parameters [12–14], but
 examples of work focused on analyzing perception change using the rule of gravity are
 lacking. Extrapolating gravity theory in physics, Willigers et al. [15] concluded that for a
 gravity model that describes the accessibility of rail transportation, a Box-Cox impedance
 function performs better than exponential and power functions. However, we argue that
 perception change falls within the scope of the gravity model, and therefore the use of this
 theory is not yet fully justified.
 Similarly, hedonic studies that apply the concept of distance–decay usually presume a
 certain type of decay curve without theoretical justification. When determining the hedonic
 prices for the accessibility of certain environmental amenities, the most commonly used
 distance variable is inverse-distance [1,16,17], which presumes that an effect increases at
 the same rate as the inverse distance increases. Other distance variables, such distance
 itself [2] or travel time [3] represent other types of distance–decay functions. The selection
 of the distance–decay function is seldom justified in the studies cited above.
 The discussion of the ‘fitness’ of a distance–decay functions is lacking. Kent et al. [18]
 and Iacano et al. [19] use different mathematical curves to fit decay in frequency as distance
 increases, based on actual crime and survey data, respectively. Martínez and Viegas [20]
 use the RSS value as a tool to determine the fitness of multiple distance–decay curves in
 mapping the relationship between the willingness to go to a destination and its actual
 distance, based on survey data. However, the number of functions used in the study are
 limited, and the rationale behind the selection of functions remains unclear.
 In summary, the theoretical work on the selection of distance–decay functions are, in
 our opinion, insufficient. This study aims to further research in this field by evaluating
 the theory of distance–decay functions and by determining an optimal distance–decay
 function for the selected dataset.

 3. Materials and Methods
 3.1. Data
 Data used in this study are all publicly available. The flood potential maps used in
 this study were published by Taipei City Government in September 2015 [21], following
 comprehensive simulation based on the condition of drainage systems and the intensity
 of precipitation. The published maps illustrate simulated flood areas following hourly
 precipitation of 78.8 mm, 100 mm, and 130 mm, respectively (shown in Figure 2b). The
 precipitation of 78.8 mm in an hour reflects a rainfall intensity that occurs once every five
 years in Taipei on average, and is used as a reference in the design of local drainage. A
 previous study has shown that only flood risk arising from an intensity of 78.8 mm of
 precipitation per hour is negatively associated with housing prices [6]. Thus, in this study,
 the discussion of distance to flood potential is limited to a flood risk stemming from a
 precipitation intensity of 78.8 mm per hour.
 In order to investigate how the effect of flood risk on residential housing value
 decays over distance, the distance to the nearest potential flood area for each sales record
 was calculated using ArcGIS. Since the flood potential maps mentioned in the previous
 paragraph only illustrate simulated flood areas in public open spaces (mainly on roads and
 streets), it makes more sense to use a distance variable rather than other common flood
 variables, such as a flood dummy or flood depth. We hypothesize that the impact of floods
 on housing prices decays relative rapidly when the distance exceeds the impact zone. Thus,
 the decay curve is expected to be best captured by either convex functions [6] or buffer
 zones [22].

Land 2021, 10, 1238 4 of 10
Land 2021, 10, x FOR PEER REVIEW 4 of 11

 Figure 2.
 Figure 2. Research area and the flood potential map. (a) Research area comparing to the boundary of Taipei
 Taipei City;
 City; (b)
 (b) flood
 flood
 potential maps
 potential maps under
 under 78.8
 78.8 mm/h
 mm/h precipitation
 precipitation intensity.
 intensity.

 The maptoofinvestigate
 In order the MRT stationhow the exits is publicly
 effect of flood available on the Government
 risk on residential housing value open data
 decays
 platform website
 over distance, the(accessed
 distance on 1 August
 to the nearest2020, data.gov.tw).
 potential Similarly,
 flood area for eachthe distance
 sales record towas
 the
 nearest
 calculatedMRT exit ArcGIS.
 using for each sales
 Sincerecord
 the floodwaspotential
 calculated via ArcGIS.
 maps mentionedAs mentioned
 in the previous previously,
 para-
 the
 grapheffect decay
 only pattern
 illustrate over distance
 simulated floodisareas
 expected to resemble
 in public a concave
 open spaces (mainlyfunction.
 on roads and
 Theitresidential
 streets), makes more property
 sense sales
 to usedata used invariable
 a distance this study werethan
 rather collected
 otherfromcommon the actual
 flood
 sales price registration system of the Department of Land Administration
 variables, such as a flood dummy or flood depth. We hypothesize that the impact of floods (DLA). A total of
 4777 sales records are listed in 2016 (the year after the disclosure
 on housing prices decays relative rapidly when the distance exceeds the impact zone. of the flood maps), which
 are
 Thus,located in seven
 the decay curvecentral districts
 is expected to of
 beTaipei (shown in
 best captured byFigure 2a). Four
 either convex surrounding
 functions [6] or
 districts are excluded
 buffer zones [22]. from the study for two principal reasons. First, mountain areas with
 low housing
 The mapdensity in these
 of the MRT fourexits
 station districts createavailable
 is publicly potentialonoutliers when analyzing
 the Government the
 open data
 data based on spatial distribution. Secondly, home buyers’ perception
 platform website (accessed on 2020/08/01, data.gov.tw). Similarly, the distance to the near-of flood risk may be
 different for urban and rural houses. Focusing on an urban area
 est MRT exit for each sales record was calculated via ArcGIS. As mentioned previously, helps to determine the
 hedonic
 the effectprice
 decay of pattern
 flood risk overin distance
 an urbanisenvironment.
 expected to resemble a concave function.
 For each sales record, variables
 The residential property sales data thatused
 are considered
 in this studytowere
 be influential
 collected fromto thethehousing
 actual
 prices
 sales price registration system of the Department of Land Administration (DLA). as
 are collected through multiple sources. Basic property characteristics such A area
 total
 size, age when the transaction occurred, building type (condominium
 of 4777 sales records are listed in 2016 (the year after the disclosure of the flood maps),or mansion), number
 of rooms
 which areand bathrooms,
 located in seventhe existence
 central of aofmanagement
 districts Taipei (shown committee, andFour
 in Figure 2a). the number
 surround- of
 parking spaces, are listed in the original dataset. To identify the
 ing districts are excluded from the study for two principal reasons. First, mountain areaseffect of floors when
 considering
 with low housing the total building
 density height,
 in these foura districts
 categorical variable
 create named
 potential floorwhen
 outliers level is created
 analyzing
 based on the relative vertical location of the property. For example, the relative height of a
 the data based on spatial distribution. Secondly, home buyers’ perception of flood risk
 third-floor property in a four-floor condominium is 0.75, thus the floor level is 3 (between
 may be different for urban and rural houses. Focusing on an urban area helps to determine
 0.6 to 0.8). Neighborhood characteristics, such as accessibility to parks, were captured using
 the hedonic price of flood risk in an urban environment.
 ArcGIS. Variables used in the regression model in this study, along with the demographic
 For each sales record, variables that are considered to be influential to the housing
 descriptive statistics, are listed in Table 1.
 prices are collected through multiple sources. Basic property characteristics such as area
 size,Methodology
 3.2. age when the transaction occurred, building type (condominium or mansion), num-
 ber of rooms and bathrooms, the existence of a management committee, and the number
 This study aims to illustrate how the perceptions of homeowners change over distance
 of parking spaces, are listed in the original dataset. To identify the effect of floors when
 with mathematical functions. As mentioned previously, functions can be categorized into
 considering the total building height, a categorical variable named floor level is created
 two groups, convex and concave functions. For convex groups, the inverse of distance,
 based on the relative vertical location of the property. For example, the relative height of
 and the negative exponential of distance were selected in this study. For concave groups,
 a third-floor
 linear decay overproperty in a squared
 distance, four-floor condominium
 decay, and ellipseisfunctions
 0.75, thus the selected.
 were floor level is 3 (be-
 Finally, an
 tween 0.6 to 0.8). Neighborhood characteristics, such as accessibility
 all-or-nothing buffer zone was also included as a comparison group to check the validity to parks, were cap-
 tured
 of using between
 selecting ArcGIS. Variables
 convex and used in the groups.
 concave regressionThe model in this study,
 mathematical along for
 definition witheach
 the
 demographic descriptive statistics, are listed in Table 1.

dist.
 sale to
 to MRT
 dist. priceflood station
 (78.8 mm)
 (logged) (m)(m) 443.63
 1919.66
 16.6 293.91
 1345.61
 0.8 3.63
 11
 13.1 2023
 5500
 20.4
 area
 dist. (m²)
 to MRT station
 dist. to flood (78.8 mm) (m) (m) 108.44
 443.63
 1919.66 84.40
 293.91
 1345.61 7.73
 3.63
 11 1149
 2023
 5500
 age
 area(yr)
 dist. (m²)
 to MRT station (m) 17.07
 108.44
 443.63 15.94
 84.40
 293.91 0.0
 7.73
 3.63 97.4
 1149
 2023
 floor
 area level
 age (yr)
 (m²) (0 to 5) 3.46
 17.07
 108.44 1.28
 15.94
 84.40 1
 0.0
 7.73 97.4
 1149 5
Land 2021, 10, 1238
 condo
 floor(yr)
 age (≤5 storeys)
 level (0 to 5) 0.21
 3.46
 17.07 0.41
 1.28
 15.94 0
 0.01 597.4
 15
 of 10
 condo (6~10
 condolevel
 floor (0storeys)
 (≤5 storeys)
 to 5) 0.35
 0.21
 3.46 0.48
 0.41
 1.28 010 151
 mansion
 condo (≤5
 condo (>10
 (6~10 storeys)
 storeys)
 storeys) 0.18
 0.35
 0.21 0.39
 0.48
 0.41 000 111
 rooms
 mansion
 condo (6~10(>10storeys)
 storeys) 2.15
 0.18
 0.35 1.30
 0.39
 0.48 000 10 11
 function
 bathrooms used in this study is listed in Table1.41 2. In each equation,
 0.79 x denotes 0 the distance 10
 rooms
 mansion (>10 storeys) 2.15
 0.18 1.30
 0.39 00 10 1
 flood area or MRT exit, y denotes the corresponding
 tomanagement distance effect. Parameters a to1e
 bathrooms committee
 rooms 0.58
 1.41
 2.15 0.49
 0.79
 1.30 000 10
 10
 denote
 parking the range of impact for each distance0.33 effect, in which
 0.66 the effect falls 000 to zero after 61
 management committee
 bathrooms 0.58
 1.41 0.49
 0.79 10
 that
 dist. distance.
 to park (m) Each distance effect y is set using
 129.70 a zero-to-one
 93.19 scale so that 00 the effects are
 519.016
 parking
 management committee 0.33
 0.58 0.66
 0.49
 directly comparable. For each function, multiple parameters are tested to0determine the
 dist. to park (m)
 parkingfunctional form.
 optimal 129.70
 0.33 93.19
 0.66 00 519.0 6
 3.2. Methodology
 dist. to park (m) 129.70 93.19 0 519.0
 3.2. Methodology
 This study
 Table aims to illustrate
 1. Demographic how of
 descriptive thevariables.
 perceptions of homeowners change over dis-
 tance with mathematical functions. As
 This study aims to illustrate how the perceptions
 3.2. Methodology mentioned previously, functions
 of homeowners can beover
 change catego-
 dis-
 rized
 tanceThisMean
 into two
 withstudy groups,
 mathematical Std.
 convex Dev.
 and
 functions. concave
 As mentioned Min
 functions. For convex
 previously, groups,
 functions Max
 the
 can inverse
 be catego- of
 aims to illustrate how the perceptions of homeowners change over dis-
 distance,
 rized into and the negative exponential of distance were selected in this study. For concave
 sale price (logged) tance withtwo groups,
 16.6
 mathematical convex and concave
 functions. functions.
 As0.8mentioned For convex
 previously, groups,can
 13.1 functions the be
 inverse20.4of
 catego-
 dist. to flood (78.8 mm) (m) groups,
 distance, linear
 and thedecay
 1919.66 over exponential
 negative distance, squared
 1345.61
 of decay,
 distance and
 were ellipse
 selected 11infunctions
 this study. were
 For selected.
 5500
 concave
 rized into two groups, convex and concave functions. For convex groups, the inverse of
 dist. to MRT station (m) Finally,
 groups, an linear 443.63
 all-or-nothing
 decay overbuffer zone
 distance, 293.91
 was also
 squared included
 decay, and as a3.63
 comparison
 ellipse group 2023
 to check
 area (m2 ) distance, and108.44
 the negative exponential of distance
 84.40 were selected
 7.73infunctions
 this study. were
 Forselected.
 concave
 1149
 the validity
 Finally, an of selecting between
 all-or-nothing convex andalso
 concave groups.aThe mathematical definition
 age (yr) groups, linear decay overbuffer
 17.07
 zonesquared
 distance, was
 15.94
 included
 decay, and as comparison
 ellipse
 0.0 functionsgroup to check
 were selected.
 97.4
 for
 the each
 validityfunction used in
 of selecting this study is listed
 between in Table 2. In each equation, x denotes the
 floor level (0 to 5) Finally, an all-or-nothing
 3.46 buffer convex
 zone 1.28
 wasand concave
 also groups.
 included as a The1 mathematical
 comparison groupdefinition
 to check 5
 distance
 for validityto flood
 each function area or MRT exit, y denotes the corresponding distance effect. Parameters
 condo (≤5 storeys) the 0.21used between
 of selecting in this study
 convex is listed in Table
 and concave
 0.41 2. In each
 groups. The equation, x denotes
 0 mathematical definitionthe
 1
 adistance
 to e denote the range orof impact for each distance effect, in which the effect falls to zero
 condo (6~10 storeys) for each to flood
 function area
 0.35used MRT
 in thisexit,
 studyy denotes
 listedthe
 0.48
 is incorresponding
 Table 2. In each distance
 0 equation, effect. Parameters
 x denotes 1
 the
 mansion (>10 storeys) after
 to ethat
 adistancedenotedistance.
 the0.18 Eachofdistance
 range impact effect
 for y isdistance
 0.39
 each set usingeffect,
 a zero-to-one
 in 0
 which scale
 the so thatfalls
 effect thetoeffects
 zero1
 to flood area or MRT exit, y denotes the corresponding distance effect. Parameters
 rooms are
 afterdirectly
 that comparable.
 distance. For eacheffect
 2.15 Each distance function,
 1.30
 y is multiple
 set using aparameters
 zero-to-one0 are tested
 scale so to determine
 that the 10
 effects
 a to e denote the range of impact for each distance effect, in which the effect falls to zero
 bathrooms the
 are optimal
 directly 1.41
 functional
 comparable. form. 0.79 multiple parameters
 For each function, 0 are tested to determine 10
 management committee after that distance. 0.58 Each distance effect0.49 y is set using a zero-to-one 0 scale so that the effects 1
 the optimal
 are directly functional
 comparable. form.
 For each function, multiple parameters
 parking Table 2. List of0.33 distance–decay functions 0.66
 tested in this study. 0 are tested to determine 6
 dist. to park (m) the optimal functional
 129.70 form. 93.19 0 519.0
 Function Group Table 2.Name
 Function List of distance–decay functions tested in this study.
 Equations Illustration
 Function Group Table 2. List of distance–decay functions
 Function tested in this study.
 Table 2. Name Equations
 List of distance–decay functions tested in this study. Illustration
 Function Group Function Name Equations Illustration
 Function Group Function
 Inverse Name
 of distance =1 1
 Equations Illustration

 Inverse of distance =1 1
 Convex Inverse of distance
 Inverse of distance y==1 1a 1 1
 Convex x
 Convex
 Convex Negative exponential = ( )

 ( )
 Negative exponential y =
 = e −( b )
 x
 Land 2021, 10, x FOR PEER REVIEW Negative exponential 6 of 11
 ( )
 Negative
 Land 2021, 10, x FOR PEER REVIEW exponential = 6 of 11

 Land 2021, 10, x FOR PEER REVIEW 6 of 11
 y =
 = 11 −− x,,
 Concave Linear decay
 Linear decay c
 y = 0 if
 ==1 1−−
 = 0 if x( ≥), c,
 Concave
 Concave Linear decay
 Squared decay =
 = 011
 ==0 if−−
 if ( ), 
 ,
 Concave Linear decay
 Squared decay 
 = =0 0 if
 if 
 y =1 1− − ( x)
2 ,,
 Squared decay
 Squared decay d
 y =
 = 00 if
 if x ≥ d
 = 1,
 Ellipse 
2 2 = 1,
 xe = + y = ,1,
 √
 Ellipse y==0 if e e−= ,1,
 
 2 x2
 Ellipse
 Ellipse y = 0 if x
 ==0 if , ≥ e

 = 0 if 
  1, ≤ 
 Buffer
 Buffer Buffer
 Buffer y = 1, x ≤ f
 y = 0, > 
 1,
 0, x ≤ > f
 Buffer Buffer y=
 1, ≤ 
 0, >
 Buffer for a distance less thanBuffer y=
 for a distance less than 1 m, the inverse of distance y =0,1/x will
 > than
 be larger
 1 Theoretically, 1 m, the inverse of distance y = 1/x will be larger 1. However,
 1 Theoretically, than since there is no
 1. However, sale there
 since recordisused
 no
 in this study located within 1 m of the flood areas or MRT exits, the value of inverse distance remains within the scale 0 to 1.
 sale record used in this study located within 1 m of the flood areas or MRT exits, the value of inverse distance remains
 1 Theoretically, for a distance less than 1 m, the inverse of distance y = 1/x will be larger than 1. However, since there is no
 within the scale 0 to 1.
 1sale record used in this study located within 1 m of the flood areas or MRT exits, the value of inverse distance remains
 Theoretically, for a distance less than 1 m, the inverse of distance y = 1/x will be larger than 1. However, since there is no
 within the scale 0 to 1.
 sale record used in this study located within
 Distance 1 m of the
 variables thatflood areas or
 represent MRT exits,
 different decaythe functions
 value of inverse distance
 for both remains
 the flood risk and
 within the scale 0 to 1. the accessibility to MRT stations are generated respectively. A hedonic price model based
 Distance variables that represent different decay functions for both the flood risk and
 on
 theordinary leasttosquare
 accessibility (OLS) regression
 MRTthat
 stations is applied
 are generated to further
 respectively. determine
 A hedonic themodel
 optimal dis-
 Distance variables represent different decay functions for bothprice
 the flood based
 risk and
 tance–decay
 on ordinary function that
 leasttosquare describes perception
 (OLS) regression changes
 is applied over
 to furtherdistance.
 determineFor example,
 themodel
 optimal con-
 dis-
 the accessibility MRT stations are generated respectively. A hedonic price based

Land 2021, 10, 1238 6 of 10

 Distance variables that represent different decay functions for both the flood risk
 and the accessibility to MRT stations are generated respectively. A hedonic price model
 based on ordinary least square (OLS) regression is applied to further determine the optimal
 distance–decay function that describes perception changes over distance. For example,
 consider a model that focuses on the hedonic price of flood risk:

 y = ln(sales price) = β 0 + β 1 (distance to f lood)1 + βX + ε 1 = ŷ1 + ε 1 (1)

 where the dependent variable y is logged sales price, (distance to flood)1 denotes the
 distance effect of flood for function 1, β1 is the regression coefficient of (distance to flood)1,
 βX denotes the coefficients and the corresponding controlling variables, ŷ1 is the estimated
 value of y with distance effect 1, and ε1 is the corresponding residual. Following the same
 logic, applying another distance–decay function, say function 2, gives:

 y = β 0 + β 2 (distance to f lood)2 + βX + ε 2 = ŷ2 + ε 2 (2)

 Comparing the RSS gives:
 ε12 > ε22 (3)

 Given that all the controlling variables for Model 1 and 2 are identical, if Model 2 re-
 sults in a smaller unexplained component, this indicates that function 2 describes the effect
 decay pattern over distance better. The distance–decay function with the minimum RSS for
Land 2021, 10, x FOR PEER REVIEW the regression model is considered to be the optimal approximation of the decaying pattern.
 7 of 11
 The illustration of conceptual framework of this study is as shown in Figure 3.

 Figure 3.
 Figure 3. Conceptual framework scheme.
 Conceptual framework scheme.

 4. Results and
 4. Results and Discussion
 Discussion
 4.1.
 4.1. Determination of
 Determination of Optimal
 Optimal Parameters
 Parameters
 This
 This study
 study uses
 uses the
 the method
 method of of minimum
 minimum residual
 residual sum
 sum of
 of squares
 squares to
 to determine
 determine the
 the
 optimal form and parameters for each distance–decay function. For each function
 optimal form and parameters for each distance–decay function. For each function type, type,
 several candidate parameters are pre-selected to determine the optimal function. For each
 several candidate parameters are pre-selected to determine the optimal function. For each
 parameter, a distance effect variable is generated based on the corresponding distance–
 parameter, a distance effect variable is generated based on the corresponding distance–
 decay equation, reflecting a decay of the perception of home buyers towards flood risk or
 decay equation, reflecting a decay of the perception of home buyers towards flood risk or
 MRT accessibility with distance. These distance variables are then added into a regression
 MRT accessibility with distance. These distance variables are then added into a regression
 model with fixed control variables. The distance variable that results in the smallest RSS
 model with fixed control variables. The distance variable that results in the smallest RSS
 for the regression model is presumed to be the optimal function to capture homebuyer
 for the regression model is presumed to be the optimal function to capture homebuyer
 perception decay patterns. This function and the corresponding parameters are consid-
 ered as the optimal decay function for a certain distance effect.
 We use the determination of the parameters for the inverse of distance for the flood
 risk as an example to further illustrate the optimization process. The regression results are

Land 2021, 10, 1238 7 of 10

 perception decay patterns. This function and the corresponding parameters are considered
 as the optimal decay function for a certain distance effect.
 We use the determination of the parameters for the inverse of distance for the flood
 risk as an example to further illustrate the optimization process. The regression results
 are listed in Table 3. In this case, two patterns reflecting how the flood effect decays
 over distance (y = x), the inverse of distance (y = 1⁄x) and the squared inverse of distance
 (y = 1⁄xˆ2), are compared. Both distance-to-flood variables are added into a hedonic price
 regression model with identical variables, including property characteristics (area, age,
 floor level, building type, number of rooms, the existence of management committee, and
 parking spaces), and neighborhood characteristics (distance-to-MRT, nearby parks). Note
 that RSS values obtained for both models with flood variables are less than that identified
 for the blank model (with no flood variable), showing that the inclusion of distance-to-flood
 effect helps to explain the difference in sales prices. The result shows that using the inverse
 of distance to describe the decay of flood effect yields the smallest RSS, which indicates
 that the optimal equation form for this group is the inverse of distance.

 Table 3. Regression results for inversed distance-to-flood variable.

 Flood Variable:
 Blank Inverse of Distance Squared Inv. of Distance
 obs. (group #) 4777 4777 4777
 R-squared 0.753 0.7587 0.7547
 flood variable - −18.430 *** −139.945 ***
 MRT effect (sq. neg.) 0.431 *** 0.449 *** 0.434 ***
 area (m2 ) 0.0062 *** 0.006 *** 0.0062 ***
 age −0.007 *** −0.007 *** −0.007 ***
 floor level 0.000 −0.001 −0.002
 floor level sq. −0.001 −0.001 −0.001
 condo (≤5 storeys) 0.172 *** 0.151 *** 0.170 ***
 condo (6~10 storeys) 0.333 *** 0.325 *** 0.330 ***
 mansion (>10 storeys) 0.337 *** 0.320 *** 0.333 ***
 rooms 0.068 *** 0.067 *** 0.067 ***
 bathrooms 0.041 *** 0.041 *** 0.041 ***
 management committee −0.011 −0.020 −0.010
 parking −0.074 *** −0.069 *** −0.073 ***
 dist. to park (1/m) 0.003 0.003 0.001
 constant 15.351 *** 15.381 *** 15.354 ***
 residual sum of squares (RSS) 673.736 658.860 669.866
 * p < 0.10, ** p < 0.05, *** p < 0.01.

 Note that most of the controlling variables included in the model are statistically signif-
 icant, and the R-squared value is relatively high even for the blank model. Both shows that
 the model has good explanation power. In addition, the significances are consistent among
 models, indicating the stability of the model. The meaning of coefficients for each variable
 is not addressed here because OLS regression for spatial data usually raises the concern for
 endogeneity and heterogeneity [6], which might bias the coefficients. The purpose of this
 study is to determine the optimal distance–decay function that fit home buyers’ perception
 changes through RSS changes. Theoretically, endogeneity and heterogeneity do not alter
 the relative explanation power of the model. Thus, OLS regression model still serves well
 for the purpose of this study.
 Following the same procedure, the optimal equation forms for each function type, and
 for both distance-to flood and MRT accessibility effect are listed in Table 4. Note that while
 determining the optimal function type for one of the variables of interest, the function type
 for the other variable must be holding constant so that the RSS of different models can be
 directly comparable. For the convex group (inverse of distance and negative exponential),
 the optimal form for the flood effect and MRT accessibility are identical. For the concave
 group and the buffer zone, optimal parameters for the flood effect and MRT accessibility

Land 2021, 10, 1238 8 of 10

 vary significantly. The result clearly indicates that the range of impact for MRT accessibility
 is generally smaller than that for the flood effect. For example, the optimal parameter for
 describing the decay of flood effect as an ellipse is 2000, which means that the impact of
 flood risk on housing prices does not fade away until the flood area is 2000 m away. On
 the other hand, the premium of being close to MRT exits fades quickly after 800 m if the
 decay curve is described as an ellipse. This result is inconsistent with our expectations.
 One possible explanation is the degree of uncertainty. The location of flood occurrence is
 much more uncertain compared with the location of MRT exits. Facing a risk of uncertainty,
 people tend to err on the side of caution and locate themselves further away. The benefit of
 access to public transportation is relatively certain. The impact range for public transport
 found in this study is generally consistent with that identified in other studies [23,24].

 Table 4. Optimal functions and parameters used in this study.

 Function Name Equations Tested Parameters Optimal Form for Flood Optimal Form for MRT
 Inverse of distance 1 a = 1, 2 1 1
 y= xa y= x y= x
 x x
 Negative exponential y=e −( xb ) b = 1, 10 y= −(
 e 10 ) y= −(
 e 10 )
 c = 100, 500, 800,
 x x x
 Linear decay y = 1− c 1200, 2000, 2500, y = 1− 2500 y = 1− 2500
 3000
 d = 100, 500, 800,
 x 2
 
 x
 
2 x
 
2
 Squared decay y = 1− d
 1200, 2000, 2500, y = 1− 2000 y = 1− 1200
 3000

 √ e = 100, 500, 800, √ √
 Ellipse e2 − x 2 1200, 2000, 2500, 20002 − x2 8002 − x2
 y= e , y= 2000 y= 800
 3000
  f = 50, 100, 500,  
 Buffer 1, x ≤ f 800, 1200, 2000, 1, x ≤ 50 1, x ≤ 800
 y= y= y=
 0, x > f 2500, 3000 0, x > 50 0, x > 800

 4.2. Comparison of Distance–Decay Functions
 This study aims to further determine the optimal distance–decay function for each
 distance effect by comparing RSS values of function types. The results for the effect of
 distance-to-flood and MRT accessibility are listed in Tables 5 and 6, respectively. As shown
 in Table 5, the coefficients for all distance variables are significant and negative, indicating
 that the effect of flood risk on housing prices decays over distance, no matter how the
 decay pattern is described. However, the result of RSS reduction indicates that concave
 functions are more suitable when describing the pattern of how flood risk decays over
 distance. Compared with the blank model (with no distance-to-flood variable), RSS for
 models with concave distance–decay variables are significantly reduced by more than
 10%. For example, by including a variable which describes the distance-to-flood effect as a
 squared decay curve that vanishes over 2000 m, the unexplained residual is reduced by
 10.7%. This is a significant reduction for a model with thirteen controlling variables. As a
 comparison, describing the distance–decay pattern as a negative exponential curve only
 improves the RSS by 0.2%. This result clearly indicates that the decay pattern over distance
 does matter in terms of the explanation power of the model. For the case of flood effect
 decay over distance in Taipei, using concave functions is clearly more suitable than other
 function types.

Land 2021, 10, 1238 9 of 10

 Table 5. RSS comparison of functions for distance-to-flood effect.

 RSS
 Distance Variable p-Value R2 RSS
 Improved
 blank - 0.7532 673.736 -
 convex group inverse dist. 0 0.7587 658.860 2.2%
 neg. exponential 0 0.7546 670.056 0.5%
 linear decay (2500 m) 0 0.7785 604.709 10.2%
 concave group squared decay (2000 m) 0 0.7795 601.956 10.7%
 ellipse (2000 m) 0 0.7785 604.655 10.3%
 buffer (50 m) 0 0.7551 668.535 0.8%

 Table 6. RSS comparison of functions for MRT accessibility effect.

 RSS
 Distance Variable p-Value R2 RSS
 Improved
 blank - 0.7544 667.889 -
 convex group inverse dist. 0.003 0.7569 661.277 1.0%
 neg. exponential 0.017 0.7547 667.099 0.1%
 linear decay (2500 m) 0 0.7788 605.267 9.4%
 concave group squared decay (1200 m) 0 0.7795 604.709 9.5%
 ellipse (800 m) 0 0.7782 607.767 9.0%
 buffer (800 m) 0 0.7734 620.340 7.1%

 The results in Table 6 show similar trends. The positive effect of MRT accessibility
 on housing prices significantly decays over distance, no matter how the decay pattern is
 described. Similarly, the explanation power for models with concave distance variables
 are significantly better than those for other function types. In the case of MRT accessibility
 in Taipei, it is more suitable to describe the premium decay of accessing MRT using a
 concave function.

 5. Conclusions
 This study illustrates the pattern of the perception of home buyers decaying over
 distance. A process based on ordinary least square regression is tested in the case of Taipei
 to determine the optimal perception decay curve. For both tested distance effects, the
 flood risk, and the mass rapid transit (MRT) accessibility, the perception decay pattern
 can be best represented by concave curves, where the effect decays slowly at first and
 then decreases rapidly after a certain distance. The result for the flood perception is not
 as expected because the perception change pattern towards natural hazards is usually
 recognized as either buffer zone or convex curves, in which the effect falls off drastically
 over a short distance.
 In addition, the result further indicates that the impact range varies between flood risk
 and public transportation accessibility. The impact range for flood risk is generally larger
 than that for MRT exits. This result implicitly identifies the spillover of risk perception of
 homebuyers, because the perception impact range of 2 km is much larger than the actual
 impact range of flood events.
 These results have policy and methodological implications. On the policy side, the
 results support the need for the Government to improve infrastructure and protect against
 natural hazards. Since the perception impact range of flood risk is larger than expected,
 the benefit of mitigating flood events is larger than expected in terms of social benefit;
 this can be quantified as the prevention of loss in residential property values. On the
 methodological side, the optimization process developed in this study is a useful tool when
 determining a more accurate distance variable in a model.
 There is limitation for this study. The developed method is only tested in a specific
 area within a short period of time. The policy suggestion based on the results of this

Land 2021, 10, 1238 10 of 10

 study is not necessarily applicable to other cities or contexts. Further studies based on this
 methodology are required before the proposed optimization process can be fully validated.

 Funding: This work was supported by the Ministry of Science and Technology, Taiwan [Grant
 Number 107-2119-M-033-004- and 109-2221-E-033-004-MY2].
 Data Availability Statement: The residential property sales data is available at: https://lvr.land.
 moi.gov.tw/ on 1 August 2018. The flood potential map is available at: https://data.gov.tw/dataset/
 121550 on 1 August 2016.
 Acknowledgments: I wish to thank Uni-edit (www.uni-edit.net, 30 September 2021) for editing and
 proofreading this manuscript.
 Conflicts of Interest: The author declares no conflict of interest.

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