Spatial Data Science

👩🏻‍💻 Dr Thiyanga S. Talagala @ University of Sri Jayewardenepura

• ttalagala@sjp.ac.lk
• @thiyangt
• github.com/thiyangt
• https://thiyanga.netlify.app/

Outline

• Introduction

• Three types of spatial processes

• Isotropy

• Spatial kriging

Introduction

A spatial stochastic process is a family of random variables \[\{Z(s): s \in D\ \subset \mathbb{R}^d \}.\]

Here,

\(D\) is the spatial domain in which observations could be made and

\(Z(s)\) is a random variable representing the quantity that you measured at location \(s\).

Daily temperature in Canada

Locations of Canadian weather stations: \(s_i\)

library(sf)
library(geoFourierFDA)
library(rnaturalearth)
canada$m_coord

            N.latitude W.longitude
Regina           50.25     -104.39
Sherbrooke       45.24      -71.54
Winnipeg         49.53      -97.09
Resolute         74.41      -94.54
Yellowknife      62.27     -114.21
Edmonton         53.33     -113.28
Dawson           64.04     -139.25
Ottawa           45.25      -75.22
Fredericton      45.58      -66.39
Churchill        58.46      -94.10
Whitehorse       60.43     -135.03
Sydney           46.09      -60.11
Scheffervll      54.47      -64.49
Montreal         45.31      -73.34

Example: \(s_1 = (50.25, -104.39)\)

Daily temperature in Canada

Observed daily temperature value: \(z(s_i)\)

canada$m_data[1, ]

     Regina  Sherbrooke    Winnipeg    Resolute Yellowknife    Edmonton 
      -15.7       -10.1       -17.9       -30.7       -24.5       -12.6 
     Dawson      Ottawa Fredericton   Churchill  Whitehorse      Sydney 
      -28.0        -9.4        -7.9       -24.9       -17.6        -3.8 
Scheffervll    Montreal 
      -22.5        -8.7 

Example:

\[s_1 = (50.25, -104.39)\] \[z(s_1) = -15.7\]

Visualization

df <- data.frame(canada$m_coord)
df

            N.latitude W.longitude
Regina           50.25     -104.39
Sherbrooke       45.24      -71.54
Winnipeg         49.53      -97.09
Resolute         74.41      -94.54
Yellowknife      62.27     -114.21
Edmonton         53.33     -113.28
Dawson           64.04     -139.25
Ottawa           45.25      -75.22
Fredericton      45.58      -66.39
Churchill        58.46      -94.10
Whitehorse       60.43     -135.03
Sydney           46.09      -60.11
Scheffervll      54.47      -64.49
Montreal         45.31      -73.34

Visualization (cont.)

df$location <- attr(canada$m_coord, "dimnames")[[1]]
df

            N.latitude W.longitude    location
Regina           50.25     -104.39      Regina
Sherbrooke       45.24      -71.54  Sherbrooke
Winnipeg         49.53      -97.09    Winnipeg
Resolute         74.41      -94.54    Resolute
Yellowknife      62.27     -114.21 Yellowknife
Edmonton         53.33     -113.28    Edmonton
Dawson           64.04     -139.25      Dawson
Ottawa           45.25      -75.22      Ottawa
Fredericton      45.58      -66.39 Fredericton
Churchill        58.46      -94.10   Churchill
Whitehorse       60.43     -135.03  Whitehorse
Sydney           46.09      -60.11      Sydney
Scheffervll      54.47      -64.49 Scheffervll
Montreal         45.31      -73.34    Montreal

Visualisation (cont.)

df <- st_as_sf(df, coords=c("W.longitude", "N.latitude"))
df

Simple feature collection with 14 features and 1 field
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: -139.25 ymin: 45.24 xmax: -60.11 ymax: 74.41
CRS:           NA
First 10 features:
               location              geometry
Regina           Regina POINT (-104.39 50.25)
Sherbrooke   Sherbrooke  POINT (-71.54 45.24)
Winnipeg       Winnipeg  POINT (-97.09 49.53)
Resolute       Resolute  POINT (-94.54 74.41)
Yellowknife Yellowknife POINT (-114.21 62.27)
Edmonton       Edmonton POINT (-113.28 53.33)
Dawson           Dawson POINT (-139.25 64.04)
Ottawa           Ottawa  POINT (-75.22 45.25)
Fredericton Fredericton  POINT (-66.39 45.58)
Churchill     Churchill   POINT (-94.1 58.46)

Visualisation (cont.)

st_crs(df) <- 4326
df

Simple feature collection with 14 features and 1 field
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: -139.25 ymin: 45.24 xmax: -60.11 ymax: 74.41
Geodetic CRS:  WGS 84
First 10 features:
               location              geometry
Regina           Regina POINT (-104.39 50.25)
Sherbrooke   Sherbrooke  POINT (-71.54 45.24)
Winnipeg       Winnipeg  POINT (-97.09 49.53)
Resolute       Resolute  POINT (-94.54 74.41)
Yellowknife Yellowknife POINT (-114.21 62.27)
Edmonton       Edmonton POINT (-113.28 53.33)
Dawson           Dawson POINT (-139.25 64.04)
Ottawa           Ottawa  POINT (-75.22 45.25)
Fredericton Fredericton  POINT (-66.39 45.58)
Churchill     Churchill   POINT (-94.1 58.46)

Add temperature data

df$temp <- canada$m_data[1, ]
df

Simple feature collection with 14 features and 2 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: -139.25 ymin: 45.24 xmax: -60.11 ymax: 74.41
Geodetic CRS:  WGS 84
First 10 features:
               location              geometry  temp
Regina           Regina POINT (-104.39 50.25) -15.7
Sherbrooke   Sherbrooke  POINT (-71.54 45.24) -10.1
Winnipeg       Winnipeg  POINT (-97.09 49.53) -17.9
Resolute       Resolute  POINT (-94.54 74.41) -30.7
Yellowknife Yellowknife POINT (-114.21 62.27) -24.5
Edmonton       Edmonton POINT (-113.28 53.33) -12.6
Dawson           Dawson POINT (-139.25 64.04) -28.0
Ottawa           Ottawa  POINT (-75.22 45.25)  -9.4
Fredericton Fredericton  POINT (-66.39 45.58)  -7.9
Churchill     Churchill   POINT (-94.1 58.46) -24.9

Getting map data

map <- rnaturalearth::ne_states("Canada", returnclass = "sf")
map

Simple feature collection with 13 features and 121 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: -141.0056 ymin: 41.72947 xmax: -52.61661 ymax: 83.11652
Geodetic CRS:  WGS 84
First 10 features:
                         featurecla scalerank adm1_code diss_me iso_3166_2
1237 Admin-1 states provinces lakes         2   CAN-633     633      CA-BC
1240 Admin-1 states provinces lakes         2   CAN-632     632      CA-AB
1241 Admin-1 states provinces lakes         2   CAN-631     631      CA-SK
1243 Admin-1 states provinces lakes         2   CAN-630     630      CA-MB
1245 Admin-1 states provinces lakes         2   CAN-682     682      CA-ON
1250 Admin-1 states provinces lakes         2   CAN-683     683      CA-QC
1254 Admin-1 states provinces lakes         2   CAN-684     684      CA-NB
1615 Admin-1 states provinces lakes         2   CAN-636     636      CA-YT
2173 Admin-1 states provinces lakes         2   CAN-634     634      CA-NU
2174 Admin-1 states provinces lakes         2   CAN-686     686      CA-NL
                                                  wikipedia iso_a2 adm0_sr
1237          http://en.wikipedia.org/wiki/British_Columbia     CA       3
1240                   http://en.wikipedia.org/wiki/Alberta     CA       1
1241              http://en.wikipedia.org/wiki/Saskatchewan     CA       1
1243                  http://en.wikipedia.org/wiki/Manitoba     CA       1
1245                   http://en.wikipedia.org/wiki/Ontario     CA       1
1250                    http://en.wikipedia.org/wiki/Quebec     CA       6
1254             http://en.wikipedia.org/wiki/New_Brunswick     CA       5
1615                     http://en.wikipedia.org/wiki/Yukon     CA       5
2173                   http://en.wikipedia.org/wiki/Nunavut     CA       5
2174 http://en.wikipedia.org/wiki/Newfoundland_and_Labrador     CA       6
                          name                                         name_alt
1237          British Columbia               Colombie britannique|New Caledonia
1240                   Alberta                                             <NA>
1241              Saskatchewan                                             <NA>
1243                  Manitoba                                             <NA>
1245                   Ontario                                     Upper Canada
1250                    Québec                                     Lower Canada
1254             New Brunswick                         Nouveau-Brunswick|Acadia
1615                     Yukon  Yukon Territory|Territoire du Yukon|Yukon|Yuk¢n
2173                   Nunavut                                             <NA>
2174 Newfoundland and Labrador Newfoundland|Terre-Neuve|Terre-Neuve-et-Labrador
     name_local       type   type_en code_local code_hasc note hasc_maybe
1237       <NA>   Province  Province       <NA>     CA.BC <NA>       <NA>
1240       <NA>   Province  Province       <NA>     CA.AB <NA>       <NA>
1241       <NA>   Province  Province       <NA>     CA.SK <NA>       <NA>
1243       <NA>   Province  Province       <NA>     CA.MB <NA>       <NA>
1245       <NA>   Province  Province       <NA>     CA.ON <NA>       <NA>
1250       <NA>   Province  Province       <NA>     CA.QC <NA>       <NA>
1254       <NA>   Province  Province       <NA>     CA.NB <NA>       <NA>
1615       <NA> Territoire Territory       <NA>     CA.YT <NA>       <NA>
2173       <NA> Territoire Territory       <NA>     CA.NU <NA>       <NA>
2174       <NA>   Province  Province       <NA>     CA.NF <NA>       <NA>
              region      region_cod provnum_ne gadm_level check_me datarank
1237  Western Canada            <NA>          2          1       20        2
1240  Western Canada            <NA>         12          1       20        2
1241  Western Canada            <NA>         13          1       20        2
1243  Western Canada            <NA>         11          1       20        2
1245  Eastern Canada            <NA>          7          1       20        2
1250  Eastern Canada            <NA>          4          1       20        2
1254  Eastern Canada            <NA>         10          1       20        2
1615 Northern Canada Northern Canada          8          1       20        2
2173 Northern Canada Northern Canada          1          1       20        2
2174  Eastern Canada            <NA>          6          1       20        2
     abbrev postal area_sqkm sameascity labelrank name_len mapcolor9 mapcolor13
1237   B.C.     BC         0        -99         2       16         2          2
1240  Alta.     AB         0        -99         2        7         2          2
1241  Sask.     SK         0        -99         2       12         2          2
1243   Man.     MB         0        -99         2        8         2          2
1245   Ont.     ON         0        -99         2        7         2          2
1250   Que.     QC         0        -99         2        6         2          2
1254   N.B.     NB         0        -99         2       13         2          2
1615   Yuk.     YT         0        -99         2        5         2          2
2173   Nun.     NU         0        -99         2        7         2          2
2174   N.L.     NL         0        -99         2       25         2          2
     fips fips_alt   woe_id                             woe_label
1237 CA02     <NA>  2344916          British Columbia, CA, Canada
1240 CA01     <NA>  2344915                   Alberta, CA, Canada
1241 CA11     <NA>  2344925              Saskatchewan, CA, Canada
1243 CA03     <NA>  2344917                  Manitoba, CA, Canada
1245 CA08     <NA>  2344922                   Ontario, CA, Canada
1250 CA10     <NA>  2344924                    Quebec, CA, Canada
1254 CA04     <NA>  2344918             New Brunswick, CA, Canada
1615 CA12     <NA>  2344926           Yukon Territory, CA, Canada
2173 CA14     CA14 20069920                   Nunavut, CA, Canada
2174 CA05     <NA>  2344919 Newfoundland and Labrador, CA, Canada
                      woe_name latitude longitude sov_a3 adm0_a3 adm0_label
1237          British Columbia  54.6943 -124.6620    CAN     CAN          2
1240                   Alberta  55.2816 -115.0000    CAN     CAN          2
1241              Saskatchewan  54.4965 -105.6820    CAN     CAN          2
1243                  Manitoba  54.8500  -97.3828    CAN     CAN          2
1245                   Ontario  50.5244  -84.7943    CAN     CAN          2
1250                    Québec  52.2593  -73.7168    CAN     CAN          2
1254             New Brunswick  46.5822  -66.4558    CAN     CAN          2
1615                     Yukon  63.6088 -135.7000    CAN     CAN          2
2173                   Nunavut  64.3853  -97.1443    CAN     CAN          2
2174 Newfoundland and Labrador  48.6598  -56.2169    CAN     CAN          2
      admin geonunit gu_a3   gn_id                   gn_name  gns_id
1237 Canada   Canada   CAN 5909050          British Columbia -561661
1240 Canada   Canada   CAN 5883102                   Alberta -559990
1241 Canada   Canada   CAN 6141242              Saskatchewan -573211
1243 Canada   Canada   CAN 6065171                  Manitoba -568643
1245 Canada   Canada   CAN 6093943                   Ontario -570663
1250 Canada   Canada   CAN 6115047                    Quebec -571854
1254 Canada   Canada   CAN 6087430             New Brunswick -570089
1615 Canada   Canada   CAN 6185811                     Yukon -576336
2173 Canada   Canada   CAN 6091732                   Nunavut  448560
2174 Canada   Canada   CAN 6354959 Newfoundland and Labrador -570103
                                   gns_name gn_level gn_region gn_a1_code
1237          British Columbia, Province of        1      <NA>      CA.02
1240                   Alberta, Province d'        1      <NA>      CA.01
1241              Saskatchewan, Province de        1      <NA>      CA.11
1243                  Manitoba, Province de        1      <NA>      CA.03
1245                   Ontario, Province d'        1      <NA>      CA.08
1250                    Quebec, Province du        1      <NA>      CA.10
1254             New Brunswick, Province of        1      <NA>      CA.04
1615                                  Yukon        1      <NA>      CA.12
2173                                Nunavut        1      <NA>      CA.14
2174 Newfoundland and Labrador, Province of        1      <NA>      CA.05
           region_sub sub_code gns_level gns_lang gns_adm1 gns_region min_label
1237 British Columbia     <NA>         1      eng     CA02       <NA>       3.5
1240         Prairies     <NA>         1      fra     CA01       <NA>       3.5
1241         Prairies     <NA>         1      fra     CA11       <NA>       3.5
1243         Prairies     <NA>         1      fra     CA03       <NA>       3.5
1245          Ontario     <NA>         1      fra     CA08       <NA>       3.5
1250           Québec     <NA>         1      fra     CA10       <NA>       3.5
1254  Atlantic Canada     <NA>         1      eng     CA04       <NA>       3.5
1615             <NA>     <NA>         1      por     CA12       <NA>       3.5
2173             <NA>     <NA>         1      iku     CA14       <NA>       3.5
2174  Atlantic Canada     <NA>         1      eng     CA05       <NA>       3.5
     max_label min_zoom wikidataid              name_ar                 name_bn
1237       7.5        2      Q1974  كولومبيا البريطانية         ব্রিটিশ কলাম্বিয়া
1240       7.5        2      Q1951               ألبرتا               অ্যালবার্টা
1241       7.5        2      Q1989           ساسكاتشوان              সাসক্যাচুয়ান
1243       7.5        2      Q1948             مانيتوبا               ম্যানিটোবা
1245       7.5        2      Q1904             أونتاريو                 অন্টারিও
1250       7.5        2       Q176                 كيبك                   কেবেক
1254       7.5        2      Q1965         نيو برونزويك            নিউ ব্রান্সউইক
1615       7.5        2      Q2009                يوكون                    ইউকন
2173       7.5        2      Q2023              نونافوت                   নুনাভুট
2174       7.5        2      Q2003 نيوفندلاند ولابرادور নিউফাউন্ডল্যান্ড ও লাব্রাডর
                      name_de                   name_en              name_es
1237         British Columbia          British Columbia   Columbia Británica
1240                  Alberta                   Alberta              Alberta
1241             Saskatchewan              Saskatchewan         Saskatchewan
1243                 Manitoba                  Manitoba             Manitoba
1245                  Ontario                   Ontario              Ontario
1250                   Québec                    Quebec               Quebec
1254            New Brunswick             New Brunswick      Nuevo Brunswick
1615                    Yukon                     Yukon                Yukón
2173                  Nunavut                   Nunavut              Nunavut
2174 Neufundland und Labrador Newfoundland and Labrador Terranova y Labrador
                     name_fr               name_el             name_hi
1237    Colombie-Britannique    Βρετανική Κολομβία     ब्रिटिश कोलम्बिया
1240                 Alberta              Αλμπέρτα              अल्बर्टा
1241            Saskatchewan           Σασκάτσουαν              सैस्कैचेवेन
1243                Manitoba             Μανιτόμπα            मानिटोबा
1245                 Ontario               Οντάριο            ओण्टारियो
1250                  Québec                Κεμπέκ                क्यूबेक
1254       Nouveau-Brunswick      Νιου Μπράνσγουικ           न्यू ब्रंसविक
1615                   Yukon               Γιούκον           युकॉन प्रांत
2173                 Nunavut             Νούναβουτ               नुनावुत
2174 Terre-Neuve-et-Labrador Νέα Γη και Λαμπραντόρ न्यूफाउंडलैंड और लैब्राडोर
                     name_hu                   name_id              name_it
1237           Brit Columbia          British Columbia  Columbia Britannica
1240                 Alberta                   Alberta              Alberta
1241            Saskatchewan              Saskatchewan         Saskatchewan
1243                Manitoba                  Manitoba             Manitoba
1245                 Ontario                   Ontario              Ontario
1250                  Québec                    Quebec               Québec
1254            Új-Brunswick             New Brunswick      Nuovo Brunswick
1615                   Yukon                     Yukon                Yukon
2173                 Nunavut                   Nunavut              Nunavut
2174 Új-Fundland és Labrador Newfoundland dan Labrador Terranova e Labrador
                                  name_ja            name_ko
1237           ブリティッシュコロンビア州   브리티시컬럼비아
1240                         アルバータ州             앨버타
1241                     サスカチュワン州         서스캐처원
1243                           マニトバ州           매니토바
1245                         オンタリオ州           온타리오
1250                           ケベック州               퀘벡
1254           ニュー・ブランズウィック州         뉴브런즈윅
1615                         ユーコン準州            유콘 준
2173                         ヌナブト準州        누나부트 준
2174 ニューファンドランド・ラブラドール州 뉴펀들랜드래브라도
                      name_nl                    name_pl               name_pt
1237           Brits-Columbia         Kolumbia Brytyjska    Colúmbia Britânica
1240                  Alberta                    Alberta               Alberta
1241             Saskatchewan               Saskatchewan          Saskatchewan
1243                 Manitoba                   Manitoba              Manitoba
1245                  Ontario                    Ontario               Ontário
1250                   Quebec                     Quebec              Quebeque
1254            New Brunswick             Nowy Brunszwik        Novo Brunswick
1615                    Yukon                      Jukon                 Yukon
2173                  Nunavut                    Nunavut               Nunavut
2174 Newfoundland en Labrador Nowa Fundlandia i Labrador Terra Nova e Labrador
                     name_ru                   name_sv                  name_tr
1237     Британская Колумбия          British Columbia     Britanya Kolumbiyası
1240                Альберта                   Alberta                  Alberta
1241              Саскачеван              Saskatchewan             Saskatchewan
1243                Манитоба                  Manitoba                 Manitoba
1245                 Онтарио                   Ontario                  Ontario
1250                  Квебек                    Québec                   Québec
1254            Нью-Брансуик             New Brunswick            New Brunswick
1615                    Юкон                     Yukon                    Yukon
2173                 Нунавут                   Nunavut                  Nunavut
2174 Ньюфаундленд и Лабрадор Newfoundland och Labrador Newfoundland ve Labrador
                      name_vi         name_zh      ne_id             name_he
1237         British Columbia  不列颠哥伦比亚 1159307717    קולומביה הבריטית
1240                  Alberta      阿尔伯塔省 1159308775              אלברטה
1241             Saskatchewan      萨斯喀彻温 1159308773            ססקצ'ואן
1243                 Manitoba        曼尼托巴 1159308777             מניטובה
1245                  Ontario        安大略省 1159309687             אונטריו
1250                   Québec          魁北克 1159308703              קוויבק
1254            New Brunswick      新不伦瑞克 1159308785        ניו ברנזוויק
1615                    Yukon            育空 1159309667               יוקון
2173                  Nunavut        努纳武特 1159307715             נונאווט
2174 Newfoundland và Labrador 纽芬兰-拉布拉多 1159308713 ניופאונדלנד ולברדור
                     name_uk                     name_ur               name_fa
1237     Британська Колумбія                برٹش کولمبیا         بریتیش کلمبیا
1240                Альберта                      البرٹا                آلبرتا
1241              Саскачеван                   ساسکچیوان               سسکچوان
1243                Манітоба                    مانیٹوبا              مانیتوبا
1245                 Онтаріо                     انٹاریو               انتاریو
1250                  Квебек                      کیوبیک             استان کبک
1254            Нью-Брансвік                 نیو برنزویک           نیوبرانزویک
1615                    Юкон                       يوكون                 یوکان
2173                 Нунавут                       نناوت               نوناووت
2174 Ньюфаундленд і Лабрадор نیو فاؤنڈ لینڈ اور لیبراڈور نیوفاندلند و لابرادور
             name_zht FCLASS_ISO FCLASS_US FCLASS_FR FCLASS_RU FCLASS_ES
1237 不列顛哥倫比亞省       <NA>      <NA>      <NA>      <NA>      <NA>
1240         亞伯達省       <NA>      <NA>      <NA>      <NA>      <NA>
1241       薩斯喀徹溫       <NA>      <NA>      <NA>      <NA>      <NA>
1243       曼尼托巴省       <NA>      <NA>      <NA>      <NA>      <NA>
1245         安大略省       <NA>      <NA>      <NA>      <NA>      <NA>
1250           魁北克       <NA>      <NA>      <NA>      <NA>      <NA>
1254       新不倫瑞克       <NA>      <NA>      <NA>      <NA>      <NA>
1615             育空       <NA>      <NA>      <NA>      <NA>      <NA>
2173         努納福特       <NA>      <NA>      <NA>      <NA>      <NA>
2174  紐芬蘭-拉布拉多       <NA>      <NA>      <NA>      <NA>      <NA>
     FCLASS_CN FCLASS_TW FCLASS_IN FCLASS_NP FCLASS_PK FCLASS_DE FCLASS_GB
1237      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1240      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1241      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1243      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1245      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1250      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1254      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1615      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
2173      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
2174      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
     FCLASS_BR FCLASS_IL FCLASS_PS FCLASS_SA FCLASS_EG FCLASS_MA FCLASS_PT
1237      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1240      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1241      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1243      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1245      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1250      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1254      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1615      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
2173      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
2174      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
     FCLASS_AR FCLASS_JP FCLASS_KO FCLASS_VN FCLASS_TR FCLASS_ID FCLASS_PL
1237      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1240      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1241      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1243      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1245      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1250      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1254      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
1615      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
2173      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
2174      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>
     FCLASS_GR FCLASS_IT FCLASS_NL FCLASS_SE FCLASS_BD FCLASS_UA FCLASS_TLC
1237      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>       <NA>
1240      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>       <NA>
1241      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>       <NA>
1243      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>       <NA>
1245      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>       <NA>
1250      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>       <NA>
1254      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>       <NA>
1615      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>       <NA>
2173      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>       <NA>
2174      <NA>      <NA>      <NA>      <NA>      <NA>      <NA>       <NA>
                           geometry
1237 MULTIPOLYGON (((-117.0382 4...
1240 MULTIPOLYGON (((-109.9993 4...
1241 MULTIPOLYGON (((-104.0476 4...
1243 MULTIPOLYGON (((-97.22609 4...
1245 MULTIPOLYGON (((-89.58175 4...
1250 MULTIPOLYGON (((-73.35134 4...
1254 MULTIPOLYGON (((-67.17601 4...
1615 MULTIPOLYGON (((-139.0602 6...
2173 MULTIPOLYGON (((-102.5728 6...
2174 MULTIPOLYGON (((-64.53141 6...

Visualisation of the spatial domain \(D\)

This represents \(D\) - spatial domain (geographical region which the observations could be made)

ggplot(map) + geom_sf()

A spatial stochastic process is a family of random variables \[\{Z(s): s \in D\}.\]

Visualisation of the spatial domain (\(D\)), and \(z(s_i)\) values

library(viridis)
ggplot(map) + geom_sf() + geom_sf(data=df, aes(color = temp), size=3) + scale_color_viridis()

Three types of spatial processes

1. Geostatistical processes:

   In theory the process could be observed at infinitely many locations in \(S \in D\).

   Example: Air temperature in Sri Lanka

2. Areal process

   Here \(D\) is divided into on-overlapping sub regions such as \(D = {B_1, B_2,...B_n}\) where \(B_i\) is the \(i^{th}\) sub region.

   Example: Total number of dengue cases in each district in Sri Lanka

3. Point process

   The process can be observed only at specific locations in \(S \in D\).

   Example: Tree volume of sandalwood trees in Sri Lanka

Tobler’s First Law of Geography

Everything is related to everything else, but near things are more related than distant things.

library(viridis)
ggplot(map) + geom_sf() +
  geom_sf(data=df, aes(color = temp), size=3) +
  scale_color_viridis()

Spatial Neighbours

Spatial Neighbours in GIS and spatial analysis refer to locations or features that are close to or related to a specific point/area, defined by proximity, contiguity (touching), or other spatial criteria.

Types of Spatial Neighbours

1. Contiguity-based neighbours

Two regions are neighbours if they share a common boundary (or vertex).

Rook criterion: Share a common edge.

Queen criterion: Share an edge or a vertex.

Example: Two adjacent districts in a country.

2. Distance-based neighbours

Two locations are neighbours if the distance between them is below a threshold.

Example: Weather stations within 10 km of each other.

3. k-nearest neighbours

Each location is connected to its k closest locations, regardless of actual distance threshold.

Example: Each city connected to its 3 nearest cities.

Modelling Spatial Neighbours (cont.)

To incorporate these relationships into models, we create spatial weights, which quantify the influence of each neighbour.

Geostatistical Theory

• Mean

• Variance

• Autocovariance

• Autocorrelation

Spatial Continuity

• Spatial continuity is the tendency for spatially nearby observations to be similar, creating smooth and structured variation across space rather than random noise.

• Correlation between values over distance.

• Spatial continuity explains why independence is often violated in spatial datasets.

Why it matters

Spatial continuity is the foundation for:

Spatial autocorrelation measures (e.g., Moran’s I)

Spatial smoothing and interpolation

Spatial regression models

Geostatistics (variograms, kriging)

Ignoring spatial continuity can lead to:

Biased estimates

Overconfident inference

Misleading conclusions

Weakly Stationary

In-class

Strictly Stationary

In-class

Isotropy

direction-independent

Anisotropic

direction-independent

1. What is isotropic?

   The correlation between any two observations depends only on the distance between those locations and not on their relative direction (directionally invariant).

2. Why is isotropic important in spatial data analysis?

   This assumption simplifies analysis and modeling processes.

Spatial Interpolation

Spatial interpolation is the activity of estimating values of spatially continuous variables (fields) for spatial locations where they have not been observed, based on observations.

Spatial Interpolation Methods

• Inverse Distance Weighting (IDW): It estimates values for unmeasured locations by considering the values and distances of nearby measured points.

• Kriging: A geostatistical technique that provides optimal predictions and quantifies uncertainty by analyzing spatial autocorrelation.

• Spline Interpolation: It fits a smooth curve through measured points to estimate values at unmeasured locations.

• Nearest-neighbor interpolation: It assigns the value of the closest measured point to the unmeasured location.

Statial Interpolation Methods (cont.)

• Natural Neighbor: Similar to nearest neighbor but considers multiple nearby points and their influence.

• Radial Basis Function (RBF): It constructs a continuous function that satisfies given data values and radial symmetry properties.

• Deep Learning Model for Spatial Interpolation

• Image-driven spatial interpolation

• Bayesian deep learning approaches

• Ensemble approaches

Inverse Distance Weighting (IDW)

• Nonparametric interpolation method

• Based on weighted averages of the observations.

• Weights are calculated based on the inverse of the distances between the known data points and the location where the estimation is being made

• Relies on Tobler’s first law of geography

Inverse Distance Weighting (IDW)

Suppose \(s_0\) is the point that we need to interpolate. Then,

\[\hat{Z}(s_0) = \frac{\sum_{i=1}^{n}w(s_i)Z(s_i)}{\sum_{i=1}^nw(s_i)}\] and \(w_i\) is defined as

\[w_i = \frac{1}{||s_i - s_o||^p}\]

• We can tune the value of \(p\) using a cross validation method

• We can limit the amount of neighbours to take into consideration to speed up the computations.

Your turn

x	y	z
2	2	10
3	7	11
9	9	15
6	5	9
5	3	8

We want to find z value at x=5, y=5.

Answer

x	y	z	dis_to_55	inv_dis	weight
2	2	10	4.242641	0.2357023	0.1040154
3	7	11	2.828427	0.3535534	0.1560231
9	9	15	5.656854	0.1767767	0.0780115
6	5	9	1.000000	1.0000000	0.4413000
5	3	8	2.000000	0.5000000	0.2206500

Predicted value at (5, 5)

\(z(5, 5) = 10\times 0.104 + 11 \times 0.156 + 15 \times 0.078 + \\ 9 \times 0.441 + 8 \times 0.220\)

IDW - Application

Question: Interpolate Mean \(NO_2\) concentrations in air in Germany, in 2017

Step 1: Load necessary packages

library(gstat)
library(tidyverse)

Step 2: Read data

no2 <- read_csv(system.file("external/no2.csv", 
    package = "gstat"), show_col_types = FALSE)
no2

# A tibble: 74 × 21
   station_european_code station_local_code country_iso_code country_name
   <chr>                 <chr>              <chr>            <chr>       
 1 DENI063               DENI063            DE               Germany     
 2 DEBY109               DEBY109            DE               Germany     
 3 DEBE056               DEBE056            DE               Germany     
 4 DEBE062               DEBE062            DE               Germany     
 5 DEBE032               DEBE032            DE               Germany     
 6 DEHE046               DEHE046            DE               Germany     
 7 DEBY122               DEBY122            DE               Germany     
 8 DESL019               DESL019            DE               Germany     
 9 DENW081               DENW081            DE               Germany     
10 DESH008               DESH008            DE               Germany     
# ℹ 64 more rows
# ℹ 17 more variables: station_name <chr>, station_start_date <date>,
#   station_end_date <lgl>, type_of_station <chr>,
#   station_ozone_classification <chr>, station_type_of_area <chr>,
#   station_subcat_rural_back <chr>, street_type <chr>,
#   station_longitude_deg <dbl>, station_latitude_deg <dbl>,
#   station_altitude <dbl>, station_city <chr>, lau_level1_code <dbl>, …

head(data.frame(no2), 3)

  station_european_code station_local_code country_iso_code country_name
1               DENI063            DENI063               DE      Germany
2               DEBY109            DEBY109               DE      Germany
3               DEBE056            DEBE056               DE      Germany
        station_name station_start_date station_end_date type_of_station
1         Altes Land         1999-02-11               NA      Background
2 Andechs/Rothenfeld         2003-04-17               NA      Background
3  B Friedrichshagen         1994-02-01               NA      Background
  station_ozone_classification station_type_of_area station_subcat_rural_back
1                        rural                rural                   unknown
2                        rural                rural                  regional
3                        rural                rural                 near city
  street_type station_longitude_deg station_latitude_deg station_altitude
1        <NA>              9.685031             53.52418                3
2        <NA>             11.220172             47.96875              700
3        <NA>             13.647050             52.44770               35
  station_city lau_level1_code lau_level2_code lau_level2_name EMEP_station
1         <NA>              NA         3359028            Jork           no
2         <NA>              NA         9188117         Andechs           no
3         <NA>              NA        11000000   Berlin, Stadt           no
        NO2
1 13.102806
2  7.135128
3 12.799011

Create a Coordinate Reference System (CRS) Object

• st_crs: used to create a coordinate reference system (CRS) object.

• EPSG:32632 is a specific CRS identifier. In this case, it refers to a CRS known as “WGS 84 / UTM zone 32N”. This CRS is commonly used for mapping and spatial analysis in Europe, particularly in the northern regions.

• The following line of code creates a CRS object named crs representing the coordinate reference system “WGS 84 / UTM zone 32N”.

crs <- st_crs("EPSG:32632")
crs

Coordinate Reference System:
  User input: EPSG:32632 
  wkt:
PROJCRS["WGS 84 / UTM zone 32N",
    BASEGEOGCRS["WGS 84",
        ENSEMBLE["World Geodetic System 1984 ensemble",
            MEMBER["World Geodetic System 1984 (Transit)"],
            MEMBER["World Geodetic System 1984 (G730)"],
            MEMBER["World Geodetic System 1984 (G873)"],
            MEMBER["World Geodetic System 1984 (G1150)"],
            MEMBER["World Geodetic System 1984 (G1674)"],
            MEMBER["World Geodetic System 1984 (G1762)"],
            MEMBER["World Geodetic System 1984 (G2139)"],
            MEMBER["World Geodetic System 1984 (G2296)"],
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ENSEMBLEACCURACY[2.0]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4326]],
    CONVERSION["UTM zone 32N",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",9,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",0.9996,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",500000,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",0,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Navigation and medium accuracy spatial referencing."],
        AREA["Between 6°E and 12°E, northern hemisphere between equator and 84°N, onshore and offshore. Algeria. Austria. Cameroon. Denmark. Equatorial Guinea. France. Gabon. Germany. Italy. Libya. Liechtenstein. Monaco. Netherlands. Niger. Nigeria. Norway. Sao Tome and Principe. Svalbard. Sweden. Switzerland. Tunisia. Vatican City State."],
        BBOX[0,6,84,12]],
    ID["EPSG",32632]]

convert no2 to spatial object

no2.sf <- st_as_sf(no2, crs = "OGC:CRS84", coords = 
    c("station_longitude_deg", "station_latitude_deg")) |>
    st_transform(crs) 
no2.sf

Simple feature collection with 74 features and 19 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 304638.2 ymin: 5263687 xmax: 900829.7 ymax: 6086661
Projected CRS: WGS 84 / UTM zone 32N
# A tibble: 74 × 20
   station_european_code station_local_code country_iso_code country_name
 * <chr>                 <chr>              <chr>            <chr>       
 1 DENI063               DENI063            DE               Germany     
 2 DEBY109               DEBY109            DE               Germany     
 3 DEBE056               DEBE056            DE               Germany     
 4 DEBE062               DEBE062            DE               Germany     
 5 DEBE032               DEBE032            DE               Germany     
 6 DEHE046               DEHE046            DE               Germany     
 7 DEBY122               DEBY122            DE               Germany     
 8 DESL019               DESL019            DE               Germany     
 9 DENW081               DENW081            DE               Germany     
10 DESH008               DESH008            DE               Germany     
# ℹ 64 more rows
# ℹ 16 more variables: station_name <chr>, station_start_date <date>,
#   station_end_date <lgl>, type_of_station <chr>,
#   station_ozone_classification <chr>, station_type_of_area <chr>,
#   station_subcat_rural_back <chr>, street_type <chr>, station_altitude <dbl>,
#   station_city <chr>, lau_level1_code <dbl>, lau_level2_code <dbl>,
#   lau_level2_name <chr>, EMEP_station <chr>, NO2 <dbl>, …

EPSG: 32632 vs OGC: CRS84

• EPSG:32632

  Commonly used for regional mapping and spatial analysis in Europe, particularly in the northern regions.

• OGC:CRS84

  Often used in web mapping applications and geospatial data exchange due to its simplicity and widespread support.

• EPSG:32632 is a regional projected CRS optimized for mapping specific areas with minimal distortion, while OGC:CRS84 is a global geographic CRS commonly used for general-purpose mapping and data exchange.

Load country boundary

map <- read_sf("de_nuts1.gpkg") |> st_transform(crs) 
map

Simple feature collection with 16 features and 16 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 280741.3 ymin: 5235822 xmax: 921330.5 ymax: 6101239
Projected CRS: WGS 84 / UTM zone 32N
# A tibble: 16 × 17
    ID_0 ISO   NAME_0   ID_1 NAME_1      VARNAME_1 NL_NAME_1 HASC_1 CC_1  TYPE_1
 * <int> <chr> <chr>   <int> <chr>       <chr>     <chr>     <chr>  <chr> <chr> 
 1    60 DEU   Germany   753 Baden-Wurt… <NA>      <NA>      DE.BW  <NA>  Land  
 2    60 DEU   Germany   754 Bayern      Bavaria   <NA>      DE.BY  <NA>  Land  
 3    60 DEU   Germany   755 Berlin      <NA>      <NA>      DE.BE  <NA>  Land  
 4    60 DEU   Germany   756 Brandenburg <NA>      <NA>      DE.BR  <NA>  Land  
 5    60 DEU   Germany   757 Bremen      <NA>      <NA>      DE.HB  <NA>  Land  
 6    60 DEU   Germany   758 Hamburg     <NA>      <NA>      DE.HH  <NA>  Land  
 7    60 DEU   Germany   759 Hessen      Hesse     <NA>      DE.HE  <NA>  Land  
 8    60 DEU   Germany   760 Mecklenbur… Mecklenb… <NA>      DE.MV  <NA>  Land  
 9    60 DEU   Germany   761 Niedersach… Lower Sa… <NA>      DE.NI  <NA>  Land  
10    60 DEU   Germany   762 Nordrhein-… North Rh… <NA>      DE.NW  <NA>  Land  
11    60 DEU   Germany   763 Rheinland-… Rhinelan… <NA>      DE.RP  <NA>  Land  
12    60 DEU   Germany   764 Saarland    <NA>      <NA>      DE.SL  <NA>  Land  
13    60 DEU   Germany   765 Sachsen     Saxony    <NA>      DE.SN  <NA>  Land  
14    60 DEU   Germany   766 Sachsen-An… Saxony-A… <NA>      DE.ST  <NA>  Land  
15    60 DEU   Germany   767 Schleswig-… <NA>      <NA>      DE.SH  <NA>  Land  
16    60 DEU   Germany   768 Thuringen   Thuringia <NA>      DE.TH  <NA>  Land  
# ℹ 7 more variables: ENGTYPE_1 <chr>, VALIDFR_1 <chr>, VALIDTO_1 <chr>,
#   REMARKS_1 <chr>, Shape_Leng <dbl>, Shape_Area <dbl>,
#   geom <MULTIPOLYGON [m]>

Data source: https://github.com/edzer/sdsr/tree/main/data (select de_nuts1.gpkg)

Plot map

ggplot() + geom_sf(data = map)

Add observed values to the map

ggplot() + geom_sf(data = map) + 
geom_sf(data = no2.sf, mapping = aes(col = NO2)) + 
  scale_color_viridis()

IDW

Step 1: Create a regular grid (\(10km \times 10km\))

library(stars)
grid <- st_bbox(map) |>
  st_as_stars(dx = 10000) |> st_crop(map) 
grid

stars object with 2 dimensions and 1 attribute
attribute(s):
        Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
values     0       0      0    0       0    0 2076
dimension(s):
  from to  offset  delta                refsys x/y
x    1 65  280741  10000 WGS 84 / UTM zone 32N [x]
y    1 87 6101239 -10000 WGS 84 / UTM zone 32N [y]

• st_bbox(map): Calculate the bounding box of the spatial object map.

library(stars)
grid <- st_bbox(map) |>
  st_as_stars(dx = 10000) |> st_crop(map) 
grid

• st_as_stars(dx = 10000): Convert the bounding box obtained from st_bbox(map) into a raster object using the stars package. The dx parameter sets the cell size (in the x-direction) for the resulting raster, with a value of 10000 specified here.

• st_crop(map): This function crops the raster grid obtained from st_as_stars(dx = 10000) to match the extent of the map object. This ensures that the resulting grid covers only the area within the spatial extent of the map object.

IDW

Step 2: Interpolation

library(gstat)
interpolated.values <- idw(NO2~1, no2.sf, grid)

[inverse distance weighted interpolation]

interpolated.values 

stars object with 2 dimensions and 2 attributes
attribute(s):
               Min.  1st Qu.   Median     Mean  3rd Qu.     Max. NA's
var1.pred  2.168083 7.397774 8.323239 8.571584 9.422718 20.47053 2076
var1.var         NA       NA       NA      NaN       NA       NA 5655
dimension(s):
  from to  offset  delta                refsys x/y
x    1 65  280741  10000 WGS 84 / UTM zone 32N [x]
y    1 87 6101239 -10000 WGS 84 / UTM zone 32N [y]

IDW

Step 3: Plot values

ggplot() + geom_stars(data = interpolated.values, 
                      aes(fill = var1.pred, x = x, y = y)) + 
    geom_sf(data = st_cast(map, "MULTILINESTRING")) + 
    geom_sf(data = no2.sf, col="red") + scale_fill_viridis() 

Comparison

library(patchwork)
p1 <- ggplot() + geom_sf(data = map) +  geom_sf(data = no2.sf, mapping = aes(col = NO2)) + scale_color_viridis()
p2 <- ggplot() + geom_stars(data = interpolated.values, 
                      aes(fill = var1.pred, x = x, y = y)) + geom_sf(data = st_cast(map, "MULTILINESTRING")) + geom_sf(data = no2.sf, col="red") + scale_fill_viridis() 
p1|p2

Variogram Modelling and Krigging

Geostatistical processes are generally modelled using additive decomposition:

\[Z(s) = \mu(s) + \epsilon(s)\]

Where,

\(\mu(s)\) - large-scale variation/ spatial-trend/ spatially varying mean that captures the long range (nonstationary) spatial variation

\(\epsilon(s)\) - small-scale variation/ innovations/ spatial geostatistical error

• To fully capture the characteristics of the geostatistical process, we need to model both \(\mu(s)\) and \(\epsilon(s)\).

Overall modeeling approach

\[Z(s) = \mu(s) + \epsilon(s)\]

First we model \(\mu(s)\) using OLS or some other trend model via covariates and the residuals are used to model small-scale variation.

Modelling Large Scale Variation

• Regression

• Machine-learning models

Modelling Small Scale Variation

Let’s learn some theories we need to model small scale variations.

Investigate Spatial Autocorrelation Using Empirical Variogram/ Variogram cloud

\[\gamma(s, t) = \frac{1}{2}Var\{Z(s_i) - Z(s_j)\}\] An estimator for this is

\[\gamma(s, t) = \frac{1}{2}\{Z(s_i) - Z(s_j)\}^2\]

Proof: In-class

Variogram Cloud/ Empirical Variogram

vcloud <- variogram(NO2~1, no2.sf, cloud=TRUE)
vcloud

           dist        gamma dir.hor dir.ver   id left right
1    291778.272 4.614575e-02       0       0 var1    3     1
2    260594.879 8.088801e-01       0       0 var1    4     1
3     33040.195 4.686251e-01       0       0 var1    4     3
4    264858.840 6.302227e-01       0       0 var1    5     1
5     28838.008 3.352990e-01       0       0 var1    5     3
6     20620.764 1.113277e-02       0       0 var1    5     4
7    238444.197 8.682758e+00       0       0 var1    6     1
8    329024.871 4.191332e+00       0       0 var1    6     4
9    317286.345 4.634489e+00       0       0 var1    6     5
10    79080.277 6.548094e-03       0       0 var1    7     2
11   280433.822 4.239450e-02       0       0 var1    8     6
12   265057.003 1.566442e+01       0       0 var1    9     1
13   149968.341 4.767188e+01       0       0 var1    9     6
14   272931.248 4.487102e+01       0       0 var1    9     8
15    73123.661 7.773505e+00       0       0 var1   10     1
16   291856.123 6.621795e+00       0       0 var1   10     3
17   258897.927 3.567274e+00       0       0 var1   10     4
18   268547.646 3.976973e+00       0       0 var1   10     5
19   309101.508 2.513857e-02       0       0 var1   10     6
20   335637.582 4.550759e+01       0       0 var1   10     9
21   302277.749 9.980953e-01       0       0 var1   11     2
22   227742.767 9.035571e+00       0       0 var1   11     3
23   239718.000 5.388715e+00       0       0 var1   11     4
24   219190.447 5.889710e+00       0       0 var1   11     5
25   247800.772 7.512323e-02       0       0 var1   11     6
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1002 111884.710 4.645348e+00       0       0 var1   61    12
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1005 132204.469 3.691171e+00       0       0 var1   61    16
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1009 242684.884 2.524352e+00       0       0 var1   61    23
1010  69041.681 2.002426e-01       0       0 var1   61    24
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1016 328556.009 3.820586e-02       0       0 var1   61    31
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1018 290079.121 5.572102e+01       0       0 var1   61    34
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1021 272859.366 5.559852e-01       0       0 var1   61    38
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1023 257276.990 6.977639e-01       0       0 var1   61    41
1024 321793.255 1.451308e+00       0       0 var1   61    43
1025 257281.352 1.653175e+00       0       0 var1   61    44
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1028 238796.686 2.524352e+00       0       0 var1   61    50
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1030 224767.549 1.819869e+01       0       0 var1   61    55
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1032 307979.320 4.057576e-01       0       0 var1   61    57
1033 246375.085 7.797357e-02       0       0 var1   61    58
1034 128143.184 2.673640e+00       0       0 var1   61    60
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1038 334282.637 2.159807e+01       0       0 var1   62     4
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1042 270833.798 7.873622e+00       0       0 var1   62     8
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1055 327058.957 2.782844e-01       0       0 var1   62    24
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1062 262795.105 8.584322e-01       0       0 var1   62    36
1063 330171.039 4.294012e+01       0       0 var1   62    37
1064 132183.805 5.260932e-02       0       0 var1   62    38
1065 278624.493 1.028488e+02       0       0 var1   62    39
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1069 113659.187 1.118596e-01       0       0 var1   62    46
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1073  61543.944 3.767621e-01       0       0 var1   62    50
1074 250628.804 2.944476e+00       0       0 var1   62    51
1075 338128.015 1.655333e+00       0       0 var1   62    52
1076 244999.937 8.215346e-01       0       0 var1   62    53
1077 259369.421 3.137349e-02       0       0 var1   62    54
1078 173056.345 2.746812e+01       0       0 var1   62    55
1079 137308.070 4.938035e+00       0       0 var1   62    56
1080  53207.434 2.598552e+00       0       0 var1   62    57
1081 327471.590 4.841013e-01       0       0 var1   62    58
1082 221385.673 8.380256e-01       0       0 var1   62    59
1083 150998.722 4.357524e-01       0       0 var1   62    60
1084 262381.838 9.506472e-01       0       0 var1   62    61
1085 117327.055 5.180362e+00       0       0 var1   63     1
1086 177003.819 4.248649e+00       0       0 var1   63     3
1087 147529.327 1.895202e+00       0       0 var1   63     4
1088 149162.334 2.196843e+00       0       0 var1   63     5
1089 228542.444 4.497194e-01       0       0 var1   63     6
1090 316334.495 3.886115e+01       0       0 var1   63     9
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1096 133996.699 1.402173e-01       0       0 var1   63    16
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1098 184896.165 7.451660e-01       0       0 var1   63    19
1099 209366.640 1.512622e+01       0       0 var1   63    22
1100 261387.331 1.508949e+01       0       0 var1   63    23
1101  40602.588 7.525052e+00       0       0 var1   63    24
1102 112884.359 4.621399e+00       0       0 var1   63    25
1103 197249.506 2.993205e-01       0       0 var1   63    26
1104 243309.628 8.734634e+00       0       0 var1   63    27
1105 217106.511 1.604648e-01       0       0 var1   63    29
1106 248217.593 5.723870e+00       0       0 var1   63    30
1107  99548.998 5.656473e+00       0       0 var1   63    33
1108 319451.300 2.671808e+01       0       0 var1   63    34
1109 215713.493 5.483104e+00       0       0 var1   63    35
1110 127050.223 5.495242e+00       0       0 var1   63    36
1111 304321.238 9.249763e+00       0       0 var1   63    38
1112 126727.738 1.588205e+01       0       0 var1   63    40
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1114 308034.714 1.190271e+00       0       0 var1   63    43
1115 276390.277 1.019978e+00       0       0 var1   63    44
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1117 267294.656 3.245118e+00       0       0 var1   63    47
1118 257651.423 1.508949e+01       0       0 var1   63    50
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1120 257329.669 3.882058e+00       0       0 var1   63    55
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1122 333882.278 2.751309e+00       0       0 var1   63    57
1123 229286.331 6.630293e+00       0       0 var1   63    58
1124 144276.019 1.545138e+01       0       0 var1   63    60
1125  32635.070 5.270230e+00       0       0 var1   63    61
1126 286461.572 1.069754e+01       0       0 var1   63    62
1127 221354.387 6.717310e+00       0       0 var1   64     6
1128  95324.586 7.826995e+00       0       0 var1   64     8
1129 180866.511 9.017895e+01       0       0 var1   64     9
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1131 297478.735 2.273026e+01       0       0 var1   64    14
1132 285885.608 6.937679e+00       0       0 var1   64    17
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1142 118671.127 1.026801e+02       0       0 var1   64    39
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1164 285951.047 1.145564e+00       0       0 var1   65    61
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1177 137650.400 1.564286e-01       0       0 var1   66    21
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1187 139341.796 7.247825e+01       0       0 var1   66    39
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1193 317287.035 1.507868e+01       0       0 var1   66    49
1194 197810.232 5.025814e+00       0       0 var1   66    50
1195 329094.571 8.495014e+00       0       0 var1   66    52
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1199 152179.150 3.530094e-01       0       0 var1   66    56
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1230 279476.190 1.864779e+01       0       0 var1   67    49
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1233 254745.121 1.338230e+00       0       0 var1   67    53
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1238 270008.450 7.416471e+00       0       0 var1   67    60
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1258 158661.192 4.952619e+01       0       0 var1   68    34
1259 179455.295 1.375486e+00       0       0 var1   68    38
1260 225682.893 7.637467e+01       0       0 var1   68    39
1261 300364.665 1.593880e+00       0       0 var1   68    41
1262 333153.845 8.910616e+00       0       0 var1   68    42
1263  42285.491 2.519173e+00       0       0 var1   68    45
1264 331115.786 1.140035e+00       0       0 var1   68    46
1265  99019.351 4.454362e+01       0       0 var1   68    48
1266 173849.598 1.337571e+01       0       0 var1   68    49
1267 284036.414 4.064208e+00       0       0 var1   68    50
1268 182679.399 7.229517e+00       0       0 var1   68    52
1269 184484.720 2.458108e-01       0       0 var1   68    53
1270 154841.940 1.500755e+00       0       0 var1   68    54
1271 246384.966 1.473658e+01       0       0 var1   68    55
1272 300381.366 6.723845e-01       0       0 var1   68    56
1273 172528.061 4.402609e-02       0       0 var1   68    57
1274 224029.422 1.966105e+00       0       0 var1   68    62
1275 114592.166 1.942839e+00       0       0 var1   68    64
1276 157126.123 5.100579e-02       0       0 var1   68    66
1277 115651.371 4.369554e-01       0       0 var1   68    67
1278 248395.740 2.281618e+01       0       0 var1   69     1
1279 296088.632 2.081014e+01       0       0 var1   69     3
1280 285460.841 1.503308e+01       0       0 var1   69     4
1281 271498.922 1.586240e+01       0       0 var1   69     5
1282  60928.270 3.348815e+00       0       0 var1   69     6
1283 309840.173 4.144792e+00       0       0 var1   69     8
1284 210727.098 7.629078e+01       0       0 var1   69     9
1285 313227.575 3.954245e+00       0       0 var1   69    10
1286 187316.157 2.420796e+00       0       0 var1   69    11
1287  81249.293 3.804126e+00       0       0 var1   69    12
1288  62050.437 2.106628e+00       0       0 var1   69    13
1289 208996.513 1.604664e+01       0       0 var1   69    14
1290 225604.227 2.620201e+00       0       0 var1   69    15
1291 158809.619 4.520438e+00       0       0 var1   69    16
1292  91506.152 3.504934e+00       0       0 var1   69    17
1293 264949.946 1.000576e+02       0       0 var1   69    18
1294 289958.749 1.131527e+01       0       0 var1   69    19
1295 194420.269 1.571042e+00       0       0 var1   69    21
1296 103216.250 1.915265e+00       0       0 var1   69    23
1297 248508.443 5.885222e-02       0       0 var1   69    24
1298 321054.321 1.230915e-01       0       0 var1   69    25
1299 324216.039 3.816108e+00       0       0 var1   69    26
1300 143507.169 2.068916e-01       0       0 var1   69    27
1301 251417.434 6.314073e-01       0       0 var1   69    28
1302 280701.167 8.416780e+00       0       0 var1   69    29
1303  54062.782 1.169146e-02       0       0 var1   69    30
1304 151433.111 1.602827e-01       0       0 var1   69    31
1305 163820.688 5.788763e+01       0       0 var1   69    32
1306 297298.281 1.494604e-02       0       0 var1   69    33
1307 113989.588 5.882187e+01       0       0 var1   69    34
1308 217061.081 2.445932e-02       0       0 var1   69    36
1309 117853.057 2.924151e-01       0       0 var1   69    38
1310 249481.087 8.782159e+01       0       0 var1   69    39
1311 303098.812 2.204159e+00       0       0 var1   69    40
1312 130016.504 3.974454e-01       0       0 var1   69    41
1313 306735.909 1.308358e+01       0       0 var1   69    42
1314 320716.056 2.222033e+00       0       0 var1   69    44
1315 264154.269 9.122796e-01       0       0 var1   69    45
1316  76297.487 1.898060e-01       0       0 var1   69    46
1317 273261.812 1.850728e+01       0       0 var1   69    47
1318 185823.414 5.337994e+01       0       0 var1   69    48
1319  99549.906 1.915265e+00       0       0 var1   69    50
1320 268576.887 8.945873e-01       0       0 var1   69    51
1321 330069.120 1.856788e-02       0       0 var1   69    53
1322 255356.506 3.516440e-01       0       0 var1   69    54
1323 117247.405 1.998879e+01       0       0 var1   69    55
1324  48834.451 2.108442e+00       0       0 var1   69    56
1325 128125.034 7.087637e-01       0       0 var1   69    57
1326 304326.360 5.527552e-03       0       0 var1   69    58
1327 285360.834 2.111672e-02       0       0 var1   69    59
1328  97292.278 2.045577e+00       0       0 var1   69    60
1329 180861.884 4.197990e-02       0       0 var1   69    61
1330  89028.907 5.930870e-01       0       0 var1   69    62
1331 208328.488 6.252941e+00       0       0 var1   69    63
1332 265210.216 5.803401e-01       0       0 var1   69    64
1333 139140.605 7.359953e-01       0       0 var1   69    66
1334 173244.841 1.672064e+00       0       0 var1   69    67
1335 272594.860 3.994961e-01       0       0 var1   69    68
1336 206235.311 2.775516e+01       0       0 var1   70     1
1337 221657.089 2.553787e+01       0       0 var1   70     3
1338 208758.865 1.908763e+01       0       0 var1   70     4
1339 195865.280 2.002071e+01       0       0 var1   70     5
1340 122218.401 5.390126e+00       0       0 var1   70     6
1341 257944.029 8.512179e+01       0       0 var1   70     9
1342 260982.761 6.151472e+00       0       0 var1   70    10
1343 169878.577 4.192577e+00       0       0 var1   70    11
1344   4518.646 2.127891e+00       0       0 var1   70    12
1345 136164.774 3.775678e+00       0       0 var1   70    13
1346 258152.687 2.022762e+01       0       0 var1   70    14
1347 173851.422 4.453751e+00       0       0 var1   70    15
1348  88665.889 6.852981e+00       0       0 var1   70    16
1349 134096.600 5.587714e+00       0       0 var1   70    17
1350 336823.399 1.101360e+02       0       0 var1   70    18
1351 260835.292 1.486492e+01       0       0 var1   70    19
1352 266156.406 3.045374e+00       0       0 var1   70    21
1353 127272.166 7.960980e-01       0       0 var1   70    23
1354 179188.025 6.204729e-02       0       0 var1   70    24
1355 249224.472 7.098598e-01       0       0 var1   70    25
1356 249555.727 5.978875e+00       0       0 var1   70    26
1357 130054.122 1.356770e-03       0       0 var1   70    27
1358 329329.785 9.176278e-02       0       0 var1   70    28
1359 266194.450 1.151147e+01       0       0 var1   70    29
1360 128703.454 3.597777e-01       0       0 var1   70    30
1361 228686.022 7.957369e-01       0       0 var1   70    31
1362 233276.393 6.561129e+01       0       0 var1   70    32
1363 221301.407 3.769244e-01       0       0 var1   70    33
1364 191708.054 6.660566e+01       0       0 var1   70    34
1365 315213.014 4.233753e-01       0       0 var1   70    35
1366 142431.588 4.200109e-01       0       0 var1   70    36
1367 192081.673 2.407521e-03       0       0 var1   70    38
1368 310002.142 9.727887e+01       0       0 var1   70    39
1369 225248.410 9.859563e-01       0       0 var1   70    40
1370 144641.608 1.925017e-02       0       0 var1   70    41
1371 333585.389 1.688232e+01       0       0 var1   70    42
1372 292035.397 3.614866e+00       0       0 var1   70    43
1373 316254.321 3.929657e+00       0       0 var1   70    44
1374  50475.865 3.138290e-03       0       0 var1   70    46
1375 224522.057 2.297953e+01       0       0 var1   70    47
1376 260795.094 6.080640e+01       0       0 var1   70    48
1377 123294.170 7.960980e-01       0       0 var1   70    50
1378 233880.050 2.066446e+00       0       0 var1   70    51
1379 326557.301 1.026333e-02       0       0 var1   70    54
1380 171714.944 2.462710e+01       0       0 var1   70    55
1381  73263.647 3.778107e+00       0       0 var1   70    56
1382 196644.965 1.778406e+00       0       0 var1   70    57
1383 238029.547 1.741728e-01       0       0 var1   70    58
1384 292190.023 4.057736e-01       0       0 var1   70    59
1385  31624.153 8.808745e-01       0       0 var1   70    60
1386 116403.296 4.852204e-01       0       0 var1   70    61
1387 147963.650 7.752534e-02       0       0 var1   70    62
1388 138620.624 8.953714e+00       0       0 var1   70    63
1389 214603.762 1.821391e+00       0       0 var1   70    66
1390 250489.641 3.185407e+00       0       0 var1   70    67
1391  77912.802 2.417568e-01       0       0 var1   70    69
1392 143755.819 2.280302e+01       0       0 var1   71     1
1393 168797.621 2.079757e+01       0       0 var1   71     3
1394 143999.775 1.502239e+01       0       0 var1   71     4
1395 139961.972 1.585143e+01       0       0 var1   71     5
1396 201082.684 3.343773e+00       0       0 var1   71     6
1397 304578.657 7.626671e+01       0       0 var1   71     9
1398 178060.895 3.948766e+00       0       0 var1   71    10
1399 232287.956 2.416509e+00       0       0 var1   71    11
1400  96166.251 3.809504e+00       0       0 var1   71    12
1401 234791.467 2.102629e+00       0       0 var1   71    13
1402 191167.425 2.615741e+00       0       0 var1   71    15
1403  97414.992 4.514579e+00       0       0 var1   71    16
1404 232616.244 3.499775e+00       0       0 var1   71    17
1405 209784.546 1.130600e+01       0       0 var1   71    19
1406 240558.367 1.932195e+00       0       0 var1   71    22
1407 221202.878 1.919081e+00       0       0 var1   71    23
1408  79945.642 5.952281e-02       0       0 var1   71    24
1409 148752.075 1.221264e-01       0       0 var1   71    25
1410 192803.802 3.810725e+00       0       0 var1   71    26
1411 203261.583 2.081472e-01       0       0 var1   71    27
1412 234584.964 8.408785e+00       0       0 var1   71    29
1413 217503.246 1.139532e-02       0       0 var1   71    30
1414 324775.762 1.591810e-01       0       0 var1   71    31
1415 333564.152 5.786666e+01       0       0 var1   71    32
1416 124541.612 1.461096e-02       0       0 var1   71    33
1417 286759.491 5.880073e+01       0       0 var1   71    34
1418 231227.214 2.484000e-02       0       0 var1   71    35
1419 104465.211 2.403013e-02       0       0 var1   71    36
1420 276583.411 2.939075e-01       0       0 var1   71    38
1421 139294.197 2.208253e+00       0       0 var1   71    40
1422 232873.758 3.991850e-01       0       0 var1   71    41
1423 285952.448 1.983067e+00       0       0 var1   71    43
1424 292625.386 2.217926e+00       0       0 var1   71    44
1425 141742.658 1.910087e-01       0       0 var1   71    46
1426 237991.726 1.849543e+01       0       0 var1   71    47
1427 217502.422 1.919081e+00       0       0 var1   71    50
1428 267463.937 8.919821e-01       0       0 var1   71    51
1429 236383.131 1.997646e+01       0       0 var1   71    55
1430 148084.098 2.104442e+00       0       0 var1   71    56
1431 296960.323 7.064450e-01       0       0 var1   71    57
1432 211752.219 5.734382e-03       0       0 var1   71    58
1433 103921.514 2.049522e+00       0       0 var1   71    60
1434  36369.513 4.141704e-02       0       0 var1   71    61
1435 248344.187 5.952117e-01       0       0 var1   71    62
1436  40485.995 6.246050e+00       0       0 var1   71    63
1437 320833.804 7.513397e-01       0       0 var1   71    65
1438 300258.860 7.336325e-01       0       0 var1   71    66
1439 173746.802 1.899422e-06       0       0 var1   71    69
1440 100475.956 2.431140e-01       0       0 var1   71    70
1441 242281.034 1.036697e+01       0       0 var1   72     1
1442 324483.431 9.029801e+00       0       0 var1   72     3
1443 311482.698 5.384259e+00       0       0 var1   72     4
1444 298818.052 5.885052e+00       0       0 var1   72     5
1445  25109.710 7.459796e-02       0       0 var1   72     6
1446 289963.649 2.294654e-01       0       0 var1   72     8
1447 175070.835 5.151807e+01       0       0 var1   72     9
1448 310923.806 1.863457e-01       0       0 var1   72    10
1449 222716.104 9.214178e-07       0       0 var1   72    11
1450 105282.872 1.230093e+01       0       0 var1   72    12
1451  66963.533 1.111503e-02       0       0 var1   72    13
1452 225654.366 5.997477e+00       0       0 var1   72    14
1453 260566.291 3.825783e-03       0       0 var1   72    15
1454 189088.645 3.240815e-01       0       0 var1   72    16
1455 117928.436 9.941174e-02       0       0 var1   72    17
1456 261356.246 7.133544e+01       0       0 var1   72    18
1457 276520.856 3.265131e+00       0       0 var1   72    19
1458 192549.873 9.207483e-02       0       0 var1   72    21
1459 134616.519 8.648196e+00       0       0 var1   72    23
1460 258763.149 3.238004e+00       0       0 var1   72    24
1461 332432.494 1.454452e+00       0       0 var1   72    25
1462 179011.724 4.046953e+00       0       0 var1   72    27
1463 232598.675 5.529374e+00       0       0 var1   72    28
1464 261252.319 1.807191e+00       0       0 var1   72    29
1465  28348.178 2.098800e+00       0       0 var1   72    30
1466 139350.759 1.337486e+00       0       0 var1   72    31
1467 165981.211 3.662115e+01       0       0 var1   72    32
1468 313855.672 2.058067e+00       0       0 var1   72    33
1469 100861.345 3.736499e+01       0       0 var1   72    34
1470 245223.334 1.961277e+00       0       0 var1   72    36
1471  89391.437 4.399946e+00       0       0 var1   72    38
1472 214246.144 6.106589e+01       0       0 var1   72    39
1473 324256.742 9.250671e+00       0       0 var1   72    40
1474 162212.312 4.784208e+00       0       0 var1   72    41
1475 331467.041 4.244721e+00       0       0 var1   72    42
1476 296737.800 4.382724e-03       0       0 var1   72    44
1477 254177.370 6.310063e+00       0       0 var1   72    45
1478 111199.793 3.970127e+00       0       0 var1   72    46
1479 308677.904 7.535896e+00       0       0 var1   72    47
1480 178904.459 3.305453e+01       0       0 var1   72    48
1481 131251.901 8.648196e+00       0       0 var1   72    50
1482 304471.712 3.733508e-01       0       0 var1   72    51
1483 335242.290 2.018066e+00       0       0 var1   72    53
1484 224066.004 4.621839e+00       0       0 var1   72    54
1485  81438.443 8.491579e+00       0       0 var1   72    55
1486  43416.681 1.098366e-02       0       0 var1   72    56
1487 133337.565 5.111815e-01       0       0 var1   72    57
1488 336706.384 2.660807e+00       0       0 var1   72    58
1489 316183.856 1.992430e+00       0       0 var1   72    59
1490 127482.081 8.922691e+00       0       0 var1   72    60
1491 189781.110 1.827798e+00       0       0 var1   72    61
1492 106736.098 5.414799e+00       0       0 var1   72    62
1493 219871.996 8.906404e-01       0       0 var1   72    63
1494 237386.042 5.376143e+00       0       0 var1   72    64
1495 112395.107 4.885296e-01       0       0 var1   72    66
1496 150316.340 6.957331e-02       0       0 var1   72    67
1497 257247.992 8.552431e-01       0       0 var1   72    68
1498  35902.032 2.423784e+00       0       0 var1   72    69
1499 102952.895 4.196509e+00       0       0 var1   72    70
1500 189320.728 2.419494e+00       0       0 var1   72    71
1501 223599.231 3.124461e+01       0       0 var1   73     1
1502 229895.905 2.888925e+01       0       0 var1   73     3
1503 202230.270 2.199901e+01       0       0 var1   73     4
1504 221201.638 2.299991e+01       0       0 var1   73     5
1505 166127.633 7.848890e+00       0       0 var1   73    10
1506 325617.975 1.293584e+00       0       0 var1   73    12
1507 294660.195 8.638842e+00       0       0 var1   73    16
1508 264945.860 1.744630e+01       0       0 var1   73    19
1509  42250.271 3.313122e-01       0       0 var1   73    22
1510 154733.531 3.254290e-01       0       0 var1   73    24
1511  81234.990 1.354669e+00       0       0 var1   73    25
1512 225583.912 7.653770e+00       0       0 var1   73    26
1513 317546.893 1.379548e+01       0       0 var1   73    29
1514 117093.855 8.748097e-01       0       0 var1   73    33
1515 143463.227 9.448694e-01       0       0 var1   73    35
1516 249911.334 9.398399e-01       0       0 var1   73    36
1517 145706.776 4.510229e-01       0       0 var1   73    40
1518 296563.583 5.456941e-01       0       0 var1   73    58
1519 328643.193 3.809089e-01       0       0 var1   73    60
1520 223756.955 1.036219e+00       0       0 var1   73    61
1521 194116.281 1.098025e+01       0       0 var1   73    63
1522 289716.314 2.741746e-03       0       0 var1   73    65
1523 329876.218 1.032793e-01       0       0 var1   73    70
1524 229518.085 6.633074e-01       0       0 var1   73    71
1525 191255.162 1.847835e+01       0       0 var1   74     3
1526 216300.147 1.306159e+01       0       0 var1   74     4
1527 197338.388 1.383538e+01       0       0 var1   74     5
1528 109816.821 1.671131e+00       0       0 var1   74    11
1529 248870.684 4.899961e+00       0       0 var1   74    12
1530 302923.285 1.411948e+00       0       0 var1   74    13
1531 284481.930 1.400748e+01       0       0 var1   74    14
1532  82310.274 1.837475e+00       0       0 var1   74    15
1533 178889.062 3.470634e+00       0       0 var1   74    16
1534 239573.474 2.588811e+00       0       0 var1   74    17
1535 200020.526 2.846171e+00       0       0 var1   74    20
1536 209957.410 2.712936e+00       0       0 var1   74    23
1537 329800.330 2.557965e-01       0       0 var1   74    24
1538 204889.599 2.857174e+00       0       0 var1   74    26
1539 147694.043 5.155553e-01       0       0 var1   74    27
1540 335004.259 2.403784e-02       0       0 var1   74    30
1541 309553.828 1.985689e-02       0       0 var1   74    33
1542 312578.654 1.085414e-02       0       0 var1   74    35
1543 182661.110 1.140063e-02       0       0 var1   74    36
1544 292268.306 3.046578e+01       0       0 var1   74    37
1545 272858.129 3.054838e+00       0       0 var1   74    40
1546 187050.176 7.985230e-01       0       0 var1   74    41
1547 252770.964 1.124901e+01       0       0 var1   74    42
1548  93421.691 1.314288e+00       0       0 var1   74    43
1549 213158.471 4.883714e-01       0       0 var1   74    46
1550  43542.185 1.631223e+01       0       0 var1   74    47
1551 210926.917 2.712936e+00       0       0 var1   74    50
1552  21796.700 4.660216e-01       0       0 var1   74    51
1553 312704.323 1.413434e+00       0       0 var1   74    56
1554 315464.056 3.349082e-01       0       0 var1   74    57
1555 131609.703 1.139170e-01       0       0 var1   74    58
1556 167717.677 1.388922e-02       0       0 var1   74    59
1557 215829.418 2.867622e+00       0       0 var1   74    60
1558 309468.967 3.396385e-03       0       0 var1   74    61
1559 271322.540 1.067688e+00       0       0 var1   74    62
1560 305085.664 5.006046e+00       0       0 var1   74    63
1561 285974.789 6.925765e-02       0       0 var1   74    69
1562 247198.415 5.698078e-01       0       0 var1   74    70
1563 273466.914 6.853415e-02       0       0 var1   74    71
1564 321865.999 1.673613e+00       0       0 var1   74    72

Illustration: gamma value calculation

right <- no2[1,]$NO2
right

[1] 13.10281

left <- no2[3,]$NO2
left

[1] 12.79901

(right-left)^2/2

[1] 0.04614575

Illustration: dist value calculation

Your turn

Variogram Cloud: Method 1

ggplot(data=vcloud, aes(x=dist, y=gamma)) + geom_point()

Variogram Cloud: Method 2

plot(vcloud)

Binned Empirical Semi-Variogram/ Sample variogram/ Sample Experimental Variogram

\[\hat{\gamma}(h) = \frac{1}{N_h}\sum_{i=1}^{N_h}(Z(s_i)-Z(s_i+h))^2\] where

\(h\) is the spatial distance between observations

\(N_h\) is the number of pairs that are within \(h\) distance.

Binned Empirical Semi-Variogram/ Sample variogram/ Sample Experimental Variogram

v1 <- variogram(NO2~1, no2.sf)
v1

    np      dist     gamma dir.hor dir.ver   id
1    5  13346.97  0.708610       0       0 var1
2   33  35120.03  9.590258       0       0 var1
3   45  57128.34 11.906498       0       0 var1
4   67  79156.08 14.268556       0       0 var1
5   85 101762.57  9.809778       0       0 var1
6  118 124431.83 12.441531       0       0 var1
7  138 146492.14 17.669501       0       0 var1
8   99 170131.95 14.815625       0       0 var1
9  132 192564.66 18.209828       0       0 var1
10 148 215025.57 15.654895       0       0 var1
11 121 237646.91 19.447882       0       0 var1
12 147 260395.54 16.009047       0       0 var1
13 142 281756.17 14.352454       0       0 var1
14 129 305033.38 16.201970       0       0 var1
15 155 328255.00 13.319879       0       0 var1

cutoff: one-third of the length of the bounding box diagonal

width: cutoff divided by 15

Plot Sample Experimental Variogram

plot(v1, plot.numbers = TRUE, xlab = "distance h [m]",
     ylab = expression(gamma(h)),
     xlim = c(0, 1.055 * max(v1$dist)))

Change default values

v2 <- variogram(NO2~1, no2.sf, cutoff = 100000, width = 10000)
plot(v2, plot.numbers = TRUE, xlab = "distance h [m]",
     ylab = expression(gamma(h)),
     xlim = c(0, 1.055 * max(v2$dist)))

Fitting variogram models

vgm()

   short                                      long
1    Nug                              Nug (nugget)
2    Exp                         Exp (exponential)
3    Sph                           Sph (spherical)
4    Gau                            Gau (gaussian)
5    Exc        Exclass (Exponential class/stable)
6    Mat                              Mat (Matern)
7    Ste Mat (Matern, M. Stein's parameterization)
8    Cir                            Cir (circular)
9    Lin                              Lin (linear)
10   Bes                              Bes (bessel)
11   Pen                      Pen (pentaspherical)
12   Per                            Per (periodic)
13   Wav                                Wav (wave)
14   Hol                                Hol (hole)
15   Log                         Log (logarithmic)
16   Pow                               Pow (power)
17   Spl                              Spl (spline)
18   Leg                            Leg (Legendre)
19   Err                   Err (Measurement error)
20   Int                           Int (Intercept)

Variogram models available in R’s gstat package

show.vgms(par.strip.text=list(cex=0.75))

Variogram

Resources: https://surferhelp.goldensoftware.com/vario/variogram_model.htm

https://csegrecorder.com/articles/view/the-variogram-basics-a-visual-intro-to-useful-geostatistical-concepts

Variogram modelling

v.m <- fit.variogram(v1, vgm(psill=20, model = "Exp", range = 20000, nugget = 1))

We need to provide partial sill, range, and nugget

Visualize the fit

plot(v1, v.m)

Assess isotropy assumption

v1.ani <- variogram(NO2~1, alpha = c(0, 45, 90, 135), no2.sf)
v.m <- fit.variogram(v1, vgm(psill=20, model = "Exp", range = 20000, nugget = 1))
plot(v1.ani, v.m)

plot(v1.ani)

Addressing anisotropy parameters

v1.ani <- variogram(NO2~1, alpha = c(0, 45, 90, 135), no2.sf)
fit.ani <- vgm(psill=20, model = "Exp", range = 25000, nugget = 3, anis = c(30, 10, 0, 0.5, 0.3))
plot(v1.ani, fit.ani)

Your turn: Fit a suitable model (I will proceed with v.m )

Krigging

1. Simple kriging

2. Ordinary kriging

3. Universal kriging

4. Regression kriging

5. Others: Staratified Kriging, Indicator Kriging, Block kriging

Ordinary Kriging

v1 <- variogram(NO2~1, no2.sf)
v.m <- fit.variogram(v1, vgm(psill=20, model = "Exp", range = 20000, nugget = 1))
krigOK <- krige(NO2~1, no2.sf, grid, v.m)

[using ordinary kriging]

krigOK

stars object with 2 dimensions and 2 attributes
attribute(s):
               Min.  1st Qu.    Median      Mean   3rd Qu.     Max. NA's
var1.pred  2.369868 7.233308  8.410634  8.754844  9.983278 20.13452 2076
var1.var   0.264067 8.052095 10.510333 10.135407 12.446936 15.82107 2076
dimension(s):
  from to  offset  delta                refsys x/y
x    1 65  280741  10000 WGS 84 / UTM zone 32N [x]
y    1 87 6101239 -10000 WGS 84 / UTM zone 32N [y]

Visualise kriging predictions

ggplot() + geom_stars(data = krigOK, aes(fill = var1.pred, x = x, y = y)) + 
    xlab(NULL) + ylab(NULL) +
    geom_sf(data = st_cast(map, "MULTILINESTRING")) + 
    geom_sf(data = no2.sf) +
    coord_sf(lims_method = "geometry_bbox") + scale_fill_viridis()

Validation

krig.ok.cv <- krige.cv(NO2~1, no2.sf, v.m)
krig.ok.cv 

Simple feature collection with 74 features and 6 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 304638.2 ymin: 5263687 xmax: 900829.7 ymax: 6086661
Projected CRS: WGS 84 / UTM zone 32N
First 10 features:
   var1.pred  var1.var  observed    residual      zscore fold
1   9.064467 14.151031 13.102806  4.03833865  1.07351642    1
2   9.074518 15.283772  7.135128 -1.93938966 -0.49607780    2
3  10.187623  7.825023 12.799011  2.61138737  0.93353044    3
4  10.291637  7.796963 11.830894  1.53925685  0.55124968    4
5  10.992385  7.354472 11.980110  0.98772545  0.36421725    5
6   8.689390  8.426194  8.935611  0.24622086  0.08482212    6
7   8.013070 15.678342  7.020690 -0.99238068 -0.25062722    7
8   7.132927 13.703602  9.226796  2.09386890  0.56562958    8
9  11.392578 15.191786 18.700023  7.30744545  1.87482686    9
10  9.252903 15.106029  9.159836 -0.09306688 -0.02394528   10
                   geometry
1  POINT (545413.7 5930802)
2  POINT (665710.6 5315213)
3  POINT (815741.2 5820995)
4  POINT (790544.1 5842367)
5  POINT (786923.1 5822067)
6  POINT (495006.7 5697747)
7  POINT (605720.5 5263687)
8  POINT (322457.1 5476682)
9  POINT (353646.6 5747825)
10 POINT (581136.7 5994606)

RMSE

round(sqrt(mean(krig.ok.cv$residual^2)), 2)

[1] 3.87

Regression Kriging

There is a trend in \(\mu(s)\). This trend is estimated using covariates other than coordinates.

plot( no2.sf$station_altitude, no2.sf$NO2,)

Sample Variogram

v.rk <- variogram(NO2~station_altitude, no2.sf)
plot(v.rk)

Fit Variogram

v.m.rk <- fit.variogram(v.rk, vgm(psill=15, model = "Exp", range = 20000, nugget = 1))
plot(v1, v.m)

Assess isotropy assumption

v1.ani.rk <- variogram(NO2~station_altitude, alpha = c(0, 45, 90, 135), no2.sf)
v.m.rk <- fit.variogram(v1.ani.rk, vgm(psill=15, model = "Exp", range = 20000, nugget = 1))
plot(v1.ani.rk, v.m.rk)

Regression Kriging

Your turn: Complete regression kriging

Using covariate

vr <- variogram(NO2~sqrt(pop_dens), no2.sf)
vr.m <- fit.variogram(vr, vgm(1, "Exp", 50000, 1))

Your turn

library(sp)
library(gstat)
library(sf)
library(mapview)

data(meuse)
data(meuse.grid)

meuse <- st_as_sf(meuse, coords = c("x", "y"), crs = 28992)
mapview(meuse, zcol = "zinc",  map.types = "CartoDB.Voyager")

meuse.grid <- st_as_sf(meuse.grid, coords = c("x", "y"),
                       crs = 28992)
mapview(meuse.grid,  map.types = "CartoDB.Voyager")

Perform EDA

meuse

Simple feature collection with 155 features and 12 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 178605 ymin: 329714 xmax: 181390 ymax: 333611
Projected CRS: Amersfoort / RD New
First 10 features:
   cadmium copper lead zinc  elev       dist   om ffreq soil lime landuse
1     11.7     85  299 1022 7.909 0.00135803 13.6     1    1    1      Ah
2      8.6     81  277 1141 6.983 0.01222430 14.0     1    1    1      Ah
3      6.5     68  199  640 7.800 0.10302900 13.0     1    1    1      Ah
4      2.6     81  116  257 7.655 0.19009400  8.0     1    2    0      Ga
5      2.8     48  117  269 7.480 0.27709000  8.7     1    2    0      Ah
6      3.0     61  137  281 7.791 0.36406700  7.8     1    2    0      Ga
7      3.2     31  132  346 8.217 0.19009400  9.2     1    2    0      Ah
8      2.8     29  150  406 8.490 0.09215160  9.5     1    1    0      Ab
9      2.4     37  133  347 8.668 0.18461400 10.6     1    1    0      Ab
10     1.6     24   80  183 9.049 0.30970200  6.3     1    2    0       W
   dist.m              geometry
1      50 POINT (181072 333611)
2      30 POINT (181025 333558)
3     150 POINT (181165 333537)
4     270 POINT (181298 333484)
5     380 POINT (181307 333330)
6     470 POINT (181390 333260)
7     240 POINT (181165 333370)
8     120 POINT (181027 333363)
9     240 POINT (181060 333231)
10    420 POINT (181232 333168)

Variogram Modelling with Pyhon

Click here to view the tutorial

Variogram Interpretations

Resources 1

Resources 2

Accuracy Measures and Cross-validation Strategies for Data with Temporal and/ or Spatial Structures

Loss function

• Function that calculates loss for a single data point

\(e_i = y - \hat{y}\)

\(e_i^2 = (y - \hat{y})^2\)

Cost function

• Calculates loss for the entire data sets

\[ME = \frac{1}{n}\sum_{i=1}^n e_i\]

Numeric outcome: Evaluations

Prediction accuracy measures (cost functions)

Mean Error

\[ME = \frac{1}{n}\sum_{i=1}^n e_i\]

• Error can be both negative and positive. So they can cancel each other during the summation.

Mean Absolute Error (L1 loss)

\[MAE = \frac{1}{n}\sum_{i=1}^n |e_i|\]

Mean Squared Error (L2 loss)

\[MSE = \frac{1}{n}\sum_{i=1}^n e^2_i\]

Mean Percentage Error

\[MPE = \frac{1}{n}\sum_{i=1}^n \frac{e_i}{y_i}\]

Mean Absolute Percentage Error

\[MAPE = \frac{1}{n}\sum_{i=1}^n |\frac{e_i}{y_i}|\]

Root Mean Squared Error

\[RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^n e^2_i}\]

Mean Absolute Scaled Error

In-class discussion

https://robjhyndman.com/papers/mase.pdf

Visualizaion of error distribution

Graphical representations reveal more than metrics alone.

Accuracy Measures on Training Set vs Test Set

Accuracy measure on training set: Tells about the model fit

Accuracy measure on test set: Model ability to predict new data

Evaluate Classifier Against Benchmarks

Naive approach: approach relies soley on \(Y\)

Outcome: Numeric

Naive Benchmark: Average (\(\bar{Y}\))

Time series cross-validation

Image credit: Professor Rob Hyndman

LOOCV

Blue - Training set

Orange - Test set

Not good for time series data. Why?

Evaluation on a rolling forecasting origin

• There are a series of test sets, each consisting of a single observation.

• The corresponding training set consists only of observations that occurred prior to the observation that forms the test set.

Training set - blue, Test set - red

The forecast accuracy is computed by averaging over the test sets.

Source: Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: principles and practice. OTexts.

Multi-step ahead forecast

Training set - blue, Test set - red

Source: Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: principles and practice. OTexts.

Cross-validation in Space

Source: Roberts, D. R., Bahn, V., Ciuti, S., Boyce, M. S., Elith, J., Guillera‐Arroita, G., … & Dormann, C. F. (2017). Cross‐validation strategies for data with temporal, spatial, hierarchical, or phylogenetic structure. Ecography, 40(8), 913-929

Case 1: Unbalanced data

Case 2

Source: Ploton, P., Mortier, F., Réjou-Méchain, M., Barbier, N., Picard, N., Rossi, V., … & Pélissier, R. (2020). Spatial validation reveals poor predictive performance of large-scale ecological mapping models. Nature communications, 11(1), 1-11.