2b:null 2c:["$","title",null,{"children":"Local Spatial Autocorrelation Interactive Tutorial | michelleeesi | Observable"}] 31:T60f,But what does this statistic really mean? **If this statistic is significant, we are rejecting the null hypothesis that the observed similarity between neighboring locations (clustering) can happen just randomly.** Yet unlike many datasets used in regressions, we cannot use the analytical p-value to determine whether our statistic is truly significant or not. Especially when spatial datasets are smaller, the assumption of normality of the variable does not hold (scroll back up to see the histogram for your chosen variable - it most likely isn't a normal distribution!). Luckily, we have a solution. We can create a reference distribution of the null hypothesis by randomly permuting the values over all the locations and then computing the statistic. This process repeats for a certain amount of permutations, and for easy visualization a histogram of the permuted Moran statistics are generated. Permutation does not require the analytical p-value's assumptions about the data, which makes it useful for our interpretation. The reference distribution created by the randomized values is used to calculate a so-called pseudo p-value: ${tex `p=\frac{R+1}{M+1}`} where *R* is the number of times the computed Moran’s I from the spatial random data sets (the permuted data sets) is equal to or more extreme than the observed statistic. *M* equals the number of permutations. The latter is typically taken as 99, 999, etc., to yield nicely rounded pseudo p-values. Below, you will find a simulation of the calculation of the pseudo p-value.32:T9c1,permutation_chart = { const svg = d3.create("svg") .attr("viewBox", [10, -10, 600, 410]); svg.append("g") .attr("fill", "black") .selectAll("rect") .data(bins) .join("rect") .attr("x", d => permElem.x(d.x0) + 1) .attr("width", d => Math.max(0, permElem.x(d.x1) - permElem.x(d.x0) - 1)) .attr("y", d => permElem.y(d.length)) .attr("fill", "gray") .attr("height", d => permElem.y(0) - permElem.y(d.length)); svg.append("text") .attr("x", 200) .attr("y", 10) .attr("text-anchor", "middle") .style("font-size", "12px") .text("Pseudo significance: " + pseudoPval.toFixed(3)); svg.append("text") .attr("x", 200) .attr("y", 30) .attr("text-anchor", "middle") .style("font-size", "9px") .text("Permuted Dist. Mean: " + ss.mean(permButton).toFixed(3) + ", Actual Moran's: " + gmi.toFixed(3)); svg.append("g").selectAll("line") .data([ { y1: permElem.y(0), y2: 100, x1: permElem.x(ss.mean(permButton)), x2: permElem.x(ss.mean(lisa_perm)) } ]) .enter().append("line") .attr("class", "specification-line") .attr("y1", d => d.y1) .attr("y2", d => d.y2) .attr("x1", d => d.x1) .attr("x2", d => d.x2); svg.append("g").selectAll("line") .data([ { y1: permElem.y(0), y2: 100, x1: permElem.x(gmi), x2: permElem.x(gmi) } ]) .enter().append("line") .attr("stroke", "green") .attr("stroke-width", 2) .attr("y1", d => d.y1) .attr("y2", d => d.y2) .attr("x1", d => d.x1) .attr("x2", d => d.x2); svg.append('rect') .attr('x',350) .attr('y',30) .attr('width', 10) .attr('height',10) .attr('fill', "red"); svg.append('text') .attr('x',365) .attr('y',35) .attr("dy", ".35em") .text(() => "Permutation Distribution Mean") .attr('font-family', 'serif') .attr('font-size', 10); svg.append('rect') .attr('x',350) .attr('y',50) .attr('width', 10) .attr('height',10) .attr('fill', "green"); svg.append('text') .attr('x',365) .attr('y',55) .attr("dy", ".35em") .text(() => "Actual Statistic") .attr('font-family', 'serif') .attr('font-size', 10); svg.append("g") .call(permElem.xAxis); svg.append("g") .call(permElem.yAxis); return svg.node(); }33:T5d3,Now we have a significant global Moran statistic. How does this relate to clusters? A great way to visualize this statistic is the Moran Scatterplot. Intuitively, the Moran Scatterplot is ${html`an illustration of a variable’s relationship with its neighboring values `} (original variable on x-axis and spatially lagged variable on y-axis). Because the variables are standardized, this creates four distinct categories of spatial autocorrelation. Since the plot is centered on the mean (of zero), all points to the right of the mean are higher than average (classified as 'high') and all points to the left are lower than average (classified as 'low'). Similarly, we can classify the values for the spatial lag above and below the mean as high and low. With these naming conventions, we can easily separate our observations into four categories. The upper-right quadrant corresponds with high-high, lower left with low-low, lower-right with high-low, and upper-left with low-high. High-high clusters are locations that have a high value and are surrounded by high values, and low-low locations have a low value and are surrounded by more low values. These are regarded as positive spatial autocorrelation (values surrounded by similar values). Low-high refers to a location where the surrounding values are on average higher than the variable itself, and high-low vice versa. These are examples of negative spatial autocorrelation—values surrounded by dissimilar values. 34:Tbde,scatterplot = { // thanks to https://observablehq.com/@harrystevens/introducing-d3-regression const svg = d3.create("svg") .attr("viewBox", [0, 0, height+100, width+20]); //d3.select(DOM.svg(innerWidth + margin.left + margin.right, innerHeight + margin.top + margin.bottom)) const g = svg.append("g") .attr("transform", `translate(${margin.left}, ${margin.top})`); g.append("g") .attr("class", "axis") .call(scatElem.xAxisLinear) .selectAll(".tick") .classed("baseline", d => d == 0); g.append("g") .attr("class", "axis") .attr("transform", `translate(${scatElem.inWidth})`) .call(scatElem.yAxisLinear) .selectAll(".tick") .classed("baseline", d => d == 0); g.append("text") .attr("x", width/2-30) .attr("y", height-15) .attr("text-anchor", "middle") .style("font-size", "9px") .text(column + " (Standardized)"); g.append("text") .attr("transform", "rotate(-90)") .attr("x", -height/2-20) .attr("y", -margin.left+10) .attr("fill", "currentColor") .attr("text-anchor", "start") .style("font-size", "9px") .text("Spatially Lagged " + column); g.append("text") .attr("x", width-103) .attr("y", 20) .attr("text-anchor", "middle") .style("font-size", "12px") .style('fill', lisa_colors[1]) .text("High-High"); g.append("text") .attr("x", 30) .attr("y", height-60) .attr("text-anchor", "middle") .style("font-size", "12px") .style('fill', lisa_colors[2]) .text("Low-Low"); g.append("text") .attr("x", width-103) .attr("y", height-60) .attr("text-anchor", "middle") .style("font-size", "12px") .style('fill', lisa_colors[4]) .text("High-Low"); g.append("text") .attr("x", 30) .attr("y", 20) .attr("text-anchor", "middle") .style("font-size", "12px") .style('fill', lisa_colors[3]) .text("Low-High"); g.append("text") .attr("x", width/2-30) .attr("y", 0 - (margin.top/3)) .attr("text-anchor", "middle") .style("font-size", "12px") .text("Moran's I: " + bestFit.slope.toFixed(3)); g.append("g") .attr("stroke", "gray") .attr("stroke-width", 1) .selectAll("circle") .data(scat_data) .join("circle") .attr("cx", d => scatElem.xScaleLinear(d.x)) .attr("cy", d => scatElem.yScaleLinear(d.y)) .attr("r", 3) .attr("fill", d => d.sigcolor); g.append("line") .attr("class", "regression") .datum(d3linearRegression(scat_data)) .attr("x1", d => scatElem.xScaleLinear(d[0][0])) .attr("x2", d => scatElem.xScaleLinear(d[1][0])) .attr("y1", d => scatElem.yScaleLinear(d[0][1])) .attr("y2", d => scatElem.yScaleLinear(d[1][1])); return svg.node(); }35:T41a,Just like the Global Moran's, we cannot use the analytical approximation of a p-value because our dataset is too small to satisfy the assumption of normality. We again use permutations (or the pseudo-p), this time called the **conditional permutation method**. This is similar to the permutation approach in the global statistic, except that the value of each ${tex `z_{i}`} is held fixed at its location *i*. The remaining n-1 z-values are then randomly permuted and the local statistic is calculated over and over again to give us a null reference distribution, the distribution we would see if there truly was spatial randomness. The p-value is then calculated as before, one for every location (satisfying the first LISA condition). Below, you may select any ID number to check its local pseudo-significance as well as visualize the reference distribution. You'll find many to be not significant at all (shaded gray on the scatterplot), and those that are significant would be colored according to their classification (High-High, Low-Low, etc.)36:Tb0d,lisa_permutation_chart = { const svg = d3.create("svg") .attr("viewBox", [10, -10, 600, 410]); svg.append("g") .attr("fill", "black") .selectAll("rect") .data(bins) .join("rect") .attr("x", d => permElem.x(d.x0) + 1) .attr("width", d => Math.max(0, permElem.x(d.x1) - permElem.x(d.x0) - 1)) .attr("y", d => permElem.y(d.length)) .attr("fill", "gray") .attr("height", d => permElem.y(0) - permElem.y(d.length)); svg.append("text") .attr("x", 200) .attr("y", 10) .attr("text-anchor", "middle") .style("font-size", "12px") .text("Pseudo significance: " + localPseudoPval.toFixed(3)); svg.append("text") .attr("x", 200) .attr("y", 30) .attr("text-anchor", "middle") .style("font-size", "9px") .text("Permuted Dist. Mean: " + ss.mean(lisa_perm).toFixed(3) + ", Actual Lisa: " + lm.lisaValues[arrIndex].toFixed(3)); svg.append("g").selectAll("line") .data([ { y1: permElem.y(0), y2: 100, x1: permElem.x(ss.mean(lisa_perm)), x2: permElem.x(ss.mean(lisa_perm)) } ]) .enter().append("line") .attr("class", "specification-line") .attr("y1", d => d.y1) .attr("y2", d => d.y2) .attr("x1", d => d.x1) .attr("x2", d => d.x2); svg.append("g").selectAll("line") .data([ { y1: permElem.y(0), y2: 100, x1: permElem.x(lm.lisaValues[arrIndex]), x2: permElem.x(lm.lisaValues[arrIndex]) } ]) .enter().append("line") .attr("stroke", "green") .attr("stroke-width", 2) .attr("y1", d => d.y1) .attr("y2", d => d.y2) .attr("x1", d => d.x1) .attr("x2", d => d.x2); svg.append('rect') .attr('x',350) .attr('y',30) .attr('width', 10) .attr('height',10) .attr('fill', "red"); svg.append('text') .attr('x',365) .attr('y',35) .attr("dy", ".35em") .text(() => "Conditional Permutation Distribution Mean") .attr('font-family', 'serif') .attr('font-size', 10); svg.append('rect') .attr('x',350) .attr('y',50) .attr('width', 10) .attr('height',10) .attr('fill', "green"); svg.append('text') .attr('x',365) .attr('y',55) .attr("dy", ".35em") .text(() => "Actual Statistic") .attr('font-family', 'serif') .attr('font-size', 10); // legendNode.append('text') // .attr('x',30) // .attr('y',i*25+45) // .attr("dy", ".35em") // .text(() => ordinalLabels[i]) // .attr('font-family', 'sans-serif') // .attr('font-size', 10); // }) svg.append("g") .call(permElem.xAxis); svg.append("g") .call(permElem.yAxis); return svg.node(); }37:T44b,For a clearer understanding of what clusters mean, let's take a look at the Natural Breaks map and the Cluster Map side by side. As you can see, red regions (high-high) are usually relatively dark on the natural breaks map and are also surrounded by relatively dark values (keep in mind that high and low are in relation to the mean, not absolute). Blue regions demonstrate the concept very well - in the south and southeast of France, significantly low-low regions are light and surrounded by more light regions. If you select the pink region on the Cluster Map, the corresponding region on the Natural Breaks map is relatively dark compared to its surrounding neighbors, and vice versa for the light blue regions. It is important to note that because the Local Moran statistic is calculated using the spatial lag, which is a weighted AVERAGE, a very dark region (like the one surrounding the light blue region in the north (ID 50) can pull the average up and create a significant low-high region in this case, even though Location 50's southeast neighbor (ID 53) is actually lighter than it.38:T84f,scatterplot_copy = { // thanks to https://observablehq.com/@harrystevens/introducing-d3-regression const svg = d3.create("svg") .attr("viewBox", [0, 0, height+100, width+20]); //d3.select(DOM.svg(innerWidth + margin.left + margin.right, innerHeight + margin.top + margin.bottom)) const g = svg.append("g") .attr("transform", `translate(${margin.left}, ${margin.top})`); g.append("g") .attr("class", "axis") .call(scatElem.xAxisLinear) .selectAll(".tick") .classed("baseline", d => d == 0); g.append("g") .attr("class", "axis") .attr("transform", `translate(${scatElem.inWidth})`) .call(scatElem.yAxisLinear) .selectAll(".tick") .classed("baseline", d => d == 0); g.append("text") .attr("x", width/2-30) .attr("y", height-15) .attr("text-anchor", "middle") .style("font-size", "9px") .text(column + " (Standardized)"); g.append("text") .attr("transform", "rotate(-90)") .attr("x", -height/2-20) .attr("y", -margin.left+10) .attr("fill", "currentColor") .attr("text-anchor", "start") .style("font-size", "9px") .text("Spatially Lagged " + column); g.append("text") .attr("x", width/2-30) .attr("y", 0 - (margin.top/3)) .attr("text-anchor", "middle") .style("font-size", "12px") .text("Moran's I: " + bestFit.slope.toFixed(3)); g.append("g") .attr("stroke", "gray") .attr("stroke-width", 1) .selectAll("circle") .data(scat_data) .join("circle") .attr("cx", d => scatElem.xScaleLinear(d.x)) .attr("cy", d => scatElem.yScaleLinear(d.y)) .attr("r", 3) .attr("fill", d => d.sigcolor); g.append("line") .attr("class", "regression") .datum(d3linearRegression(scat_data)) .attr("x1", d => scatElem.xScaleLinear(d[0][0])) .attr("x2", d => scatElem.xScaleLinear(d[1][0])) .attr("y1", d => scatElem.yScaleLinear(d[0][1])) .attr("y2", d => scatElem.yScaleLinear(d[1][1])); return svg.node(); }39:T419,scatElem = { const maxx = Math.max(Math.round(d3.extent(scat_data, d => d.x)[0]),Math.round(d3.extent(scat_data, d => d.x)[1])) const domx = [0-maxx-1, maxx+1]; const maxy = Math.max(Math.round(d3.extent(scat_data, d => d.y)[0]),Math.round(d3.extent(scat_data, d => d.y)[1])) const domy = [0-maxy-1, maxy+1] const base = Math.min(width, 400); const inWidth = base - margin.left - margin.right; const inHeight = base - margin.top - margin.bottom; const yScaleLinear = d3.scaleLinear() .domain(domy) .range([inHeight, 0]) const yAxisLinear = d3.axisLeft(yScaleLinear) .tickSize(inWidth + 10) const xScaleLinear = d3.scaleLinear() .domain(domx) .range([0, inWidth]) const xAxisLinear = d3.axisBottom(xScaleLinear) .tickSize(inHeight + 10); return { // Shorthand trick for you! If the object key name is the same as the thing you're returning, you can just include it directly! base, inWidth, inHeight, yScaleLinear, xScaleLinear, yAxisLinear, xAxisLinear: xAxisLinear, } }3a:T98b,linkedElem = { const encoding = ({ x: 'stdVal', y: 'spatialLag', c: column, s: column, }) const encoding_defaults = ({ x: 'stdVal', y: 'spatialLag', c: column, s: column, }) const is_narrow = width <= 640; const map_aspect_ratio = 587.09 / 590.5; const chart_width = 400; const chart_height = 300; const map_margin = is_narrow ? ({ top: 60, right: 2, bottom: 2, left: 2 }) : ({ top: 14, right: 8, bottom: 2, left: 0 }); const chart_margin = is_narrow ? ({ top: 20, right: chart_width * 0.04, bottom: 20, left: 20 }) : ({ top: 20, right: 30, bottom: 20, left: 30 }); const chart_offset = is_narrow ? ({ x: 15, y: 140 }) : ({ x: 35, y: 14 }); const height = 450; const x = d3.scaleLinear() .range([chart_margin.left, chart_width - chart_margin.right]) .domain(d3.extent(data, d => d[encoding_defaults.x])) const y = d3.scaleLinear() .range([chart_height - chart_margin.bottom, chart_margin.top]) .domain(d3.extent(data, d => d[encoding_defaults.y])) const linear_regression = d3.regressionLinear() .x(d => d[encoding_defaults.x]) .y(d => d[encoding_defaults.y]) const voronoi = d3.voronoi() .extent([[0, 0], [chart_width, chart_height]]) .x(d => x(d[encoding_defaults.x])) .y(d => y(d[encoding_defaults.y])) const style_axis = (axis, x, y) => { const text = axis.selectAll('.tick text:first-of-type'); text.style('text-anchor', 'start'); if (x !== undefined) text.attr('x', x); if (y !== undefined) text.attr('y', y); axis.selectAll('.tick line') .attr('color', 'lightgrey') .attr('stroke-width', 1) .attr('stroke-linecap', 'round'); axis.selectAll('.domain').remove(); return axis; } const x_axis = format => g => g .attr('transform', `translate(0, ${chart_height})`) .attr('class', 'axis x') .call(d3.axisBottom(x) .ticks(5, format).tickSize(-chart_height).tickPadding(0)) .call(el => style_axis(el, '0.5em', '-1em')) const y_axis = format => g => g .attr('class', 'axis y') .call(d3.axisLeft(y).ticks(5, format).tickSize(-chart_width).tickPadding(0)) .call(el => style_axis(el, undefined, '-0.7em')) return { is_narrow, map_aspect_ratio, map_margin, chart_width, chart_height, chart_margin, chart_offset, height, x, y, linear_regression, voronoi, x_axis, y_axis, encoding, encoding_defaults, style_axis, } }3b:T6c5,function deckLineMap({ initialViewState={ latitude:0, longitude:0, zoom:0 }, geography={}, controller=true, width=500, height=500, customDeckParams={}, strokeColor="#999999", title="", connectivityData={}, }){ const container = html `

` const deckgl = new deck.DeckGL({ container: container, map: false, // Viewport settings initialViewState, controller, glOptions: { preserveDrawingBuffer: true // https://developer.mozilla.org/en-US/docs/Web/API/HTMLCanvasElement/getContext } }); const layers = [new deck.GeoJsonLayer({ id: `geojson-${Math.random()}`, data: geography, pickable: true, getFillColor: [255,255,255], getLineColor: [40,40,40], lineWidthScale: 1, lineWidthMinPixels: 1, getLineWidth: 1, autoHighlight: true, ...customDeckParams }), connectivityData.arcs && new deck.LineLayer({ data: connectivityData.arcs, pickable: false, getWidth: d => d.value, getSourcePosition: d => d.source, getTargetPosition: d => d.target, getColor: [0,0,0,120] }), connectivityData.targets && new deck.ScatterplotLayer({ data: connectivityData.targets, opacity: 0.8, stroked: false, filled: true, radiusScale: 6, radiusMinPixels: 5, radiusMaxPixels: 10, getPosition: d => d.position, getRadius: d => 1, getFillColor: d => [255, 140, 0], getLineColor: d => [0, 0, 0] }) ] deckgl.setProps({layers: layers}); return html`

${title}

${container}

` }3c:[{"id":"602dde7c42338e81f502e2dc7d76e911439faf63cc6e6b83a67a4d82e46c154a1f729feeabe2e981b92e1286af2e7036e51f22aed9d1bf7bf3c8cc17dd4e19f9","href":"https://static.observableusercontent.com/files/602dde7c42338e81f502e2dc7d76e911439faf63cc6e6b83a67a4d82e46c154a1f729feeabe2e981b92e1286af2e7036e51f22aed9d1bf7bf3c8cc17dd4e19f9","name":"guerry_wgs84@2.geojson","type":"application/geo+json","lastModified":1625693832394.88,"size":1955249},{"id":"602dde7c42338e81f502e2dc7d76e911439faf63cc6e6b83a67a4d82e46c154a1f729feeabe2e981b92e1286af2e7036e51f22aed9d1bf7bf3c8cc17dd4e19f9","href":"https://static.observableusercontent.com/files/602dde7c42338e81f502e2dc7d76e911439faf63cc6e6b83a67a4d82e46c154a1f729feeabe2e981b92e1286af2e7036e51f22aed9d1bf7bf3c8cc17dd4e19f9","name":"guerry_wgs84.geojson","type":"application/geo+json","lastModified":1624467930661.065,"size":1955249},{"id":"602dde7c42338e81f502e2dc7d76e911439faf63cc6e6b83a67a4d82e46c154a1f729feeabe2e981b92e1286af2e7036e51f22aed9d1bf7bf3c8cc17dd4e19f9","href":"https://static.observableusercontent.com/files/602dde7c42338e81f502e2dc7d76e911439faf63cc6e6b83a67a4d82e46c154a1f729feeabe2e981b92e1286af2e7036e51f22aed9d1bf7bf3c8cc17dd4e19f9","name":"guerry_wgs84@3.geojson","type":"application/geo+json","lastModified":1625693886888.892,"size":1955249}] 2f:{"id":"07adcce179451a46","slug":"local-spatial-autocorrelation-interactive-tutorial","title":"Local Spatial Autocorrelation Interactive Tutorial","thumbnail":"e0a6a8a06d73a85c7a86f0cba3bdadab7c036a207243568b9cef301cf8dbfa00","access_level":"public","create_time":"$D2021-06-23T17:05:30.661Z","update_time":"$D2022-10-26T20:41:08.720Z","view_time":null,"first_public_time":"$D2021-07-12T19:59:13.882Z","fork":true,"stars":33,"starred":false,"owner":{"type":"team","id":"15b3ab74dfe1e757","login":"michelleeesi","name":"","avatar_url":"https://avatars-next.observableusercontent.com/521e99898bc8a5a4f7b2843746e29253e7305c565383c4eb1d1ec2c236b696c9","tier":"starter_2024"},"version":2640,"publish_version":2640,"role":null,"authored":false,"authors":[{"type":"individual","id":"22090083f3a72824","login":"michelleeesi","name":"","avatar_url":"https://avatars-next.observableusercontent.com/521e99898bc8a5a4f7b2843746e29253e7305c565383c4eb1d1ec2c236b696c9","tier":null}],"members":[{"type":"individual","id":"b7ea45ffedaecfd3","login":"jkoschinsky","name":"","avatar_url":"https://avatars.githubusercontent.com/u/13679743?v=4","tier":null,"role":"editor"},{"type":"individual","id":"22090083f3a72824","login":"michelleeesi","name":"","avatar_url":"https://avatars-next.observableusercontent.com/521e99898bc8a5a4f7b2843746e29253e7305c565383c4eb1d1ec2c236b696c9","tier":null,"role":"editor"},{"type":"individual","id":"54c23d6cacd69e20","login":"nofurtherinformation","name":"Dylan","avatar_url":"https://avatars-next.observableusercontent.com/a494170e1e25c8f92f54468c30fecf4bcf81095a7268e37af8e633e7fd8d22ba","tier":null,"role":"editor"}],"comments":[{"id":"bc494b1235d9f89f","reply_to_id":null,"content":"Strong LateX game!!!\n\nAlso, realizing here, it might be good to add earlier in the text simply what spatial autocorrelation means?","create_time":"$D2021-07-01T01:58:17.672Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"54c23d6cacd69e20","login":"nofurtherinformation","name":"Dylan","avatar_url":"https://avatars-next.observableusercontent.com/a494170e1e25c8f92f54468c30fecf4bcf81095a7268e37af8e633e7fd8d22ba","tier":null}},{"id":"61a6601107016b12","reply_to_id":null,"content":"Legend could be good here, maybe top right or on the bottom, also different colors for permuted mean vs actual?","create_time":"$D2021-07-01T02:00:58.206Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"54c23d6cacd69e20","login":"nofurtherinformation","name":"Dylan","avatar_url":"https://avatars-next.observableusercontent.com/a494170e1e25c8f92f54468c30fecf4bcf81095a7268e37af8e633e7fd8d22ba","tier":null}},{"id":"4f39cce4dc32d57d","reply_to_id":null,"content":"just threw the function for this in the bottom of the notebook, adapted from https://codesandbox.io/s/4spatial-weights-owi84?from-embed=&file=/src/index.js:2251-2369","create_time":"$D2021-07-01T02:25:58.583Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"54c23d6cacd69e20","login":"nofurtherinformation","name":"Dylan","avatar_url":"https://avatars-next.observableusercontent.com/a494170e1e25c8f92f54468c30fecf4bcf81095a7268e37af8e633e7fd8d22ba","tier":null}},{"id":"579cf692b3db96b5","reply_to_id":null,"content":"Thank you for this notebook. This is a new topic to me, and I found the statement in bold somewhat confusing. Thinking of a chess board, I would imagine that rook contiguity would imply that each square is only contiguous with squares of the opposite color, e.g., a white square is only rook contiguity neighbors with black squares, which made me confused about the statement \"black squares would count as neighbors with other black squares under the rook criterion\".\nBest,\nEvan","create_time":"$D2021-07-14T13:37:38.721Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"9dc9f884d5f95f34","login":"gallowayevan","name":"Evan Galloway","avatar_url":"https://avatars-next.observableusercontent.com/ec81d47e8647af7f76ea8f171c5d7792d26d3e9a0b353b893529f22882a121c4","tier":null}},{"id":"4142494df3a39c94","reply_to_id":"579cf692b3db96b5","content":"Thanks Evan! Great catch :)","create_time":"$D2021-07-14T16:27:35.200Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"22090083f3a72824","login":"michelleeesi","name":"","avatar_url":"https://avatars-next.observableusercontent.com/521e99898bc8a5a4f7b2843746e29253e7305c565383c4eb1d1ec2c236b696c9","tier":null}}],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":11,"collection":null,"type":"notebook","body":{"title":"Local Spatial Autocorrelation Interactive Tutorial","cells":[{"id":0,"value":"# Local Spatial Autocorrelation Interactive Tutorial\n*Text in this notebook is adapted from [Luc Anselin](https://spatial.uchicago.edu/directory/luc-anselin-phd)'s GeoDa workbook, available [here](https://geodacenter.github.io/documentation.html). GeoDa is a free and open-source program for exploratory spatial data analysis and basic spatial regression modeling that you can download [here](https://geodacenter.github.io/download.html).*","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1692,"value":"When we look at maps, we spot concentrations and groupings of high and low values, start to recognize patterns, and construct stories about what might be going on. But how can we statistically measure where hotspots and coldspots really exist?\n\nThis is where local spatial autocorrelation comes into play. In this interactive tutorial, we’ll be focusing on a common implementation in the form of the Local Moran statistic, developed by [Luc Anselin (1995)](https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1538-4632.1995.tb00338.x). \n\nYou may use any data set for geographic areas or points you like, but we will be using the classic Guerry data set on moral statistics in 1830 France. Other sample datasets are hosted by the Center for Spatial Data Science [here](https://geodacenter.github.io/data-and-lab/).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1695,"value":"## I. Setting It Up\n*All original documentation with detailed explanations are linked in each section heading.*","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2299,"value":"### What is spatial autocorrelation?\nEssentially, it is a measure for ${html `how similar one object is to its neighboring objects.`} How do we get there? First up, spatial weights.\n\n### [Spatial Weights](https://geodacenter.github.io/workbook/4a_contig_weights/lab4a.html#spatial-weights---basic-concepts)\nSpatial weights are how we determine geographical neighbors. They are expressed as a matrix, with each row corresponding to the location itself and columns as potential neighbors. If two locations are neighbors, their corresponding entry in the matrix would be 1 in the binary case. If not, 0. For example, if Location 3 had Locations 1 and 4 bordering it, row 3 of the spatial weights matrix would look like this:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1699,"value":"\\mathbf{W}_{3}=\\left[\\begin{array}{cccc}1 & 0 & 0 & 1 \\end{array}\\right]","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":29,"value":"weights = weight_type==='Queen' ? geoda.getQueenWeights(data) : geoda.getRookWeights(data)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2431,"value":"### [Contiguity](https://geodacenter.github.io/workbook/4a_contig_weights/lab4a.html#contiguity-weights)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1726,"value":"How do we determine who gets to be whose neighbor? **Contiguity.** Contiguity means two locations share a connection (using some criteria such as border or distance), that they follow a set of mathematical criteria that make them neighbors. Queen contiguity says that neighbors must share a common border or vertex. Rook says border only. **On a chess board, a square would be contiguous with squares of the opposite color under the rook criterion, while all 8 surrounding squares would count as neighbors under the queen criterion**. We’ll be using queen contiguity weights as a demo, but feel free to experiment with other types. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1444,"value":"viewof weight_type = Inputs.select(['Queen','Rook'], {label: 'Weight type',value: 'Queen'})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1448,"value":"conn = geoda.getConnectivity(weights, data)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2574,"value":"contiguityLinkedMap.renderGraphic()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1733,"value":"You’ll also need to choose an ID variable, which is a critical element to make sure that the weights are connected to the correct observations in the data table. The ID variable is a key that links the data to the weights.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":587,"value":"viewof id_var = Inputs.select(columnnames, {label: 'ID variable',value: 'dept'})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1839,"value":"We will focus on the variable Donatns (charitable donations per capita), which has an interesting spatial distribution (see map below). Select other variables with the drop-down menu to explore other distributions. Here are [descriptions](https://geodacenter.github.io/data-and-lab/Guerry/) of the variables.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":430,"value":"viewof column = Inputs.select(columnnames, {label: 'Spatial distribution variable', value: 'Donatns'})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":45,"value":"column_selected = geoda.getColumn(data, column)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1848,"value":"

\n ${viewof breaksMapDeck}\n

\n\n

\n Mean: ${ss.mean(column_selected).toFixed(3)}\n
Stdev: ${ss.standardDeviation(column_selected).toFixed(3)}\n
Median: ${ss.median(column_selected).toFixed(3)}\n
Range: [${ss.min(column_selected).toFixed(3)}, ${ss.max(column_selected).toFixed(3)}]\n ${viewof valCharts}\n

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":1897,"value":"### [Spatial Lag](https://geodacenter.github.io/workbook/4d_weights_applications/lab4d.html#spatially-lagged-variables)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1886,"value":"The spatial lag is another important concept used for autocorrelation statistics. In this case, it is simply the weighted average of a spatial unit's neighboring values.\n\nLet's use our previous example of row 3 of our spatial weights matrix:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2448,"value":"\\mathbf{W}_{3}=\\left[\\begin{array}{cccc}1 & 0 & 0 & 1 \\end{array}\\right]","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":2451,"value":"To demonstrate this concept, let's assume that location 1 has a value of 15, 2 a value of 25, 3 a value of 30, and 4 a value of 35. The spatial lag of location 3 would be:\n ${tex` lag_{3} = 1*15+0*25+0*30+1*35 = 50`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1911,"value":"The spatial distribution variable will usually be standardized as the number of standard deviations from the mean. The spatial lag will then be calculated from these standardized values.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":772,"value":"std_vals = standardize(column_selected)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":758,"value":"spatial_lags = geoda.spatialLag(weights, std_vals)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2356,"value":"RECAP: Spatial weights define each location's neighbors, and queen/rook contiguity are ways to determine what these neighbors are (sharing edges/vertices). A spatial lag is the weighted average of a location's neighbors' values, or an approximation of the 'value' of the neighbors compared to the value of the location.","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":1919,"value":"## II. [Moran's I](https://geodacenter.github.io/workbook/5a_global_auto/lab5a.html#morans-i)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1922,"value":"Let’s dive into the spatial autocorrelation statistic. Today, we'll be looking into the Moran’s I statistic, one of the most commonly used indicators of spatial autocorrelation. Global Moran's I uses the product of a variable and its spatial lag to determine similarity (variable is expressed in number of standard deviations from its mean). For an observation at location i, this is expressed as ${tex `z_{i}=x_{i}-\\bar{x}`}, where ${tex `\\bar{x}`} is the mean of variable *x*.\n\nMathematically, Moran's I statistic is the following:\n\n${tex `\nI=\\frac{\\sum_{i} \\sum_{j} w_{i j} z_{i} \\cdot z_{j} / S_{0}}{\\sum_{i} z_{i}^{2} / n}\n`}\nwith ${tex `w_{i j}`} as the elements of the spatial weights matrix, ${tex `S_{0}=\\sum_{i} \\sum_{j} w_{i j}`} as the sum of all the weights, and *n* as the number of observations.\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1945,"value":"### [Permutation Inference](https://geodacenter.github.io/workbook/5a_global_auto/lab5a.html#permutation-inference)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1948,"value":"$31","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1168,"value":"viewof pseudo_perms = Inputs.range([500,2000], {label: 'Number of permutations to check pseudo-significance', step: 1, value: 999})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1152,"value":"viewof permButton = Inputs.button(\"Run\", {value:pseudop(), reduce: ()=>pseudop()})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1206,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1960,"value":"### [Moran Scatterplot](https://geodacenter.github.io/workbook/5a_global_auto/lab5a.html#moran-scatter-plot)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1958,"value":"$33","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1989,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1993,"value":"Remember that the variable *z* equals the observed variable given in standard deviations from the mean. With row-standardized weights, the sum of all the weights ${tex`\\left(S_{0}\\right)`} equals the number of obsevations *n*. As a result, the expression for Moran's I simplifies to:\n\n${tex`\nI=\\frac{\\sum_{i} \\sum_{j} w_{i j} z_{i} \\cdot z_{j}}{\\sum_{i} z_{i}^{2}}=\\frac{\\sum_{i}\\left(z_{i} \\times \\sum_{j} w_{i j} z_{j}\\right)}{\\sum_{i} z_{i}^{2}}\n`}\n**This turns out to be the slope of a regression of a spatial lag on an observed value, or the slope of the best fit line of the scatterplot.** The colored points indicate the locations that are significant, and the classification of the spatial autocorrelation into four types begins to illuminate the connection between global and local spatial autocorrelation.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":800,"value":"bestFit = linearRegression(spatial_lags,std_vals)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1230,"value":"gmi = bestFit.slope","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2065,"value":"## III. [LISA Principle](https://geodacenter.github.io/workbook/6a_local_auto/lab6a.html#lisa-principle)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2068,"value":"A significant global Moran statistic tells us there is clustering ... but where are these clusters? We need **local indicators of spatial association**, or LISA, to find out. \n\nThe concept of a LISA was developed by [Anselin (1995)](https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1538-4632.1995.tb00338.x):","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2483,"value":"*As an operational definition, I suggest that a local indicator of spatial association (LISA) is any statistic that satisfies the following two requirements:*\n\n1. *the LISA for each observation gives an indication of the extent of significant spatial clustering of similar values around that observations;*\n2. *the sum of LISAs for all observations is proportional to a global indicator of spatial association.*","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2082,"value":"To break down the global statistic, it is simply a product of some function that measures similarity between two values and a measure of locational similarity (spatial weights):\n\n${tex`\\sum_{i} \\sum_{j} w_{i j} f\\left(x_{i}, x_{j}\\right)`}\n\nTo make this statistic local, we just take this above function but at a specific location *i* with its neighborhood *j*.\n\n${tex`\n\\sum_{j} w_{i j} f\\left(x_{i}, x_{j}\\right)\n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2092,"value":"### [Local Moran's I](https://geodacenter.github.io/workbook/6a_local_auto/lab6a.html#local-moran)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2100,"value":"We can see how the Local Moran satisfies the second property of a LISA—because it is simply the inner part of the double sum, all of the LISAs summed together (for all *i*) will sum to a value proportional to the global Moran statistic. To simplify our expression, we will make the constant denominator in the global Moran expression *c*, and we obtain the following:\n\n${tex`\nI_{i}=c . z_{i} \\sum_{j} w_{i j} z_{j}\n`}\nTo break it down, it is a constant times the value of the location itself, which is then multiplied by the spatial lag (the summation expression).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2106,"value":"$35","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2025,"value":"viewof id = Inputs.range([ss.min(id_variables),ss.max(id_variables)],{label: 'ID number of location to check pseudo-significance', step: 1, value: id_variables[0]})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2037,"value":"viewof lisa_perm = Inputs.button(\"Run\", {value:localPseudop(id), reduce: ()=>localPseudop(id)})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2018,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2031,"value":"arrIndex = fulldata.find(o => o[id_var]===id).arrIndex","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2044,"value":"id_variables = geoda.getColumn(data, id_var)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2367,"value":"RECAP: If there is global spatial autocorrrelation, that means there exists clustering of similar values in certain places. Because this doesn't give us WHERE these clusters are, then we look at local indicators of spatial autocorrelation (LISA). In this case, the Moran LISA measures the similarity between a location and its neighbors by multiplying the location's value and its neighbors' values (spatial lag). ","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":2118,"value":"Below, you will find the LISA Cluster Map, highlighting the regions according to classification. Feel free to adjust the parameters to see significance comparisons! You may also scroll back up to the beginning of the notebook to adjust the spatial distribution parameter, the weights, or the ID variable. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":187,"value":"**Select parameters:**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":168,"value":"viewof significance_cutoff = Inputs.range([0.001,1], {label: 'Significance cutoff', step: 0.001, value: 0.05})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":183,"value":"viewof permutations = Inputs.range([500,9999], {label: 'Number of permutations', step: 1, value: 999})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":49,"value":"lm = geoda.localMoran(weights, column_selected, permutations, 'lookup', significance_cutoff)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1970,"value":"

\n ${scatterplot_copy}\n \n ${viewof sigMap1_copy}\n\n \n

\n\n

\n ${viewof clusterMap}\n

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":2504,"value":"### [Clusters](https://geodacenter.github.io/workbook/6a_local_auto/lab6a.html#clusters-and-spatial-outliers)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2158,"value":"$37","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2594,"value":"mapsInteractive.renderGraphic()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2375,"value":"## Putting it all together!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2207,"value":"Below, the interactive map can demonstrate the conceptual link. Again, the colored points on the scatterplot represent significant locations. If you click on them, it will show you the core and the neighbors on the natural breaks map, clearly showing what High-High, Low-Low, Low-High, and High-Low clusters mean.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1877,"value":"classInteractive.renderGraphic()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2373,"value":"If you hover over either a point or a location, you will see all the statistics of that location, from LISA value to pseudo p-value to cluster information. We encourage you to go back to the conditional permutation simulator to see where the p-value in the information box comes from. The outline of the hovered-over location will be the color of the cluster, corresponding to the scatterplot. Notice that those that are light-blue are lighter locations surrounded by darker locations. Those that are outlined in red are dark surrounded by generally more dark locations, blue are light surrounded by more light, and pink are dark surrounded by light. With this interactive, the types of hotspots become clear!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2536,"value":"### Thanks for reading! \n*For more information and resources, visit the [GeoDa](https://geodacenter.github.io/index.html) homepage and download GeoDa [here](https://geodacenter.github.io/download.html).*","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2530,"value":"

","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2121,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2638,"value":"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":834,"value":"viewof breaksMapDeck = deckMap({\n data: colValues,\n geography: geo,\n column: column,\n colVals: column_selected,\n topographyBorder: border,\n breaks: breaks_col, // breaks for column attribute array\n colorBins: breaks_colors,\n strokeColor: '#999999',\n colorVariable: column,\n geographyIdVariable: id_var,\n dataIdVariable: id_var, \n initialViewState: initialViewState,\n title: 'Natural Breaks Map',\n controller:false,\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":91,"value":"viewof clusterMap = deckMap({\n data: clusterValues,\n geography: geo,\n mapId: data,\n column: column,\n colVals: column_selected,\n topographyBorder: border, // pre-computed border\n breaks: [1,2,3,4,5,6,7], // breaks for lisa values\n colorBins: lisa_colors,\n ordinalLabels: lisa_labels,\n strokeColor: '#999999',\n colorVariable: 'lisa',\n geographyIdVariable: id_var,\n dataIdVariable: id_var, \n controller: false,\n initialViewState: initialViewState,\n title: 'Local Moran Cluster Map',\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2130,"value":"viewof sigMap1_copy = deckMap({\n data: pValues,\n geography: geo,\n column: column,\n colVals: column_selected,\n topographyBorder: border, // pre-computed border\n breaks: [0.0001,0.001,0.01,0.05,1], // breaks for p values\n colorBins: pval_map_colors,\n strokeColor: '#999999',\n colorVariable: 'pval',\n geographyIdVariable: id_var,\n dataIdVariable: id_var,\n ordinalLabels: pval_labels,\n controller: false,\n initialViewState: initialViewState,\n title: 'Local Moran Significance Map'\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1595,"value":"viewof valCharts = {\n \n const valBox = vl.markPoint({ type: 'boxplot', opacity: 0.8, ticks: true })\n .data(colValues)\n .encode(\n vl.x().fieldQ(column).scale({ zero: false })\n )\n .width(400)\n .height(150);\n \n const valHist = vl.markBar()\n .data(colValues)\n .encode(\n vl.x().fieldQ(column).bin(true),\n vl.y().count()\n )\n\n /*\n const lisaBox = vl.markPoint({ type: 'boxplot', opacity: 0.8, ticks: true })\n .data(lisaValues)\n .encode(\n vl.x().fieldQ(\"lisa\").scale({ zero: false })\n )\n .width(400)\n .height(150);\n \n const lisaHist = vl.markBar()\n .data(lisaValues)\n .encode(\n vl.x().fieldQ(\"lisa\").bin(true),\n vl.y().count()\n )\n */\n \n return vl.vconcat(valHist, valBox).render()\n //vl.hconcat(vl.vconcat(valHist, valBox), vl.vconcat(lisaHist, lisaBox)).render();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2562,"value":"#### Elements","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1271,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1222,"value":"permElem = {\n \n const x = d3.scaleLinear()\n .domain([bins[0].x0, gmi+0.2])\n .range([margin.left, width - margin.right])\n \n const y = d3.scaleLinear()\n .domain([0, d3.max(bins, d => d.length)]).nice()\n .range([height - margin.bottom, margin.top])\n \n const xAxis = g => g\n .attr(\"transform\", `translate(0,${height - margin.bottom})`)\n .call(d3.axisBottom(x).ticks(width / 80 ).tickSizeOuter(0))\n .call(g => g.append(\"text\")\n .attr(\"x\", width - margin.right)\n .attr(\"y\", -4)\n .attr(\"fill\", \"currentColor\")\n .attr(\"text-anchor\", \"end\")\n .text(data.x))\n \n const yAxis = g => g\n .attr(\"transform\", `translate(${margin.left},0)`)\n .call(d3.axisLeft(y).ticks(height / 40))\n .call(g => g.select(\".domain\").remove())\n .call(g => g.select(\".tick:last-of-type text\").clone()\n .attr(\"x\", 4)\n .attr(\"text-anchor\", \"start\")\n .text(data.y))\n \n return {\n xAxis: xAxis,\n yAxis: yAxis,\n x: x,\n y: y,\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1858,"value":"$3a","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1749,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":892,"value":"html`\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2313,"value":"style = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1207,"value":"bins = d3.bin().thresholds(40)(permButton)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1681,"value":"mutable currId = null","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1539,"value":"mutable hoverTarget = null","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2346,"value":"contiguityLinkedMap = new LinkedContiguityMap({\n data: fulldata,\n geojson: geo,\n geojson1: geo,\n weightType: weight_type,\n conn: conn,\n breaks: breaks_col, // breaks for lisa values\n colorBins: breaks_colors,\n colorSig: lm.colors,\n column: column,\n ordinalLabels: breaks_labels,\n colorVariable: column,\n sigVar: 'cluster',\n id_var: id_var,\n chartTitle: 'Global Moran: ' + gmi.toFixed(3)\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2591,"value":"mapsInteractive = new LinkedMap({\n data: fulldata,\n geojson: geo,\n geojson1: geo,\n breaks: breaks_col, \n breaks1: [1,2,3,4,5,6,7],\n colorBins: breaks_colors,\n colorBins1: lisa_colors,\n colorSig: lisa_colors,\n column: column,\n ordinalLabels: breaks_labels,\n ordinalLabels1: lisa_labels,\n colorVariable: column,\n colorVariable1: 'cluster',\n sigVar: 'cluster',\n id_var: id_var,\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":271,"value":"**Imports**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":58,"value":"import {fileInput} from \"@mbostock/file-input\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":506,"value":"function jsonInput({initialUrl, init, accept=\".geojson, .json\"}) {\n return fileInput({\n initialUrl,\n accept,\n load(url) {\n return fetch(url, init).then(response => {\n if (!response.ok) throw new Error(response.status + \" \" + response.statusText);\n const responseClone = response.clone();\n return [response.json(), responseClone.arrayBuffer()];\n });\n }\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":844,"value":"height = 400;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":846,"value":"width = 400;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1309,"value":"margin = ({left: 50, right: 20, top: 25, bottom: 25});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":842,"value":"initialViewState = getViewPort(geoda, data, height, width)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":82,"value":"import {choroplethMap} from \"@nofurtherinformation/reusable-choropleth-map\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":811,"value":"import {deckMap} from \"@michelleeesi/reusable-deckgl-map\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1870,"value":"import {LinkedInteractiveMap, LinkedContiguityMap, LinkedMap} from '@michelleeesi/linked-interactive-class'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1561,"value":"import {neighborsChoroplethMap} from \"@michelleeesi/neighbors-map\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":115,"value":"import {legend, swatches} from \"@d3/color-legend\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":857,"value":"d3 = require('d3@5', 'd3-array@2', 'd3-regression@1')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":117,"value":"topojson = require('topojson@3')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":4,"value":"jsgeoda = import('https://cdn.skypack.dev/jsgeoda')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":6,"value":"geoda = await jsgeoda.New()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":767,"value":"ss = require('simple-statistics')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":851,"value":"mathgl = import('https://cdn.skypack.dev/pin/@math.gl/web-mercator@v3.4.3-OM1oEZTp6s3D0fWnTwXF/mode=imports,min/optimized/@math.gl/web-mercator.js')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":770,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1537,"value":"import {callout} from '@d3/line-chart-with-tooltip'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1874,"value":"classInteractive = new LinkedInteractiveMap({\n data: fulldata,\n geojson: geo,\n breaks: breaks_col, // breaks for lisa values\n colorBins: breaks_colors,\n colorSig: lm.colors,\n column: column,\n ordinalLabels: breaks_labels,\n colorVariable: column,\n sigVar: 'cluster',\n id_var: id_var,\n chartTitle: 'Global Moran: ' + gmi.toFixed(3)\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1602,"value":"import {vl} from \"@vega/vega-lite-api\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2302,"value":"deck = require.alias({\n h3: {},\n s2Geometry: {}\n})('deck.gl@^8.0.0/dist.min.js')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":274,"value":"**Calculating Geospatial Statistics**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":9,"value":"attachment_array = await FileAttachment(\"guerry_wgs84@2.geojson\").arrayBuffer()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":87,"value":"geo = await FileAttachment(\"guerry_wgs84@3.geojson\").json()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":446,"value":"columnnames = Object.getOwnPropertyNames(geo.features[0].properties)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1865,"value":"data_inc = d3.groups(geo.features, d => d.properties[id_var])\n .map(([, [{ properties }]]) => properties)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1868,"value":"fulldata = {\n for (let i=0; i=' + breaks_col[breaks_col.length-1])\n return temp\n} ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1535,"value":"neighbors = getNeighbors(geo, id_var, weights)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1362,"value":"pseudoPval = {\n let count = 0;\n for (let i=0; i=gmi) {count++}\n }\n return (count+1) / (pseudo_perms+1)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2029,"value":"localPseudoPval = {\n let count = 0;\n for (let i=0; i=lm.lisaValues[arrIndex]) {count++}\n }\n return (count+1) / (pseudo_perms+1)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":786,"value":"**Functions**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":777,"value":"function standardize(arr){ // used to standardize column values as # of stdev away from the mean\n const avg = ss.mean(arr);\n const stdev = ss.standardDeviation(arr);\n return arr.map(val => {\n return ss.zScore(val, avg, stdev);\n })\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":849,"value":"function getViewPort(geoda, id, height, width){\n const bounds = geoda.getBounds(id)\n return mathgl.fitBounds({\n height: height,\n width: width,\n bounds: [[bounds[0],bounds[2]],[bounds[1],bounds[3]]]\n }) \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":798,"value":"function linearRegression(y,x){ // line of best fit function - simple-statistics linreg output does not match MI on GeoDa\n let lr = {};\n let n = y.length;\n let sum_x = 0, sum_y = 0, sum_xy = 0, sum_xx = 0, sum_yy = 0;\n\n for (let i = 0; i < y.length; i++) {\n\n sum_x += x[i];\n sum_y += y[i];\n sum_xy += (x[i]*y[i]);\n sum_xx += (x[i]*x[i]);\n sum_yy += (y[i]*y[i]);\n } \n\n lr['slope'] = (n * sum_xy - sum_x * sum_y) / (n*sum_xx - sum_x * sum_x);\n lr['intercept'] = (sum_y - lr.slope * sum_x)/n;\n lr['r2'] = Math.pow((n*sum_xy - sum_x*sum_y)/Math.sqrt((n*sum_xx-sum_x*sum_x)*(n*sum_yy-sum_y*sum_y)),2);\n\n return lr;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":871,"value":"d3linearRegression = d3.regressionLinear()\n .x(d => d.x)\n .y(d => d.y)\n .domain([(-1)*d3.extent(scat_data, d => d.x)[1]-1,d3.extent(scat_data, d => d.x)[1]+1])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1131,"value":"function pseudop(){ // pseudo p-val model \n //shuffle col values, attach them back to the ids, calculate morans, put morans values into array\n let perm_morans = [];\n let q_w = weights;\n let col = column_selected;\n for (let i=0;i o[id_var]===id).arrIndex\n let origVal = col[arrIndex]\n for (let i=0;i geojsonData.features[i].properties[idCol]))]\n }\n \n return neighborsObj\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2293,"value":"$3b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":267,"value":"**Values, Colors, and Labels**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1499,"value":"sig_indices = pValues.map(obj => {\n if (obj.pval<=significance_cutoff) {return obj.index} \n }).filter(val => val)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":148,"value":"clusterValues = { // parsing clusters into collection -- js array of objects\n let temp = []\n\n for (let i=0; i=0.001 && lm.pvalues[i]<=0.0011) ? 0.0009999 : lm.pvalues[i],\n index: i\n })\n }\n\n return temp\n \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":648,"value":"colValues = { // parsing clusters into collection -- js array of objects\n let temp = []\n\n for (let i=0; i0.0001 && arr[i]<=0.001) {p001++;}\n else if (arr[i]>0.001 && arr[i]<=0.01) {p01++;}\n else if (arr[i]>0.01 && arr[i]<=0.05) {p05++;}\n else {notsig++;}\n\n if (arr[i]<=significance_cutoff) {pcut++;}\n }\n\n return [p0001, p001, p01, p05, notsig, pcut, arr.length-pcut]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":720,"value":"cluster_count = { //counting number of departments per cluster\n let notsig=0, hh=0, ll=0, hl=0, lh=0;\n\n let arr = lm.clusters\n\n for (let i=0; i