2e:null 2f:["$","title",null,{"children":"TTT : Visualizing W. Tobler's \"first law of geography\" ? - Spatial autocorrelation | Laurent Jégou | Observable"}] 34:T11cb,moran_i = function(v, typepoids, seuil) { var va = get_var(v); const n = depts.features.length; const moyv = maths.mean(va); var wi = []; var numerateur = 0; var denominateur = 0; if (typepoids === undefined) {typepoids = vpoids.value;} if (seuil === undefined) {seuil = seuil_poids;} var S0 = 0; var S1 = 0; var S2 = 0; var A = 0; var B = 0; var C = 0; var D = 0; var EI = 0; var EI2 = 0; var VI = 0; var ZI = 0; // Calcul de S0 (somme des poids) for (var i=0; i (d[0])) .y(d => (d[1])); for (var i=0; i 0 && i == 23) { data_lines.push(laproj([turf.centroid(depts.features[i]).geometry.coordinates[0], turf.centroid(depts.features[i]).geometry.coordinates[1]])); data_lines.push(laproj([turf.centroid(depts.features[j]).geometry.coordinates[0], turf.centroid(depts.features[j]).geometry.coordinates[1]])); } } } carteg.append("path") .data([data_lines]) .attr("d", line) .attr("fill", "none") .attr("stroke", "white"); } // Titre var titre =""; var sous_titre = ""; var titre_legende = ""; switch(vrepval) { case "centre": titre = "Test Centre"; sous_titre = ""; titre_legende = "Test"; break; case "sudouest": titre = "Test South-West"; sous_titre = ""; titre_legende = "Test"; break; case "damier": titre = "Test Checkers"; sous_titre = ""; titre_legende = "Test"; break; case "res2": titre = "Prop. of second homes"; sous_titre = "in 2019"; titre_legende = 'housing %'; break; case "resp": titre = "Prop. of primary homes"; sous_titre = "in 2019"; titre_legende = 'housing %'; break; case "med_niv_vie": titre = "Median standard of living"; sous_titre = "en 2020"; titre_legende = 'euros'; break; case "part_retraites_rev": titre = "Prop. of pensions in available income"; sous_titre = "en 2020"; titre_legende = '% of income'; break; case "part_mini_soc_rev": titre = "Prop. of social minima in available income"; sous_titre = "en 2020"; titre_legende = '% of income'; break; } if (letyperep == "Moran I local") { sous_titre += " - local Moran's I"; titre_legende = "Indicator"; } if (letyperep == "M.I.L. Z-Score") { sous_titre += " - local I Z-Score"; titre_legende = "Indicator"; } if (letyperep == "Getis&Ord local") { sous_titre += " - local Getis & Ord"; titre_legende = "Indicator"; } if (letyperep == "Geary C local") { sous_titre += " - local Geary's C"; titre_legende = "Indicator"; } carteg.append("g") .attr("transform", "translate("+decalage+", -20)") .append("text") .text(titre) .attr("font-size", 20); carteg.append("g") .attr("transform", "translate("+decalage+", 0)") .append("text") .text(sous_titre) .attr("font-size", 16); // Légende carteg.append("g") .attr("transform", "translate("+decalage+", 480)") .append(() => legend({color: colorQtvi, title: titre_legende, width: 420, tickFormat: ".2r"})); // Sources carteg.append("g") .attr("transform", "translate("+decalage+", 540)") .append("text") .text("Sources : INSEE, IGN") .attr("font-size", 10) .attr("font-family", "sans-serif"); }39:T2302,calcul_il = function(v, letypepoids) { var lavar = ''; var d = []; var g; var mi = []; if (v === undefined) { switch (typerep) { case 'Variable': lavar = vrep.value; break; case 'Moran I local': lavar = vrep.value+'_il'; break; case 'M.I.L. Z-Score': lavar = vrep.value+'_iz'; break; case 'Geary C local': lavar = vrep.value+'_cl'; break; case 'G local': lavar = vrep.value+'_gg'; break; case 'Getis&Ord local': lavar = vrep.value + '_go'; break; } } else { lavar = v; } if (letypepoids === undefined) { letypepoids = vpoids.value; } switch (lavar) { case 'centre': d.push(maths.min(centre)); d.push(maths.max(centre)); g = d3.interpolateYlOrRd; break; case 'centre_il': mi = moran_il('centre', letypepoids).il; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'centre_iz': mi = moran_il('centre', letypepoids).ilz; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'centre_cl': mi = geary_cl('centre', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'centre_gg': mi = g_statl('centre', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'centre_go': mi = go('centre'); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'sudouest': d.push(maths.min(sudouest)); d.push(maths.max(sudouest)); g = d3.interpolateYlOrRd; break; case 'sudouest_il': mi = moran_il('sudouest', letypepoids).il; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'sudouest_iz': mi = moran_il('sudouest', letypepoids).ilz; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'sudouest_cl': mi = geary_cl('sudouest', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'sudouest_gg': mi = g_statl('sudouest', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; break; case 'sudouest_go': mi = go('sudouest'); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'damier': d.push(maths.min(damier)); d.push(maths.max(damier)); g = d3.interpolateYlOrRd; break; case 'damier_il': mi = moran_il('damier', letypepoids).il; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'damier_iz': mi = moran_il('damier', letypepoids).ilz; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'damier_cl': mi = geary_cl('damier', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'damier_gg': mi = g_statl('damier', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'damier_go': mi = go('damier', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'resp': d.push(maths.min(resp)); d.push(maths.max(resp)); g = d3.interpolateYlOrRd; break; case 'resp_il': mi = moran_il('resp', letypepoids).il; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'resp_iz': mi = moran_il('resp', letypepoids).ilz; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'resp_cl': mi = geary_cl('resp', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'resp_gg': mi = g_statl('resp', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'resp_go': mi = go('resp', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'res2': d.push(maths.min(res2)); d.push(maths.max(res2)); g = d3.interpolateYlOrRd; break; case 'res2_il': mi = moran_il('res2', letypepoids).il; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'res2_iz': mi = moran_il('res2', letypepoids).ilz; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'res2_cl': mi = geary_cl('res2', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'res2_gg': mi = g_statl('res2', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'res2_go': mi = go('res2', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'med_niv_vie': d.push(maths.min(med_niv_vie)); d.push(maths.max(med_niv_vie)); g = d3.interpolateYlOrRd; break; case 'med_niv_vie_il': mi = moran_il('med_niv_vie', letypepoids).il; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'med_niv_vie_iz': mi = moran_il('med_niv_vie', letypepoids).ilz; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'med_niv_vie_cl': mi = geary_cl('med_niv_vie', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'med_niv_vie_gg': mi = g_statl('med_niv_vie', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'med_niv_vie_go': mi = go('med_niv_vie', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'part_retraites_rev': d.push(maths.min(part_retraites_rev)); d.push(maths.max(part_retraites_rev)); g = d3.interpolateYlOrRd; break; case 'part_retraites_rev_il': mi = moran_il('part_retraites_rev', letypepoids).il; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'part_retraites_rev_iz': mi = moran_il('part_retraites_rev', letypepoids).ilz; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'part_retraites_rev_cl': mi = geary_cl('part_retraites_rev', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'part_retraites_rev_gg': var mi = g_statl('part_retraites_rev', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'part_retraites_rev_go': var mi = go('part_retraites_rev', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'part_mini_soc_rev': d.push(maths.min(part_mini_soc_rev)); d.push(maths.max(part_mini_soc_rev)); g = d3.interpolateYlOrRd; break; case 'part_mini_soc_rev_il': var mi = moran_il('part_mini_soc_rev', letypepoids).il; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'part_mini_soc_rev_iz': var mi = moran_il('part_mini_soc_rev', letypepoids).ilz; d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'part_mini_soc_rev_cl': var mi = geary_cl('part_mini_soc_rev'); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'part_mini_soc_rev_gg': var mi = g_statl('part_mini_soc_rev', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; case 'part_mini_soc_rev_go': var mi = go('part_mini_soc_rev', letypepoids); d.push(maths.max(mi)); d.push(maths.min(mi)); g = d3.interpolateRdYlGn; break; } return {domain: d, gradient: g, table_il: mi}; }3a:[{"id":"ca80c518573bf0f33279673a0a552a6a5aa3ca62a31ce00f8153944182a19863f20b182b52c27d1f30ee185ce1a19295c0978bb793e781d4db5efa465e4db8b1","href":"https://static.observableusercontent.com/files/ca80c518573bf0f33279673a0a552a6a5aa3ca62a31ce00f8153944182a19863f20b182b52c27d1f30ee185ce1a19295c0978bb793e781d4db5efa465e4db8b1","name":"fr_deptsresrev@2.json","type":"application/json","lastModified":1678692657330.11,"size":149396}] 32:{"id":"e309030588d19e37","slug":"spatial_autocorrelation","title":"TTT : Visualizing W. Tobler's \"first law of geography\" ? - Spatial autocorrelation","thumbnail":"87b5f8ca4b0f4c9002e287d30aabce901201139dfcd578ed59db16cae43a9a1f","access_level":"public","create_time":"$D2023-03-13T07:30:57.330Z","update_time":"$D2023-03-14T08:22:22.734Z","view_time":null,"first_public_time":"$D2023-03-14T08:22:59.581Z","has_unpublished_edits":null,"fork":{"id":"7a3737fc833471e0","slug":"autocorrelation_spatiale","title":"TTT : Visualiser la \"première loi de la géographie\" de W. Tobler ? - L'autocorrélation spatiale","owner":{"id":"1936870012656af6","login":"ljegou","name":"Laurent Jégou"}},"stars":8,"starred":false,"owner":{"type":"team","id":"1936870012656af6","login":"ljegou","name":"Laurent Jégou","avatar_url":"https://avatars-next.observableusercontent.com/447aa44a872789c3c13d2505fdb5a34daaaa3a44590422f4f782b6ea9a4bd3ac","tier":"starter_2024"},"version":1990,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"dc56e6af58493346","login":"ljegou","name":"Laurent Jégou","avatar_url":"https://avatars-next.observableusercontent.com/447aa44a872789c3c13d2505fdb5a34daaaa3a44590422f4f782b6ea9a4bd3ac","tier":null}],"members":[{"type":"individual","id":"dc56e6af58493346","login":"ljegou","name":"Laurent Jégou","avatar_url":"https://avatars-next.observableusercontent.com/447aa44a872789c3c13d2505fdb5a34daaaa3a44590422f4f782b6ea9a4bd3ac","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":2,"collection":null,"type":"notebook","body":{"title":"TTT : Visualizing W. Tobler's \"first law of geography\" ? - Spatial autocorrelation","cells":[{"id":0,"value":"# TTT : Visualizing W. Tobler's \"first law of geography\" ? - Spatial autocorrelation","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1602,"value":"\n\"I invoke the first law of geography: everything is related to everything else, but near things are more related than distant things.\"\n\nTobler, 1970.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1989,"value":"French version: https://observablehq.com/@ljegou/autocorrelation_spatiale","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1911,"value":"toc();","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1615,"value":"W. Tobler introduced this idea somewhat jokingly at an *International Geographical Union* meeting in 1969, on the first ideas about using a computer in geography, and then explored it further in a 1970 paper. This hypothesis has become a fertile research question, which has produced many other publications. It had already appeared with R.A. Fisher in 1935, but had not had the favorable context of geomatics to develop.\n\nThis \"law\" is the source of concepts as **spatial dependancy**, **effect distance ponderation** and other analysis of the role of geographical space and distance. Here, we will focus on the family of indicators which are directly linked to the concept of spatial relations : **spatial autocorrelation**.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1617,"value":"Spatial autocorrelation is a measure of the effect of distance to the intensity of the relation between objects. It implies a **spatial variable** and a **distance metric**, a mean to take into account the distance between objects, wich can take several forms.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1619,"value":"Several indicators were proposed by the research, we will present and visualize them using a statistical dataset about housing on the french metropolitan départements (source INSEE and IGN). At the end of the notepad, on can find an interactive map to test different variables and parameters.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1626,"value":"## First application example: global indicators","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1630,"value":"To start our analysis, let's represent a spatially heterogenous variable: **the proportion of second homes** in 2019's total housing:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1540,"value":"{\n const svg = d3.create(\"svg\")\n .attr(\"viewBox\", [0, 0, 975, 580]);\n \n carte(svg, \"gauche\", \"res2\", \"Variable\");\n \n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1636,"value":"Quite clearly, the departments with higher values are those with amenities for tourism: coastline, mountains. The departments are sometimes grouped into areas with similar values, such as the Northeast (low %) or the Alps and Corsica (high %).\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1653,"value":"One can assess the spatial autocorrelation by calculating **global indicators**, producing a single value for the entire dataset.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1741,"value":"### Geary's contiguity indicator","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1692,"value":"The contiguity indicator of Geary, or **Geary's C**, was developed by Roy C. Geary in 1954. It measures the difference between the values of the spatial entities (the departments), weighted by a distance metric and related to the deviations of the entity values from the mean. If less than 1, it expresses a positive spatial autocorrelation: close objects present values more similar than distant objects.\n\n(Cf. the Wikipedia article: https://en.wikipedia.org/wiki/Geary%27s_C).\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1747,"value":"It's a global indicator, qualifying the entire dataset, but it is relatively sensible to local autocorrelation groups.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1699,"value":"Below, we calculate it with a distance weighting metric based on the inverse distance of neighboring entities (adjacent). So, **the further an object is from the one considered, the less its value will be taken into account**. Moreover, only *adjacent* départements are used. The distances are measured from the outline of the départements, which can bias the calculation for small entities (Paris region). Other types of distance metrics can alleviate this bias (a specified number of neighbors).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1668,"value":"Gearys_C_invd = parseFloat(geary_c(\"res2\", \"poids_idv\").toFixed(3));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1704,"value":"With a inverse squared distance (like gravity in physics), the Geary's C indicator is higher, the autocorrelation is lesser :","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1709,"value":"Gearys_C_g = parseFloat(geary_c(\"res2\", \"poids_g\").toFixed(3));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1726,"value":"By using a weighting metric based on the inverse distance neighbors with a distance threshold, on can observe the variation of the Gear'y C indicator:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1721,"value":"viewof seuil_geary = Inputs.range([maths.min(matrice_distances), maths.max(matrice_distances)], {value: parseFloat((etendue_md/2).toPrecision(2)), step: parseFloat((etendue_md/100).toPrecision(2)), label: \"Seuil poids inv/distance km\"});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1717,"value":"Gearys_C_s = parseFloat(geary_c(\"res2\", \"poids_idseuil\", seuil_geary).toFixed(3));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1738,"value":"Below, one can visualize the distance threshold for the neighbors of the Creuse département:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1730,"value":"{\n const svg = d3.create(\"svg\")\n .attr(\"viewBox\", [0, 0, 975, 580]);\n\n carte(svg, \"gauche\", \"res2\", \"Variable\", null, seuil_geary, true);\n\n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1785,"value":"### Moran's I spatial autocorrelation indice","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1750,"value":"In 1950, **Patrick Moran** develops a global autocorrelation indice, the **I**, which is less sensible to local groups of values. It is based on the sum of the deviations to the mean of the elements of the dataset, weighted by a distance metric, related to the sum of the weights. Cf. the Wikipedia article: https://en.wikipedia.org/wiki/Moran%27s_I","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1758,"value":"The Moran's I fluctuates between -1 to 1, with a negative spatial autocorrelation below 0 (close entities have very different values).\n\nUsed on the proportion of second homes, with the inverse distance weighting metric, it shows a value near 0.5:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1659,"value":"Moran_I_idv = parseFloat(moran_i(\"res2\", \"poids_idv\").moran_i.toFixed(3));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1762,"value":"With the gravity weighting metric, the value decreases a bit:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1766,"value":"Moran_I_g = parseFloat(moran_i(\"res2\", \"poids_g\").moran_i.toFixed(3));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1800,"value":"One can, here too, test the indicator with a distance threshold for the inverse distance weighting:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1773,"value":"viewof seuil_moran = Inputs.range([maths.min(matrice_distances), maths.max(matrice_distances)], {value: parseFloat((etendue_md/2).toPrecision(2)), step: parseFloat((etendue_md/100).toPrecision(2)), label: \"Weighting threshold inv/distance km\"});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1776,"value":"Moran_I_g_th = parseFloat(moran_i(\"res2\", \"poids_idseuil\", seuil_moran).moran_i.toFixed(3));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1787,"value":"The calculation of this indicator allows also the production of expected values, therefore also deviations from them, to express the significance of the results:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1791,"value":"Moran_I_g_th_expected = parseFloat(moran_i(\"res2\", \"poids_idseuil\", seuil_moran).EI.toFixed(3));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1795,"value":"Moran_I_g_th_zscore = parseFloat(moran_i(\"res2\", \"poids_idseuil\", seuil_moran).ZI.toFixed(3));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1789,"value":"These global indices are interesting, but obviously very synthetic. It is therefore useful to evaluate **locally** the autocorrelation values, i.e. for a subset of local values close to the entity being evaluated, defined by a distance metric. These are the so-called **local indicators of spatial autocorrelation**.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1805,"value":"## Local indicators of spatial autocorrelation","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1807,"value":"**Luc Anselin** developed the field of spatial econometrics and proposed in 1995 the idea of Local Indicators of Spatial Association, **LISA** (Anselin, 1995), to further *exploratory data analysis* with the help of computers.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1842,"value":"### Local Moran's I ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1811,"value":"The principal LISA indicator is **the local version of the Moran's** I: for each entity of the dataset, we will calculate a value of the I, using a selection of neighbors entities and a weighting metric to set the range of the local autocorrelation measure.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1817,"value":"{\n const svg = d3.create(\"svg\")\n .attr(\"viewBox\", [0, 0, 975, 580]);\n \n carte(svg, \"gauche\", \"res2\", \"Variable\");\n carte(svg, \"droite\", \"res2\", \"Moran I local\", \"poids_idv\");\n \n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1821,"value":"As for the global Moran's I, the local I measures the spatial autocorrelations between -1 and 1, negatives and positives spatial correlations of local values. In our example on the proportion of second homes, as the values are quite widely fluctuating from a département to another, the indicators presents small values (in red). The patches of relatively high values are identifying small groups of high percentages, as in the Alps or the Paris region. In this last case, the small size of the Paris départements explain part of the high local I values.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1824,"value":"The heterogenous regions are marked by the reddest colors : the Rhône valley, Haute-Garonne and Tarn, Creuse...","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1826,"value":"To evaluate the *significativity* of the local indicator, one can calculate and represent the deviation from the expected value, the Z-Score :","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1594,"value":"{\n const svg = d3.create(\"svg\")\n .attr(\"viewBox\", [0, 0, 975, 580]);\n\n carte(svg, \"gauche\", \"res2\", \"Moran I local\", \"poids_idv\");\n carte(svg, \"droite\", \"res2\", \"M.I.L. Z-Score\", \"poids_idv\");\n \n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1844,"value":"### Other local indicators : local Geary's C and local Getis&Ord *","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1847,"value":"The local version of the **Geary's C** ratio is directly derived from the global version and inherits its characteristics. The *** (star) indicator from Getis and Ord** is more a *hotspots* indicator, ie. of spatial concentrations of high values, with a longer range for larger zones. It's a Z-Score, which can be analyzed directly, cf. Getis and Ord, 1992.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1829,"value":"{\n const svg = d3.create(\"svg\")\n .attr(\"viewBox\", [0, 0, 975, 580]);\n\n carte(svg, \"gauche\", \"res2\", \"Geary C local\", \"poids_idv\");\n carte(svg, \"droite\", \"res2\", \"Getis&Ord local\", \"poids_idv\");\n \n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1832,"value":"# Interactive test map: variables, indicators, distance metrics","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1859,"value":"Below is a \"sandbox\" interactive map, to test different variables, indicators and their parameters (weighting distance metrics, thresholds.\n\nOne can, notably, choose to use *standardized* distances matrices (lines sums as 1) and to display the lines between neighboring départements of the Creuse (center of France) for distance metrics using a *threshold*.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1565,"value":"{\n const svg = d3.create(\"svg\")\n .attr(\"viewBox\", [0, 0, 975, 580]);\n\n carte (svg);\n\n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1028,"value":"viewof nb_classes = Inputs.range([1, 24], {value: 10, step: 1, label: \"Nb. of classes\"});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":300,"value":"viewof vrep = Inputs.radio(les_variables, {label: \"Variable :\", value: les_variables[0], format: x => html`${x.label}`});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":911,"value":"viewof typerep = Inputs.radio(['Variable','Moran I local','M.I.L. Z-Score','Getis&Ord local','Geary C local'], {label: \"Type carto. :\", value: 'Variable'});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1062,"value":"viewof type_matrice = Inputs.radio(['Distances km', 'Normalized distances'], {label: 'Type de matrice :', value: 'Distances km'});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":518,"value":"viewof vpoids = Inputs.radio(les_poids, {label: \"Pondération distance :\", value: les_poids[0], format: x => x.label});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":787,"value":"viewof seuil_poids = Inputs.range([maths.min(matrice_distances), maths.max(matrice_distances)], {value: parseFloat((etendue_md/2).toPrecision(2)), step: parseFloat((etendue_md/100).toPrecision(2)), label: \"Seuil poids inv/distance\"});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1299,"value":"viewof aff_creuse = Inputs.radio([\"Yes\", \"No\"], {label: 'Distances from Creuse', value: 'No'});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1891,"value":"# Calculations, algorithms and formulas","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":64,"value":"## Distances matrixes","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":72,"value":"matrice_distances = {\n const n = depts.features.length;\n var d = 0;\n var di = [];\n for (var i=0; i {return o / aj});\n di.push(djp);\n break;\n case 'Distances km':\n di.push(dj);\n break;\n }\n }\n return di;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1081,"value":"etendue_md = maths.max(matrice_distances) - maths.min(matrice_distances)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":245,"value":"matrice_adj = {\n const n = depts.features.length;\n var di = [];\n for (var i=0; i {\n var pt = turf.centroid(d);\n pt.properties = d.properties;\n return pt;\n }\n );\n return centres;\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":614,"value":"get_var = function(nom) {\n var v;\n switch (nom) {\n case 'resp':\n v = resp;\n break;\n case 'res2':\n v = res2;\n break;\n case 'med_niv_vie':\n v = med_niv_vie;\n break;\n case 'part_retraites_rev':\n v = part_retraites_rev;\n break;\n case 'part_mini_soc_rev':\n v = part_mini_soc_rev;\n break;\n case 'centre':\n v = centre;\n break;\n case 'sudouest':\n v = sudouest;\n break;\n case 'damier':\n v = damier;\n break;\n }\n return v;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":213,"value":"poids_g = function (i,j) {\n // Fonction gravitaire\n return 1/Math.pow(matrice_distances[i][j], 2);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":282,"value":"poids_id = function (i,j) {\n // Fonciton inverse de la distance\n return 1/matrice_distances[i][j];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":532,"value":"poids_idv = function (i,j) {\n // Fonciton inverse de la distance des voisins immédiats seulement\n return matrice_adj[i][j] * (1 / matrice_distances[i][j]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":563,"value":"poids_v = function (i,j) {\n // Fonction qui ne garde que les voisins immédiats\n return matrice_adj[i][j];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":784,"value":"poids_idseuil = function(i,j, s) {\n var v = 0;\n var d = matrice_distances[i][j];\n if (d <= s){\n return 1/d;\n } else {\n return 0;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1165,"value":"poids_seuil = function(i,j, s) {\n var v = 0;\n var d = matrice_distances[i][j];\n if (d <= s){\n return 1;\n } else {\n return 0;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":117,"value":"res2 = depts.features.map( d => d.properties.res2);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1106,"value":"res2_cr = {\n var m = maths.mean(res2);\n var sd = maths.std(res2, 'uncorrected');\n return res2.map( d => d/sd - m);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":133,"value":"resp = depts.features.map( d => d.properties.resp);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":572,"value":"med_niv_vie = depts.features.map( d => d.properties.med_niv_vie);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":575,"value":"part_retraites_rev = depts.features.map( d => d.properties.part_retraites_rev);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":578,"value":"part_mini_soc_rev = depts.features.map( d => d.properties.part_mini_soc_rev);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":872,"value":"sudouest = depts.features.map( d => d.properties.sudouest);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1148,"value":"centre = depts.features.map( d => d.properties.centre);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":875,"value":"damier = depts.features.map( d => d.properties.damier);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":502,"value":"les_variables = [\n {value: \"centre\", label: \"Test Centre \"},\n {value: \"sudouest\", label: \"Test Sud-Ouest \"},\n {value: \"damier\", label: \"Test Damier \"},\n {value: \"res2\", label: \"Second homes\"},\n {value: \"resp\", label: \"Primary homes \"},\n {value: \"med_niv_vie\", label: \"Median std of living\"},\n {value: \"part_retraites_rev\", label: \"Prop. of pensions in available income\"},\n {value: \"part_mini_soc_rev\", label: \"Prop. of social minima in available income\"},\n];","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":524,"value":"les_poids = [\n {value: \"poids_id\", label:\"Inv. distance\"},\n {value:\"poids_g\", label: \"Inv.distance²\"},\n {value: \"poids_idv\", label:\"Inv. distance of contiguous\"},\n {value: \"poids_v\", label:\"Contiguous\"},\n {value: \"poids_seuil\", label:\"1 if <= threshold in km, 0 otherwise\"},\n {value: \"poids_idseuil\", label:\"Inv. distance if <= threshold in km\"},];","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1337,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1344,"value":"cil = calcul_il();","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":168,"value":"colorQt = {\n var d = cil.domain;\n var g = cil.gradient;\n return d3.scaleQuantile()\n .domain(d)\n .range(d3.quantize(g, nb_classes));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"pathLeft = d3.geoPath(projectionLeft)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1480,"value":"pathRight = d3.geoPath(projectionRight)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":20,"value":"// Projection Albers sur la France, équivalente, carte de gauche\nprojectionLeft = d3.geoAlbers()\n\t.rotate([-2, 0])\n\t.center([2, 46])\n\t.parallels([29.5, 45.5])\n\t.scale(2800)\n\t.translate([975 / 2 - 180, 610 / 2 - 60])\n\t.precision(.1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1483,"value":"// Projection Albers sur la France, équivalente, carte de droite\nprojectionRight = d3.geoAlbers()\n\t.rotate([-2, 0])\n\t.center([2, 46])\n\t.parallels([29.5, 45.5])\n\t.scale(2800)\n\t.translate([975 / 2 + 280, 610 / 2 - 60])\n\t.precision(.1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":31,"value":"echelleLeft = d3.geoScaleBar()\n .projection(projectionLeft)\n .size([140, 460])\n .left(10)\n .top(400)\n .distance(200) // The distance of the scale bar\n .label(\"200km\") // The label on top of the scale bar\n .labelAnchor(\"middle\") // The position of the label (you can also pass \"start\" and \"end\")\n .tickSize(null) // The height of the bar\n .tickValues(null); // The tick values below the bar will be removed","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1486,"value":"echelleRight = d3.geoScaleBar()\n .projection(projectionRight)\n .size([660, 460])\n .left(10)\n .top(20)\n .distance(200) // The distance of the scale bar\n .label(\"200km\") // The label on top of the scale bar\n .labelAnchor(\"middle\") // The position of the label (you can also pass \"start\" and \"end\")\n .tickSize(null) // The height of the bar\n .tickValues(null); // The tick values below the bar will be removed","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":770,"value":"## Imports and styles","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":9,"value":"topojson = require(\"topojson-client@3\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":24,"value":"d3 = require(\"d3@5\", \"d3-geo-scale-bar@1.0.2\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":27,"value":"import {legend} from \"@d3/color-legend\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":66,"value":"turf = require(\"@turf/turf\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":112,"value":"maths = require(\"https://unpkg.com/mathjs@11.5.1\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":738,"value":"styles = html`\n `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":741,"value":"// ToolTip\ndiv = d3.select(\"body\").append(\"div\")\n\t.attr(\"class\", \"tooltip\")\n .style(\"position\", \"absolute\")\n .style(\"visibility\", \"hidden\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1908,"value":"// Source : https://observablehq.com/@mbostock/toc\nfunction toc(selector = \"h1,h2,h3\", heading = \"Table of Contents\") {\n return Generators.observe(notify => {\n let headings = [];\n\n function observed() {\n const h = Array.from(document.querySelectorAll(selector));\n if (h.length !== headings.length || h.some((h, i) => headings[i] !== h)) {\n notify(html`${heading}

${DOM.text(h.textContent)}`,\n {onclick: e => (e.preventDefault(), h.scrollIntoView())}\n );\n })}`);\n }\n }\n\n const observer = new MutationObserver(observed);\n observer.observe(document.body, {childList: true, subtree: true});\n observed();\n return () => observer.disconnect();\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1605,"value":"# Bibliography\n- Tobler, W. R. (1970). A computer movie simulating urban growth in the Detroit region. Economic geography, 46(sup1), 234-240, online : https://www.jstor.org/stable/pdf/143141.pdf\n- Anselin, L. (1995). Local indicators of spatial association—LISA. Geographical analysis, 27(2), 93-115. online : https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1538-4632.1995.tb00338.x\n- Getis, A., & Ord, J. K. (1992). The analysis of spatial association by use of distance statistics. Geographical analysis, 24(3), 189-206., online : https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1538-4632.1992.tb00261.x","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"}],"files":"$W3a","stdlib":"1","titling":"auto"}} 33:"$32:body"