problems 6-7
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"id": "88ac965b",
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"metadata": {},
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"source": [
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"## Problems 6, etc: coming soon"
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"## Problem 6\n",
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"\n",
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"Show that the pseudo-inverse $A^+$ obeys $AA^+A=A$ and $A^+ A A^+ = A^+$."
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]
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},
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{
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"attachments": {
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"image.png": {
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truncated
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}
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},
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"cell_type": "markdown",
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"id": "c918611f",
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"metadata": {},
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"source": [
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"## Problem 7\n",
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"\n",
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"In this problem you will use **principal components analysis (PCA)** and the SVD to classify a images.\n",
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"\n",
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"The code below downloads a [collection of 80 monochrome/grayscale drawings](https://math.mit.edu/~stevenj/snow.zip) of [snowpeople](https://en.wikipedia.org/wiki/Snowman) and [snowflakes](https://en.wikipedia.org/wiki/Snowflake) found by a google image search in Spring 2023, shown in miniature here:\n",
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"\n",
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"\n",
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"\n",
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"Each image is resized to a common size of $156 \\times 128$, and then reshaped into a vector of $156\\times 156 = 19968$ numbers (the grayscale intensity of each pixel), and stored in a matrix $X$ of \"measurements\" for 80 images:\n",
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"$$\n",
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"X = 19968 \\times 80 \\mbox{ matrix} = \\mbox{(data per image)} \\times \\mbox{(# images)}\n",
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"$$\n",
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"Think of this as 80 points in a **19968-dimensional space**, i.e. each image is a point in $\\mathbb{R}^{19968}$!!\n",
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"\n",
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"PCA allows us to figure out which combinations of these 19968 variables explain most of the variations, and allows us to *project* from $\\mathbb{R}^{19968}$ to a *much* lower-dimensional space in order to help us *classify* the data and identify *clusters* of similar images. In this problem (parts a and b, below), you will perform the PCA clustering analysis yourself.\n",
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"\n",
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"You will need to install some Julia packages to help extract the image data. \n",
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"```jl\n",
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"import Pkg\n",
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"Pkg.add(\"ZipFile\")\n",
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"Pkg.add(\"Images\")\n",
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"Pkg.add(\"FileIO\")\n",
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"```\n",
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"(or type `] add ZipFile Images FileIO`).\n",
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"\n",
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"You can then run the following code to download the data and reshape it into our $X$ matrix:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "5f33349c",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"\"/var/folders/q9/2vfl1wt51558xwpl4nv4wt580000gn/T/jl_aH2TS1Y9Mx\""
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"using ZipFile, Images, FileIO, Statistics, LinearAlgebra, PyPlot\n",
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"\n",
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"# download the files as a zip archive:\n",
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"imagezip = download(\"https://math.mit.edu/~stevenj/snow.zip\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "e1fca19a",
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"id": "7be7e0f5",
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"metadata": {},
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"outputs": [],
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"source": []
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"source": [
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"rows, cols = 156,128 # common size to resize all of the images to\n",
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"\n",
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"# extract the images from the zip file, load them, resize them, and stick them into a big array Xdata\n",
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"Xdata = Float64[]\n",
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"r = ZipFile.Reader(imagezip)\n",
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"for f in r.files\n",
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" # load image and rescale to rows x cols\n",
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" image = imresize(load(IOBuffer(read(f))), rows, cols)\n",
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" imagedata = Float64.(channelview(image))\n",
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" if length(imagedata) == rows*cols # grayscale\n",
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" # convert to RGB\n",
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" bw = vec(imagedata)\n",
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" imagedata = permutedims([bw bw bw])\n",
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" elseif size(imagedata, 1) == 4 # RGBA\n",
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" α = reshape(imagedata[4, :, :], 1, rows, cols) # alpha (transparency)\n",
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" # convert to RGB, with transparent pixels converted to white:\n",
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" imagedata = imagedata[1:3, :, :] .* α .+ (1 .- α)\n",
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" end\n",
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" # convert back to grayscale (average RGB channels) and collect in Xdata\n",
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" append!(Xdata, reshape(mean(imagedata, dims=1), rows, cols))\n",
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"end\n",
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"\n",
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"# array of file names, and indices of snowflakes vs snowpeople\n",
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"names = [f.name for f in r.files]\n",
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"snowperson_indices = findall(startswith(\"s\"), names)\n",
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"snowflake_indices = findall(startswith(\"f\"), names)\n",
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"\n",
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"# reshape into 19968x80 matrix X\n",
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"X = reshape(Xdata, :, length(r.files)) # reshape to (imagesize) x (numimages) matrix\n",
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"@show size(X)\n",
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"\n",
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"# a useful function to reshape a vector into a BW image and plot it\n",
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"function plotimg(imgdata, rows=rows, cols=cols)\n",
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" u = copy(imgdata)\n",
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" u .-= minimum(u)\n",
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" u ./= maximum(u)\n",
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" imshow(reshape(u, rows, cols), cmap=\"gray\")\n",
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" axis(\"off\")\n",
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"end"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "14b8590f",
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"metadata": {},
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"outputs": [],
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"source": [
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"# first, let's plot all of the images to make sure they look okay:\n",
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"figure(figsize=(6.3, 3.15))\n",
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"r, c = 7, 14\n",
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"for i = 1:r, j=1:c\n",
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" k = (i-1)*c + j\n",
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" k > size(X,2) && break\n",
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" subplot(r, c, k)\n",
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" plotimg(X[:,k])\n",
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"end"
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]
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},
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{
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"cell_type": "markdown",
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"id": "4917563f",
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"metadata": {},
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"source": [
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"**(a)** Fill in the code below to compute the matrix `A` with the *mean of each pixel* (over all images) subtracted, and take its SVD to plot the singular values.\n",
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"\n",
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"(Be careful: some treatments of PCA put the different samples in the rows of $X$ and $A$, and some put the samples in the columns. Which are we doing here?)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "776c2100",
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"metadata": {},
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"outputs": [],
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"source": [
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"A = ??? # subtract means from X\n",
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"\n",
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"U, σ, V = svd(A);\n",
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"\n",
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"plot(σ, \"b.-\")\n",
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"title(\"Problem 7(a): Singular values of A\");\n",
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"xlabel(L\"k\");\n",
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"legend([L\"\\sigma_k\"]);"
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]
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},
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{
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"cell_type": "markdown",
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"id": "52307e61",
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"metadata": {},
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"source": [
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"**(b)** Compute the projections `c1[i]` and `c2[i]` of each image\n",
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"onto the dominant two principal components (vectors u or v in this case?). That is, reduce the 19968-dimensional data to **2 dimensions**.\n",
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"\n",
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"(Does it matter much whether you do the projection to the images in $A$ or $X$? Why or why not?)\n",
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"\n",
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"Run the code below to plot `c2` vs `c1` for every image, coloring the points corresponding to snowpeople blue and the points corresponding to snowflakes red. Hopefully, in this 2d projection, you will see\n",
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"different types of images separated into mostly distinct coordinate regions, with similar images clustered together."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "4f260943",
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"metadata": {},
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"outputs": [],
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"source": [
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"c1 = ???\n",
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"c2 = ???"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "16f9fdc0",
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"metadata": {},
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"outputs": [],
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"source": [
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"figure(figsize=(20,15))\n",
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"plot(c1[snowperson_indices], c2[snowperson_indices], \"b.\")\n",
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"plot(c1[snowflake_indices], c2[snowflake_indices], \"r.\")\n",
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"text.(c1, c2, string.(1:length(c1)), fontsize=12);\n",
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"\n",
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"title(\"Problem 7(b): Compressed data, visualized on a plane\", fontsize=20);\n",
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"xlabel(L\"$c_1$\", fontsize=20);\n",
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"ylabel(L\"$c_2$\", fontsize=20);"
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]
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}
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],
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"metadata": {
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@@ -300,7 +491,7 @@
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"file_extension": ".jl",
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"mimetype": "application/julia",
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"name": "julia",
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"version": "1.8.3"
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"version": "1.8.5"
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}
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},
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"nbformat": 4,
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