# MLlib: Basic Statistics and Exploratory Data Analysis¶

So far we have used different map and aggregation functions, on simple and key/value pair RDD's, in order to get simple statistics that help us understand our datasets. In this notebook we will introduce Spark's machine learning library MLlib through its basic statistics functionality in order to better understand our dataset. We will use the reduced 10-percent KDD Cup 1999 datasets through the notebook.

## Getting the data and creating the RDD¶

As we did in our first notebook, we will use the reduced dataset (10 percent) provided for the KDD Cup 1999, containing nearly half million network interactions. The file is provided as a Gzip file that we will download locally.

In [1]:
import urllib
f = urllib.urlretrieve ("http://kdd.ics.uci.edu/databases/kddcup99/kddcup.data_10_percent.gz", "kddcup.data_10_percent.gz")

In [2]:
data_file = "./kddcup.data_10_percent.gz"
raw_data = sc.textFile(data_file)


## Local vectors¶

A local vector is often used as a base type for RDDs in Spark MLlib. A local vector has integer-typed and 0-based indices and double-typed values, stored on a single machine. MLlib supports two types of local vectors: dense and sparse. A dense vector is backed by a double array representing its entry values, while a sparse vector is backed by two parallel arrays: indices and values.

For dense vectors, MLlib uses either Python lists or the NumPy array type. The later is recommended, so you can simply pass NumPy arrays around.

For sparse vectors, users can construct a SparseVector object from MLlib or pass SciPy scipy.sparse column vectors if SciPy is available in their environment. The easiest way to create sparse vectors is to use the factory methods implemented in Vectors.

### An RDD of dense vectors¶

Let's represent each network interaction in our dataset as a dense vector. For that we will use the NumPy array type.

In [3]:
import numpy as np

def parse_interaction(line):
line_split = line.split(",")
# keep just numeric and logical values
symbolic_indexes = [1,2,3,41]
clean_line_split = [item for i,item in enumerate(line_split) if i not in symbolic_indexes]
return np.array([float(x) for x in clean_line_split])

vector_data = raw_data.map(parse_interaction)


## Summary statistics¶

Spark's MLlib provides column summary statistics for RDD[Vector] through the function colStats available in Statistics. The method returns an instance of MultivariateStatisticalSummary, which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the total count.

In [4]:
from pyspark.mllib.stat import Statistics
from math import sqrt

# Compute column summary statistics.
summary = Statistics.colStats(vector_data)

print "Duration Statistics:"
print " Mean: {}".format(round(summary.mean()[0],3))
print " St. deviation: {}".format(round(sqrt(summary.variance()[0]),3))
print " Max value: {}".format(round(summary.max()[0],3))
print " Min value: {}".format(round(summary.min()[0],3))
print " Total value count: {}".format(summary.count())
print " Number of non-zero values: {}".format(summary.numNonzeros()[0])

Duration Statistics:
Mean: 47.979
St. deviation: 707.746
Max value: 58329.0
Min value: 0.0
Total value count: 494021
Number of non-zero values: 12350.0


### Summary statistics by label¶

The interesting part of summary statistics, in our case, comes from being able to obtain them by the type of network attack or 'label' in our dataset. By doing so we will be able to better characterise our dataset dependent variable in terms of the independent variables range of values.

If we want to do such a thing we could filter our RDD containing labels as keys and vectors as values. For that we just need to adapt our parse_interaction function to return a tuple with both elements.

In [5]:
def parse_interaction_with_key(line):
line_split = line.split(",")
# keep just numeric and logical values
symbolic_indexes = [1,2,3,41]
clean_line_split = [item for i,item in enumerate(line_split) if i not in symbolic_indexes]
return (line_split[41], np.array([float(x) for x in clean_line_split]))

label_vector_data = raw_data.map(parse_interaction_with_key)


The next step is not very sophisticated. We use filter on the RDD to leave out other labels but the one we want to gather statistics from.

In [6]:
normal_label_data = label_vector_data.filter(lambda x: x[0]=="normal.")


Now we can use the new RDD to call colStats on the values.

In [7]:
normal_summary = Statistics.colStats(normal_label_data.values())


And collect the results as we did before.

In [8]:
print "Duration Statistics for label: {}".format("normal")
print " Mean: {}".format(normal_summary.mean()[0],3)
print " St. deviation: {}".format(round(sqrt(normal_summary.variance()[0]),3))
print " Max value: {}".format(round(normal_summary.max()[0],3))
print " Min value: {}".format(round(normal_summary.min()[0],3))
print " Total value count: {}".format(normal_summary.count())
print " Number of non-zero values: {}".format(normal_summary.numNonzeros()[0])

Duration Statistics for label: normal
Mean: 216.657322313
St. deviation: 1359.213
Max value: 58329.0
Min value: 0.0
Total value count: 97278
Number of non-zero values: 11690.0


Instead of working with a key/value pair we could have just filter our raw data split using the label in column 41. Then we can parse the results as we did before. This will work as well. However having our data organised as key/value pairs will open the door to better manipulations. Since values() is a transformation on an RDD, and not an action, we don't perform any computation until we call colStats anyway.

But lets wrap this within a function so we can reuse it with any label.

In [9]:
def summary_by_label(raw_data, label):
label_vector_data = raw_data.map(parse_interaction_with_key).filter(lambda x: x[0]==label)
return Statistics.colStats(label_vector_data.values())


Let's give it a try with the "normal." label again.

In [10]:
normal_sum = summary_by_label(raw_data, "normal.")

print "Duration Statistics for label: {}".format("normal")
print " Mean: {}".format(normal_sum.mean()[0],3)
print " St. deviation: {}".format(round(sqrt(normal_sum.variance()[0]),3))
print " Max value: {}".format(round(normal_sum.max()[0],3))
print " Min value: {}".format(round(normal_sum.min()[0],3))
print " Total value count: {}".format(normal_sum.count())
print " Number of non-zero values: {}".format(normal_sum.numNonzeros()[0])

Duration Statistics for label: normal
Mean: 216.657322313
St. deviation: 1359.213
Max value: 58329.0
Min value: 0.0
Total value count: 97278
Number of non-zero values: 11690.0


Let's try now with some network attack. We have all of them listed here.

In [11]:
guess_passwd_summary = summary_by_label(raw_data, "guess_passwd.")

print "Duration Statistics for label: {}".format("guess_password")
print " Mean: {}".format(guess_passwd_summary.mean()[0],3)
print " St. deviation: {}".format(round(sqrt(guess_passwd_summary.variance()[0]),3))
print " Max value: {}".format(round(guess_passwd_summary.max()[0],3))
print " Min value: {}".format(round(guess_passwd_summary.min()[0],3))
print " Total value count: {}".format(guess_passwd_summary.count())
print " Number of non-zero values: {}".format(guess_passwd_summary.numNonzeros()[0])

Duration Statistics for label: guess_password
Mean: 2.71698113208
St. deviation: 11.88
Max value: 60.0
Min value: 0.0
Total value count: 53
Number of non-zero values: 4.0


We can see that this type of attack is shorter in duration than a normal interaction. We could build a table with duration statistics for each type of interaction in our dataset. First we need to get a list of labels as described in the first line here.

In [12]:
label_list = ["back.","buffer_overflow.","ftp_write.","guess_passwd.",
"neptune.","nmap.","normal.","perl.","phf.","pod.","portsweep.",
"rootkit.","satan.","smurf.","spy.","teardrop.","warezclient.",
"warezmaster."]


Then we get a list of statistics for each label.

In [13]:
stats_by_label = [(label, summary_by_label(raw_data, label)) for label in label_list]


Now we get the duration column, first in our dataset (i.e. index 0).

In [14]:
duration_by_label = [
(stat[0], np.array([float(stat[1].mean()[0]), float(sqrt(stat[1].variance()[0])), float(stat[1].min()[0]), float(stat[1].max()[0]), int(stat[1].count())]))
for stat in stats_by_label]


That we can put into a Pandas data frame.

In [15]:
import pandas as pd
pd.set_option('display.max_columns', 50)

stats_by_label_df = pd.DataFrame.from_items(duration_by_label, columns=["Mean", "Std Dev", "Min", "Max", "Count"], orient='index')


And print it.

In [16]:
print "Duration statistics, by label"
stats_by_label_df

Duration statistics, by label

Out[16]:
Mean Std Dev Min Max Count
back. 0.128915 1.110062 0 14 2203
buffer_overflow. 91.700000 97.514685 0 321 30
ftp_write. 32.375000 47.449033 0 134 8
guess_passwd. 2.716981 11.879811 0 60 53
imap. 6.000000 14.174240 0 41 12
ipsweep. 0.034483 0.438439 0 7 1247
land. 0.000000 0.000000 0 0 21
loadmodule. 36.222222 41.408869 0 103 9
multihop. 184.000000 253.851006 0 718 7
neptune. 0.000000 0.000000 0 0 107201
nmap. 0.000000 0.000000 0 0 231
normal. 216.657322 1359.213469 0 58329 97278
perl. 41.333333 14.843629 25 54 3
phf. 4.500000 5.744563 0 12 4
pod. 0.000000 0.000000 0 0 264
portsweep. 1915.299038 7285.125159 0 42448 1040
rootkit. 100.800000 216.185003 0 708 10
satan. 0.040277 0.522433 0 11 1589
smurf. 0.000000 0.000000 0 0 280790
spy. 318.000000 26.870058 299 337 2
teardrop. 0.000000 0.000000 0 0 979
warezclient. 615.257843 2207.694966 0 15168 1020
warezmaster. 15.050000 33.385271 0 156 20

In order to reuse this code and get a dataframe from any variable in our dataset we will define a function.

In [17]:
def get_variable_stats_df(stats_by_label, column_i):
column_stats_by_label = [
(stat[0], np.array([float(stat[1].mean()[column_i]), float(sqrt(stat[1].variance()[column_i])), float(stat[1].min()[column_i]), float(stat[1].max()[column_i]), int(stat[1].count())]))
for stat in stats_by_label
]
return pd.DataFrame.from_items(column_stats_by_label, columns=["Mean", "Std Dev", "Min", "Max", "Count"], orient='index')


Let's try for duration again.

In [18]:
get_variable_stats_df(stats_by_label,0)

Out[18]:
Mean Std Dev Min Max Count
back. 0.128915 1.110062 0 14 2203
buffer_overflow. 91.700000 97.514685 0 321 30
ftp_write. 32.375000 47.449033 0 134 8
guess_passwd. 2.716981 11.879811 0 60 53
imap. 6.000000 14.174240 0 41 12
ipsweep. 0.034483 0.438439 0 7 1247
land. 0.000000 0.000000 0 0 21
loadmodule. 36.222222 41.408869 0 103 9
multihop. 184.000000 253.851006 0 718 7
neptune. 0.000000 0.000000 0 0 107201
nmap. 0.000000 0.000000 0 0 231
normal. 216.657322 1359.213469 0 58329 97278
perl. 41.333333 14.843629 25 54 3
phf. 4.500000 5.744563 0 12 4
pod. 0.000000 0.000000 0 0 264
portsweep. 1915.299038 7285.125159 0 42448 1040
rootkit. 100.800000 216.185003 0 708 10
satan. 0.040277 0.522433 0 11 1589
smurf. 0.000000 0.000000 0 0 280790
spy. 318.000000 26.870058 299 337 2
teardrop. 0.000000 0.000000 0 0 979
warezclient. 615.257843 2207.694966 0 15168 1020
warezmaster. 15.050000 33.385271 0 156 20

Now for the next numeric column in the dataset, src_bytes.

In [19]:
print "src_bytes statistics, by label"
get_variable_stats_df(stats_by_label,1)

src_bytes statistics, by label

Out[19]:
Mean Std Dev Min Max Count
back. 54156.355878 3159.360232 13140 54540 2203
buffer_overflow. 1400.433333 1337.132616 0 6274 30
ftp_write. 220.750000 267.747616 0 676 8
guess_passwd. 125.339623 3.037860 104 126 53
imap. 347.583333 629.926036 0 1492 12
ipsweep. 10.083400 5.231658 0 18 1247
land. 0.000000 0.000000 0 0 21
loadmodule. 151.888889 127.745298 0 302 9
multihop. 435.142857 540.960389 0 1412 7
neptune. 0.000000 0.000000 0 0 107201
nmap. 24.116883 59.419871 0 207 231
normal. 1157.047524 34226.124718 0 2194619 97278
perl. 265.666667 4.932883 260 269 3
phf. 51.000000 0.000000 51 51 4
pod. 1462.651515 125.098044 564 1480 264
portsweep. 666707.436538 21500665.866700 0 693375640 1040
rootkit. 294.700000 538.578180 0 1727 10
satan. 1.337319 42.946200 0 1710 1589
smurf. 935.772300 200.022386 520 1032 280790
spy. 174.500000 88.388348 112 237 2
teardrop. 28.000000 0.000000 28 28 979
warezclient. 300219.562745 1200905.243130 30 5135678 1020
warezmaster. 49.300000 212.155132 0 950 20

And so on. By reusing the summary_by_label and get_variable_stats_df functions we can perform some exploratory data analysis in large datasets with Spark.

## Correlations¶

Spark's MLlib supports Pearsonâ€™s and Spearmanâ€™s to calculate pairwise correlation methods among many series. Both of them are provided by the corr method in the Statistics package.

We have two options as input. Either two RDD[Double]s or an RDD[Vector]. In the first case the output will be a Double value, while in the second a whole correlation Matrix. Due to the nature of our data, we will obtain the second.

In [20]:
from pyspark.mllib.stat import Statistics
correlation_matrix = Statistics.corr(vector_data, method="spearman")


Once we have the correlations ready, we can start inspecting their values.

In [21]:
import pandas as pd
pd.set_option('display.max_columns', 50)

col_names = ["duration","src_bytes","dst_bytes","land","wrong_fragment",
"root_shell","su_attempted","num_root","num_file_creations",
"num_shells","num_access_files","num_outbound_cmds",
"srv_serror_rate","rerror_rate","srv_rerror_rate","same_srv_rate",
"diff_srv_rate","srv_diff_host_rate","dst_host_count","dst_host_srv_count",
"dst_host_same_srv_rate","dst_host_diff_srv_rate","dst_host_same_src_port_rate",
"dst_host_srv_diff_host_rate","dst_host_serror_rate","dst_host_srv_serror_rate",
"dst_host_rerror_rate","dst_host_srv_rerror_rate"]

corr_df = pd.DataFrame(correlation_matrix, index=col_names, columns=col_names)

corr_df

Out[21]:
duration src_bytes dst_bytes land wrong_fragment urgent hot num_failed_logins logged_in num_compromised root_shell su_attempted num_root num_file_creations num_shells num_access_files num_outbound_cmds is_hot_login is_guest_login count srv_count serror_rate srv_serror_rate rerror_rate srv_rerror_rate same_srv_rate diff_srv_rate srv_diff_host_rate dst_host_count dst_host_srv_count dst_host_same_srv_rate dst_host_diff_srv_rate dst_host_same_src_port_rate dst_host_srv_diff_host_rate dst_host_serror_rate dst_host_srv_serror_rate dst_host_rerror_rate dst_host_srv_rerror_rate
duration 1.000000 0.014196 0.299189 -0.001068 -0.008025 0.017883 0.108639 0.014363 0.159564 0.010687 0.040425 0.026015 0.013401 0.061099 0.008632 0.019407 -0.000019 -0.000010 0.205606 -0.259032 -0.250139 -0.074211 -0.073663 -0.025936 -0.026420 0.062291 -0.050875 0.123621 -0.161107 -0.217167 -0.211979 0.231644 -0.065202 0.100692 -0.056753 -0.057298 -0.007759 -0.013891
src_bytes 0.014196 1.000000 -0.167931 -0.009404 -0.019358 0.000094 0.113920 -0.008396 -0.089702 0.118562 0.003067 0.002282 -0.002050 0.027710 0.014403 -0.001497 0.000010 0.000019 0.027511 0.666230 0.722609 -0.657460 -0.652391 -0.342180 -0.332977 0.744046 -0.739988 -0.104042 0.130377 0.741979 0.729151 -0.712965 0.815039 -0.140231 -0.645920 -0.641792 -0.297338 -0.300581
dst_bytes 0.299189 -0.167931 1.000000 -0.003040 -0.022659 0.007234 0.193156 0.021952 0.882185 0.169772 0.026054 0.012192 -0.003884 0.034154 -0.000054 0.065776 -0.000031 0.000041 0.085947 -0.639157 -0.497683 -0.205848 -0.198715 -0.100958 -0.081307 0.229677 -0.222572 0.521003 -0.611972 0.024124 0.055033 -0.035073 -0.396195 0.578557 -0.167047 -0.158378 -0.003042 0.001621
land -0.001068 -0.009404 -0.003040 1.000000 -0.000333 -0.000065 -0.000539 -0.000076 -0.002785 -0.000447 -0.000093 -0.000049 -0.000230 -0.000150 -0.000076 -0.000211 -0.002881 0.002089 -0.000250 -0.010939 -0.010128 0.014160 0.014342 -0.000451 -0.001690 0.002153 -0.001846 0.020678 -0.019923 -0.012341 0.002576 -0.001803 0.004265 0.016171 0.013566 0.012265 0.000389 -0.001816
wrong_fragment -0.008025 -0.019358 -0.022659 -0.000333 1.000000 -0.000150 -0.004042 -0.000568 -0.020911 -0.003370 -0.000528 -0.000248 -0.001727 -0.001160 -0.000507 -0.001519 -0.000147 0.000441 -0.001869 -0.057711 -0.029117 -0.008849 -0.023382 0.000430 -0.012676 0.010218 -0.009386 0.012117 -0.029149 -0.058225 -0.049560 0.055542 -0.015449 0.007306 0.010387 -0.024117 0.046656 -0.013666
urgent 0.017883 0.000094 0.007234 -0.000065 -0.000150 1.000000 0.008594 0.063009 0.006821 0.031765 0.067437 0.000020 0.061994 0.061383 -0.000066 0.023380 0.012879 0.005162 -0.000100 -0.004778 -0.004799 -0.001338 -0.001327 -0.000705 -0.000726 0.001521 -0.001522 -0.000788 -0.005894 -0.005698 -0.004078 0.005208 -0.001939 -0.000976 -0.001381 -0.001370 -0.000786 -0.000782
hot 0.108639 0.113920 0.193156 -0.000539 -0.004042 0.008594 1.000000 0.112560 0.189126 0.811529 0.101983 -0.000400 0.003096 0.028694 0.009146 0.004224 -0.000393 -0.000248 0.463706 -0.120847 -0.114735 -0.035487 -0.034934 0.013468 0.052003 0.041342 -0.040555 0.032141 -0.074178 -0.017960 0.018783 -0.017198 -0.086998 -0.014141 -0.004706 -0.010721 0.199019 0.189142
num_failed_logins 0.014363 -0.008396 0.021952 -0.000076 -0.000568 0.063009 0.112560 1.000000 -0.002190 0.004619 0.016895 0.072748 0.010060 0.015211 -0.000093 0.005581 0.003431 -0.001560 -0.000428 -0.018024 -0.018027 -0.003674 -0.004027 0.035324 0.034876 0.005716 -0.005538 -0.003096 -0.028369 -0.015092 0.003004 -0.002960 -0.006617 -0.002588 0.014713 0.014914 0.032395 0.032151
logged_in 0.159564 -0.089702 0.882185 -0.002785 -0.020911 0.006821 0.189126 -0.002190 1.000000 0.161190 0.025293 0.011813 0.082533 0.055530 0.024354 0.072698 0.000079 0.000127 0.089318 -0.578287 -0.438947 -0.187114 -0.180122 -0.091962 -0.072287 0.216969 -0.214019 0.503807 -0.682721 0.080352 0.114526 -0.093565 -0.359506 0.659078 -0.143283 -0.132474 0.007236 0.012979
num_compromised 0.010687 0.118562 0.169772 -0.000447 -0.003370 0.031765 0.811529 0.004619 0.161190 1.000000 0.085558 0.048985 0.028557 0.031223 0.011256 0.006977 0.001048 -0.000438 -0.002504 -0.097212 -0.091154 -0.030516 -0.030264 0.008573 0.054006 0.035253 -0.034953 0.036497 -0.041615 0.003465 0.038980 -0.039091 -0.078843 -0.020979 -0.005019 -0.004504 0.214115 0.217858
root_shell 0.040425 0.003067 0.026054 -0.000093 -0.000528 0.067437 0.101983 0.016895 0.025293 0.085558 1.000000 0.233486 0.094512 0.140650 0.132056 0.069353 0.011462 -0.006602 -0.000405 -0.016409 -0.015174 -0.004952 -0.004923 -0.001104 -0.001143 0.004946 -0.004553 0.002286 -0.021367 -0.011906 0.000515 -0.000916 -0.004617 0.008631 -0.003498 -0.003032 0.002763 0.002151
su_attempted 0.026015 0.002282 0.012192 -0.000049 -0.000248 0.000020 -0.000400 0.072748 0.011813 0.048985 0.233486 1.000000 0.119326 0.053110 0.040487 0.081272 -0.018896 0.012927 -0.000219 -0.008279 -0.008225 -0.002318 -0.002295 -0.001227 -0.001253 0.002634 -0.002649 0.000348 -0.006697 -0.006288 -0.005738 0.006687 -0.005020 0.001052 0.001974 0.002893 0.003173 0.001731
num_root 0.013401 -0.002050 -0.003884 -0.000230 -0.001727 0.061994 0.003096 0.010060 0.082533 0.028557 0.094512 0.119326 1.000000 0.047521 0.034405 0.014513 0.001524 -0.002585 -0.001281 -0.054721 -0.053530 -0.016031 -0.015936 -0.008610 -0.008708 0.013881 -0.011337 0.006316 -0.078717 -0.038689 -0.038935 0.047414 -0.015968 0.061030 -0.008457 -0.007096 -0.000421 -0.005012
num_file_creations 0.061099 0.027710 0.034154 -0.000150 -0.001160 0.061383 0.028694 0.015211 0.055530 0.031223 0.140650 0.053110 0.047521 1.000000 0.068660 0.031042 -0.004081 -0.001664 0.013242 -0.036467 -0.034598 -0.009703 -0.010390 -0.005069 -0.004775 0.009784 -0.008711 0.014412 -0.049529 -0.026890 -0.021731 0.027092 -0.015018 0.030590 -0.002257 -0.004295 0.000626 -0.001096
num_shells 0.008632 0.014403 -0.000054 -0.000076 -0.000507 -0.000066 0.009146 -0.000093 0.024354 0.011256 0.132056 0.040487 0.034405 0.068660 1.000000 0.019438 -0.002592 -0.006631 -0.000405 -0.013938 -0.011784 -0.004343 -0.004740 -0.002541 -0.002572 0.004282 -0.003743 0.001096 -0.021200 -0.012017 -0.009962 0.010761 -0.003521 0.015882 -0.001588 -0.002357 -0.000617 -0.002020
num_access_files 0.019407 -0.001497 0.065776 -0.000211 -0.001519 0.023380 0.004224 0.005581 0.072698 0.006977 0.069353 0.081272 0.014513 0.031042 0.019438 1.000000 -0.001597 -0.002850 0.002466 -0.045282 -0.040497 -0.013945 -0.013572 -0.007581 0.001874 0.015499 -0.015112 0.024266 -0.023865 -0.023657 -0.021358 0.026703 -0.033288 0.011765 -0.011197 -0.011487 -0.004743 -0.004552
num_outbound_cmds -0.000019 0.000010 -0.000031 -0.002881 -0.000147 0.012879 -0.000393 0.003431 0.000079 0.001048 0.011462 -0.018896 0.001524 -0.004081 -0.002592 -0.001597 1.000000 0.822890 0.000924 -0.000076 0.000100 0.000167 0.000209 0.000536 0.000346 0.000208 0.000328 -0.000141 -0.000424 -0.000280 -0.000503 -0.000181 -0.000455 0.000288 -0.000011 -0.000372 -0.000823 -0.001038
is_hot_login -0.000010 0.000019 0.000041 0.002089 0.000441 0.005162 -0.000248 -0.001560 0.000127 -0.000438 -0.006602 0.012927 -0.002585 -0.001664 -0.006631 -0.002850 0.822890 1.000000 0.001512 0.000036 0.000064 0.000102 -0.000302 -0.000550 0.000457 -0.000159 -0.000235 -0.000360 -0.000106 0.000206 0.000229 -0.000004 0.000283 0.000538 -0.000076 -0.000007 -0.000435 -0.000529
is_guest_login 0.205606 0.027511 0.085947 -0.000250 -0.001869 -0.000100 0.463706 -0.000428 0.089318 -0.002504 -0.000405 -0.000219 -0.001281 0.013242 -0.000405 0.002466 0.000924 0.001512 1.000000 -0.062340 -0.062713 -0.017343 -0.017240 -0.008867 -0.009193 0.018042 -0.017000 -0.008878 -0.055453 -0.044366 -0.041749 0.044640 -0.038092 -0.012578 -0.001066 -0.016885 0.025282 -0.004292
count -0.259032 0.666230 -0.639157 -0.010939 -0.057711 -0.004778 -0.120847 -0.018024 -0.578287 -0.097212 -0.016409 -0.008279 -0.054721 -0.036467 -0.013938 -0.045282 -0.000076 0.000036 -0.062340 1.000000 0.950587 -0.303538 -0.308923 -0.213824 -0.221352 0.346718 -0.361737 -0.384010 0.547443 0.586979 0.539698 -0.546869 0.776906 -0.496554 -0.331571 -0.335290 -0.261194 -0.256176
srv_count -0.250139 0.722609 -0.497683 -0.010128 -0.029117 -0.004799 -0.114735 -0.018027 -0.438947 -0.091154 -0.015174 -0.008225 -0.053530 -0.034598 -0.011784 -0.040497 0.000100 0.000064 -0.062713 0.950587 1.000000 -0.428185 -0.421424 -0.281468 -0.284034 0.517227 -0.511998 -0.239057 0.442611 0.720746 0.681955 -0.673916 0.812280 -0.391712 -0.449096 -0.442823 -0.313442 -0.308132
serror_rate -0.074211 -0.657460 -0.205848 0.014160 -0.008849 -0.001338 -0.035487 -0.003674 -0.187114 -0.030516 -0.004952 -0.002318 -0.016031 -0.009703 -0.004343 -0.013945 0.000167 0.000102 -0.017343 -0.303538 -0.428185 1.000000 0.990888 -0.091157 -0.095285 -0.851915 0.828012 -0.121489 0.165350 -0.724317 -0.745745 0.719708 -0.650336 -0.153568 0.973947 0.965663 -0.103198 -0.105434
srv_serror_rate -0.073663 -0.652391 -0.198715 0.014342 -0.023382 -0.001327 -0.034934 -0.004027 -0.180122 -0.030264 -0.004923 -0.002295 -0.015936 -0.010390 -0.004740 -0.013572 0.000209 -0.000302 -0.017240 -0.308923 -0.421424 0.990888 1.000000 -0.110664 -0.115286 -0.839315 0.815305 -0.112222 0.160322 -0.713313 -0.734334 0.707753 -0.646256 -0.148072 0.967214 0.970617 -0.122630 -0.124656
rerror_rate -0.025936 -0.342180 -0.100958 -0.000451 0.000430 -0.000705 0.013468 0.035324 -0.091962 0.008573 -0.001104 -0.001227 -0.008610 -0.005069 -0.002541 -0.007581 0.000536 -0.000550 -0.008867 -0.213824 -0.281468 -0.091157 -0.110664 1.000000 0.978813 -0.327986 0.345571 -0.017902 -0.067857 -0.330391 -0.303126 0.308722 -0.278465 0.073061 -0.094076 -0.110646 0.910225 0.911622
srv_rerror_rate -0.026420 -0.332977 -0.081307 -0.001690 -0.012676 -0.000726 0.052003 0.034876 -0.072287 0.054006 -0.001143 -0.001253 -0.008708 -0.004775 -0.002572 0.001874 0.000346 0.000457 -0.009193 -0.221352 -0.284034 -0.095285 -0.115286 0.978813 1.000000 -0.316568 0.333439 0.011285 -0.072595 -0.323032 -0.294328 0.300186 -0.282239 0.075178 -0.096146 -0.114341 0.904591 0.914904
same_srv_rate 0.062291 0.744046 0.229677 0.002153 0.010218 0.001521 0.041342 0.005716 0.216969 0.035253 0.004946 0.002634 0.013881 0.009784 0.004282 0.015499 0.000208 -0.000159 0.018042 0.346718 0.517227 -0.851915 -0.839315 -0.327986 -0.316568 1.000000 -0.982109 0.140660 -0.190121 0.848754 0.873551 -0.844537 0.732841 0.179040 -0.830067 -0.819335 -0.282487 -0.282913
diff_srv_rate -0.050875 -0.739988 -0.222572 -0.001846 -0.009386 -0.001522 -0.040555 -0.005538 -0.214019 -0.034953 -0.004553 -0.002649 -0.011337 -0.008711 -0.003743 -0.015112 0.000328 -0.000235 -0.017000 -0.361737 -0.511998 0.828012 0.815305 0.345571 0.333439 -0.982109 1.000000 -0.138293 0.185942 -0.844028 -0.868580 0.850911 -0.727031 -0.176930 0.807205 0.795844 0.299041 0.298904
srv_diff_host_rate 0.123621 -0.104042 0.521003 0.020678 0.012117 -0.000788 0.032141 -0.003096 0.503807 0.036497 0.002286 0.000348 0.006316 0.014412 0.001096 0.024266 -0.000141 -0.000360 -0.008878 -0.384010 -0.239057 -0.121489 -0.112222 -0.017902 0.011285 0.140660 -0.138293 1.000000 -0.445051 0.035010 0.068648 -0.050472 -0.222707 0.433173 -0.097973 -0.092661 0.022585 0.024722
dst_host_count -0.161107 0.130377 -0.611972 -0.019923 -0.029149 -0.005894 -0.074178 -0.028369 -0.682721 -0.041615 -0.021367 -0.006697 -0.078717 -0.049529 -0.021200 -0.023865 -0.000424 -0.000106 -0.055453 0.547443 0.442611 0.165350 0.160322 -0.067857 -0.072595 -0.190121 0.185942 -0.445051 1.000000 0.022731 -0.070448 0.044338 0.189876 -0.918894 0.123881 0.113845 -0.125142 -0.125273
dst_host_srv_count -0.217167 0.741979 0.024124 -0.012341 -0.058225 -0.005698 -0.017960 -0.015092 0.080352 0.003465 -0.011906 -0.006288 -0.038689 -0.026890 -0.012017 -0.023657 -0.000280 0.000206 -0.044366 0.586979 0.720746 -0.724317 -0.713313 -0.330391 -0.323032 0.848754 -0.844028 0.035010 0.022731 1.000000 0.970072 -0.955178 0.769481 0.043668 -0.722607 -0.708392 -0.312040 -0.300787
dst_host_same_srv_rate -0.211979 0.729151 0.055033 0.002576 -0.049560 -0.004078 0.018783 0.003004 0.114526 0.038980 0.000515 -0.005738 -0.038935 -0.021731 -0.009962 -0.021358 -0.000503 0.000229 -0.041749 0.539698 0.681955 -0.745745 -0.734334 -0.303126 -0.294328 0.873551 -0.868580 0.068648 -0.070448 0.970072 1.000000 -0.980245 0.771158 0.107926 -0.742045 -0.725272 -0.278068 -0.264383
dst_host_diff_srv_rate 0.231644 -0.712965 -0.035073 -0.001803 0.055542 0.005208 -0.017198 -0.002960 -0.093565 -0.039091 -0.000916 0.006687 0.047414 0.027092 0.010761 0.026703 -0.000181 -0.000004 0.044640 -0.546869 -0.673916 0.719708 0.707753 0.308722 0.300186 -0.844537 0.850911 -0.050472 0.044338 -0.955178 -0.980245 1.000000 -0.766402 -0.088665 0.719275 0.701149 0.287476 0.271067
dst_host_same_src_port_rate -0.065202 0.815039 -0.396195 0.004265 -0.015449 -0.001939 -0.086998 -0.006617 -0.359506 -0.078843 -0.004617 -0.005020 -0.015968 -0.015018 -0.003521 -0.033288 -0.000455 0.000283 -0.038092 0.776906 0.812280 -0.650336 -0.646256 -0.278465 -0.282239 0.732841 -0.727031 -0.222707 0.189876 0.769481 0.771158 -0.766402 1.000000 -0.175310 -0.658737 -0.652636 -0.299273 -0.297100
dst_host_srv_diff_host_rate 0.100692 -0.140231 0.578557 0.016171 0.007306 -0.000976 -0.014141 -0.002588 0.659078 -0.020979 0.008631 0.001052 0.061030 0.030590 0.015882 0.011765 0.000288 0.000538 -0.012578 -0.496554 -0.391712 -0.153568 -0.148072 0.073061 0.075178 0.179040 -0.176930 0.433173 -0.918894 0.043668 0.107926 -0.088665 -0.175310 1.000000 -0.118697 -0.103715 0.114971 0.120767
dst_host_serror_rate -0.056753 -0.645920 -0.167047 0.013566 0.010387 -0.001381 -0.004706 0.014713 -0.143283 -0.005019 -0.003498 0.001974 -0.008457 -0.002257 -0.001588 -0.011197 -0.000011 -0.000076 -0.001066 -0.331571 -0.449096 0.973947 0.967214 -0.094076 -0.096146 -0.830067 0.807205 -0.097973 0.123881 -0.722607 -0.742045 0.719275 -0.658737 -0.118697 1.000000 0.968015 -0.087531 -0.096899
dst_host_srv_serror_rate -0.057298 -0.641792 -0.158378 0.012265 -0.024117 -0.001370 -0.010721 0.014914 -0.132474 -0.004504 -0.003032 0.002893 -0.007096 -0.004295 -0.002357 -0.011487 -0.000372 -0.000007 -0.016885 -0.335290 -0.442823 0.965663 0.970617 -0.110646 -0.114341 -0.819335 0.795844 -0.092661 0.113845 -0.708392 -0.725272 0.701149 -0.652636 -0.103715 0.968015 1.000000 -0.111578 -0.110532
dst_host_rerror_rate -0.007759 -0.297338 -0.003042 0.000389 0.046656 -0.000786 0.199019 0.032395 0.007236 0.214115 0.002763 0.003173 -0.000421 0.000626 -0.000617 -0.004743 -0.000823 -0.000435 0.025282 -0.261194 -0.313442 -0.103198 -0.122630 0.910225 0.904591 -0.282487 0.299041 0.022585 -0.125142 -0.312040 -0.278068 0.287476 -0.299273 0.114971 -0.087531 -0.111578 1.000000 0.950964
dst_host_srv_rerror_rate -0.013891 -0.300581 0.001621 -0.001816 -0.013666 -0.000782 0.189142 0.032151 0.012979 0.217858 0.002151 0.001731 -0.005012 -0.001096 -0.002020 -0.004552 -0.001038 -0.000529 -0.004292 -0.256176 -0.308132 -0.105434 -0.124656 0.911622 0.914904 -0.282913 0.298904 0.024722 -0.125273 -0.300787 -0.264383 0.271067 -0.297100 0.120767 -0.096899 -0.110532 0.950964 1.000000

We have used a Pandas DataFrame here to render the correlation matrix in a more comprehensive way. Now we want those variables that are highly correlated. For that we do a bit of dataframe manipulation.

In [22]:
# get a boolean dataframe where true means that a pair of variables is highly correlated
highly_correlated_df = (abs(corr_df) > .8) & (corr_df < 1.0)
# get the names of the variables so we can use them to slice the dataframe
correlated_vars_index = (highly_correlated_df==True).any()
correlated_var_names = correlated_vars_index[correlated_vars_index==True].index
# slice it
highly_correlated_df.loc[correlated_var_names,correlated_var_names]

Out[22]:
src_bytes dst_bytes hot logged_in num_compromised num_outbound_cmds is_hot_login count srv_count serror_rate srv_serror_rate rerror_rate srv_rerror_rate same_srv_rate diff_srv_rate dst_host_count dst_host_srv_count dst_host_same_srv_rate dst_host_diff_srv_rate dst_host_same_src_port_rate dst_host_srv_diff_host_rate dst_host_serror_rate dst_host_srv_serror_rate dst_host_rerror_rate dst_host_srv_rerror_rate
src_bytes False False False False False False False False False False False False False False False False False False False True False False False False False
dst_bytes False False False True False False False False False False False False False False False False False False False False False False False False False
hot False False False False True False False False False False False False False False False False False False False False False False False False False
logged_in False True False False False False False False False False False False False False False False False False False False False False False False False
num_compromised False False True False False False False False False False False False False False False False False False False False False False False False False
num_outbound_cmds False False False False False False True False False False False False False False False False False False False False False False False False False
is_hot_login False False False False False True False False False False False False False False False False False False False False False False False False False
count False False False False False False False False True False False False False False False False False False False False False False False False False
srv_count False False False False False False False True False False False False False False False False False False False True False False False False False
serror_rate False False False False False False False False False False True False False True True False False False False False False True True False False
srv_serror_rate False False False False False False False False False True False False False True True False False False False False False True True False False
rerror_rate False False False False False False False False False False False False True False False False False False False False False False False True True
srv_rerror_rate False False False False False False False False False False False True False False False False False False False False False False False True True
same_srv_rate False False False False False False False False False True True False False False True False True True True False False True True False False
diff_srv_rate False False False False False False False False False True True False False True False False True True True False False True False False False
dst_host_count False False False False False False False False False False False False False False False False False False False False True False False False False
dst_host_srv_count False False False False False False False False False False False False False True True False False True True False False False False False False
dst_host_same_srv_rate False False False False False False False False False False False False False True True False True False True False False False False False False
dst_host_diff_srv_rate False False False False False False False False False False False False False True True False True True False False False False False False False
dst_host_same_src_port_rate True False False False False False False False True False False False False False False False False False False False False False False False False
dst_host_srv_diff_host_rate False False False False False False False False False False False False False False False True False False False False False False False False False
dst_host_serror_rate False False False False False False False False False True True False False True True False False False False False False False True False False
dst_host_srv_serror_rate False False False False False False False False False True True False False True False False False False False False False True False False False
dst_host_rerror_rate False False False False False False False False False False False True True False False False False False False False False False False False True
dst_host_srv_rerror_rate False False False False False False False False False False False True True False False False False False False False False False False True False

### Conclusions and possible model selection hints¶

The previous dataframe showed us which variables are highly correlated. We have kept just those variables with at least one strong correlation. We can use as we please, but a good way could be to do some model selection. That is, if we have a group of variables that are highly correlated, we can keep just one of them to represent the group under the assumption that they convey similar information as predictors. Reducing the number of variables will not improve our model accuracy, but it will make it easier to understand and also more efficient to compute.

For example, from the description of the KDD Cup 99 task we know that the variable dst_host_same_src_port_rate references the percentage of the last 100 connections to the same port, for the same destination host. In our correlation matrix (and auxiliar dataframes) we find that this one is highly and positively correlated to src_bytes and srv_count. The former is the number of bytes sent form source to destination. The later is the number of connections to the same service as the current connection in the past 2 seconds. We might decide not to include dst_host_same_src_port_rate in our model if we include the other two, as a way to reduce the number of variables and later one better interpret our models.

Later on, in those notebooks dedicated to build predictive models, we will make use of this information to build more interpretable models.