{ "cells": [ { "cell_type": "markdown", "id": "5886e925", "metadata": { "id": "5886e925" }, "source": [ "# Lecture 8 - Preprocessing and Cleaning: Missing Data; Outliers; Numerical Normalization" ] }, { "cell_type": "markdown", "id": "6ad0a05a", "metadata": { "id": "6ad0a05a" }, "source": [ "![https://imgs.xkcd.com/comics/every_data_table.png](https://imgs.xkcd.com/comics/every_data_table.png)" ] }, { "cell_type": "code", "execution_count": 20, "id": "b788b5bb", "metadata": { "executionInfo": { "elapsed": 165, "status": "ok", "timestamp": 1675101882279, "user": { "displayName": "Scott Wehrwein", "userId": "11327482518794216604" }, "user_tz": 480 }, "id": "b788b5bb" }, "outputs": [], "source": [ "import numpy as np\n", "import pandas as pd\n", "import seaborn as sns\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f95f47be", "metadata": { "id": "f95f47be" }, "source": [ "# Announcements:\n", "\n", "Let's shuffle the tables because I'm the worst and I want you to make more friends.\n", "\n", "* Keira Allison Marcus Malik\n", "\n", "* Dylan Josh Sebastian Zachary Haden\n", "\n", "* Erika Maven Finnley Narina\n", "\n", "# Goals:\n", "* Be aware of, and get practice deciding how to handle common issues that arise before analysis:\n", " * Data types and units\n", " * Missing Data\n", " * Outliers\n", "* Know how and why to compute a few different numerical normalizations:\n", " * $z$-scores\n", " * 0-1 normalization\n", " * Exponential normalization\n" ] }, { "cell_type": "markdown", "id": "fdeeeec5", "metadata": { "id": "fdeeeec5" }, "source": [ "# Rule #1 of Data Science: GIGO" ] }, { "cell_type": "code", "execution_count": null, "id": "1ae4c260", "metadata": { "id": "1ae4c260" }, "outputs": [], "source": [ "\n", "\n", "\n", "\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "id": "65c9bf6d", "metadata": { "id": "65c9bf6d" }, "source": [ "> *On two occasions I have been asked, \"Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?\" ... I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.*\n", ">\n", "> — Charles Babbage, Passages from the Life of a Philosopher" ] }, { "cell_type": "markdown", "id": "MTOvNEPmsbz4", "metadata": { "id": "MTOvNEPmsbz4" }, "source": [ "# Data Pitfalls" ] }, { "cell_type": "markdown", "id": "5ceebe4d-c62b-4ede-9476-c88882042ea0", "metadata": {}, "source": [ "Here's a \"real\" dataset that might help us think about issues we may encounter when analyzing data." ] }, { "cell_type": "code", "execution_count": 21, "id": "4bdc4bda-fa4e-4f8f-9360-526824f903f2", "metadata": {}, "outputs": [], "source": [ "df = pd.read_csv(\"/cluster/academic/DATA311/202620/WA_Bellingham.csv\", low_memory=False)\n" ] }, { "cell_type": "code", "execution_count": 22, "id": "119b95f1-9b7a-40a8-ad7a-460bb5ec6ea6", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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'DailyAverageWetBulbTemperature',\n", " 'DailyAverageWindSpeed',\n", " 'DailyCoolingDegreeDays',\n", " 'DailyDepartureFromNormalAverageTemperature',\n", " 'DailyHeatingDegreeDays',\n", " 'DailyMaximumDryBulbTemperature',\n", " 'DailyMinimumDryBulbTemperature',\n", " 'DailyPeakWindDirection',\n", " 'DailyPeakWindSpeed',\n", " 'DailyPrecipitation',\n", " 'DailySnowDepth',\n", " 'DailySnowfall',\n", " 'DailySustainedWindDirection',\n", " 'DailySustainedWindSpeed',\n", " 'DailyWeather',\n", " 'HDSD',\n", " 'HTDD',\n", " 'HeavyFog',\n", " 'HourlyAltimeterSetting',\n", " 'HourlyDewPointTemperature',\n", " 'HourlyDryBulbTemperature',\n", " 'HourlyPrecipitation',\n", " 'HourlyPresentWeatherType',\n", " 'HourlyPressureChange',\n", " 'HourlyPressureTendency',\n", " 'HourlyRelativeHumidity',\n", " 'HourlySeaLevelPressure',\n", " 'HourlySkyConditions',\n", " 'HourlyStationPressure',\n", " 'HourlyVisibility',\n", " 'HourlyWetBulbTemperature',\n", " 'HourlyWindDirection',\n", " 'HourlyWindGustSpeed',\n", " 'HourlyWindSpeed',\n", " 'MonthlyAverageRH',\n", " 'MonthlyDaysWithGT001Precip',\n", " 'MonthlyDaysWithGT010Precip',\n", " 'MonthlyDaysWithGT32Temp',\n", " 'MonthlyDaysWithGT90Temp',\n", " 'MonthlyDaysWithLT0Temp',\n", " 'MonthlyDaysWithLT32Temp',\n", " 'MonthlyDepartureFromNormalAverageTemperature',\n", " 'MonthlyDepartureFromNormalCoolingDegreeDays',\n", " 'MonthlyDepartureFromNormalHeatingDegreeDays',\n", " 'MonthlyDepartureFromNormalMaximumTemperature',\n", " 'MonthlyDepartureFromNormalMinimumTemperature',\n", " 'MonthlyDepartureFromNormalPrecipitation',\n", " 'MonthlyDewpointTemperature',\n", " 'MonthlyGreatestPrecip',\n", " 'MonthlyGreatestPrecipDate',\n", " 'MonthlyGreatestSnowDepth',\n", " 'MonthlyGreatestSnowDepthDate',\n", " 'MonthlyGreatestSnowfall',\n", " 'MonthlyGreatestSnowfallDate',\n", " 'MonthlyMaxSeaLevelPressureValue',\n", " 'MonthlyMaxSeaLevelPressureValueDate',\n", " 'MonthlyMaxSeaLevelPressureValueTime',\n", " 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DATEHourlyDryBulbTemperatureHourlyPrecipitationHourlySkyConditions
22020-01-01T00:53:00530.00FEW:02 28 SCT:04 34 BKN:07 95
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" ], "text/plain": [ " DATE HourlyDryBulbTemperature HourlyPrecipitation \\\n", "2 2020-01-01T00:53:00 53 0.00 \n", "3 2020-01-01T01:53:00 52 0.00 \n", "4 2020-01-01T02:53:00 52 0.00 \n", "5 2020-01-01T03:53:00 52 0.00 \n", "6 2020-01-01T04:53:00 51 NaN \n", "... ... ... ... \n", "11342 2020-12-31T19:53:00 46 NaN \n", "11343 2020-12-31T20:53:00 49 NaN \n", "11344 2020-12-31T21:53:00 50 NaN \n", "11345 2020-12-31T22:53:00 49 0.01 \n", "11346 2020-12-31T23:53:00 49 0.03 \n", "\n", " HourlySkyConditions \n", "2 FEW:02 28 SCT:04 34 BKN:07 95 \n", "3 CLR:00 \n", "4 SCT:04 90 BKN:07 110 \n", "5 SCT:04 95 \n", "6 NaN \n", "... ... \n", "11342 NaN \n", "11343 NaN \n", "11344 NaN \n", "11345 NaN \n", "11346 NaN \n", "\n", "[8768 rows x 4 columns]" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.groupby(\"REPORT_TYPE\").get_group(\"FM-15\")[[\"DATE\", \"HourlyDryBulbTemperature\", \"HourlyPrecipitation\", \"HourlySkyConditions\"]]" ] }, { "cell_type": "markdown", "id": "44878180", "metadata": { "id": "44878180" }, "source": [ "A potentially insidious example: In the LCD data, there are two types of Hourly reports: FM-15 and FM-16. The latter appears to be taken more frequently than hourly, only when aviators need more frequent updates due to some interesting weather. What might this mean for if:\n", "* you investigate how often thunderstorms happen by counting hourly measurements with thunderstorms in the weather column?\n", "* you compare average wind speeds in two cities, one of which has very gusty (i.e., variable) winds often, while the other has high sustained winds more often?" ] }, { "cell_type": "markdown", "id": "4d7a5ef2", "metadata": { "id": "4d7a5ef2" }, "source": [ "# Data Cleaning Worksheet\n", "Sometimes data should be there but isn't. What would you do here?\n", "\n", "Complete the worksheet in groups of two.\n" ] }, { "cell_type": "markdown", "id": "ce9af722", "metadata": { "id": "ce9af722" }, "source": [ "What general strategies can we extract from these examples?\n", "\n", "* \n" ] }, { "cell_type": "code", "execution_count": null, "id": "77386f05", "metadata": { "id": "77386f05" }, "outputs": [], "source": [ "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "id": "6313d082", "metadata": { "id": "6313d082" }, "source": [ "What strategies for handling missing data can we extract from the above (and some others that may not have come up)?\n", "* Heuristics (birth + life expectancy for Year of Death)\n", "* Mean value (fill in the average)\n", "* Random value (fill in garbage)\n", "* Nearest Neighbor (in time, or most-similar datapoint)\n", "* Interpolation / imputation" ] }, { "cell_type": "markdown", "id": "463d8348", "metadata": { "id": "463d8348" }, "source": [ "### More things to watch out for when approaching a new dataset\n", "\n", "* Data types and units\n", " * Kilograms vs pounds\n", "* Numerical representations\n", " * 1.6e1 vs 16 vs 16.0 vs sixteen vs 32/2\n", " * Generally, do processing on floats unless (and perhaps even when) the underlying data is integral\n", "\n", "* Unification and general apples-to-apples issues\n", " * Money - inflation, economic context\n", " * Time zones\n", " * Multiple data sources - name differences (Wehrwein, S. vs Scott Wehrwein)\n" ] }, { "cell_type": "markdown", "id": "zeCeXLBRsUJR", "metadata": { "id": "zeCeXLBRsUJR" }, "source": [] }, { "cell_type": "markdown", "id": "HLf0zB5VIsEm", "metadata": { "id": "HLf0zB5VIsEm" }, "source": [ "# Outliers\n", "\n", "- An **outlier** is a datapoint that is significantly separated from the main body of observations/data\n", "- Several causes:\n", " - They can actual, valid observations/measurement. \n", " - The \"heavier tail\" the distribution that the data comes from is, the more likely these are to appear. \"Heavy tail\" has more likelihood of things far from the mean appearing.\n", " - Data entry errors; e.g., punching in the wrong numbers\n", " - Fraud; e.g., tampering with the data\n", " - Instrument error; e.g., malfunctioning sensor\n", " - Imputation gone awry\n", " \n", "Strategies for dealing with outliers that you've decided are erroneous - treat as missing data.\n", "\n", "**Be careful - could you be wrong?** How would this affect the outcomes of your analysis?" ] }, { "cell_type": "markdown", "id": "168ba014", "metadata": { "id": "168ba014" }, "source": [ "## Numerical Normalization" ] }, { "cell_type": "markdown", "id": "9716cf6b", "metadata": { "id": "9716cf6b" }, "source": [ "Let's load up the NHANES body measurement dataset." ] }, { "cell_type": "code", "execution_count": 27, "id": "e06b9dbb", "metadata": { "id": "e06b9dbb" }, "outputs": [ { "data": { "text/html": [ "
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GenderAgeWeightHeightLegArmArm CirWaist Cir
22.066.079.5158.337.036.032.0101.8
52.066.053.5150.231.830.627.488.2
62.075.088.8151.131.936.038.1113.0
81.056.062.1170.640.137.230.886.6
101.067.074.9178.639.141.430.699.7
...........................
86971.033.076.9180.145.241.032.985.5
86992.070.049.0156.534.432.625.182.2
87001.042.097.4164.938.236.640.6114.8
87012.041.069.1162.639.235.226.886.4
87031.038.0111.5175.842.538.040.0122.0
\n", "

5193 rows × 8 columns

\n", "
" ], "text/plain": [ " Gender Age Weight Height Leg Arm Arm Cir Waist Cir\n", "2 2.0 66.0 79.5 158.3 37.0 36.0 32.0 101.8\n", "5 2.0 66.0 53.5 150.2 31.8 30.6 27.4 88.2\n", "6 2.0 75.0 88.8 151.1 31.9 36.0 38.1 113.0\n", "8 1.0 56.0 62.1 170.6 40.1 37.2 30.8 86.6\n", "10 1.0 67.0 74.9 178.6 39.1 41.4 30.6 99.7\n", "... ... ... ... ... ... ... ... ...\n", "8697 1.0 33.0 76.9 180.1 45.2 41.0 32.9 85.5\n", "8699 2.0 70.0 49.0 156.5 34.4 32.6 25.1 82.2\n", "8700 1.0 42.0 97.4 164.9 38.2 36.6 40.6 114.8\n", "8701 2.0 41.0 69.1 162.6 39.2 35.2 26.8 86.4\n", "8703 1.0 38.0 111.5 175.8 42.5 38.0 40.0 122.0\n", "\n", "[5193 rows x 8 columns]" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "data_url = \"https://fw.cs.wwu.edu/~wehrwes/courses/data311_21f/data/NHANES/NHANES.csv\"\n", "cols_renamed = {\"SEQN\": \"SEQN\",\n", " \"RIAGENDR\": \"Gender\", # 1 = M, 2 = F\n", " \"RIDAGEYR\": \"Age\", # years\n", " \"BMXWT\": \"Weight\", # kg\n", " \"BMXHT\": \"Height\", # cm\n", " \"BMXLEG\": \"Leg\", # cm\n", " \"BMXARML\": \"Arm\", # cm\n", " \"BMXARMC\": \"Arm Cir\", # cm\n", " \"BMXWAIST\": \"Waist Cir\"} # cm\n", "\n", "df = pd.read_csv(data_url)\n", "df = df.rename(cols_renamed, axis='columns')\n", "df = df.drop(\"SEQN\", axis='columns')\n", "df = df[df[\"Age\"] >= 21]\n", "df" ] }, { "cell_type": "markdown", "id": "3cWdTpTaw6XM", "metadata": { "id": "3cWdTpTaw6XM" }, "source": [ "In the NHANES dataset, heights and other length measurements are given in centimeters." ] }, { "cell_type": "code", "execution_count": 28, "id": "7c9f3aed", "metadata": { "id": "7c9f3aed" }, "outputs": [ { "data": { "text/plain": [ "2 158.3\n", "5 150.2\n", "6 151.1\n", "8 170.6\n", "10 178.6\n", " ... \n", "8697 180.1\n", "8699 156.5\n", "8700 164.9\n", "8701 162.6\n", "8703 175.8\n", "Name: Height, Length: 5193, dtype: float64" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ht_col = df[\"Height\"]\n", "ht_col" ] }, { "cell_type": "markdown", "id": "FJ4Oxg6CwM7m", "metadata": { "id": "FJ4Oxg6CwM7m" }, "source": [ "\n", "**Question**: If you're 160cm tall, are you short? tall? average?\n", "**Answer:** " ] }, { "cell_type": "markdown", "id": "160f685f", "metadata": { "id": "160f685f" }, "source": [ "## $z$-scores\n", "To compute a $z$-score:\n", "1. Subtract the mean\n", "2. Divide by the standard deviation.\n", "\n", "In math:\n", "$$ \\hat{x}_i = \\frac{x_i -\\mu}{\\sigma}$$\n", "\n", "In pandas:" ] }, { "cell_type": "code", "execution_count": 29, "id": "77c14b4d", "metadata": { "id": "77c14b4d" }, "outputs": [ { "data": { "text/plain": [ "2 -0.787712\n", "5 -1.589290\n", "6 -1.500226\n", "8 0.429499\n", "10 1.221180\n", " ... \n", "8697 1.369621\n", "8699 -0.965840\n", "8700 -0.134575\n", "8701 -0.362183\n", "8703 0.944092\n", "Name: Height-z, Length: 5193, dtype: float64" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df[\"Height-z\"] = (ht_col - ht_col.mean()) / ht_col.std()\n", "df[\"Height-z\"]" ] }, { "cell_type": "code", "execution_count": 30, "id": "91f89450-c94c-4a12-9377-eddfdec5c0f6", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sns.histplot(x=\"Height\", data=df)" ] }, { "cell_type": "code", "execution_count": 31, "id": "1f3a4e4c", "metadata": { "id": "1f3a4e4c" }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sns.histplot(x=\"Height-z\", data=df)" ] }, { "cell_type": "markdown", "id": "f879632a", "metadata": { "id": "f879632a" }, "source": [ "Nice properties of $z$-scores:\n", " - $\\hat{x}_i < 0$, smaller than average\n", " - $\\hat{x}_i > 0$, greater than average\n", " - $\\hat{x}_i > 1$, more than one standard deviation above average\n", " - etc.\n", " - Can give context to how normal or anomalous a datapoint is" ] }, { "cell_type": "markdown", "id": "P53qWccSfg3T", "metadata": { "id": "P53qWccSfg3T" }, "source": [ "## 0-1 normalization\n", "$$ \\hat{x}_i = \\frac{x_i - x_{min}}{x_{max}-x_{min}}$$\n", " - Here $x_{max}$ and $x_{min}$ are the max/min values observed in the dataset -- ***or*** a theoretical min or max.\n", " - Warning: if a new datapoint comes along and you use the same mapping, can get values that are $<0$ or $>1$.\n", " - We did this with images!\n", "\n", "## Exponentiation\n", "If we need to to make values non-negative, can exponentiate:\n", "$$ \\hat{x}_i = e^{x_i}$$\n", " - $x_i \\to -\\infty$, normalized value approaches 0\n", " - $x_i \\to \\infty$, normalized value gets large quickly!" ] }, { "cell_type": "code", "execution_count": 32, "id": "aq20pzLdxSZy", "metadata": { "id": "aq20pzLdxSZy" }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = np.linspace(-5,5,num=10000)\n", "\n", "sns.lineplot(x=x, y = np.exp(x))" ] }, { "cell_type": "markdown", "id": "8fd1d104-51bf-41e4-bf11-dbbcd2425ab0", "metadata": {}, "source": [ "Common case: you have a collection of values that you want to turn into probabilities.\n", "\n", "Normalization scheme: **softmax**\n", "\n", "$$\n", "\\hat{x_i} = \\frac{e^{x_i}}{\\sum_{j=1}^{K} e^{x_j}}\n", "$$\n" ] }, { "cell_type": "code", "execution_count": 33, "id": "c801480f-9d47-42e4-aef1-f4d2f518cd42", "metadata": {}, "outputs": [], "source": [ "def softmax(x):\n", " ex = np.exp(x)\n", " return ex / np.sum(ex)\n" ] }, { "cell_type": "code", "execution_count": 34, "id": "94d0f864-34f6-4984-81a6-dbe0db4953ab", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([0.33333333, 0.33333333, 0.33333333])" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "softmax(np.array([1, 1, 1]))" ] }, { "cell_type": "code", "execution_count": 35, "id": "ea1547d3-e56a-4752-8873-d5ee0fdcbce0", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([0.01321289, 0.26538793, 0.72139918])" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "softmax(np.array([-1, 2, 3]))" ] }, { "cell_type": "code", "execution_count": null, "id": "ddb064e2-e5ff-445d-b04a-264ffbd2e5ef", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "colab": { "provenance": [] }, "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.12.13" } }, "nbformat": 4, "nbformat_minor": 5 }