36:null 37:["$","title",null,{"children":"t-SNE explain in plain javascript | Nick Strayer | Observable"}] 3d:T67c,md` ## Goal Walk through each of the steps needed to perform t-SNE. We will execute each step explicitely in plain javascript, no libraries allowed. At the end there will be a bunch of sliders that allow you to play with the algorithm and watch it attempt to visualize some simulated high-dimensional data. --- ## Calculate pairwise distances. The first step of t-SNE is to calculate the pairwise distances between all of the points in your data. Luckily we only need to do this a single time. This is because t-sne only cares about these distances and they don't change ever change. This only-once property is great in that after you have done the distance calculations, the actual data doesn't matter. So really the runtime of the algorithm is bounded not by the dimensionality of your input data but by how many samples you have. The nieve way of doing this pairwise calculation would be a simple outer loop scanning over each observation in turn and then within that outer loop an inner loop that calculates the distance between it and every other point. This works but is wasteful. Let's think about these distances. The distance between point \`i\` and point \`j\` is the same as the distance between \`j\` and point \`i\` so really we should only calculate a single one, in addition we don't really need to check between \`i\` and \`i\` as we know it's zero. To see how much time this can save us we can use the formula for calculating all _permutations_(order matters) of two numbers and all _combinations_(order doesn't matter) of two numbers. With ${n} total datapoints or observations, we can calculate the total number of permutations: `3e:T4d3,md` If you expand the returned array from above you will see we have a series of objects containing \`source\`: the index in our original data corresponding to the first point, \`target\`: the index for our second point, and \`distance\`: the euclidean distance between those two points. ## Pairwise probability calculations Now that we have the distances we can get to the fun stuff. We need to calculate the conditional probabilities between points in their original high dimensional space. ### Going from distance to probability This takes the probability seeing point \`j\` given a multi-dimensional gaussian distribution centered around point \`i\`. This is a bit to wrap your head around so let's look at it in two dimensions. In the following plot we are showing the calculation of the conditional probability of point i given a gaussian centered around point j. Play around with the position of point i and the standard deviation of the distribution to see the relationship between euclidean distances between points and their conditional probabilies.`3f:T61e,{ const pdf = linspace(-5, 5, 250).map(x => ({x, px:stats.dnorm(x, 0, orig_sd)})) const svg = d3.select(DOM.svg(width, height)); const x = d3.scaleLinear() .domain(d3.extent(pdf, d => d.x)) .range([margin.left, width - margin.right]); const y = d3.scaleLinear() .domain(d3.extent(pdf, d => d.px)) .range([height - margin.bottom, margin.top]); const line = d3.line() .x(d => x(d.x)) .y(d => y(d.px)) const p_ij = stats.dnorm(new_point, 0, orig_sd) // line showing point up to pdf svg.append('circle') .attr('cx', x(0)) .attr('cy', y(0)) .attr('r', 5) .attr('fill', 'steelblue'); svg.append('circle') .attr('cx', x(new_point)) .attr('cy', y(0)) .attr('r', 5) .attr('fill', 'orangered'); svg.append("line") .attr("fill", "none") .attr("stroke", "orangered") .attr("stroke-width", 1.5) .attr("stroke-linejoin", "round") .attr("stroke-linecap", "round") .attr("x1", x(new_point)) .attr("x2", x(new_point)) .attr("y1", y(p_ij)) .attr("y2", y(0)) svg.append("text") .html( `P(i|j) = ${p_ij.toPrecision(3)}` ) .attr('font-size', 20) .attr("x", x(new_point) + 3) .attr("y", y(p_ij) + 10) svg.append("path") .datum(pdf) .attr("fill", "none") .attr("stroke", "darkgrey") .attr("stroke-width", 1.5) .attr("stroke-linejoin", "round") .attr("stroke-linecap", "round") .attr("d", line); return svg.node(); }40:T601,md` ### Perplexity You may notice that in the above example we can tune the standard deviation, or width, of the distribution that we center over a point to calculate it's respective conditional probabilities. It's tempting to say that in the final algorithm we just set this value as a constant and sit the same distribution over every point, unfortunately that doesn't work very well. The reason is that we may have areas in the space that are very dense with points and others that are very spare. Since we car about structure at multiple scales treating data in both of these regions the same will not give us good results. The way that t-SNE deals with this is by scanning over possible standard deviations for every point's distribution trying to fit to a parameter called 'perplexity'. Perplexity a measure related to the entropy of the system of points. A valuable way of thinking about perplexity is to think of it as the effective number of neighbors for each point. Aka how many points that the distribution over point \`i\` capture with a non-trivial probability. Below we plot a point in our data and each other point's distance away from that point. The gaussian probability curve is drawn over these points based upon the current variance for the distribution drawn above the given point. As you change the variance with the slider notice how the distribution gets wider or smaller to essentially 'grab' more or less points that contribute a meaningful amount of probability to it's overall sum of probabilities. ` 41:T8a6,{ const dist_and_probs = anchor_distances .filter(d => d.distance !== 0) .map(d => ({dist: d.distance, prob:gaussian_prob(d.distance, variance)})), distance_max = d3.max(anchor_distances, d => d.distance), entropy = calc_entropy(dist_and_probs.map(d => d.prob)), prob_dist_curve = linspace(0, distance_max, 300) .map(dist => ({dist, prob:gaussian_prob(dist, variance)})), prob_extents = d3.extent(prob_dist_curve, d => d.prob); const x = d3.scaleLinear() .domain([0, distance_max]) .range([margin.left, width - margin.right]), y = d3.scaleLinear() .domain(prob_extents) .range([height - margin.bottom, margin.top]), point_color = d3.scaleLinear() .domain(prob_extents) .range(['grey','steelblue']), area = d3.area() .x(d => x(d.dist)) .y0(y(0)) .y1(d => y(d.prob)); const svg = d3.select(DOM.svg(width, height)); svg.append("text") .text(`Entropy: ${entropy.toPrecision(3)}`) .attr('x', width - 10) .attr('y', 40) .attr('text-anchor', 'end') .attr('font-size', 20) svg.append("text") .text(`Perplexity: ${exp(entropy).toPrecision(3)}`) .attr('x', width - 10) .attr('y', 70) .attr('text-anchor', 'end') .attr('font-size', 20) svg.append("text") .text(`Perplexity Goal: ${perplexity.toPrecision(3)}`) .attr('x', width - 10) .attr('y', 100) .attr('text-anchor', 'end') .attr('font-size', 20) svg.append("path") .datum(prob_dist_curve) .attr("fill", "steelblue") .attr("fill-opacity", 0.2) .attr("d", area); svg.append("circle") .attr("cx", x(0)) .attr("cy", y(0)) .attr("stroke", "grey") .attr("stroke-opacity", 0.2) .attr('fill', 'orangered') .attr('fill-opacity', 0.75) .attr("r", 7); svg.append("g") .selectAll("line") .data(dist_and_probs) .enter().append("line") .attr("x1", d => x(d.dist)) .attr("x2", d => x(d.dist)) .attr("y1", y(0) - 10) .attr("y2", y(0) + 10) .attr("stroke", d => point_color(d.prob)) .attr("stroke-width", 2) .attr("stroke-opacity", 0.5); return svg.node(); }42:T618,function variance_binary_search(i, pw_dists, entropy_tol = 1e-3, max_attempts = 40){ // grab the distances for point i const dists_from_i = pw_dists .reduce( (dists, d) => d.source === i ? [...dists, d.distance] : dists, [] ); let variance_min = 0, // lower bound variance_max = Infinity, // upper bound variance = 1, // initial value searching = true, attempts = 0, probs_given_variance, current_entropy; // number of iterations through while looking while(searching){ attempts++ // increment out attempts counter // calculate the probabilities and entropy given the current beta. probs_given_variance = calc_gaussian_probs(dists_from_i, variance); current_entropy = calc_entropy(probs_given_variance); // Entropy is below where we want, so try a higher variance if(current_entropy < target_entropy){ variance_min = variance; // move our min bound up variance = variance_max === Infinity ? variance*2 : (variance+variance_max)/2; } else { // distribution is too narrow. variance_max = variance; // move our max bound down variance = variance_min === -Infinity ? variance/2 : (variance+variance_min)/2; } // check some termination conditions const too_many_attempts = attempts > max_attempts; const close_to_target = Math.abs(current_entropy - target_entropy) < entropy_tol; searching = !(too_many_attempts || close_to_target); } return probs_given_variance }43:T547,texmd` # The mapping calculations We have now done all we need to do with the high dimensional data. Since at each iteration the high dimensional probabilities don't move we only have to do everything above once. From now on, what we need to do is take a random initial mapping of all our datapoints into two(or three) dimensions: $y_{\\{1,2, ..., ${num_points}\\}}$, and then nudge those mappings around until they best 'match' the distribution of probabilities we generated in the above sections. What 'match' means in this context we will get into in two seconds, but first we need to be able to take our mapped distances and translate them to probabilities. Just like we did in the high dimension. ## Calculating the mapping probabilities We will first make a function that calculates the conditional probabilities in the low-dimensional mapping. This function looks very similar to the one for the high dimensional mapings, but now it is the [t-distribution](https://en.wikipedia.org/wiki/Student%27s_t-distribution) with one degree of freedom (aka the [Cauchy](https://en.wikipedia.org/wiki/Cauchy_distribution) distribution). We will denote these probabilties with a $q$. $$q_{ij} = \\frac{\\{1 + (\\text{distance from } y_i \\text{ to } y_j)^2\\}^{-1}}{\\sum_{k \\ne l} \\{1 + (\\text{distance from } y_l \\text{ to } y_k)^2\\}^{-1}}$$ `44:T536,texmd` Note that we don't have any $\\sigma$ to change, as we simply set it to a constant and it cancels out. This also contributes to the fact that our normalization constant (the denomonator) is over all pairs of points, not just for a common point. Another important difference is there is no exponential in the formula. This is nice because exponentials are expensive to calculate. ## The cost function Next, we need some form of telling the model how good it is doing. What we want is our probabilities from the high dimension $P$ to match the probabilities for the low dimension $Q$ as much as possible. Luckily in statistics there is an equation for doing just this. ### Kullback-Leibler Divergence The [Kullback-Leibler Divergence](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) (or KLD) is a measure of how close two distributions are to eachother. It's not a distance in the strict definition but instead a similarity measure. It's used all over in machine learning and statistics to measure goodness of fit. The equation is fortunately very simple. $$C = \\text{KLD}(P,Q) = \\sum_i\\sum_jp_{ij}\\log\\left(\\frac{p_{ij}}{q_{ij}}\\right)$$ Below we will define a function for performing the inside of those two sums which we will then use to accumulate the full KLD when running our algorithm. `45:T590,function step_down_gradient(iter, Y, Y_step, Y_gains){ const momentum = momentums[iter > momentum_switch ? 1: 0], Q = find_Q(Y); let average_y = make_array(map_dims).map(d=>0), cost_total = 0, grad_id, step_id,step_id_new, gain_id,gain_id_new; for(let i=0; i { const low = -cluster_spread/2; const high = cluster_spread/2; const point_gen = d3.randomUniform(low, high); return () => [...(new Array(ndim))].map(point_gen); } const gen_centroids = (n, ndim) => { const center_maker = make_centroid(ndim); return [...(new Array(n))].map(center_maker) } const centroids = gen_centroids(num_clusters, num_dimensions) const sample_from_centroid = (nsamples, centroid, centroid_id) => { const rnorm = d3.randomNormal(0, random_noise_sd); const get_sample = () => ({group:centroid_id, location : centroid.map(d => d + rnorm())}); return [...(new Array(nsamples))].map(get_sample); } function gen_data_from_centroids(centroids, nsamples){ return centroids.reduce( (data, d, i) => [...data, ...sample_from_centroid(nsamples, d, i)], []) .map((d,i) => (Object.assign({id: i},d,{}))) // add in node id in addition to the centroid one. } return gen_data_from_centroids(centroids, num_samples) }47:[] 3b:{"id":"61c4a5a87867bc54","slug":"t-sne-in-plain-javascript/3","title":"t-SNE explain in plain javascript","thumbnail":null,"access_level":"public","create_time":"$D2018-02-04T17:06:54.006Z","update_time":"$D2018-02-05T22:01:45.426Z","view_time":null,"first_public_time":"$D2018-02-04T17:52:35.130Z","has_unpublished_edits":null,"fork":{"id":"b67c0ea6cabfced5","slug":"t-sne-in-plain-javascript/2","title":"t-SNE in plain javascript","owner":{"id":"c7ceaaf194433370","login":"nstrayer","name":"Nick Strayer"}},"stars":0,"starred":false,"owner":{"type":"team","id":"c7ceaaf194433370","login":"nstrayer","name":"Nick Strayer","avatar_url":"https://avatars-next.observableusercontent.com/56ce9e43c9ae8d967575f2641fd299251ce5e022874e0c8fe8d04becea060719","tier":"starter_2024","bio":"Engineer on the Shiny team at RStudio. Biostats Ph.D. from Vanderbilt.","following":false},"version":1869,"publish_version":1869,"role":null,"authored":false,"authors":[{"type":"individual","id":"e253fe2c017ea3f1","login":"nstrayer","name":"Nick Strayer","avatar_url":"https://avatars-next.observableusercontent.com/56ce9e43c9ae8d967575f2641fd299251ce5e022874e0c8fe8d04becea060719","tier":null,"bio":"Engineer on the Shiny team at RStudio. Biostats Ph.D. from Vanderbilt.","following":false}],"members":[{"type":"individual","id":"e253fe2c017ea3f1","login":"nstrayer","name":"Nick Strayer","avatar_url":"https://avatars-next.observableusercontent.com/56ce9e43c9ae8d967575f2641fd299251ce5e022874e0c8fe8d04becea060719","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":8,"collection":null,"type":"notebook","body":{"titling":"auto","title":"t-SNE explain in plain javascript","cells":[{"id":0,"value":"md`# t-SNE explain in plain javascript`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":54,"value":"$3d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":835,"value":"viewof n = slider3({\n message:'Set number of observations in your dataset',\n min: 1, \n max: 100,\n start: 15,\n step: 1, \n int:true})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":785,"value":"tex`\n\\begin{aligned}\nn^2 = ${n}^2 = \\boxed{${squared(n)} \\text{ pairs}}\n\\end{aligned}`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":789,"value":"md`\nNow if we're efficient about it and only look at the combinations of the data we can cut this down. The combination equation is a bit more complicated...\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":798,"value":"tex`\n\\begin{aligned}\n \\frac{n!}{2!(n - 2)!} =& \\frac{${n}!}{2\\cdot${n - 2}!} \\\\\n& \\frac{ ... ${n - 3} \\cdot ${n - 2} \\cdot ${n - 1} \\cdot ${n}}{2\\cdot(...${n - 3} \\cdot ${n - 2})} \\\\\n& \\frac{${n - 1} \\cdot ${n}}{2} \\\\\n& \\boxed{${n*(n - 1)/2}\\text{ pairs}}\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":797,"value":"md`So we can see that by taking advantage of this combination vs permutation distinction we are saving ourselves __${squared(n) - (n*(n - 1)/2)}__ total distance calculations. Kinda nice, especially when the data gets larger. \n\nNow that we've gotten some superfluous math out of the way let's actually implement this in code. A couple things to note:\n\n1. The function \\`calc_distance\\` simply takes two arrays of some dimension and calculates their euclidean distance(sometimes called the _L2_ norm) from eachother. The actual code to implement it is at the bottom of this notebook if you're curious.\n2. We are filling an array that has every simple permutation of the data in it, but if you note the second for loop's bounds we are only calculating the combinations. We simple fill in two spots in our distances array for each distance. This is because we later need the array to be the size of all the distances due to the quirk of non-symmetry in the conditional probability step. (We'll get to it, don't worry.)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":902,"value":"md`The first thing we are going to do is make a helper function that will take an i and j coordinate and point us to the corresponding position in a javascript array. This will make our lives a lot easier as it can be tricky to think about mapping two dimensional coordinates to a 1-d array`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":895,"value":"function mat_ind(i,j){\n //Get index in a 1-d array from a 2d matrix reference point. (Assumes our matrix is square)\n //This function will auto update when we change the number of points so we don't need to pass \n //that as a parameter thanks to the beauty of reactive programing!\n return i*num_points + j\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":906,"value":"md`Okay, now we can get into the pairwise distances calculation itself. `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":62,"value":"pairwise_distances = {\n // initialize some variables we use inside our loops\n let x_i, x_j, distance;\n // make_array is another helper function that just generates a new empty array of a given size.\n const pw_dists = make_array(squared(num_points));\n for(let i = 0; i < num_points; i++){\n \tx_i = data[i].location;\n // deal with the point itself here.\n pw_dists[mat_ind(i,i)] = {source: i, target: i, distance:0};\n for(let j = i + 1; j < num_points; j++){\n x_j = data[j].location;\n distance = calc_distance(x_i, x_j);\n pw_dists[mat_ind(i,j)] = {source: i, target: j, distance};\n pw_dists[mat_ind(j,i)] = {source: j, target: i, distance};\n }\n }\n return pw_dists\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":59,"value":"$3e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":918,"value":"$3f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":967,"value":"viewof new_point = slider2('Set distance of point j from point i', '', -5, 5, 0.333, 3)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1022,"value":"viewof orig_sd = slider2('Set the standard deviation of the distribution', '', 0.1, 3, 1, 3)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1062,"value":"$40","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1250,"value":"$41","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1503,"value":"viewof variance = slider3({\n message: 'Set the variance for above', \n min: 1,start:2, max: 50, step: 0.1, precision: 3})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1268,"value":"anchor_distances = {\n const anchor_point = 5;\n const anchor_distances = pairwise_distances\n \t.filter(d => d.source === anchor_point);\n return anchor_distances\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1255,"value":"md`\nIn order to find the optimal variance for each point in our dataset we use a searching technique called [binary search](https://en.wikipedia.org/wiki/Binary_search_algorithm). What binary search is doing here is goes point by point and searching in a disciplined way for the variance value that gets the perplexity as close to the desired perplexity as possible. Just like you did with the slider above.\n\nTo do this we first need to find our goal entropy for our points. Do do this we simply take the log of our goal perplexity.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1274,"value":"target_entropy = log(perplexity) // entropy level we're shooting for given perplexity","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1278,"value":"md`\nNow that we have the entropy we just need a couple of functions to make our lives easier.\n\nThe first is one that will simply perform the gaussian probability calculation of a point \\`j\\` given its distance from point \\`i\\` and point \\`i\\`'s variance...\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1318,"value":"tex`\np_{j|i} = \\exp\\{-(\\text{distance from } i \\text{ to } j)^2 / 2\\sigma_i\\}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1095,"value":"function gaussian_prob(distance, variance){\n return exp(-squared(distance) / (2*variance))\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1292,"value":"md`\nNext is a function that takes an array of distances from point \\`i\\` and converts them to an array containing both the distance and the probability according to the guassian model.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1427,"value":"tex`\nP_{i|j} = \\frac{\\exp(-[\\text{distance between } i \\text{ and } j]^2 / 2\\sigma_i)}{\\sum_{k \\ne i} \\exp(-[\\text{distance between } i \\text{ and } k]^2 / 2\\sigma_i)}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1429,"value":"md`This can be thought of as the numerator being the un-normalized probability of \\`j\\` given \\`i\\`, which is then divided by the sum of all the probabilities given point \\`i\\` to normalize it.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1170,"value":"function calc_gaussian_probs(distances, variance){\n let sum_probs = 0;\n\n // calc un-normalized probs and accumulate sum of probs for normalization\n return distances.map( (dist, i) => {\n const curr_prob = dist !== 0 ? gaussian_prob(dist, variance): 0;\n sum_probs += curr_prob; // accumulate the sums \t\n return curr_prob; // send probability to our array \n }).map(p => p/sum_probs) // return the prob vector after normalizing it using sum.\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1300,"value":"md`\nLast, we need a function that can take our array of probabilities and calculate the entropy for point \\`i\\` given its current variance. To do this we need to normalize the probabilities and then sum them using the formula for entropy...`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1305,"value":"tex`\n(\\text{entropy for point } i) = \\sum_{j \\ne i} -p_{j|i}\\log(p_{j|i})\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1169,"value":"function calc_entropy(probs){\n return probs.reduce( \n (sum, p) => sum - (p>1e-7 ? p*log(p): 0), \n 0\n )\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1071,"value":"md`\nLastly, we set up a function for doing the binary search for the optimal variance for a given point. I won't go too far into this other than to leave the code here. What matters is that the algorithm bounces around checking the perplexity that results from different variance values in a disciplined way in an effort to make the entropy match our goal, it then returns the vector of probabilities that correspond to the optimal variance.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1531,"value":"$42","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":273,"value":"md`Now that we have our binary search function in place, we can calculate the non-symmetric probabilities. 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margin.right]);\n const dim2 = d3.scaleLinear().domain(dim2_domain).range([margin.top, height - margin.bottom]);\n const group_color = d3.scaleOrdinal()\n \t.domain(make_array(num_clusters).map((d,i)=>i))\n .range(d3_color.schemeCategory10);\n \n const vis_data = data.map((d,i) => ({...d, map: Y[i]}));\n \n svg.append(\"g\")\n .attr(\"stroke\", \"#000\")\n .attr(\"stroke-opacity\", 0.2)\n .selectAll(\"circle\")\n .data(vis_data)\n .enter().append(\"circle\")\n .attr(\"cx\", d => dim1(d.map[0]))\n .attr(\"cy\", d => dim2(d.map[1]))\n .attr(\"fill\", d => group_color(d.group))\n .attr(\"r\", 5);\n \n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":729,"value":"{\n const svg = d3.select(DOM.svg(width, height));\n \n const x = d3.scaleLinear().domain([0, costs.length]).range([margin.left, width - margin.right]);\n const y = d3.scaleLinear().domain(d3.extent(costs)).range([height - 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