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alexbepple/correlation-matrix-binary-variables.ipynb

Created November 26, 2018 13:09
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correlation-matrix-binary-variables.ipynb
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"cell_type": "markdown",
"source": [
"# Setup"
],
"metadata": {}
},
{
"cell_type": "code",
"source": [
"import pandas as pd\n",
"import numpy as np\n",
"\n",
"import matplotlib.pyplot as plt\n",
"plt.style.use('seaborn-darkgrid')\n",
"%matplotlib inline\n",
"%config InlineBackend.figure_format = 'svg'\n",
"\nimport seaborn as sns"
],
"outputs": [],
"execution_count": 1,
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"cell_type": "code",
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"def invert(zero_or_one): return zero_or_one * -1 + 1\n",
"def invert_first(array_like):\n",
" foo = np.copy(array_like)\n",
" foo[0] = invert(foo[0])\n",
" return foo"
],
"outputs": [],
"execution_count": 2,
"metadata": {
"collapsed": false,
"outputHidden": false,
"inputHidden": false
}
},
{
"cell_type": "markdown",
"source": [
"## No of measurements to simulate per variable"
],
"metadata": {}
},
{
"cell_type": "code",
"source": [
"count_samples = 24"
],
"outputs": [],
"execution_count": 3,
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"cell_type": "markdown",
"source": [
"# Demo with controlled variables \n",
"\n",
"### in order to explain the mechanism\n",
"\nLet’s define a couple of variables. We start with _positive,_ a randomly generated series of 0 and 1, and proceed from there."
],
"metadata": {}
},
{
"cell_type": "code",
"source": [
"positive = np.random.random(count_samples).round()\n",
"negative = invert(positive)\n",
"demo = pd.DataFrame.from_dict({\n",
" 'positive':positive, \n",
" 'negative':negative,\n",
" 'close to positive':invert_first(positive),\n",
" 'close to negative':invert_first(negative)\n",
"})\n",
"\ndemo.head()"
],
"outputs": [
{
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"execution_count": 4,
"data": {
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" positive negative close to positive close to negative\n",
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"cell_type": "markdown",
"source": [
"## Correlation matrix\n",
"\n",
"Not sure how familiar you are with correlation. It is a measure of similarity. (But not causality!) \n",
"I believe that a good way to describe it is that correlation reflects to which degree two variables move together.\n",
"\n",
"If two variables are perfectly correlated, the correlation coefficient is 1. A variable is perfectly correlated with itself.\n",
"\nIf a variable perfectly mirrors the movement of another, but in the opposite direction, the correlation coefficient is -1."
],
"metadata": {}
},
{
"cell_type": "code",
"source": [
"corr_mat_demo = demo.corr(method='pearson')\n",
"corr_mat_demo"
],
"outputs": [
{
"output_type": "execute_result",
"execution_count": 5,
"data": {
"text/plain": [
" positive negative close to positive close to negative\n",
"positive 1.000000 -1.000000 0.919866 -0.919866\n",
"negative -1.000000 1.000000 -0.919866 0.919866\n",
"close to positive 0.919866 -0.919866 1.000000 -1.000000\n",
"close to negative -0.919866 0.919866 -1.000000 1.000000"
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"As we can expect, _positive_ and _close to positive_ are strongly positively correlated, just as _negative_ and _close to negative._\n",
"\n_Positive_ and _close to negative_ are strongly negatively correlated. They almost perfectly mirror each other’s movements."
],
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"cell_type": "markdown",
"source": [
"This can be visualized with a heatmap:"
],
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"cell_type": "code",
"source": [
"_ = sns.heatmap(corr_mat_demo, annot=True)"
],
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"And we can make the information even more digestable with a clustermap (sorted heatmap).\n",
"\n",
"Here we can, just by looking at the colors, easily discern that the variables in the top-left corner are strongly positively correlated with each other. Same with the variables in the bottom-right corner.\n",
"\nAlso, the top-left cluster is strongly negatively correlated with the bottom-right cluster."
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"# How to measure similarity?\n",
"\n",
"The above approach uses Pearson’s $\\rho$ to measure similarity. Two other common measures are Spearman’s $\\rho$ and Kendall’s $\\tau$. But in the case of binary variables (as you described to me), these three are equivalent.\n",
"\n",
"There are a ton of other similarity measures. I even found a doctoral thesis on this subject:\n",
"\n",
"> Matthijs J. Warrens (2008) Similarity Coefficients for Binary Data. \n",
"Properties of Coefficients, Coefficient Matrices, Multi-way Metrics and Multivariate Coefficients \n",
"https://openaccess.leidenuniv.nl/handle/1887/12987\n",
"\n",
"Not knowing anything about your domain, I had little to go on in order to determine whether you might need something specialized.\n",
"\nYou might. For instance, when measuring the distance between strings (their non-similarity), one measure is the Levenshtein distance, which counts insertions, deletions and substitutions. So the Levenshtein distance between _abc_ and _acb_ is 2. But the Damerau–Levenshtein distance also considers transpositions as edit operations, which would make the distance between these two strings only 1. Thus, in some domains it is vital to look closer at the measure of similarity."
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"I don’t have a better solution right now. If you don’t have variables with all 0s or all 1s, this is not even a problem. Depending on whether you might need a similarity measure other than Pearson’s $\\rho$, this might also be a non-issue."
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"# Demo with random variables"
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"count_variables = 20\n",
"demo_random = pd.DataFrame(np.random.random((count_samples,count_variables)).round())\n",
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"_ = sns.clustermap(demo_random.corr(method='pearson'), annot=True, figsize=(12,12))"
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" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(514.02218 166.049984)scale(0.1 -0.1)\">\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(541.546272 166.049984)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-32\"/>\n",
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" <!-- 0.071 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(210.14154 300.118949)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-32\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(267.970194 300.118949)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-32\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(321.353534 300.118949)scale(0.1 -0.1)\">\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(375.286094 300.118949)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(405.590655 300.118949)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-34\"/>\n",
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" <!-- -0.2 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(434.230372 300.118949)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <!-- -0.27 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(458.973996 300.118949)scale(0.1 -0.1)\">\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(486.498088 300.118949)scale(0.1 -0.1)\">\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <!-- -0.2 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(158.989449 407.374122)scale(0.1 -0.1)\">\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(183.733073 407.374122)scale(0.1 -0.1)\">\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-37\"/>\n",
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" </g>\n",
" <g id=\"text_278\">\n",
" <!-- -0.015 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(593.813988 434.187915)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-30\"/>\n",
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" <g id=\"text_279\">\n",
" <!-- 0.2 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(628.563862 434.187915)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(653.307485 434.187915)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(130.349732 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
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" <!-- 0.071 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(155.093356 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-37\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(182.617448 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-30\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(208.476696 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-30\"/>\n",
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" <!-- -0.071 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(236.000789 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-37\"/>\n",
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" <!-- 0.12 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(267.970194 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
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" <!-- -0.27 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(293.829442 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-32\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(320.237909 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(350.54247 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-32\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(376.401719 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(401.145342 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <!-- 1 -->\n",
" <g style=\"fill:#262626;\" transform=\"translate(467.588839 461.001708)scale(0.1 -0.1)\">\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(488.162932 461.001708)scale(0.1 -0.1)\">\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <!-- 0.071 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(127.569263 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <!-- -0.14 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(156.208981 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <!-- -0.099 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(180.952604 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(208.476696 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <!-- -0.029 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(236.000789 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"172.314453\" xlink:href=\"#ArialMT-32\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(267.970194 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
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" <!-- -0.31 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(293.829442 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-33\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(321.353534 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(348.877627 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(376.401719 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(403.925811 487.815501)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g id=\"text_316\">\n",
" <!-- 0.044 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(540.430647 487.815501)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-34\"/>\n",
" <use x=\"194.628906\" xlink:href=\"#ArialMT-34\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_317\">\n",
" <!-- 0.14 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(570.735208 487.815501)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-34\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_318\">\n",
" <!-- -0.14 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(596.594457 487.815501)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-34\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_319\">\n",
" <!-- -0.029 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(621.33808 487.815501)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-32\"/>\n",
" <use x=\"227.929688\" xlink:href=\"#ArialMT-39\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_320\">\n",
" <!-- 0.24 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(653.307485 487.815501)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-32\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-34\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_321\">\n",
" <!-- -0.099 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(125.90442 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-39\"/>\n",
" <use x=\"227.929688\" xlink:href=\"#ArialMT-39\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_322\">\n",
" <!-- -0.31 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(156.208981 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-33\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-31\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_323\">\n",
" <!-- -0.099 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(180.952604 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-39\"/>\n",
" <use x=\"227.929688\" xlink:href=\"#ArialMT-39\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_324\">\n",
" <!-- -0.58 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(211.257165 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-35\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-38\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_325\">\n",
" <!-- -0.37 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(238.781257 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-33\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-37\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_326\">\n",
" <!-- 0.12 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(267.970194 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-32\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_327\">\n",
" <!-- 0.029 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(292.713817 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-32\"/>\n",
" <use x=\"194.628906\" xlink:href=\"#ArialMT-39\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_328\">\n",
" <!-- -0.3 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(324.134003 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-33\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_329\">\n",
" <!-- -0.14 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(348.877627 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-34\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_330\">\n",
" <!-- -0.071 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(373.62125 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-37\"/>\n",
" <use x=\"227.929688\" xlink:href=\"#ArialMT-31\"/>\n",
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" </g>\n",
" <g id=\"text_331\">\n",
" <!-- 0.2 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(408.371124 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-32\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_332\">\n",
" <!-- 0.2 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(435.895216 514.629294)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-32\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_333\">\n",
" <!-- -0.071 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(456.193527 514.629294)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-37\"/>\n",
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" <!-- 0.14 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(488.162932 514.629294)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-34\"/>\n",
" </g>\n",
" </g>\n",
" <g id=\"text_335\">\n",
" <!-- 1 -->\n",
" <g style=\"fill:#262626;\" transform=\"translate(522.637024 514.629294)scale(0.1 -0.1)\">\n",
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" </g>\n",
" </g>\n",
" <g id=\"text_336\">\n",
" <!-- 0.22 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(543.211116 514.629294)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-32\"/>\n",
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" </g>\n",
" <g id=\"text_337\">\n",
" <!-- 0.49 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(570.735208 514.629294)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-34\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-39\"/>\n",
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" </g>\n",
" <g id=\"text_338\">\n",
" <!-- 0.029 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(595.478832 514.629294)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-32\"/>\n",
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" </g>\n",
" <g id=\"text_339\">\n",
" <!-- 0.14 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(625.783393 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
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" </g>\n",
" <g id=\"text_340\">\n",
" <!-- 0.41 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(653.307485 514.629294)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-34\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-31\"/>\n",
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" </g>\n",
" <g id=\"text_341\">\n",
" <!-- -0.19 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(128.684888 541.443087)scale(0.1 -0.1)\">\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-31\"/>\n",
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" </g>\n",
" </g>\n",
" <g id=\"text_342\">\n",
" <!-- -0.22 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(156.208981 541.443087)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-32\"/>\n",
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" <g id=\"text_343\">\n",
" <!-- 0.15 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(185.397917 541.443087)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
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" <g id=\"text_344\">\n",
" <!-- 0.022 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(210.14154 541.443087)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-32\"/>\n",
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" <g id=\"text_345\">\n",
" <!-- 0.044 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(237.665632 541.443087)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-34\"/>\n",
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" <!-- 0.18 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(267.970194 541.443087)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-38\"/>\n",
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" <g id=\"text_347\">\n",
" <!-- 0.13 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(295.494286 541.443087)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-33\"/>\n",
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" <g id=\"text_348\">\n",
" <!-- 6.1e-17 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(315.792597 541.443087)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-36\"/>\n",
" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-65\"/>\n",
" <use x=\"194.628906\" xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"227.929688\" xlink:href=\"#ArialMT-31\"/>\n",
" <use x=\"283.544922\" xlink:href=\"#ArialMT-37\"/>\n",
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" <g id=\"text_349\">\n",
" <!-- -0.22 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(348.877627 541.443087)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-32\"/>\n",
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" <g id=\"text_350\">\n",
" <!-- -0.15 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(376.401719 541.443087)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-31\"/>\n",
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" <g id=\"text_351\">\n",
" <!-- -0.39 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(403.925811 541.443087)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-33\"/>\n",
" <use x=\"172.314453\" xlink:href=\"#ArialMT-39\"/>\n",
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" <g id=\"text_352\">\n",
" <!-- -0.12 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(431.449903 541.443087)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-31\"/>\n",
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" <!-- -0.15 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(458.973996 541.443087)scale(0.1 -0.1)\">\n",
" <use xlink:href=\"#ArialMT-2d\"/>\n",
" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"116.699219\" xlink:href=\"#ArialMT-31\"/>\n",
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" <!-- 0.044 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(485.382463 541.443087)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-34\"/>\n",
" <use x=\"194.628906\" xlink:href=\"#ArialMT-34\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(515.687024 541.443087)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-32\"/>\n",
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" <!-- 1 -->\n",
" <g style=\"fill:#262626;\" transform=\"translate(550.161116 541.443087)scale(0.1 -0.1)\">\n",
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" <!-- 0.39 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(570.735208 541.443087)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-33\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(598.259301 541.443087)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-31\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
" <use x=\"139.013672\" xlink:href=\"#ArialMT-37\"/>\n",
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" <use x=\"33.300781\" xlink:href=\"#ArialMT-30\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(236.000789 568.25688)scale(0.1 -0.1)\">\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(266.30535 568.25688)scale(0.1 -0.1)\">\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <!-- 0.2 -->\n",
" <g style=\"fill:#ffffff;\" transform=\"translate(298.274755 568.25688)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(324.134003 568.25688)scale(0.1 -0.1)\">\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(348.877627 568.25688)scale(0.1 -0.1)\">\n",
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" <use x=\"88.916016\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(378.066563 568.25688)scale(0.1 -0.1)\">\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(569.070365 595.070673)scale(0.1 -0.1)\">\n",
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" <use x=\"83.398438\" xlink:href=\"#ArialMT-30\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(128.684888 621.884467)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(212.922009 621.884467)scale(0.1 -0.1)\">\n",
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" <use x=\"55.615234\" xlink:href=\"#ArialMT-2e\"/>\n",
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" <g style=\"fill:#ffffff;\" transform=\"translate(321.353534 621.884467)scale(0.1 -0.1)\">\n",
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"source": [
"In this particular instance we can see, for instance, in the top-left corner, that the variables 10, 14, and 18 are comparatively strongly positively correlated.\n",
"\nBut this data is random. So it’s not surprising that we do not see many patterns. ;)"
],
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},
{
"cell_type": "markdown",
"source": [
"I hope that this helps a bit. If you want to dive in more deeply, get in touch."
],
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}
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