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 -73.34Spatial 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 \[\{Z(s): s \in D\}.\]
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: \(s_i\)
Example: \(s_1 = (50.25, -104.39)\)
Daily temperature in Canada
Observed daily temperature value: \(z(s_i)\)
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
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 -73.34Visualization (cont.)
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 MontrealVisualisation (cont.)
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.)
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
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) -24.9Getting map data
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)
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
Three types of spatial processes
Geostatistical processes:
In theory the process could be observed at infinitely many locations in \(S \in D\).
Example: Air temperature in Sri Lanka
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
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.
Weakly Stationary
Strictly Stationary
Isotropy
direction-independent
Anisotropic
direction-independent
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).
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
Step 2: Read data
# 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>, … 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 12.799011Create a Coordinate Reference System (CRS) Object
st_crs: used to create a coordinate reference system (CRS) object.
EPSG:32632is 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”.
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)"],
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.sfSimple 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
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
Add observed values to the map
IDW
Step 1: Create a regular grid (\(10km \times 10km\))
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 objectmap.
st_as_stars(dx = 10000): Convert the bounding box obtained fromst_bbox(map)into a raster object using the stars package. Thedxparameter 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
[inverse distance weighted interpolation]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
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
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
26 202066.014 4.525315e+01 0 0 var1 12 1
27 219675.929 4.240914e+01 0 0 var1 12 3
28 206223.935 3.396172e+01 0 0 var1 12 4
29 193640.088 3.520263e+01 0 0 var1 12 5
30 123962.871 1.429138e+01 0 0 var1 12 6
31 258065.531 1.141666e+02 0 0 var1 12 9
32 256540.245 1.551529e+01 0 0 var1 12 10
33 173001.612 1.229419e+01 0 0 var1 12 11
34 306687.068 1.105700e+01 0 0 var1 13 1
35 336696.701 8.002720e-01 0 0 var1 13 2
36 337867.345 5.884644e+00 0 0 var1 13 4
37 322273.312 6.407685e+00 0 0 var1 13 5
38 82333.968 1.433032e-01 0 0 var1 13 6
39 256694.483 3.415857e-01 0 0 var1 13 8
40 217225.223 5.304262e+01 0 0 var1 13 9
41 195245.096 1.091355e-02 0 0 var1 13 11
42 140033.354 1.157251e+01 0 0 var1 13 12
43 177037.634 1.189548e+01 0 0 var1 14 2
44 241876.889 4.734317e+00 0 0 var1 14 6
45 214265.356 1.246021e+01 0 0 var1 14 7
46 272981.806 3.880700e+00 0 0 var1 14 8
47 187543.709 6.002180e+00 0 0 var1 14 11
48 262671.278 3.547682e+01 0 0 var1 14 12
49 159659.285 6.524973e+00 0 0 var1 14 13
50 334903.705 9.972493e+00 0 0 var1 15 1
51 134732.569 8.661895e+00 0 0 var1 15 3
52 151531.011 5.101038e+00 0 0 var1 15 4
53 131037.224 5.588778e+00 0 0 var1 15 5
54 284515.210 4.463643e-02 0 0 var1 15 6
55 95613.911 3.945451e-03 0 0 var1 15 11
56 174762.017 1.273862e+01 0 0 var1 15 12
57 255937.721 2.798286e-02 0 0 var1 15 13
58 281883.158 5.698351e+00 0 0 var1 15 14
59 239736.621 7.025133e+00 0 0 var1 16 1
60 138101.308 5.932543e+00 0 0 var1 16 3
61 132129.447 3.066418e+00 0 0 var1 16 4
62 115523.342 3.447079e+00 0 0 var1 16 5
63 210028.260 8.770794e-02 0 0 var1 16 6
64 274410.505 1.893480e-02 0 0 var1 16 10
65 138577.783 3.251753e-01 0 0 var1 16 11
66 87894.286 1.661825e+01 0 0 var1 16 12
67 206958.971 4.552328e-01 0 0 var1 16 13
68 288495.957 3.533247e+00 0 0 var1 16 14
69 97652.941 2.574838e-01 0 0 var1 16 15
70 332893.032 8.436016e+00 0 0 var1 17 1
71 294803.590 1.730023e+00 0 0 var1 17 2
72 309241.370 7.234305e+00 0 0 var1 17 3
73 308350.641 4.020444e+00 0 0 var1 17 4
74 290426.182 4.454701e+00 0 0 var1 17 5
75 140194.979 1.778308e-03 0 0 var1 17 6
76 338409.793 1.949440e+00 0 0 var1 17 7
77 302161.784 2.680727e-02 0 0 var1 17 8
78 284053.586 4.709133e+01 0 0 var1 17 9
79 130107.297 1.000180e-01 0 0 var1 17 11
80 138612.212 1.461199e+01 0 0 var1 17 12
81 68763.425 1.770088e-01 0 0 var1 17 13
82 124304.904 4.552584e+00 0 0 var1 17 14
83 202739.659 6.423353e-02 0 0 var1 17 15
84 177081.839 6.450852e-02 0 0 var1 17 16
85 241497.224 8.922763e+01 0 0 var1 18 2
86 264406.127 6.679637e+01 0 0 var1 18 6
87 227936.069 9.076293e+01 0 0 var1 18 7
88 140202.306 6.347318e+01 0 0 var1 18 8
89 328274.538 1.608895e+00 0 0 var1 18 9
90 322212.616 7.135165e+01 0 0 var1 18 11
91 202916.355 7.312745e+01 0 0 var1 18 13
92 144861.519 3.596468e+01 0 0 var1 18 14
93 217525.496 6.610885e+01 0 0 var1 18 17
94 67580.859 1.996034e+00 0 0 var1 19 1
95 327324.291 2.636137e-01 0 0 var1 19 4
96 332162.777 3.830931e-01 0 0 var1 19 5
97 267020.523 2.352669e+00 0 0 var1 19 6
98 254503.089 2.884379e+01 0 0 var1 19 9
99 98868.656 1.891422e+00 0 0 var1 19 10
100 257026.546 2.824109e+01 0 0 var1 19 12
101 304107.038 1.531866e+00 0 0 var1 19 16
102 195849.609 3.923311e+00 0 0 var1 20 2
103 274764.059 3.609296e+00 0 0 var1 20 7
104 204307.772 8.879101e+00 0 0 var1 20 11
105 232909.549 2.948182e+01 0 0 var1 20 14
106 263911.734 9.257384e+00 0 0 var1 20 15
107 283540.027 1.086387e+01 0 0 var1 20 17
108 257285.775 4.852080e-01 0 0 var1 21 2
109 197798.332 3.324268e-01 0 0 var1 21 6
110 264650.262 6.044892e-01 0 0 var1 21 7
111 166014.500 6.122497e-01 0 0 var1 21 8
112 281230.817 5.596607e+01 0 0 var1 21 9
113 264795.856 9.149320e-02 0 0 var1 21 11
114 270288.164 1.026452e+01 0 0 var1 21 12
115 132370.211 3.920815e-02 0 0 var1 21 13
116 110916.126 7.575778e+00 0 0 var1 21 14
117 325591.236 7.616400e-01 0 0 var1 21 16
118 150369.845 3.828325e-01 0 0 var1 21 17
119 70727.265 7.655321e+01 0 0 var1 21 18
120 256158.476 3.801074e+01 0 0 var1 22 1
121 209895.718 3.540808e+01 0 0 var1 22 3
122 186028.622 2.772978e+01 0 0 var1 22 4
123 206198.644 2.885214e+01 0 0 var1 22 5
124 203978.307 1.140537e+01 0 0 var1 22 10
125 335094.113 3.155756e-01 0 0 var1 22 12
126 336983.540 9.044210e+00 0 0 var1 22 15
127 293343.858 1.235373e+01 0 0 var1 22 16
128 302172.533 2.258601e+01 0 0 var1 22 19
129 331840.290 3.795250e+01 0 0 var1 23 1
130 296404.389 3.766605e+00 0 0 var1 23 2
131 284223.175 3.535187e+01 0 0 var1 23 3
132 285298.666 2.768004e+01 0 0 var1 23 4
133 266753.031 2.880141e+01 0 0 var1 23 5
134 158574.308 1.032920e+01 0 0 var1 23 6
135 331516.964 1.169508e+01 0 0 var1 23 8
136 306069.609 1.023818e+02 0 0 var1 23 9
137 100792.862 8.642551e+00 0 0 var1 23 11
138 131715.831 3.209064e-01 0 0 var1 23 12
139 95278.684 8.039230e+00 0 0 var1 23 13
140 134902.850 2.904947e+01 0 0 var1 23 14
141 173724.792 9.015814e+00 0 0 var1 23 15
142 155921.065 1.232054e+01 0 0 var1 23 16
143 29847.845 1.060204e+01 0 0 var1 23 17
144 242279.528 1.296595e+02 0 0 var1 23 18
145 265620.926 1.596840e-03 0 0 var1 23 20
146 177052.200 6.955579e+00 0 0 var1 23 21
147 114239.235 2.519261e+01 0 0 var1 24 1
148 180866.687 2.308234e+01 0 0 var1 24 3
149 148617.232 1.697313e+01 0 0 var1 24 4
150 155672.458 1.785365e+01 0 0 var1 24 5
151 266103.228 4.295553e+00 0 0 var1 24 6
152 115053.624 4.977910e+00 0 0 var1 24 10
153 305405.814 3.234550e+00 0 0 var1 24 11
154 174756.446 2.916657e+00 0 0 var1 24 12
155 310424.728 2.869695e+00 0 0 var1 24 13
156 249171.760 3.464431e+00 0 0 var1 24 15
157 166920.485 5.610867e+00 0 0 var1 24 16
158 312341.278 4.472132e+00 0 0 var1 24 17
159 179540.860 1.300621e+01 0 0 var1 24 19
160 172979.335 1.313456e+00 0 0 var1 24 22
161 301111.453 1.302648e+00 0 0 var1 24 23
162 160597.332 1.958757e+01 0 0 var1 25 1
163 185717.330 1.773226e+01 0 0 var1 25 3
164 153527.947 1.243555e+01 0 0 var1 25 4
165 168634.541 1.319084e+01 0 0 var1 25 5
166 123586.697 2.682010e+00 0 0 var1 25 10
167 244918.076 5.295802e+00 0 0 var1 25 12
168 287078.662 1.607468e+00 0 0 var1 25 15
169 220724.411 3.151648e+00 0 0 var1 25 16
170 214840.785 9.078012e+00 0 0 var1 25 19
171 100838.157 3.025860e+00 0 0 var1 25 22
172 73843.832 3.521696e-01 0 0 var1 25 24
173 309446.771 7.970135e+00 0 0 var1 26 1
174 28169.938 6.803370e+00 0 0 var1 26 3
175 49765.519 3.700872e+00 0 0 var1 26 4
176 54617.976 4.117965e+00 0 0 var1 26 5
177 303677.578 1.227957e-03 0 0 var1 26 10
178 251092.902 1.580821e-01 0 0 var1 26 11
179 247503.068 1.524046e+01 0 0 var1 26 12
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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 72Illustration: gamma value calculation
Illustration: dist value calculation
Your turn
Variogram Cloud: Method 1
Variogram Cloud: Method 2
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
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 var1cutoff: one-third of the length of the bounding box diagonal
width: cutoff divided by 15
Plot Sample Experimental Variogram
Change default values
Fitting variogram models
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
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
We need to provide partial sill, range, and nugget
Visualize the fit
Assess isotropy assumption
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
Simple kriging
Ordinary kriging
Universal kriging
Regression kriging
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]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
Validation
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
Regression Kriging
There is a trend in \(\mu(s)\). This trend is estimated using covariates other than coordinates.
Sample Variogram
Fit Variogram
Assess isotropy assumption
Regression Kriging
Your turn: Complete regression kriging
Using covariate
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
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
Variogram Interpretations
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
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.
