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Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

    +
  • Interpolation in Spark
  • Logistic Regression
  • Spark and Visualisation
  • Data binning
    • diff --git a/objects.inv b/objects.inv index 5cbda91fae10b37d41a5759d6bffae6deb97da8f..5c2d0c18675bf0a189befc650d01dba362486414 100644 GIT binary patch delta 1272 zcmVMWIPBW(p|sx555LI7*bUrGUygD%)#2WD(p&S~lX!el)$7X?_AK%e%@ zf{}va0cwkji+#+B=9nP+|1kML(9)gfIVoI<{Re|Bz$S(T2#tT{a2v99E@ONX;;wTc zR5LsNP4i&L#h-*uwQ&f&n+RV>#l;=iLL5k7SY5Hf#Lhd(n+-bnz?PcAHa!#2Z!&gV z&_(`cW$X?bvR>4s8DW1Bv1=&m7`-AS&TfM%48-gls zA2%(Nn~TC~23mjbFJdGEQ4enK&ftZJFE|fbe8AmaMR>--j>+E+(S5=eOx+@6v$|bZ zc-}rW!}sOGlqtntrt7-21&N8K8EfX$twCRmzp!St$e6h}g^*=7YVO}IuxLtcI-Cy} z(90zIcVYU7Qg>W%EPm}i9B}U-X%f3muMQi+7R;(zJAQvNt>n5p5^izZZz!fXu8yjw ze|BGE?hq-l%Y1Q>V6-b>nV?r8u$zRb5GaOUmIipH&Gou{euLEyTUmHPJ35sLOvl^w zuMZiWl$dCPuHBYwaq;^f%rsH?`W|7yCF39DHXC+yB2Nw5N5uNc+ij7b`$Y@!hyipx z(J#xatO|d;U@dBuK2gch8+WJiGL23$0TMX9hE!9zVnF1XIpP2<7}3Y%qCxyXaVB`4 zm$qa-y9W?l_ydKTe0U}GCNtox3`x(Dg|zO$Q`o^fwe-zdfe6W=hJ%E(Y6S_hdG zUR3oMpiJez0g;7y??9V08aA5n*7EX9?-O=f?;N<=^dpD-pqP9(8SC&I8KT^*D0J z1x_x=J z<-C8P@K>>ds7+;(SJP`XDC#3A?Yag(D_ypa=a`xh^#IC7=TPo((Pvs>y52LCohYva zM3xj*Myi1dM@%kO))?wdRRlsZhnE6|NKg0lXLjCW#lAB6p+He zlxvIfDUMp|`#`6@+#6wE2;nl~(j=fe(e;1aW~MX>zQjDTHI@&~$<8H!BmGW|D$3K5 za~z$@RWdl?H}e@~UV6bdlL3e=vT1>5ydqp(u}->iYg;BkN1sK4Lg9}g0~LRQr*{%E z#xbqnqc?j9$gY7~_iRYHywZ7);?j!dnRBL;tj-nQDm3)4oFtQo&l;4isJAf^Vc5E~B3ZI(*5k+zI7)fP5j zA%HdJFQowYtqb<77HoJ1~Cc+m| zan2uGhyw`>tIPOJ0K1dC*(HMyY`rIJ(=!46O1I;Je(EXkOT|-gF z=oKM*b-Rq|D;qfdnS8cBbAnr8o}0T3zIzk4A*kXuQ`0iJxhO0tp!NPDMlulf;P&ne zUWoXr?~uhuz3qQhgl8=5nEdS!-Dgd~{3${B13W9onby>zaB7p9LWb;kwA;@9rO0rw6v z7qRQ~>aZbf!K|va<2TbvuDc`Q7PtM7VJg(>sG{~~_ep=`4v`YO%ok_sqg?@W^HEGf znwOsOK2vr4F~LNgzo{M!C8D*0S>Z*MfdR@?ju;SGm>UkX**7LLuKEFQNXAlJ`P*c+ zavuQ=l=&wL_(^O}h%hY4CI>3QKDQx`=2Z-SBZ8jF_9eZChtOM*q>2-T9j zW~jm>tWIqb_y}+-<+2ysU_VDXCmnH85$KsnJ5>6OA%-H}g&zuH(9er+ty&xYpf}x5 zbjnb|<-}IH7Z&-N>rgLH{ECQ;OI}0auVMv3mC7Wqrq^mv)CW4+bq#)cxojWLF*On7 z0hE7@&Y|4nn#{DsbU9`yJ5l`zh%Bjlj8p>!h?rcg>M+!sO4vg(hiB?Tq^YbqBoj-Q zM>;lm9esa@$vJ%6GIEk53dk^E%C&{-6h|%feV|ic5{|;dj!t1I8JvHDoB7l*FTLQq!T`k9z_h?~P!X=KSSK5} zwJnpNqfZAxq3{QTfr>xn(mO2}FnOkvyz}a!oEZRzqWwXDC>MW9 zIRd8b4u?(5om8IeQ#^W>YLV88%?|2F)~#tgy>=(a7sa|9oiOUC1061nY@a8g+D3Tq ziwQ5S6)vt09#aq0R6V}j5oL4G#2!lac!q(OYjJWsGQzlH5kx(Rok9bQF{eM;bkhKL~ BXoCO% diff --git a/pyspark-intro/f-col.html b/pyspark-intro/f-col.html index 0a70869f..d53f8039 100644 --- a/pyspark-intro/f-col.html +++ b/pyspark-intro/f-col.html @@ -238,6 +238,7 @@

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

    +
  • Interpolation in Spark
  • Logistic Regression
  • Spark and Visualisation
  • Data binning
    • diff --git a/raw-notebooks/checkpoint-staging/checkpoint-staging.html b/raw-notebooks/checkpoint-staging/checkpoint-staging.html index 2b1a5574..52ce263c 100644 --- a/raw-notebooks/checkpoint-staging/checkpoint-staging.html +++ b/raw-notebooks/checkpoint-staging/checkpoint-staging.html @@ -235,6 +235,7 @@

Analysis in Spark

Analysis in Spark

    +
  • Interpolation in Spark
  • Logistic Regression
  • Spark and Visualisation
  • Data binning
    • diff --git a/raw-notebooks/creating-dataframes/creating-dataframes.html b/raw-notebooks/creating-dataframes/creating-dataframes.html index 91637bd4..290e1eca 100644 --- a/raw-notebooks/creating-dataframes/creating-dataframes.html +++ b/raw-notebooks/creating-dataframes/creating-dataframes.html @@ -235,6 +235,7 @@

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

    +
  • Interpolation in Spark
  • Logistic Regression
  • Spark and Visualisation
  • Data binning
    • diff --git a/raw-notebooks/interpolation/interpolation.html b/raw-notebooks/interpolation/interpolation.html new file mode 100644 index 00000000..b367b89e --- /dev/null +++ b/raw-notebooks/interpolation/interpolation.html @@ -0,0 +1,1000 @@ + + + + + + + + + + + + Interpolation in Spark — Spark at the ONS + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    Book version 2022.6

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    +

    Interpolation in Spark

    + +
    + +
    +
    + + + + +
    + +
    +

    Interpolation in Spark#

    +

    In this article, we will impute some missing values in time series data. The PySpark guidance below has been adapted from another source.

    +

    Please note that there are many imputation methods to choose from, for more information see the Awareness in Editing and Imputation course on the Learning Hub. Before using this code in your work please consult other team members or methodologists on the suitablility of this method for your use case.

    +

    We don’t want to spend too much time on the methodology here, the purpose is to give you inspiration on how to solve your own PySpark/SparklyR problem and maybe learn about some new functions.

    +

    It’s also worth mentioning that there is an interpolate function in pandas, so if your DataFrame is small enough just use pandas!

    +
    +

    What is interpolation?#

    +

    Wikipedia says “In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing new data points based on the range of a discrete set of known data points.”

    +

    The DataFrame we will create consists of the columns area_code, period and count. To impute the missing count values we will use linear interpolation, which means to draw a straight line between two known points and use this gradient line to fill in the missing values.

    +

    The linear interpolation equation we will use is:

    +
    +\[ +y = y_1 + m(x - x_1) +\]
    +

    where

    +
    +\[ +m = \frac{(y_2 - y_1)}{(x_2 - x_1)} +\]
    +

    is the gradient between points 1 and 2,
    +\(y\) = linear interpolation value (count in this example) between \(y_1\) and \(y_2\),
    +\(x\) = independent variable (period in this example) between \(x_1\) and \(x_2\),
    +\(x_1, y_1\) = known values at point 1,
    +\(x_2, y_2\) = known values at point 2.

    +
    +
    +

    Create DataFrame and plot#

    +

    Let’s start with some imports and creating a spark session

    +
    +
    +
    from pyspark.sql import SparkSession
+from pyspark.sql import functions as F
+from pyspark.sql.window import Window
+import pandas as pd
+import datetime
+
+spark = (
+    SparkSession.builder.master("local[2]").appName("interpolate")
+    .getOrCreate()
+)
+
    +
    +
    +
    +
    library(sparklyr)
+library(dplyr)
+library(broom)
+
+# set up spark session
+sc <- sparklyr::spark_connect(
+  master = "local[2]",
+  app_name = "interpolation",
+  config = sparklyr::spark_config())
+
    +
    +

    As mentioned previously we will create three columns, area_code, period and count. Note that count contains some missing values to impute and period contains some mixed frequency points to make it more interesting.

    +
    +
    +
    df = spark.createDataFrame([
+    ["A", "20210101", 100],
+    ["A", "20210106", None],
+    ["A", "20210111", None],
+    ["A", "20210116", 130],
+    ["A", "20210121", 100],
+    ["A", "20210126", 120],
+    ["A", "20210131", None],
+    ["A", "20210205", 130],
+    ["B", "20210101", None],
+    ["B", "20210111", 85],
+    ["B", "20210119", 82],
+    ["B", "20210131", 75],
+    ["B", "20210210", None],
+    ["B", "20210215", 85],
+    ["B", "20210220", None],
+    ["B", "20210225", 75]
+    ],
+    ["area_code", "period", "count"])
+
    +
    +
    +
    +

    Let’s take a quick look at some charts to see what this looks like

    +
    +
    +
    pdf = df.toPandas()
+pdf.set_index(pd.to_datetime(pdf["period"]), inplace=True)
+
    +
    +
    +
    +
    +
    +
    pdf[pdf["area_code"]=="A"].plot(marker="o")
+pdf[pdf["area_code"]=="B"].plot(marker="o")
+
    +
    +
    +
    +
    <matplotlib.axes._subplots.AxesSubplot at 0x7f1eeac0d860>
+
    +
    +../../_images/85e6d09d5c46c1dd6bd494f0a400fc67dbe3291a691e58e3dcf6b0ddaa067854.png +../../_images/120095ef88ba916816bcc4556862efba553bb04888a973475d0efbbacf682bf5.png +
    +
    +

    As you can see, the lines don’t meet all the points becuase there are missing counts in between.

    +
    +
    +

    Interpolate#

    +

    The approach we will take to interpolate is to create new columns containing forward filled and backward filled counts and periods where the count is missing. Then use these columns in the formula above to calculate the gradient and interpolated counts.

    +

    Here are the steps in more detail,

    +
      +

    1. Add timestamp columns

      2. +

    2. Create ordered windows for forward and backward filling

      3. +

    3. Carry out the forward and backward fills on count and timestamp

      4. +

    4. Calculate the gradient

      5. +

    5. Calculate the interpolated counts

      6. +
    +

    In practice there is often some cleaning to do with dates, here we just have a string in the format YYYYMMDD.

    +

    Before we impute any values we will create a flag to indicate an imputed value for the end user.

    +
    +
    +
    df = df.withColumn("impute_flag",
+                   F.when(F.col("count").isNull(), 1)
+                   .otherwise(0))
+
    +
    +
    +
    +

    Next we will need to create some new timestamp columns. The function unix_timestamp() will convert the period column into seconds since 1st of January 1970. We need one version of the column with missing timestamps corresponding to the missing counts, and another version including all the periods.

    +
    +
    +
    # Add timestamp for data we have only. This will be used to interpolate
+df = df.withColumn("timestamp_with_nulls",
+                   F.when(F.col("impute_flag")!=1,
+                          F.unix_timestamp(F.col("period"), "yyyyMMdd"))
+                   .otherwise(None))
+
+# Add timestamp for all data
+df = df.withColumn("timestamp_all",
+                  F.unix_timestamp(F.col("period"), "yyyyMMdd")
+                  )
+
    +
    +
    +
    +

    Next we create the Window functions to forward fill and backward fill the missing values in the count and timestamp_with_nulls columns.

    +
    +
    +
    # Create window for forward filling
+window_ff = (
+    Window
+    .partitionBy("area_code")
+    .orderBy("period")
+    .rangeBetween(Window.unboundedPreceding, 0)
+) 
+
+# Create window for backward filling
+window_bf = (
+    Window
+    .partitionBy('area_code')
+    .orderBy("period")
+    .rangeBetween(0, Window.unboundedFollowing)
+) 
+
    +
    +
    +
    +

    To forward and backward fill the missing values we use the F.last() and F.first() functions respectively. These will grab the preceding and following values of a missing value within our defined windows.

    +

    Once assembled we can put them in .withColumn() to add the forward and backward fill columns.

    +
    +
    +
    # create the series containing the filled values
+count_last = F.last(F.col("count"), ignorenulls=True).over(window_ff)
+count_first = F.first(F.col("count"), ignorenulls=True).over(window_bf)
+
+period_last = F.last(F.col("timestamp_with_nulls"), ignorenulls=True).over(window_ff)
+period_first = F.first(F.col("timestamp_with_nulls"), ignorenulls=True).over(window_bf)
+
+# add the columns to the dataframe
+df = (
+    df.withColumn('count_ff', count_last)
+    .withColumn('count_bf', count_first)
+    .withColumn('period_ff', period_last)
+    .withColumn('period_bf', period_first)
+)
+
+df.show()
+
    +
    +
    +
    +
    +---------+--------+-----+-----------+--------------------+-------------+--------+--------+----------+----------+
+|area_code|  period|count|impute_flag|timestamp_with_nulls|timestamp_all|count_ff|count_bf| period_ff| period_bf|
++---------+--------+-----+-----------+--------------------+-------------+--------+--------+----------+----------+
+|        B|20210101| null|          1|                null|   1609459200|    null|      85|      null|1610323200|
+|        B|20210111|   85|          0|          1610323200|   1610323200|      85|      85|1610323200|1610323200|
+|        B|20210119|   82|          0|          1611014400|   1611014400|      82|      82|1611014400|1611014400|
+|        B|20210131|   75|          0|          1612051200|   1612051200|      75|      75|1612051200|1612051200|
+|        B|20210210| null|          1|                null|   1612915200|      75|      85|1612051200|1613347200|
+|        B|20210215|   85|          0|          1613347200|   1613347200|      85|      85|1613347200|1613347200|
+|        B|20210220| null|          1|                null|   1613779200|      85|      75|1613347200|1614211200|
+|        B|20210225|   75|          0|          1614211200|   1614211200|      75|      75|1614211200|1614211200|
+|        A|20210101|  100|          0|          1609459200|   1609459200|     100|     100|1609459200|1609459200|
+|        A|20210106| null|          1|                null|   1609891200|     100|     130|1609459200|1610755200|
+|        A|20210111| null|          1|                null|   1610323200|     100|     130|1609459200|1610755200|
+|        A|20210116|  130|          0|          1610755200|   1610755200|     130|     130|1610755200|1610755200|
+|        A|20210121|  100|          0|          1611187200|   1611187200|     100|     100|1611187200|1611187200|
+|        A|20210126|  120|          0|          1611619200|   1611619200|     120|     120|1611619200|1611619200|
+|        A|20210131| null|          1|                null|   1612051200|     120|     130|1611619200|1612483200|
+|        A|20210205|  130|          0|          1612483200|   1612483200|     130|     130|1612483200|1612483200|
++---------+--------+-----+-----------+--------------------+-------------+--------+--------+----------+----------+
+
    +
    +
    +
    +

    We’re now ready to calculate the gradient between the known points and interpolate.

    +
    +
    +
    # Create a gradient column
+df = df.withColumn("gradient",
+                   F.when(F.col("impute_flag")==1,
+                          (F.col("count_bf") - F.col("count_ff")) /
+                          (F.col("period_bf") - F.col("period_ff")))
+                   .otherwise(0)
+                  )
+
+# Create the imputed values
+df = df.withColumn("count_final",
+                  F.when(F.col("impute_flag")==1,
+                        F.col("count_ff") +
+                        F.col("gradient") *
+                        (F.col("timestamp_all") - F.col("period_ff"))
+                        )
+                   .otherwise(F.col("count"))
+                  )
+
    +
    +
    +
    +
    +
    +

    View the results#

    +

    To view the results we will select the columns of interest and convert to pandas.

    +
    +
    +
    pdf = df.select("area_code", "period", "count", "impute_flag", "count_final").toPandas()
+pdf = pdf.set_index(pd.to_datetime(pdf["period"])).drop("period", axis=1)
+pdf
+
    +
    +
    +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    area_codecountimpute_flagcount_final
    period
    2021-01-01BNaN1NaN
    2021-01-11B85.0085.000000
    2021-01-19B82.0082.000000
    2021-01-31B75.0075.000000
    2021-02-10BNaN181.666667
    2021-02-15B85.0085.000000
    2021-02-20BNaN180.000000
    2021-02-25B75.0075.000000
    2021-01-01A100.00100.000000
    2021-01-06ANaN1110.000000
    2021-01-11ANaN1120.000000
    2021-01-16A130.00130.000000
    2021-01-21A100.00100.000000
    2021-01-26A120.00120.000000
    2021-01-31ANaN1125.000000
    2021-02-05A130.00130.000000
    +
    +
    +

    Note that the first value for area_code B could not be interpolated because there is no preceding count available. In other words, we would need to extrapolate to estimate for this value.

    +

    Once again, we will plot the series with pandas. The original data are plotted as red circles and the final series in blue.

    +
    +
    +
    pdf[pdf["area_code"] == "A"].drop("impute_flag", axis=1).plot(style={"count":"ro", "count_final":"b+-"}, alpha=0.5)
+pdf[pdf["area_code"] == "B"].drop("impute_flag", axis=1).plot(style={"count":"ro", "count_final":"b+-"}, alpha=0.5)
+
    +
    +
    +
    +
    <matplotlib.axes._subplots.AxesSubplot at 0x7f1eead5d2b0>
+
    +
    +../../_images/f8a7b760a6925ef345cd30ed9f9b67dd03a4e092a941441f5af4bd95a5ab88a0.png +../../_images/07aaa09f9b3b2e12f9b9c2da0d1a05cb457684f024f209d52e080d6aa61045f7.png +
    +
    +
    +
    +

    Further resources#

    + +
    +
    + + + + +
    + + + + + + +
    + +
    +
    +
    + +
    + + + +
    + +
    +
    + Contents +
    +
    + +
    + + +
    +
    + +
    + +
    + +

    +By Analysis Standards and Pipelines in Quality and Improvement, Office for National Statistics +

    + +
    + +
    + + +
    + +
    + +
    + +
    + +
    + +
    +
    + + +
    +
    +
    + + + + + +
    +
    + + \ No newline at end of file diff --git a/raw-notebooks/join-concepts/join-concepts.html b/raw-notebooks/join-concepts/join-concepts.html index e55e276e..9435b6d6 100644 --- a/raw-notebooks/join-concepts/join-concepts.html +++ b/raw-notebooks/join-concepts/join-concepts.html @@ -235,6 +235,7 @@

Analysis in Spark

    +
  • Interpolation in Spark
  • Logistic Regression
  • Spark and Visualisation
  • Data binning
    • diff --git a/raw-notebooks/logistic-regression/logistic-regression.html b/raw-notebooks/logistic-regression/logistic-regression.html index 701343f9..9c67e79b 100644 --- a/raw-notebooks/logistic-regression/logistic-regression.html +++ b/raw-notebooks/logistic-regression/logistic-regression.html @@ -235,6 +235,7 @@

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

    +
  • Interpolation in Spark
  • Logistic Regression
  • Spark and Visualisation
  • Data binning
    • diff --git a/raw-notebooks/reading-and-writing-data/reading-and-writing-data-spark.html b/raw-notebooks/reading-and-writing-data/reading-and-writing-data-spark.html index a1fb2994..1028626b 100644 --- a/raw-notebooks/reading-and-writing-data/reading-and-writing-data-spark.html +++ b/raw-notebooks/reading-and-writing-data/reading-and-writing-data-spark.html @@ -235,6 +235,7 @@

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

    +
  • Interpolation in Spark
  • Logistic Regression
  • Spark and Visualisation
  • Data binning
    • diff --git a/raw-notebooks/sparklyr-functions/sparklyr-functions.html b/raw-notebooks/sparklyr-functions/sparklyr-functions.html index 3e8ca696..81a73948 100644 --- a/raw-notebooks/sparklyr-functions/sparklyr-functions.html +++ b/raw-notebooks/sparklyr-functions/sparklyr-functions.html @@ -235,6 +235,7 @@

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

Analysis in Spark

    +
  • Interpolation in Spark
  • Logistic Regression
  • Spark and Visualisation
  • Data binning
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68, 69, 72, 73, 74, 75], "view_nam": 0, "so": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 31, 32, 35, 36, 37, 38, 39, 40, 41, 43, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 58, 60, 61, 63, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75], "follow": [0, 2, 3, 6, 8, 10, 13, 16, 17, 21, 23, 26, 28, 30, 31, 32, 34, 35, 36, 37, 38, 43, 45, 46, 49, 50, 58, 61, 65, 67, 70, 73, 75], "IF": [0, 9, 39], "code": [0, 1, 2, 3, 4, 6, 9, 12, 13, 14, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 53, 54, 55, 57, 58, 60, 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75], "check": [0, 1, 3, 6, 7, 8, 16, 17, 21, 22, 26, 27, 28, 30, 33, 35, 37, 38, 43, 44, 45, 46, 48, 49, 53, 56, 58, 60, 71, 73, 74], "whether": [0, 6, 8, 21, 23, 26, 30, 35, 38, 42, 45, 58, 67, 71, 73], "actual": [0, 1, 5, 6, 7, 15, 16, 19, 27, 29, 30, 43, 48, 66, 72, 73, 74, 75], "remov": [0, 2, 3, 7, 8, 10, 21, 25, 27, 28, 32, 34, 35, 37, 38, 41, 42, 44, 45, 47, 48, 49, 61, 69], "document": [0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 39, 40, 43, 44, 45, 46, 47, 48, 50, 51, 53, 55, 56, 57, 58, 60, 61, 63, 65, 66, 67, 68, 69, 72, 73, 74, 75], "lead": [1, 16, 18, 19, 21, 25, 26, 31, 36, 43, 45, 47, 54, 58, 74], "alreadi": [1, 6, 9, 11, 13, 19, 23, 27, 30, 35, 39, 48, 63, 65, 67, 71, 73, 75], "articl": [1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 35, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75], "demonstr": [1, 2, 8, 9, 12, 14, 16, 19, 20, 21, 23, 26, 27, 30, 33, 38, 39, 40, 43, 45, 48, 50, 51, 55, 58, 66, 67, 73], "why": [1, 6, 8, 9, 13, 16, 19, 21, 22, 23, 27, 28, 30, 37, 38, 39, 43, 45, 48, 49, 56, 65, 66, 67, 73], "should": [1, 2, 4, 6, 8, 16, 17, 19, 25, 26, 27, 28, 30, 31, 35, 36, 37, 38, 41, 43, 44, 47, 48, 49, 53, 58, 60, 68, 69, 70, 73, 74, 75], "care": [1, 3, 6, 7, 9, 10, 11, 13, 16, 19, 20, 22, 24, 25, 27, 30, 32, 34, 35, 39, 43, 47, 48, 55, 56, 57, 60, 61, 63, 65, 66, 73], "when": [1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 44, 46, 47, 48, 49, 50, 51, 53, 54, 56, 57, 58, 61, 63, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75], "occur": [1, 2, 6, 10, 12, 16, 17, 19, 21, 25, 26, 27, 28, 34, 35, 37, 43, 45, 47, 48, 49, 51, 58, 74], "common": [1, 6, 10, 11, 16, 19, 21, 26, 29, 30, 31, 32, 34, 35, 36, 43, 45, 46, 50, 58, 61, 63, 69, 71, 72, 73, 75], "us": [1, 2, 6, 7, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 28, 31, 33, 34, 35, 36, 37, 40, 41, 45, 47, 49, 50, 51, 53, 54, 55, 56, 57, 60, 63, 65, 66, 68, 70], "sql": [1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 63, 65, 66, 68, 69, 71, 72, 74], "overwrit": [1, 6, 9, 17, 19, 21, 23, 30, 35, 39, 45, 67, 73], "e": [1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 30, 31, 32, 34, 36, 37, 38, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 56, 58, 60, 61, 63, 65, 67, 69, 73, 74, 75], "g": [1, 3, 5, 6, 8, 11, 13, 14, 15, 16, 19, 21, 22, 25, 26, 27, 28, 30, 31, 32, 36, 37, 38, 40, 43, 44, 45, 46, 47, 48, 49, 50, 56, 58, 61, 63, 65, 69, 73, 74, 75], "sum": [1, 6, 7, 10, 12, 17, 19, 21, 22, 30, 32, 34, 35, 37, 45, 51, 56, 61, 69, 73], "instead": [1, 2, 5, 6, 7, 9, 10, 11, 13, 15, 16, 18, 19, 20, 21, 23, 27, 30, 34, 35, 37, 39, 43, 45, 46, 48, 54, 55, 63, 65, 67, 68, 71, 73, 74, 75], "good": [1, 7, 9, 13, 16, 19, 21, 22, 23, 24, 25, 28, 30, 31, 36, 37, 39, 43, 44, 45, 46, 47, 49, 56, 57, 65, 66, 67, 68, 69, 73, 74, 75], "practic": [1, 3, 6, 9, 13, 16, 19, 27, 30, 31, 36, 39, 43, 44, 46, 48, 65, 68, 73, 75], "f": [1, 2, 3, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 54, 55, 56, 57, 58, 60, 61, 63, 65, 67, 74, 75], "prefix": [1, 17, 19, 29, 30, 35, 72, 73], "give": [1, 5, 6, 8, 10, 13, 15, 16, 17, 18, 19, 21, 23, 26, 28, 30, 32, 34, 35, 37, 38, 43, 45, 49, 54, 58, 60, 61, 65, 66, 67, 68, 69, 71, 73, 74, 75], "sensibl": [1, 5, 6, 15, 19, 21, 27, 45, 46, 48, 68, 69, 75], "too": [1, 6, 7, 13, 16, 18, 19, 20, 21, 25, 26, 30, 31, 32, 36, 43, 44, 45, 46, 47, 54, 55, 58, 61, 65, 66, 68, 70, 71, 73, 75], "long": [1, 6, 7, 9, 13, 18, 19, 21, 25, 27, 30, 39, 45, 46, 47, 48, 54, 65, 66, 71, 73, 74], "clean": [1, 8, 9, 19, 21, 38, 39, 45, 46], "section": [1, 4, 6, 9, 13, 16, 19, 21, 22, 23, 26, 27, 28, 30, 32, 35, 37, 39, 41, 42, 43, 44, 45, 48, 49, 52, 56, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75], "qa": [1, 19, 27, 48, 71, 75], "analysi": [1, 4, 16, 17, 18, 19, 21, 26, 27, 31, 36, 43, 44, 45, 46, 48, 54, 58, 71, 75], "research": [1, 4, 19, 27, 44, 48, 71, 75], "user": [1, 3, 6, 7, 9, 16, 17, 19, 24, 26, 27, 30, 31, 35, 36, 39, 41, 43, 46, 48, 57, 58, 66, 68, 69, 73, 74, 75], "defin": [1, 5, 6, 10, 13, 15, 17, 18, 19, 21, 24, 27, 28, 30, 32, 33, 34, 35, 37, 45, 48, 49, 54, 57, 60, 61, 65, 68, 69, 70, 71, 73, 74, 75], "word": [1, 19, 27, 28, 37, 48, 49], "reserv": [1, 19], "instanc": [1, 2, 5, 6, 7, 8, 9, 12, 13, 15, 19, 20, 25, 26, 27, 29, 30, 32, 35, 37, 38, 39, 47, 48, 51, 55, 58, 61, 65, 68, 69, 72, 73, 74, 75], "cannot": [1, 2, 5, 7, 11, 13, 15, 19, 21, 22, 27, 29, 35, 44, 45, 48, 56, 63, 65, 66, 69, 72, 74], "def": [1, 2, 8, 10, 17, 19, 21, 34, 38, 45, 60, 74], "lambda": [1, 7, 19, 21, 45], "These": [1, 4, 6, 13, 16, 19, 23, 27, 28, 30, 32, 35, 37, 43, 44, 48, 49, 61, 65, 67, 68, 69, 71, 73, 75], "even": [1, 2, 6, 7, 9, 12, 16, 19, 20, 21, 23, 24, 26, 27, 30, 32, 35, 39, 43, 44, 45, 46, 48, 51, 55, 57, 58, 61, 66, 67, 69, 73, 74, 75], "compil": [1, 13, 19, 65, 74, 75], "try": [1, 2, 5, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 19, 21, 24, 27, 28, 30, 34, 37, 39, 40, 42, 43, 44, 45, 46, 48, 49, 51, 57, 60, 65, 68, 70, 73, 74, 75], "rais": [1, 6, 9, 10, 17, 19, 34, 39, 46, 74, 75], "syntaxerror": [1, 19], "gener": [1, 2, 6, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 23, 25, 26, 27, 28, 30, 33, 34, 35, 37, 38, 39, 43, 45, 46, 47, 48, 49, 53, 58, 63, 65, 66, 67, 69, 71, 73, 74, 75], "list": [1, 4, 6, 7, 8, 11, 13, 16, 17, 18, 19, 21, 22, 23, 26, 30, 35, 38, 43, 44, 45, 46, 50, 54, 56, 58, 60, 63, 65, 67, 69, 73, 74, 75], "kwlist": [1, 19], "print": [1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 32, 34, 35, 37, 38, 39, 40, 43, 45, 47, 48, 49, 51, 54, 55, 56, 57, 58, 60, 61, 63, 65, 74], "none": [1, 7, 9, 11, 16, 17, 19, 26, 28, 32, 35, 37, 39, 43, 49, 50, 53, 58, 60, 61, 63, 74], "true": [1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 45, 46, 47, 48, 49, 51, 53, 54, 55, 58, 60, 65, 66, 67, 68, 69, 73, 74, 75], "assert": [1, 19], "break": [1, 8, 9, 12, 19, 32, 34, 38, 39, 51, 61, 74], "class": [1, 6, 10, 17, 19, 22, 30, 34, 35, 56, 70, 73, 75], "continu": [1, 19, 33, 74], "del": [1, 19], "elif": [1, 19], "els": [1, 16, 17, 19, 27, 28, 37, 43, 48, 49, 71, 74], "except": [1, 5, 6, 9, 10, 12, 15, 17, 19, 34, 39, 46, 51, 60, 75], "final": [1, 5, 9, 12, 15, 16, 17, 19, 21, 25, 26, 27, 28, 31, 32, 35, 36, 37, 39, 41, 43, 45, 46, 47, 48, 49, 51, 58, 60, 61, 68, 69, 71, 74], "global": [1, 19, 75], "nonloc": [1, 19], "pass": [1, 10, 17, 18, 19, 23, 31, 34, 35, 36, 41, 54, 67, 75], "return": [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 27, 29, 30, 31, 32, 34, 35, 36, 38, 39, 40, 41, 43, 45, 46, 47, 48, 50, 51, 54, 55, 56, 57, 60, 61, 63, 72, 73, 74, 75], "while": [1, 6, 10, 11, 17, 19, 23, 26, 27, 28, 30, 34, 35, 37, 44, 48, 49, 53, 58, 63, 67, 69, 73], "yield": [1, 19], "base": [1, 2, 10, 11, 12, 13, 14, 16, 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24, 25, 26, 27, 28, 32, 33, 35, 37, 38, 39, 40, 41, 43, 45, 47, 48, 49, 50, 51, 55, 57, 58, 60, 61, 63, 65], "double_id": [1, 19], "id": [1, 6, 9, 10, 11, 12, 13, 14, 19, 21, 24, 25, 26, 27, 28, 30, 33, 34, 37, 39, 40, 45, 47, 48, 49, 51, 57, 58, 63, 65, 73, 74], "4": [1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 30, 32, 33, 34, 35, 37, 38, 39, 40, 43, 45, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 61, 63, 65, 67, 72, 73, 74, 75], "8": [1, 2, 9, 10, 11, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 32, 33, 34, 35, 38, 39, 40, 43, 45, 47, 48, 51, 54, 55, 56, 57, 58, 61, 63, 73, 74], "loop": [1, 3, 9, 19, 21, 39, 45], "through": [1, 4, 6, 9, 19, 21, 28, 30, 37, 39, 44, 45, 49, 53, 69, 71, 73, 75], "column": [1, 2, 3, 7, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 32, 33, 34, 35, 37, 39, 40, 43, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 57, 61, 65, 67, 72, 74, 75], "control": [1, 6, 13, 16, 19, 21, 30, 34, 43, 45, 65, 73], "If": [1, 3, 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33, 35, 37, 38, 40, 41, 42, 45, 46, 47, 48, 49, 50, 52, 53, 55, 57, 58, 59, 60, 62, 63, 65, 67, 68, 69, 70, 71, 74, 75, 76, 77], "lead": [1, 17, 19, 20, 22, 26, 27, 32, 38, 45, 47, 49, 56, 60, 76], "alreadi": [1, 6, 9, 11, 13, 20, 24, 28, 31, 37, 41, 50, 65, 67, 69, 73, 75, 77], "articl": [1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 59, 60, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77], "demonstr": [1, 2, 8, 9, 12, 14, 17, 20, 21, 22, 24, 27, 28, 31, 34, 40, 41, 42, 45, 47, 50, 52, 53, 57, 60, 68, 69, 75], "why": [1, 6, 8, 9, 13, 17, 20, 22, 23, 24, 28, 29, 31, 39, 40, 41, 45, 47, 50, 51, 58, 67, 68, 69, 75], "should": [1, 2, 4, 6, 8, 17, 18, 20, 26, 27, 28, 29, 31, 32, 37, 38, 39, 40, 43, 45, 46, 49, 50, 51, 55, 60, 62, 70, 71, 72, 75, 76, 77], "care": [1, 3, 6, 7, 9, 10, 11, 13, 17, 20, 21, 23, 25, 26, 28, 31, 33, 35, 37, 41, 45, 49, 50, 57, 58, 59, 62, 63, 65, 67, 68, 75], "when": [1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 46, 48, 49, 50, 51, 52, 53, 55, 56, 58, 59, 60, 63, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77], "occur": [1, 2, 6, 10, 12, 17, 18, 20, 22, 26, 27, 28, 29, 35, 37, 39, 45, 47, 49, 50, 51, 53, 60, 76], "common": [1, 6, 10, 11, 17, 20, 22, 27, 30, 31, 32, 33, 35, 37, 38, 45, 47, 48, 52, 60, 63, 65, 71, 73, 74, 75, 77], "us": [1, 2, 6, 7, 10, 11, 12, 13, 14, 16, 18, 19, 20, 21, 22, 23, 25, 26, 29, 32, 34, 35, 36, 37, 38, 39, 42, 43, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 62, 65, 67, 68, 70, 72], "sql": [1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 65, 67, 68, 70, 71, 73, 74, 76], "overwrit": [1, 6, 9, 18, 20, 22, 24, 31, 37, 41, 47, 69, 75], "e": 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50, 51, 53, 55, 56, 57, 58, 59, 60, 62, 63, 65, 67, 69, 76, 77], "prefix": [1, 18, 20, 30, 31, 37, 74, 75], "give": [1, 5, 6, 8, 10, 13, 15, 16, 17, 18, 19, 20, 22, 24, 27, 29, 31, 33, 35, 36, 37, 39, 40, 45, 47, 51, 56, 60, 62, 63, 67, 68, 69, 70, 71, 73, 75, 76, 77], "sensibl": [1, 5, 6, 15, 20, 22, 28, 47, 48, 50, 70, 71, 77], "too": [1, 6, 7, 13, 16, 17, 19, 20, 21, 22, 26, 27, 31, 32, 33, 36, 38, 45, 46, 47, 48, 49, 56, 57, 60, 63, 67, 68, 70, 72, 73, 75, 77], "long": [1, 6, 7, 9, 13, 19, 20, 22, 26, 28, 31, 41, 47, 48, 49, 50, 56, 67, 68, 73, 75, 76], "clean": [1, 8, 9, 16, 20, 22, 36, 40, 41, 47, 48], "section": [1, 4, 6, 9, 13, 17, 20, 22, 23, 24, 27, 28, 29, 31, 33, 37, 39, 41, 43, 44, 45, 46, 47, 50, 51, 54, 58, 60, 61, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77], "qa": [1, 20, 28, 50, 73, 77], "analysi": [1, 4, 16, 17, 18, 19, 20, 22, 27, 28, 32, 36, 38, 45, 46, 47, 48, 50, 56, 60, 73, 77], "research": [1, 4, 20, 28, 46, 50, 73, 77], "user": [1, 3, 6, 7, 9, 16, 17, 18, 20, 25, 27, 28, 31, 32, 36, 37, 38, 41, 43, 45, 48, 50, 59, 60, 68, 70, 71, 75, 76, 77], "defin": [1, 5, 6, 10, 13, 15, 16, 18, 19, 20, 22, 25, 28, 29, 31, 33, 34, 35, 36, 37, 39, 47, 50, 51, 56, 59, 62, 63, 67, 70, 71, 72, 73, 75, 76, 77], "word": [1, 16, 20, 28, 29, 36, 39, 50, 51], "reserv": [1, 20], "instanc": [1, 2, 5, 6, 7, 8, 9, 12, 13, 15, 20, 21, 26, 27, 28, 30, 31, 33, 37, 39, 40, 41, 49, 50, 53, 57, 60, 63, 67, 70, 71, 74, 75, 76, 77], "cannot": [1, 2, 5, 7, 11, 13, 15, 20, 22, 23, 28, 30, 37, 46, 47, 50, 58, 65, 67, 68, 71, 74, 76], "def": [1, 2, 8, 10, 18, 20, 22, 35, 40, 47, 62, 76], "lambda": [1, 7, 20, 22, 47], "These": [1, 4, 6, 13, 16, 17, 20, 24, 28, 29, 31, 33, 36, 37, 39, 45, 46, 50, 51, 63, 67, 69, 70, 71, 73, 75, 77], "even": [1, 2, 6, 7, 9, 12, 17, 20, 21, 22, 24, 25, 27, 28, 31, 33, 37, 41, 45, 46, 47, 48, 50, 53, 57, 59, 60, 63, 68, 69, 71, 75, 76, 77], "compil": [1, 13, 20, 67, 76, 77], "try": [1, 2, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20, 22, 25, 28, 29, 31, 35, 39, 41, 42, 44, 45, 46, 47, 48, 50, 51, 53, 59, 62, 67, 70, 72, 75, 76, 77], "rais": [1, 6, 9, 10, 18, 20, 35, 41, 48, 76, 77], "syntaxerror": [1, 20], "gener": [1, 2, 6, 7, 8, 9, 10, 11, 13, 17, 18, 20, 22, 24, 26, 27, 28, 29, 31, 34, 35, 37, 39, 40, 41, 45, 47, 48, 49, 50, 51, 55, 60, 65, 67, 68, 69, 71, 73, 75, 76, 77], "list": [1, 4, 6, 7, 8, 11, 13, 17, 18, 19, 20, 22, 23, 24, 27, 31, 37, 40, 45, 46, 47, 48, 52, 56, 58, 60, 62, 65, 67, 69, 71, 75, 76, 77], "kwlist": [1, 20], "print": [1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 33, 35, 36, 37, 39, 40, 41, 42, 45, 47, 49, 50, 51, 53, 56, 57, 58, 59, 60, 62, 63, 65, 67, 76], "none": [1, 7, 9, 11, 16, 17, 18, 20, 27, 29, 33, 36, 37, 39, 41, 45, 51, 52, 55, 60, 62, 63, 65, 76], "true": [1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 55, 56, 57, 60, 62, 67, 68, 69, 70, 71, 75, 76, 77], "assert": [1, 20], "break": [1, 8, 9, 12, 20, 33, 35, 40, 41, 53, 63, 76], "class": [1, 6, 10, 18, 20, 23, 31, 35, 37, 58, 72, 75, 77], "continu": [1, 20, 34, 76], "del": [1, 20], "elif": [1, 20], "els": [1, 17, 18, 20, 28, 29, 39, 45, 50, 51, 73, 76], "except": [1, 5, 6, 9, 10, 12, 15, 18, 20, 35, 41, 48, 53, 62, 77], "final": [1, 5, 9, 12, 15, 16, 17, 18, 20, 22, 26, 27, 28, 29, 32, 33, 36, 37, 38, 39, 41, 43, 45, 47, 48, 49, 50, 51, 53, 60, 62, 63, 70, 71, 73, 76], "global": [1, 20, 77], "nonloc": [1, 20], "pass": [1, 10, 18, 19, 20, 24, 32, 35, 37, 38, 43, 56, 69, 77], "return": [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 30, 31, 32, 33, 35, 37, 38, 40, 41, 42, 43, 45, 47, 48, 49, 50, 52, 53, 56, 57, 58, 59, 62, 63, 65, 74, 75, 76, 77], "while": [1, 6, 10, 11, 18, 20, 24, 27, 28, 29, 31, 35, 37, 39, 46, 50, 51, 55, 60, 65, 69, 71, 75], "yield": [1, 20], "base": [1, 2, 10, 11, 12, 13, 14, 16, 17, 18, 23, 26, 27, 30, 31, 33, 35, 36, 37, 42, 43, 45, 49, 53, 55, 58, 60, 63, 65, 67, 68, 74, 75, 76, 77], "_": [1, 22, 47], "r_": 1, "e_": 1, "s_": 1, "v_": 1, "d": [1, 22, 26, 27, 36, 47, 49, 60], "w_": 1, "o_": 1, "d_": 1, "s": [1, 6, 8, 9, 16, 18, 22, 24, 28, 29, 32, 34, 36, 37, 38, 39, 40, 41, 46, 47, 48, 50, 51, 52, 55, 62, 68, 69, 71, 73, 75, 77], "i_": 1, "n": [1, 7, 8, 10, 12, 14, 18, 19, 21, 22, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 47, 49, 50, 51, 53, 56, 57, 58, 60, 62, 63, 74, 75], "c_": 1, "p_": 1, "t_": 1, "The": [1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 62, 63, 65, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77], "parser": 1, "repeat": [1, 8, 9, 10, 24, 28, 33, 35, 37, 40, 41, 46, 50, 63, 69, 76], "null": [1, 2, 6, 9, 11, 12, 13, 16, 17, 18, 28, 29, 31, 33, 36, 37, 39, 41, 45, 47, 49, 50, 51, 52, 53, 62, 63, 65, 67, 75], "inf": [1, 23, 36, 37, 58], "nan": [1, 6, 11, 16, 18, 36, 65], "na": [1, 8, 11, 17, 18, 19, 23, 31, 36, 37, 40, 45, 51, 53, 56, 58, 63, 65, 75, 77], "na_integer_": 1, "na_real_": 1, "na_complex_": 1, "na_character_": 1, "1": [1, 2, 3, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 33, 34, 36, 37, 39, 40, 41, 42, 43, 45, 47, 49, 51, 52, 53, 56, 57, 59, 60, 62, 63, 65, 67, 68, 69, 70, 71, 74, 77], "etc": [1, 18, 28, 31, 37, 46, 50, 75, 77], "argument": [1, 2, 6, 7, 8, 11, 12, 13, 18, 19, 21, 22, 23, 24, 27, 30, 31, 33, 36, 37, 40, 47, 48, 52, 53, 56, 57, 58, 60, 63, 65, 67, 69, 74, 75], "down": [1, 4, 9, 24, 25, 27, 28, 29, 36, 39, 41, 50, 51, 55, 59, 60, 69, 71], "see": [1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 24, 26, 27, 28, 29, 31, 33, 35, 36, 37, 39, 40, 41, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 60, 62, 63, 65, 67, 68, 69, 70, 71, 73, 75, 76, 77], "a_": 1, "l_": 1, "outsid": [1, 9, 34, 41, 70, 73, 76], "quot": [1, 31, 75], "alwai": [1, 4, 5, 9, 11, 13, 14, 15, 17, 21, 22, 24, 28, 29, 30, 31, 37, 39, 41, 42, 45, 46, 47, 50, 51, 57, 65, 67, 69, 70, 71, 74, 75, 76], "pars": [1, 27, 28, 34, 50, 60], "object": [1, 2, 5, 6, 7, 11, 13, 14, 15, 20, 22, 28, 37, 39, 42, 43, 46, 47, 48, 50, 65, 67, 76, 77], "link": [1, 6, 8, 10, 12, 13, 16, 17, 18, 21, 22, 26, 27, 28, 29, 31, 32, 35, 36, 37, 38, 39, 40, 45, 46, 47, 48, 49, 50, 51, 53, 55, 57, 60, 61, 67, 70, 72, 73, 75, 76, 77], "descript": [1, 2, 6, 7, 8, 9, 22, 26, 29, 31, 39, 40, 41, 46, 47, 49, 51, 75, 76], "henc": [1, 6, 8, 22, 28, 34, 40, 47, 48, 50], "thei": [1, 2, 3, 6, 7, 8, 9, 11, 12, 13, 17, 21, 22, 24, 25, 27, 28, 30, 31, 32, 33, 37, 38, 40, 41, 45, 47, 48, 50, 52, 53, 57, 59, 60, 63, 65, 67, 69, 70, 71, 72, 74, 75, 76, 77], "allow": [1, 2, 3, 6, 8, 9, 10, 12, 19, 24, 28, 30, 31, 33, 35, 37, 40, 41, 50, 53, 56, 63, 69, 70, 71, 72, 73, 74, 75, 77], "syntact": 1, "make": [1, 2, 6, 7, 8, 9, 10, 13, 16, 17, 21, 22, 23, 27, 28, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 55, 57, 58, 60, 62, 63, 67, 71, 72, 73, 74, 75, 76, 77], "non": [1, 5, 6, 7, 9, 14, 15, 17, 18, 20, 29, 33, 37, 39, 41, 42, 45, 51, 63, 76], "insid": [1, 6, 17, 19, 21, 22, 28, 29, 31, 33, 39, 45, 47, 50, 51, 56, 57, 63, 75], "backtick": 1, "might": [1, 8, 13, 20, 21, 22, 23, 28, 29, 30, 37, 39, 40, 44, 47, 48, 50, 51, 57, 58, 62, 67, 71, 72, 73, 74, 76], "notic": [1, 6, 20, 24, 31, 32, 37, 38, 48, 69, 75, 77], "quit": [1, 7, 20, 22, 23, 24, 33, 34, 37, 47, 48, 58, 63, 69, 73], "short": [1, 4, 20, 21, 22, 31, 47, 55, 57, 75], "possibl": [1, 2, 4, 11, 12, 13, 20, 22, 23, 26, 27, 28, 31, 33, 37, 44, 46, 47, 48, 49, 50, 53, 58, 60, 63, 65, 67, 72, 75, 76, 77], "obvious": [1, 20, 22, 26, 47, 49, 77], "surprisingli": [1, 20], "easi": [1, 6, 7, 17, 20, 21, 22, 27, 28, 31, 37, 45, 47, 50, 57, 60, 70, 75, 76], "hard": [1, 6, 20, 76], "debug": [1, 4, 6, 9, 20, 28, 29, 31, 39, 41, 48, 50, 51, 72, 75, 76], "know": [1, 3, 6, 8, 17, 20, 22, 24, 28, 29, 31, 39, 40, 45, 46, 47, 48, 50, 51, 69, 72, 73, 75, 76], "error": [1, 5, 6, 9, 10, 12, 15, 17, 18, 20, 24, 28, 32, 35, 37, 38, 41, 45, 48, 50, 53, 62, 69, 77], "look": [1, 4, 6, 7, 8, 9, 10, 13, 14, 16, 17, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 55, 57, 58, 60, 62, 67, 69, 70, 71, 72, 75, 76, 77], "help": [1, 4, 6, 9, 17, 20, 22, 28, 29, 31, 32, 38, 39, 41, 45, 46, 47, 48, 50, 51, 55, 71, 75, 76], "builtin": [1, 20], "iter": [1, 2, 9, 17, 20, 41, 45, 48], "valu": [1, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 33, 34, 35, 36, 41, 45, 47, 49, 50, 52, 53, 56, 57, 60, 62, 63, 65, 67, 69, 70, 71, 74, 75, 76, 77], "default": [1, 3, 6, 7, 11, 12, 13, 17, 20, 22, 23, 24, 26, 27, 28, 31, 33, 34, 37, 43, 45, 46, 47, 48, 49, 50, 53, 58, 60, 63, 65, 67, 69, 72, 75, 76, 77], "plu": [1, 20, 77], "number": [1, 2, 6, 7, 8, 9, 10, 11, 13, 14, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 55, 56, 58, 59, 60, 62, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77], "empti": [1, 7, 9, 12, 20, 22, 29, 37, 41, 46, 47, 48, 51, 52, 53, 62], "intend": [1, 6, 9, 20, 31, 41, 75], "specif": [1, 4, 6, 10, 12, 13, 14, 20, 22, 23, 24, 28, 31, 34, 35, 42, 47, 50, 53, 55, 58, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77], "numer": [1, 6, 10, 16, 18, 19, 20, 21, 27, 31, 35, 36, 37, 56, 57, 60, 75], "reject": [1, 20, 71], "ad": [1, 8, 17, 18, 19, 20, 22, 23, 27, 28, 29, 37, 39, 40, 44, 45, 47, 50, 51, 55, 56, 58, 60, 62, 71, 76, 77], "three": [1, 2, 5, 6, 7, 8, 9, 15, 16, 17, 19, 20, 21, 22, 23, 24, 27, 29, 31, 33, 36, 37, 39, 40, 41, 45, 47, 48, 51, 52, 56, 57, 58, 60, 62, 63, 68, 69, 70, 75, 77], "integ": [1, 6, 7, 13, 18, 20, 21, 24, 25, 28, 37, 50, 57, 59, 62, 67, 69], "togeth": [1, 2, 5, 6, 15, 20, 29, 39, 43, 51, 62, 77], "3": [1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 33, 35, 36, 37, 39, 40, 41, 42, 45, 47, 49, 50, 51, 52, 53, 55, 56, 57, 59, 60, 62, 63, 65, 67, 68, 69, 70, 74, 77], "6": [1, 2, 6, 8, 9, 10, 11, 17, 18, 20, 21, 22, 25, 26, 28, 29, 33, 34, 35, 36, 37, 39, 40, 41, 45, 47, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 63, 65, 71, 75, 76, 77], "spark": [1, 3, 4, 5, 7, 9, 12, 15, 18, 20, 21, 22, 26, 28, 41, 43, 44, 47, 49, 50, 52, 53, 57, 62], "recommend": [1, 4, 5, 6, 11, 12, 15, 17, 20, 23, 25, 26, 27, 28, 31, 45, 46, 49, 50, 53, 58, 59, 60, 65, 70, 71, 75, 76, 77], "whenev": [1, 6, 20, 22, 47], "sometim": [1, 6, 8, 9, 12, 13, 14, 20, 21, 22, 25, 26, 28, 29, 31, 39, 40, 41, 42, 47, 48, 49, 50, 51, 53, 55, 57, 59, 67, 71, 72, 73, 75, 76, 77], "best": [1, 2, 5, 6, 15, 17, 20, 22, 26, 37, 45, 47, 49, 68, 73, 77], "someth": [1, 5, 7, 10, 15, 20, 24, 27, 28, 29, 35, 39, 44, 50, 51, 60, 69, 72, 77], "differ": [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 20, 22, 23, 24, 27, 28, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 45, 47, 50, 51, 53, 56, 58, 60, 63, 65, 67, 69, 70, 72, 74, 75, 76, 77], "method": [1, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 16, 18, 20, 22, 24, 28, 31, 34, 35, 36, 37, 41, 47, 48, 50, 53, 65, 67, 69, 73, 75, 76, 77], "bad": [1, 20, 29, 39, 46, 51], "idea": [1, 5, 6, 9, 15, 17, 20, 22, 23, 24, 25, 31, 41, 43, 45, 47, 58, 59, 69, 71, 75, 76], "case": [1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 19, 20, 21, 22, 24, 28, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 45, 47, 50, 51, 53, 56, 57, 63, 65, 68, 69, 70, 71, 72, 74, 75, 76], "all": [1, 3, 5, 6, 7, 8, 13, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 58, 59, 60, 62, 63, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77], "attempt": [1, 10, 20, 24, 26, 35, 49, 69, 76], "attributeerror": [1, 5, 15, 20], "nonetyp": [1, 20], "attribut": [1, 5, 15, 20], "_jvm": [1, 20], "take": [1, 2, 3, 6, 8, 9, 10, 12, 16, 17, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 35, 36, 37, 39, 40, 41, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 60, 62, 69, 71, 73, 75, 77], "col": [1, 2, 7, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 67], "aggreg": [1, 8, 12, 20, 22, 27, 28, 29, 39, 40, 47, 48, 50, 51, 53, 60, 73], "express": [1, 6, 10, 20, 23, 27, 31, 34, 35, 58, 60, 75, 76], "versionad": [1, 20], "overwritten": [1, 9, 20, 41, 48], "kei": [1, 4, 6, 11, 12, 14, 17, 20, 22, 24, 26, 28, 30, 31, 37, 42, 45, 46, 47, 49, 50, 53, 62, 65, 69, 71, 72, 74, 75, 77], "individu": [1, 6, 8, 20, 29, 31, 37, 39, 40, 51, 52, 75, 77], "own": [1, 3, 6, 16, 20, 26, 31, 36, 49, 73, 75, 76, 77], "often": [1, 3, 6, 7, 8, 11, 13, 14, 16, 17, 20, 21, 24, 28, 29, 31, 32, 36, 37, 38, 39, 40, 42, 45, 48, 50, 51, 52, 57, 65, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77], "unintention": [1, 20], "master": [1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76], "local": [1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 65, 67, 69, 71, 72, 74, 75, 76, 77], "creat": [1, 2, 4, 5, 8, 9, 12, 13, 14, 15, 17, 18, 19, 20, 23, 25, 26, 28, 33, 40, 41, 42, 45, 46, 48, 49, 50, 53, 55, 56, 58, 59, 62, 63, 67, 68, 70, 72, 73, 76, 77], "small": [1, 2, 3, 6, 7, 8, 9, 11, 14, 16, 17, 20, 22, 23, 24, 26, 27, 28, 29, 31, 32, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 58, 60, 65, 69, 70, 72, 73, 75, 77], "datafram": [1, 2, 3, 6, 7, 9, 12, 13, 17, 18, 20, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 37, 38, 39, 41, 43, 45, 46, 49, 51, 53, 58, 59, 60, 63, 67, 69, 70, 71, 74, 75, 76], "sdf": [1, 8, 20, 22, 25, 34, 36, 40, 47, 59, 76], "rang": [1, 6, 9, 11, 12, 13, 14, 16, 20, 22, 25, 26, 27, 28, 35, 36, 41, 42, 47, 49, 50, 53, 59, 60, 65, 67], "withcolumn": [1, 2, 5, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 25, 26, 27, 28, 29, 33, 34, 36, 37, 39, 40, 41, 42, 43, 45, 47, 49, 50, 51, 52, 53, 57, 59, 60, 62, 63, 65, 67], "double_id": [1, 20], "id": [1, 6, 9, 10, 11, 12, 13, 14, 20, 22, 25, 26, 27, 28, 29, 31, 34, 35, 39, 41, 42, 47, 49, 50, 51, 53, 59, 60, 65, 67, 75, 76], "4": [1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 45, 47, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 63, 65, 67, 69, 74, 75, 76, 77], "8": [1, 2, 9, 10, 11, 14, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 33, 34, 35, 36, 37, 40, 41, 42, 45, 47, 49, 50, 53, 56, 57, 58, 59, 60, 63, 65, 75, 76], "loop": [1, 3, 9, 20, 22, 41, 47], "through": [1, 4, 6, 9, 20, 22, 29, 31, 39, 41, 46, 47, 51, 55, 71, 73, 75, 77], "column": [1, 2, 3, 7, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 33, 34, 35, 36, 37, 39, 41, 42, 45, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 59, 63, 67, 69, 74, 76, 77], "control": [1, 6, 13, 17, 20, 22, 31, 35, 45, 47, 67, 75], "If": [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 58, 59, 60, 62, 63, 65, 67, 69, 70, 71, 72, 73, 75, 76, 77], "reassign": [1, 6, 20, 31, 75], "triple_id": [1, 20], "typeerror": [1, 10, 20, 35], "str": [1, 9, 20, 41, 76], "callabl": [1, 18, 20], "solv": [1, 11, 16, 20, 29, 36, 39, 48, 51, 65], "problem": [1, 7, 9, 14, 16, 17, 20, 29, 36, 37, 39, 41, 42, 45, 51, 55, 76, 77], "9": [1, 2, 6, 9, 10, 11, 13, 14, 18, 19, 20, 21, 22, 26, 27, 28, 33, 34, 35, 36, 37, 40, 41, 42, 47, 49, 50, 53, 56, 57, 58, 60, 63, 65, 67, 68, 75, 77], "12": [1, 2, 3, 6, 9, 10, 14, 17, 18, 20, 22, 26, 28, 33, 35, 36, 37, 41, 42, 45, 47, 49, 50, 52, 53, 58, 63, 68, 70, 74, 75, 77], "It": [1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 35, 36, 37, 38, 39, 40, 41, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 58, 59, 60, 65, 67, 68, 69, 70, 71, 73, 75, 76, 77], "advis": [1, 2, 20, 27, 60, 70, 73], "directli": [1, 3, 5, 6, 7, 10, 13, 15, 20, 21, 30, 31, 32, 33, 35, 37, 38, 57, 63, 67, 70, 74, 75, 77], "filter": [1, 3, 7, 8, 12, 14, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 30, 33, 37, 40, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 55, 56, 57, 58, 63, 69, 74], "both": [1, 2, 6, 10, 11, 12, 13, 17, 20, 23, 24, 25, 26, 27, 28, 30, 31, 35, 37, 45, 49, 50, 53, 58, 59, 60, 62, 65, 67, 69, 71, 73, 74, 75, 76, 77], "stat": [1, 10, 18, 20, 29, 32, 35, 37, 38, 39, 51], "dplyr": [1, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 47, 49, 50, 51, 53, 56, 57, 58, 59, 60, 63, 65, 67, 69, 74, 75, 76, 77], "specifi": [1, 3, 6, 7, 8, 9, 11, 13, 14, 18, 19, 20, 21, 22, 24, 26, 27, 28, 31, 33, 36, 37, 40, 41, 42, 46, 47, 49, 50, 56, 57, 60, 63, 65, 67, 69, 70, 75, 76, 77], "exactli": [1, 19, 20, 28, 29, 39, 50, 51, 56, 72, 76], "despit": [1, 14, 20, 25, 42, 59], "commonli": [1, 5, 12, 15, 20, 21, 22, 33, 37, 47, 53, 57, 63, 70, 71, 76], "distribut": [1, 2, 3, 4, 6, 7, 14, 17, 19, 20, 22, 26, 28, 29, 31, 39, 42, 45, 46, 47, 49, 50, 51, 56, 72, 75], "fdist": 1, "h_": 1, "b_": 1, "u_": 1, "densiti": 1, "quantil": [1, 19, 56], "random": [1, 9, 11, 14, 22, 26, 27, 28, 34, 41, 42, 47, 49, 50, 60, 65], "df1": [1, 9, 41, 62], "df2": [1, 62], "degre": [1, 18, 19, 37, 56], "freedom": [1, 18, 37], "option": [1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 17, 18, 23, 26, 27, 28, 29, 31, 33, 37, 39, 40, 41, 45, 48, 49, 50, 51, 53, 58, 60, 63, 65, 67, 68, 70, 71, 72, 73, 75, 77], "central": [1, 6, 18], "paramet": [1, 10, 18, 19, 22, 33, 35, 37, 47, 48, 56, 63, 76], "ncp": 1, "g_": 1, "x": [1, 2, 7, 16, 22, 23, 27, 31, 32, 34, 35, 36, 37, 38, 46, 47, 56, 58, 60, 63, 67, 71, 74, 75], "log": [1, 19, 56, 76, 77], "pf": 1, "q": [1, 6], "lower": [1, 2, 6, 10, 18, 27, 35, 37, 60], "tail": [1, 6, 28, 31, 50, 75], "p": [1, 9, 12, 18, 22, 37, 41, 47, 53, 75], "qf": 1, "rf": 1, "m_": 1, "n_": 1, "vector": [1, 18, 31, 37, 75, 77], "probabl": [1, 19, 27, 37, 56, 60, 73], "observ": [1, 8, 19, 37, 40, 56], "length": [1, 7, 21, 57, 76], "taken": [1, 8, 9, 18, 27, 29, 37, 39, 40, 41, 48, 51, 60, 62], "requir": [1, 4, 14, 17, 18, 19, 24, 26, 28, 31, 32, 33, 37, 38, 42, 45, 49, 50, 56, 63, 68, 69, 70, 72, 75, 76, 77], "omit": [1, 17, 45], "assum": [1, 4, 6, 8, 17, 19, 31, 39, 40, 45, 56, 70, 71, 75, 77], "logic": [1, 6, 10, 12, 28, 31, 34, 35, 46, 50, 53, 75, 76, 77], "given": [1, 3, 6, 10, 13, 17, 19, 22, 23, 24, 25, 27, 29, 31, 32, 33, 35, 38, 39, 45, 47, 48, 51, 55, 56, 58, 59, 60, 63, 67, 69, 70, 71, 72, 75, 76, 77], "otherwis": [1, 9, 16, 17, 18, 22, 26, 27, 29, 31, 34, 36, 37, 39, 41, 45, 47, 49, 51, 60, 75, 76], "n1": 1, "n2": 1, "gamma": 1, "ratio": 1, 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69, 70, 75], "behaviour": [1, 7, 17, 22, 25, 28, 45, 47, 48, 50, 59, 62, 77], "extrem": [1, 22, 26, 47, 49], "princip": 1, "moder": [1, 37], "highli": [1, 22, 24, 47, 69], "accur": [1, 19, 22, 47, 56], "especi": [1, 9, 22, 28, 41, 46, 47, 48, 50, 76], "larg": [1, 3, 6, 7, 8, 9, 12, 13, 17, 19, 22, 23, 24, 27, 28, 29, 30, 31, 32, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 50, 51, 53, 56, 58, 60, 67, 69, 71, 72, 73, 74, 75, 77], "comput": [1, 4, 6, 8, 22, 28, 29, 31, 39, 40, 47, 48, 50, 51, 68, 70, 72, 73, 75], "_via_": 1, "binomi": [1, 18, 27, 37, 60], "contribut": [1, 4, 8, 40, 46, 77], "catherin": 1, "loader": 1, "dbinom": 1, "dbeta": 1, "peter": [1, 77], "ruckdeschel": 1, "pbeta": 1, "pchisq": 1, "qchisq": 1, "qbeta": 1, "f_": 1, "becker": 1, "chamber": 1, "j": 1, "wilk": 1, "1988": 1, "_the": 1, "language_": 1, "wadsworth": 1, "brook": 1, "cole": 1, "johnson": 1, "kotz": 1, "balakrishnan": 1, "1995": 1, "_continu": 1, "univari": 1, "distributions_": 1, "volum": 1, "chapter": [1, 6, 24, 27, 31, 43, 60, 69, 75, 77], "27": [1, 2, 3, 6, 10, 12, 18, 34, 35, 37, 41, 45, 53, 58, 75], "30": [1, 7, 18, 34, 35, 37, 41, 45, 58, 75], "wilei": 1, "york": 1, "o": [1, 16, 19, 36, 56], "includ": [1, 4, 6, 7, 8, 9, 10, 11, 16, 17, 18, 22, 23, 24, 27, 28, 30, 31, 32, 35, 36, 37, 38, 40, 41, 45, 47, 50, 58, 60, 65, 68, 69, 70, 73, 74, 75, 77], "dchisq": 1, "dt": 1, "t": [1, 2, 3, 6, 7, 8, 9, 16, 18, 22, 24, 29, 30, 32, 36, 37, 38, 39, 40, 41, 47, 48, 51, 62, 68, 69, 71, 74, 76], "x_": 1, "equival": [1, 6, 7, 8, 37, 40, 62, 77], "pt": 1, "nu": 1, "seq": 1, "001": [1, 19, 26, 49, 56], "len": [1, 21, 22, 47, 57], "100": [1, 2, 7, 11, 13, 16, 22, 23, 26, 27, 28, 33, 36, 47, 49, 50, 58, 60, 63, 65, 67, 68], "stopifnot": 1, "equal": [1, 4, 6, 12, 17, 26, 27, 28, 31, 33, 34, 37, 45, 49, 50, 53, 60, 63, 75, 76], "upper": [1, 5, 6, 10, 15, 18, 19, 35, 37, 56, 75], "ident": [1, 6, 7, 10, 13, 14, 17, 19, 21, 26, 27, 29, 31, 32, 33, 35, 38, 39, 42, 45, 49, 51, 56, 57, 60, 63, 67, 75, 77], "qt": 1, 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63, 65, 67, 68, 70, 71, 74, 75, 76, 77], "nativ": [2, 6, 10, 12, 18, 26, 27, 28, 31, 35, 37, 49, 50, 53, 60, 75], "function": [2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 29, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 53, 55, 56, 57, 58, 59, 60, 62, 65, 67, 69, 71, 77], "wherev": [2, 28, 50], "i": [2, 6, 8, 17, 29, 39, 40, 44, 45, 48, 51], "howev": [2, 8, 9, 10, 17, 19, 21, 22, 24, 27, 28, 29, 31, 33, 35, 37, 39, 40, 41, 45, 47, 48, 50, 51, 55, 56, 57, 60, 62, 63, 69, 70, 71, 73, 75, 76, 77], "avail": [2, 6, 13, 16, 18, 19, 24, 28, 29, 30, 31, 32, 36, 37, 38, 39, 46, 50, 51, 56, 67, 69, 71, 73, 74, 75, 77], "what": [2, 6, 8, 9, 13, 19, 23, 26, 29, 31, 33, 37, 39, 40, 41, 44, 48, 49, 51, 56, 58, 62, 63, 67, 71, 75], "second": [2, 6, 8, 9, 11, 16, 17, 19, 21, 22, 23, 28, 29, 31, 36, 37, 39, 40, 41, 45, 47, 50, 51, 56, 57, 58, 62, 65, 75], "ordinari": 2, "particularli": [2, 6, 22, 24, 29, 37, 39, 47, 51, 69, 73], "effici": 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42, 45, 47, 60, 68, 70, 72, 76], "massiv": 2, "de": [2, 22, 29, 39, 47, 51], "overhead": [2, 22, 27, 47, 60, 68, 71], "high": [2, 7, 19, 22, 37, 47, 56, 73], "copi": [2, 17, 18, 28, 32, 38, 45, 46, 48, 50, 70, 76], "memori": [2, 3, 6, 7, 8, 11, 14, 17, 22, 24, 26, 27, 28, 29, 31, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 60, 65, 68, 69, 70, 71, 72, 73, 75, 76], "slow": [2, 6, 9, 13, 17, 22, 28, 29, 31, 39, 41, 43, 45, 47, 50, 51, 55, 67, 73, 75], "known": [2, 14, 16, 17, 18, 31, 36, 37, 42, 45, 46, 68, 71, 72, 75], "vectoris": [2, 18, 34, 37, 46], "executor": [2, 8, 17, 22, 27, 29, 39, 40, 44, 45, 46, 47, 48, 51, 60, 68, 70, 71], "manner": [2, 6, 7, 28, 31, 44, 50, 75, 77], "oper": [2, 5, 6, 9, 11, 14, 15, 17, 19, 24, 28, 29, 31, 39, 41, 42, 43, 45, 48, 50, 51, 56, 65, 69, 72, 73, 75], "whole": [2, 6, 7, 13, 14, 31, 33, 39, 42, 48, 63, 67, 75], "partit": [2, 3, 6, 8, 17, 26, 29, 31, 33, 39, 40, 45, 49, 51, 63, 68, 70, 71, 72, 75], "onc": [2, 6, 8, 9, 16, 17, 22, 26, 28, 29, 31, 32, 33, 36, 37, 38, 39, 40, 41, 45, 47, 48, 49, 50, 51, 52, 55, 63, 68, 70, 73, 75], "just": [2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 22, 23, 25, 26, 28, 29, 31, 32, 33, 35, 36, 37, 38, 39, 41, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 62, 63, 65, 67, 68, 75, 76, 77], "achiev": [2, 22, 24, 26, 37, 47, 49, 69], "apach": [2, 8, 10, 13, 18, 35, 37, 40, 50, 67, 72, 76], "transfer": [2, 28, 50], "across": [2, 4, 6, 14, 19, 27, 31, 37, 42, 44, 56, 60, 72, 75], "low": [2, 48, 68, 72], "essenti": [2, 9, 17, 26, 28, 33, 37, 41, 45, 48, 49, 50, 63, 77], "act": [2, 37], "middleman": 2, "singl": [2, 6, 7, 8, 9, 18, 22, 23, 28, 29, 31, 34, 37, 39, 40, 41, 46, 47, 50, 51, 52, 58, 62, 68, 71, 72, 75, 77], "compat": [2, 6, 31, 75], "cogroup": 2, "demo": 2, "cover": [2, 4, 6, 9, 17, 24, 27, 28, 29, 31, 39, 41, 45, 50, 51, 52, 60, 69, 72, 75, 77], "support": [2, 3, 4, 13, 22, 24, 25, 37, 47, 48, 52, 59, 67, 69, 71, 72, 73, 76], "found": [2, 8, 10, 27, 35, 37, 40, 60, 76], 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76, 77], "expect": [2, 8, 11, 22, 24, 25, 27, 28, 29, 37, 39, 40, 47, 50, 51, 59, 60, 65, 69, 76, 77], "deprec": 2, "14": [2, 11, 13, 14, 18, 21, 26, 27, 28, 33, 34, 35, 36, 37, 41, 42, 47, 49, 50, 53, 57, 58, 60, 63, 65, 67, 76], "offic": [2, 4, 6, 31, 75], "usag": [2, 6, 13, 28, 31, 50, 67, 68, 73, 75, 76, 77], "guid": [2, 4, 28, 32, 37, 38, 50, 73], "issu": [2, 4, 7, 11, 12, 22, 26, 27, 29, 39, 47, 48, 49, 51, 53, 60, 65, 70, 71, 72, 73, 76, 77], "19": [2, 9, 10, 16, 18, 26, 27, 28, 33, 34, 35, 36, 37, 41, 45, 49, 50, 58, 60, 63, 74, 75, 76], "find": [2, 4, 6, 7, 8, 9, 10, 18, 19, 22, 23, 26, 28, 29, 31, 35, 39, 40, 41, 43, 46, 47, 48, 49, 50, 51, 55, 56, 58, 70, 75, 76, 77], "switch": [2, 71], "add": [2, 6, 7, 8, 9, 11, 12, 16, 17, 18, 22, 23, 27, 28, 31, 33, 36, 37, 40, 41, 45, 47, 50, 52, 53, 58, 60, 62, 63, 65, 75, 76, 77], "enabl": [2, 6, 7, 12, 27, 29, 31, 39, 51, 53, 60, 68, 70, 71, 75, 76, 77], "optimis": [2, 4, 6, 12, 22, 26, 29, 31, 39, 43, 47, 49, 51, 53, 71, 72, 75, 76], 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9, 18, 22, 29, 39, 41, 47, 51, 55, 75], "person": [6, 9, 17, 18, 23, 37, 41, 45, 58, 71, 75, 76, 77], "land": [6, 9, 17, 18, 23, 37, 41, 45, 58, 75], "hous": [6, 18, 75], "oc": 6, "dwell": [6, 18, 37, 41, 74, 75], "assi": 6, "assist": [6, 7, 9, 13, 18, 22, 29, 32, 37, 38, 39, 41, 47, 51, 55, 67, 75, 77], "e05011467": [6, 75], "crystal": [6, 18, 75], "palac": [6, 18, 75], "e09000008": [6, 75], "croydon": [6, 9, 18, 19, 41, 56, 75], "norburi": [6, 18, 75], "se19": [6, 17, 18, 19, 45, 56, 75], "rspca": [6, 7, 17, 18, 22, 29, 39, 45, 47, 51, 55, 75], "rail": [6, 7, 9, 18, 29, 39, 41, 51, 75], "outdoor": [6, 9, 18, 37, 41, 74, 75], "structur": [6, 7, 9, 13, 18, 22, 24, 28, 29, 31, 37, 39, 41, 46, 47, 48, 50, 51, 67, 69, 72, 74, 75, 76], "e05000169": [6, 75], "woodsid": [6, 18, 75], "se25": [6, 17, 18, 19, 45, 56, 75], "534785": [6, 75], "167546": [6, 75], "caught": [6, 18, 41, 55, 75], "IN": [6, 7, 9, 17, 29, 39, 41, 45, 51, 55, 75], "dra": 6, "pipe": [6, 7, 8, 18, 22, 23, 31, 37, 40, 47, 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"camelcas": [6, 31, 75, 77], "snake_cas": [6, 31, 75], "charact": [6, 7, 9, 12, 13, 17, 24, 26, 33, 37, 41, 45, 49, 53, 63, 67, 69, 76], "problemat": 6, "cal_year": [6, 7, 12, 13, 17, 18, 23, 31, 32, 33, 38, 45, 53, 58, 63, 67, 75], "job_hour": [6, 7, 18, 31, 37, 75], "engine_count": [6, 7, 18, 31, 37, 75], "noth": [6, 12, 22, 29, 39, 47, 51, 53], "until": [6, 48, 71, 76], "postcode_district": [6, 7, 8, 17, 19, 31, 40, 43, 45, 56, 75], "six": [6, 14, 17, 42, 45, 77], "were": [6, 8, 11, 12, 22, 24, 29, 31, 33, 37, 39, 40, 44, 47, 51, 52, 53, 62, 63, 65, 69, 75, 77], "eight": 6, "total": [6, 22, 27, 28, 31, 33, 37, 43, 44, 47, 50, 60, 63, 75], "solut": [6, 9, 17, 22, 24, 31, 39, 41, 45, 47, 48, 69, 73, 75, 76], "ub9": [6, 17, 18, 19, 45, 56, 75], "un": [6, 7, 75], "rm3": [6, 17, 18, 19, 45, 56, 75], "condit": [6, 12, 17, 26, 27, 31, 33, 34, 45, 49, 53, 60, 63, 75, 76], "robust": 6, "sign": [6, 8, 22, 31, 40, 47, 75], "37009101": [6, 75], "58746101": [6, 75], "212716101": [6, 75], "messi": 6, "5186091": 6, "5724091": 6, "5770091": 6, "bracket": [6, 13, 67, 71], "put": [6, 8, 13, 16, 17, 22, 28, 29, 31, 36, 39, 40, 45, 46, 47, 50, 51, 62, 67, 71, 75, 76, 77], "expensive_olympic_dog": [6, 31, 75], "750": [6, 31, 41, 75], "2012": [6, 12, 14, 22, 23, 27, 31, 33, 41, 42, 47, 53, 58, 60, 63, 67, 75], "16209121": [6, 75], "17531121": [6, 75], "20818121": [6, 75], "38636121": [6, 75], "64764121": [6, 75], "1040": [6, 23, 53, 58, 75], "match": [6, 9, 12, 17, 22, 29, 39, 41, 45, 47, 48, 51, 52, 53, 77], "isin": [6, 8, 14, 17, 23, 27, 33, 40, 42, 45, 58, 60, 63], "cleaner": 6, "cats_and_dog": 6, "tra": 6, "e11": [6, 19, 56, 75], "labrador": 6, "fallen": [6, 7, 29, 39, 51], "e12": [6, 19, 56], "stuck": [6, 7, 9, 22, 29, 39, 41, 47, 51, 76], "mud": 6, "e5": [6, 19, 56], "2015": [6, 8, 10, 12, 14, 22, 23, 33, 35, 40, 41, 42, 47, 53, 55, 58, 63, 67], "2017": [6, 8, 9, 11, 12, 14, 17, 18, 22, 23, 27, 33, 37, 40, 41, 42, 45, 47, 52, 53, 58, 60, 62, 63, 65], "500": [6, 71], 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57, 60, 65, 67, 69, 70, 71, 72, 75, 77], "deriv": [6, 28, 31, 33, 50, 63, 70, 75, 76], "took": [6, 9, 29, 31, 39, 41, 51, 75, 77], "hour": [6, 22, 31, 47, 75], "attend": [6, 7, 31, 75], "durat": [6, 8, 29, 31, 39, 40, 51, 71, 75, 77], "incident_dur": [6, 18, 31, 37, 75], "built": [6, 7, 10, 19, 24, 28, 31, 35, 37, 50, 56, 69, 72, 75, 77], "form": [6, 8, 9, 23, 27, 29, 31, 33, 39, 40, 41, 48, 51, 58, 60, 63, 71, 73, 75], "did": [6, 9, 24, 28, 31, 37, 39, 41, 50, 69, 75], "preprocess": [6, 8, 10, 27, 31, 35, 40, 60, 75], "guarante": [6, 31, 75], "descend": [6, 28, 31, 33, 37, 50, 63, 75], "cost": [6, 8, 12, 22, 29, 31, 39, 40, 47, 51, 53, 73, 75], "49189111": [6, 75], "2340": [6, 8, 18, 37, 40, 75], "82423111": [6, 75], "101755111": [6, 75], "030477": [6, 75], "09032018": [6, 75], "2296": [6, 8, 18, 37, 40, 75], "028258": [6, 75], "08032017": [6, 75], "2282": [6, 8, 18, 37, 40, 75], "expens": [6, 14, 17, 19, 27, 28, 31, 42, 43, 45, 46, 48, 50, 56, 60, 71, 75], "sens": [6, 8, 10, 22, 31, 35, 40, 47, 48, 75, 76], "principl": [6, 9, 23, 28, 31, 32, 33, 38, 41, 50, 58, 63, 75, 76, 77], "syntax": [6, 12, 23, 24, 28, 31, 33, 37, 50, 53, 58, 63, 69, 75, 76], "colleagu": [6, 28, 31, 50, 75, 76, 77], "shuffl": [6, 8, 14, 22, 26, 29, 31, 33, 39, 40, 42, 47, 49, 51, 63, 68, 70, 71, 72, 75], "readi": [6, 16, 18, 31, 36, 37, 48, 73, 75], "term": [6, 14, 17, 22, 28, 29, 31, 37, 39, 42, 45, 47, 48, 50, 51, 73, 75], "bottom": [6, 28, 29, 31, 36, 39, 44, 50, 51, 75], "anyth": [6, 28, 29, 31, 39, 50, 51, 73, 75, 76], "strang": [6, 31, 75], "top10": [6, 31, 75], "bottom10": [6, 31, 75], "2013": [6, 8, 10, 12, 14, 22, 23, 33, 35, 40, 42, 45, 47, 53, 58, 63, 67, 74, 75], "foal": 6, "river": 6, "wat": 6, "e4": [6, 8, 40], "137525091": [6, 21, 57, 75], "1300": [6, 8, 18, 23, 37, 40, 58, 75], "canal": [6, 7, 29, 39, 51], "n18": 6, "18627122": [6, 75], "ON": [6, 7, 29, 39, 45, 51, 75], "en3": [6, 22, 47], "92423121": [6, 75], "TO": [6, 22, 47], "955141": [6, 21, 57, 75], "2014": [6, 8, 10, 12, 14, 22, 23, 33, 35, 40, 42, 45, 47, 53, 58, 63, 67, 74, 75], "1450": [6, 75], "polic": [6, 17, 21, 23, 45, 57, 58], "wit": 6, "INTO": [6, 7, 29, 39, 51], "tn14": [6, 8, 40], "125704": [6, 75], "02092018": [6, 75], "1665": [6, 75], "da17": 6, "126939": [6, 75], "05092018": [6, 75], "kitten": [6, 7], "tw4": [6, 8, 40], "cr5": 6, "35453121": [6, 75], "injur": 6, "w11": 6, "4920141": [6, 75], "pidgeon": [6, 41], "sw3": 6, "36995121": [6, 75], "poni": 6, "collasp": 6, "cr0": [6, 9, 17, 19, 22, 41, 45, 47, 56, 75], "194986111": [6, 75], "2011": [6, 8, 10, 12, 14, 17, 22, 23, 33, 35, 40, 42, 45, 47, 53, 58, 63, 75], "neck": 6, "51412121": [6, 75], "e8": 6, "208663091": [6, 75], "loos": [6, 73], "ub10": 6, "18325122": [6, 75], "stu": 6, "se6": 6, "43265111": [6, 75], "distress": 6, "2602131": [6, 75], "w4": 6, "118613121": [6, 75], "se16": 6, "insight": [6, 31, 75], "raw": [6, 13, 17, 31, 32, 38, 45, 67, 75, 77], "averag": [6, 31, 75], "largest": [6, 22, 31, 37, 47, 75], "smallest": [6, 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45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 58, 62, 65, 71, 72, 75, 77], "danger": [6, 12, 31, 53, 75], "overload": [6, 7, 31, 75], "postcod": [6, 17, 19, 31, 45, 56, 75], "district": [6, 31, 75], "popul": [6, 8, 17, 19, 27, 31, 34, 40, 43, 45, 56, 60, 75], "compress": [6, 8, 24, 31, 40, 69, 75], "drawback": [6, 31, 75], "readabl": [6, 7, 14, 22, 26, 31, 33, 34, 42, 46, 47, 49, 63, 75], "interrog": [6, 31, 75], "population_path": [6, 8, 17, 19, 31, 40, 43, 45, 56, 75], "dh7": [6, 17, 45, 75], "41076": [6, 17, 45, 75], "nw3": [6, 17, 45, 75], "52376": [6, 17, 45, 75], "nr4": [6, 17, 45, 75], "22331": [6, 17, 19, 45, 56, 75], "so31": [6, 17, 45, 75], "44742": [6, 17, 45, 75], "ct18": [6, 17, 45, 75], "14357": [6, 17, 45, 75], "who": [6, 9, 17, 29, 31, 32, 38, 39, 41, 45, 51, 75], "merg": [6, 12, 18, 26, 28, 31, 37, 43, 49, 50, 53, 75], "map": [6, 10, 13, 17, 18, 22, 26, 27, 31, 35, 36, 37, 45, 47, 49, 52, 60, 67, 75], "rescue_with_pop": [6, 17, 31, 43, 45, 75], "000014": [6, 9, 17, 41, 45, 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73, "want": 73, "tool": 73, "handl": 76, "tip": 76, "recap": 76, "kei": 76, "point": 76, "except": 76, "No": 76, "captur": 76, "ignor": [76, 77], "clean": 76, "option": 76, "unit": 77, "pytest": 77, "structur": 77, "warn": 77, "mock": 77, "parametris": 77, "coverag": 77, "chispa": 77, "check": 77, "equal": 77, "testthat": 77}, "envversion": {"sphinx.domains.c": 2, "sphinx.domains.changeset": 1, "sphinx.domains.citation": 1, "sphinx.domains.cpp": 6, "sphinx.domains.index": 1, "sphinx.domains.javascript": 2, "sphinx.domains.math": 2, "sphinx.domains.python": 3, "sphinx.domains.rst": 2, "sphinx.domains.std": 2, "sphinx.ext.intersphinx": 1, "sphinx.ext.todo": 2, "sphinxcontrib.bibtex": 9, "sphinx": 56}}) \ No newline at end of file diff --git a/spark-analysis/bin-continuous-variable.html b/spark-analysis/bin-continuous-variable.html index 6191f536..3be74523 100644 --- a/spark-analysis/bin-continuous-variable.html +++ b/spark-analysis/bin-continuous-variable.html @@ -238,6 +238,7 @@

Analysis in Spark

Analysis in Spark

    +
  • Interpolation in Spark
  • Logistic Regression
  • Spark and Visualisation
  • Data binning
    • diff --git a/spark-analysis/interpolation.html b/spark-analysis/interpolation.html new file mode 100644 index 00000000..dbe5b94a --- /dev/null +++ b/spark-analysis/interpolation.html @@ -0,0 +1,1046 @@ + + + + + + + + + + + + Interpolation in Spark — Spark at the ONS + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    Book version 2022.6

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    Interpolation in Spark

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    + +
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    + +
    +

    Interpolation in Spark#

    +

    In this article, we will impute some missing values in time series data. The PySpark guidance below has been adapted from another source.

    +

    Please note that there are many imputation methods to choose from, for more information see the Awareness in Editing and Imputation course on the Learning Hub. Before using this code in your work please consult other team members or methodologists on the suitability of this method for your use case.

    +

    We don’t want to spend too much time on the methodology here, the purpose is to give you inspiration on how to solve your own PySpark/SparklyR problem and maybe learn about some new functions.

    +

    It’s also worth mentioning that there is an interpolate function in pandas, so if your DataFrame is small enough just use pandas!

    +
    +

    What is interpolation?#

    +

    Wikipedia says “In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing new data points based on the range of a discrete set of known data points.”

    +

    The DataFrame we will create consists of the columns area_code, period and count. To impute the missing count values we will use linear interpolation, which means to draw a straight line between two known points and use this gradient line to fill in the missing values.

    +

    The linear interpolation equation we will use is:

    +
    +\[ +y = y_1 + m(x - x_1) +\]
    +

    where

    +
    +\[ +m = \frac{(y_2 - y_1)}{(x_2 - x_1)} +\]
    +

    is the gradient between points 1 and 2,
    +\(y\) = linear interpolation value (count in this example) between \(y_1\) and \(y_2\),
    +\(x\) = independent variable (period in this example) between \(x_1\) and \(x_2\),
    +\(x_1, y_1\) = known values at point 1,
    +\(x_2, y_2\) = known values at point 2.

    +
    +
    +

    Create DataFrame and plot#

    +

    Let’s start with some imports and creating a spark session

    +
    +
    from pyspark.sql import SparkSession
+from pyspark.sql import functions as F
+from pyspark.sql.window import Window
+import pandas as pd
+import datetime
+
+spark = (
+    SparkSession.builder.master("local[2]").appName("interpolate")
+    .getOrCreate()
+)
+
    +
    +
    +

    As mentioned previously we will create three columns, area_code, period and count. Note that count contains some missing values to impute and period contains some mixed frequency points (points with differing time gaps between them) to make it more interesting.

    +
    +
    df = spark.createDataFrame([
+    ["A", "20210101", 100],
+    ["A", "20210106", None],
+    ["A", "20210111", None],
+    ["A", "20210116", 130],
+    ["A", "20210121", 100],
+    ["A", "20210126", 120],
+    ["A", "20210131", None],
+    ["A", "20210205", 130],
+    ["B", "20210101", None],
+    ["B", "20210111", 85],
+    ["B", "20210119", 82],
+    ["B", "20210131", 75],
+    ["B", "20210210", None],
+    ["B", "20210215", 85],
+    ["B", "20210220", None],
+    ["B", "20210225", 75]
+    ],
+    ["area_code", "period", "count"])
+
    +
    +
    +

    Let’s bring the dataframe into pandas/our local R session ready to make some plots:

    +
    +
    pdf = df.toPandas()
+pdf.set_index(pd.to_datetime(pdf["period"]), inplace=True)
+
    +
    +
    +

    Now we can take a quick look at some charts to see what this looks like:

    +
    +
    pdf[pdf["area_code"]=="A"].plot(marker="o")
+pdf[pdf["area_code"]=="B"].plot(marker="o")
+
    +
    +
    +
    +Chart showing value of count for area "A" above and area "B" below. Values are represented by dots and these are joined up by a line. Some values are not joined up to the line due to missing values in between points. +
    +
    +Chart showing value of count for area "A" above and area "B" below. Values are represented by dots and these are joined up by a line. Some values are not joined up to the line due to missing values in between points. +
    +

    Fig. 40 Plot of pdf in Python split by area code “A” (top) and area code “B” (bottom).#

    +
    +
    +
    +
    ggplot(df, mapping = aes(x = period, y = count)) +
+  geom_point() +
+  geom_line() +
+  facet_wrap(~area_code, nrow = 2)
+
    +
    +
    +
    +Chart showing value of count for area "A" above and area "B" below. Values are represented by dots and these are joined up by a line. Some values are not joined up to the line due to missing values in between points. +
    +

    Fig. 41 Plot of df in R split by area code “A” (top) and area code “B” (bottom).#

    +
    +
    +

    As you can see, the lines don’t meet all the points because there are missing counts in between.

    +
    +
    +

    Interpolate#

    +

    To carry out the interpolation, we will first create new columns containing forward filled and backward filled counts and periods where the count is missing. Then, we will use these columns in the formula above to calculate the gradient and interpolated counts.

    +

    Here are the steps in more detail:

    +
      +

    1. Add timestamp columns

      2. +

    2. Create ordered windows for forward and backward filling

      3. +

    3. Carry out the forward and backward fills on count and timestamp

      4. +

    4. Calculate the gradient

      5. +

    5. Calculate the interpolated counts

      6. +
    +

    In practice there is often some cleaning to do with dates, here we just have a string in the format YYYYMMDD.

    +

    Before we impute any values we will create a flag to indicate an imputed value for the end user.

    +
    +
    df = df.withColumn("impute_flag",
+                   F.when(F.col("count").isNull(), 1)
+                   .otherwise(0))
+
    +
    +
    +

    Next we will need to create some new timestamp columns. The function unix_timestamp() will convert the period column into seconds since 1st of January 1970. We need one version of the column with missing timestamps corresponding to the missing counts, and another version including all the periods.

    +
    +
    # Add timestamp for data we have only. This will be used to interpolate
+df = df.withColumn("timestamp_with_nulls",
+                   F.when(F.col("impute_flag")!=1,
+                          F.unix_timestamp(F.col("period"), "yyyyMMdd"))
+                   .otherwise(None))
+
+# Add timestamp for all data
+df = df.withColumn("timestamp_all",
+                  F.unix_timestamp(F.col("period"), "yyyyMMdd")
+                  )
+
    +
    +
    +

    Next we create the Window functions to forward fill and backward fill the missing values in the count and timestamp_with_nulls columns. Note that for SparklyR we can skip this step, as we can just use group_by() and mutate() in the forward and back filling step to specify our window.

    +
    +
    # Create window for forward filling
+window_ff = (
+    Window
+    .partitionBy("area_code")
+    .orderBy("period")
+    .rangeBetween(Window.unboundedPreceding, 0)
+) 
+
+# Create window for backward filling
+window_bf = (
+    Window
+    .partitionBy('area_code')
+    .orderBy("period")
+    .rangeBetween(0, Window.unboundedFollowing)
+) 
+
    +
    +
    +

    To forward and backward fill the missing values we use the F.last() and F.first() functions in PySpark, respectively. These will grab the preceding and following values of a missing value within our defined windows. Once assembled we can put them in .withColumn() to add the forward and backward fill columns.

    +

    For SparklyR, we will use the fill function with the "up" (backward filling) and "down" (forward filling) direction arguments.

    +
    +
    # create the series containing the filled values
+count_last = F.last(F.col("count"), ignorenulls=True).over(window_ff)
+count_first = F.first(F.col("count"), ignorenulls=True).over(window_bf)
+
+period_last = F.last(F.col("timestamp_with_nulls"), ignorenulls=True).over(window_ff)
+period_first = F.first(F.col("timestamp_with_nulls"), ignorenulls=True).over(window_bf)
+
+# add the columns to the dataframe
+df = (
+    df.withColumn('count_ff', count_last)
+    .withColumn('count_bf', count_first)
+    .withColumn('period_ff', period_last)
+    .withColumn('period_bf', period_first)
+)
+
+df.show()
+
    +
    +
    +
    +
        +---------+--------+-----+-----------+--------------------+-------------+--------+--------+----------+----------+
+    |area_code|  period|count|impute_flag|timestamp_with_nulls|timestamp_all|count_ff|count_bf| period_ff| period_bf|
+    +---------+--------+-----+-----------+--------------------+-------------+--------+--------+----------+----------+
+    |        B|20210101| null|          1|                null|   1609459200|    null|      85|      null|1610323200|
+    |        B|20210111|   85|          0|          1610323200|   1610323200|      85|      85|1610323200|1610323200|
+    |        B|20210119|   82|          0|          1611014400|   1611014400|      82|      82|1611014400|1611014400|
+    |        B|20210131|   75|          0|          1612051200|   1612051200|      75|      75|1612051200|1612051200|
+    |        B|20210210| null|          1|                null|   1612915200|      75|      85|1612051200|1613347200|
+    |        B|20210215|   85|          0|          1613347200|   1613347200|      85|      85|1613347200|1613347200|
+    |        B|20210220| null|          1|                null|   1613779200|      85|      75|1613347200|1614211200|
+    |        B|20210225|   75|          0|          1614211200|   1614211200|      75|      75|1614211200|1614211200|
+    |        A|20210101|  100|          0|          1609459200|   1609459200|     100|     100|1609459200|1609459200|
+    |        A|20210106| null|          1|                null|   1609891200|     100|     130|1609459200|1610755200|
+    |        A|20210111| null|          1|                null|   1610323200|     100|     130|1609459200|1610755200|
+    |        A|20210116|  130|          0|          1610755200|   1610755200|     130|     130|1610755200|1610755200|
+    |        A|20210121|  100|          0|          1611187200|   1611187200|     100|     100|1611187200|1611187200|
+    |        A|20210126|  120|          0|          1611619200|   1611619200|     120|     120|1611619200|1611619200|
+    |        A|20210131| null|          1|                null|   1612051200|     120|     130|1611619200|1612483200|
+    |        A|20210205|  130|          0|          1612483200|   1612483200|     130|     130|1612483200|1612483200|
+    +---------+--------+-----+-----------+--------------------+-------------+--------+--------+----------+----------+
+
    +
    +

    We’re now ready to calculate the gradient between the known points and interpolate.

    +
    +
    # Create a gradient column
+df = df.withColumn("gradient",
+                   F.when(F.col("impute_flag")==1,
+                          (F.col("count_bf") - F.col("count_ff")) /
+                          (F.col("period_bf") - F.col("period_ff")))
+                   .otherwise(0)
+                  )
+
+# Create the imputed values
+df = df.withColumn("count_final",
+                  F.when(F.col("impute_flag")==1,
+                        F.col("count_ff") +
+                        F.col("gradient") *
+                        (F.col("timestamp_all") - F.col("period_ff"))
+                        )
+                   .otherwise(F.col("count"))
+                  )
+
    +
    +
    +
    +
    +
    +

    View the results#

    +

    To view the results we will select the columns of interest and convert to pandas, or collect into a local R dataframe.

    +
    +
    pdf = df.select("area_code", "period", "count", "impute_flag", "count_final").toPandas()
+pdf = pdf.set_index(pd.to_datetime(pdf["period"])).drop("period", axis=1)
+pdf
+
    +
    +
    +
    +
    period     area_code	count	impute_flag	count_final
+				
+2021-01-01	B	        NaN	   1	        NaN
+2021-01-11	B	        85.0	 0	       85.000000
+2021-01-19	B	        82.0	 0	       82.000000
+2021-01-31	B	        75.0	 0	       75.000000
+2021-02-10	B	        NaN	   1	       81.666667
+2021-02-15	B	        85.0	 0	       85.000000
+2021-02-20	B	        NaN	   1	       80.000000
+2021-02-25	B	        75.0	 0	       75.000000
+2021-01-01	A	        100.0	 0	       100.000000
+2021-01-06	A	        NaN	   1	       110.000000
+2021-01-11	A	        NaN	   1	       120.000000
+2021-01-16	A	        130.0	 0	       130.000000
+2021-01-21	A	        100.0	 0	       100.000000
+2021-01-26	A	        120.0	 0	       120.000000
+2021-01-31	A	        NaN	   1	       125.000000
+2021-02-05	A	        130.0	 0	       130.000000
+
    +
    +
    +

    Note that the first value for area_code B could not be interpolated because there is no preceding count available. In other words, we would need to extrapolate to estimate for this value.

    +

    Once again, we will plot the series with pandas/ggplot. The original data are plotted as red circles and the final series in blue.

    +
    +
    pdf[pdf["area_code"] == "A"].drop("impute_flag", axis=1).plot(style={"count":"ro", "count_final":"b+-"}, alpha=0.5)
+pdf[pdf["area_code"] == "B"].drop("impute_flag", axis=1).plot(style={"count":"ro", "count_final":"b+-"}, alpha=0.5)
+
    +
    +
    +
    +Chart showing value of count for area "A" above and area "B" below. All values are now connected by a line. +
    +
    +Chart showing value of count for area "A" above and area "B" below. All values are now connected by a line. +
    +

    Fig. 42 Interpolated plot of pdf in Python split by area code “A” (top) and area code “B” (bottom).#

    +
    +
    +
    +
    ggplot(df, mapping = aes(x = period, y = count_final)) +
+  geom_point(aes(colour = factor(impute_flag))) +
+  scale_fill_manual(values = c("red", "blue")) +
+  geom_line() +
+  labs(colour = "impute_flag") +
+  facet_wrap(~area_code, nrow = 2)
+
    +
    +
    +
    +Chart showing value of count for area "A" above and area "B" below. All values are now connected by a line. +
    +

    Fig. 43 Interpolated plot of df in R split by area code “A” (top) and area code “B” (bottom).#

    +
    +
    +
    +
    +

    Further resources#

    +

    Spark at the ONS articles:

    + +

    Other resources:

    + +
    +
    + + + + +
    + + + + + + + + +
    + + + +
    + +
    +
    + Contents +
    +
    + +
    + + +
    +
    + +
    + +
    + +

    +By Analysis Standards and Pipelines in Quality and Improvement, Office for National Statistics +

    + +
    + +
    + + +
    + +
    + +
    + +
    + +
    + +
    +
    + + +
    +
    +
    + + + + + +
    +
    + + \ No newline at end of file diff --git a/spark-analysis/logistic-regression.html b/spark-analysis/logistic-regression.html index 5459b0ac..0ec191dc 100644 --- a/spark-analysis/logistic-regression.html +++ b/spark-analysis/logistic-regression.html @@ -76,7 +76,7 @@ - + @@ -238,6 +238,7 @@

Analysis in Spark