Data Science - Part III - EDA & Model Selection

 Introduction to Exploratory Data Analysis
 Dataset Features
 Data Munging
 Descriptive Statistics
 Data Transformations
 Variable Selection Procedures
 Model Selection Procedures

The “law of the instrument” developed by
Abraham Kaplan in 1964 and Abraham
Maslow’s hammer in 1966 state: “If all you
have is a hammer, everything looks like a
nail.”

There is no panacea for advanced analytics;
data scientists need to know what the correct
tool is required for a particular task to be most
effective and offer the proper solution.

 Whenever we engage in a predictive modeling activity, we need to first understand
the data for which we are working from. This is called Exploratory Data Analysis or
EDA.
 The primary purpose for the EDA is to better understand the data we are using, how
to transform the data, if necessary, and how to assess limitations and underlying
assumptions inherent in the data structure.
 Data scientists need to know how the various pieces of data fit together and nuances
in the underlying structures in order to decide what the best approach to the
modeling task.
 Any type of method to look at data that does not include formal statistical modeling
and inference generally falls under the EDA.

Here are some of the main reasons why we utilize EDA:
 Detection of mistakes.
 Checking of assumptions.
 Preliminary selection of appropriate models and tools.
 Determining relationships of the explanatory variables (independent).
 Detecting the direction and size of relationships between variables.

Observations
Variables
Qualitative Variables
 Categorical
Quantitative Variables
 Numeric

Dependent Variable Independent Variables
The dependent variable is the variable that we are interested in predicting
and the independent variables are the variables which may or may not help
to predict the dependent variable.

 Data Munging is the transformation
of raw data to a useable format.
 Many datasets are not readily
available for analysis.
 Data needs to be transformed or
cleaned first.
 This process is often the most
difficult and the most time
consuming.

Data Munging tasks include:
 Renaming Variables
 Data Type Conversion
 Encoding, Decoding, recoding data.
 Merging Datasets
 Transforming Data
 Handling Missing Data (Imputation)
 Handling Anomalous values
 These data munging tasks are an iterate process and can occur at any stage
throughout the overall EDA procedure.

Missing Values
Outlier Error
Non-Numeric
Observation: The first row should contain
variable names and all of the data should be
completely filled after the data munging
process is complete.

Renaming Variables
 The names of variables should make intuitive sense to non-practitioners and
does not have to conform to IT protocols and standards.
Data Type Conversion
 Depending upon the modeling task at hand and the software, the data may
need to be expressed in a specific format in order to process correctly.
SQL: Text String
Varchar (max)
SQL: Date Value
Datetime

Encoding Data
 There are times when we need to change the underlying contents in a variable
to prepare them for analytics. Ex. Qualitative to Quantitative.
 If we are using categorical variables, we need to clean them to get rid of non
response categories like “I don’t know”, “no answer”, “n/a”, etc… We also need
to order the encoding of categories (potentially reverse valence) to ensure that
models are built and interpreted correctly.
 Usually non response categories is coded with values like 999. If this was a
value in the variable “Age”, this will skew the results and should be turned to
NULL and reviewed further.

Merging Datasets
 It is quite rare that you will have a dataset readily constructed for analysis. This
may require some data manipulation and merging in order to get the data in
the correct form.
Observation: The datasets will need to have a common
ID as the link to join the data. After the data has been
merged, the ID may not be necessary to retain for
model building.

Transforming Variables
 There may be times where a variable will need to be transformed in order to
achieve linearity. This will aid in strengthening the results of parametric based
methods.
Natural Log
Transformation

Imputation
 If there are missing values in a column, these cannot be left unattended. We
must decided if we want to:
 Remove the observation from the dataset
 Calculate a value for the null (impute). This usually is determined with the
mean or median, however, a more advanced version can use a multiple
linear regression formula.
Mean = 58,660
Median = 60,500

Handling Anomalous Values
 Depending upon the analytic task, we need
to assess points which exhibit a great deal of
influence on the model.
 Outliers are data points that deviate
significantly from the spread or distribution
of other similar data points. These can
typically be detected through the use of
scatterplots.
 Many times we will delete the entry with an
outlier to achieve normality in the dataset.
 In some instances, an outlier can be imputed
but this must be approached with caution.
 Important: The drivers of outlying data
points need to first be understood prior to
devising an approach to dealing with them.
They can hold the clues to new insights.

 Describes the data in a qualitative
or quantitative manner.
 Provides a summary of the shape of
the data.
 These statistics help us to
understand when transformations,
imputations, and removal of outliers
are necessary prior to model
building.

 Descriptive Statistics are a collection of
measurements of two things: Location and
variability.
 Location tells you of the central value of your
variable.
 Variability refers to the spread of the data from
the center value.
 Statistics is essentially the study of what causes
variability in the data.

Mean
 The sum of the observations divided by the total
number of observations. It is the most common
indicator of central tendency of a variable.
Median
 To get the median, we need to sort the data from
lowest to highest. The median is the number in
the middle of the data
Mode
 Refers to the most frequent of commonly
occurring number within the variable.

Variance
 Measures the dispersion of the data from the
mean.
Standard Deviation
 The Standard Deviation is the Squared Root of
the Variance. This indicates how close the data is
to the mean.
Range
 Range is a measure of dispersion. It is the simple
difference between the largest and smallest
values.

 Histograms are a graphical display of data using bars of different heights. This
allows us to evaluate the shape of the underlying distribution. Essentially, a
histogram is a bar chart that groups numbers into ranges or bins.

 Data can be distributed (spread out) in many different ways.
It can be spread out
more on the left
Or more on the right Or all jumbled up.

 But there are many cases where the data tends to be around a central value with
no bias left or right, and it gets close to a "Normal Distribution" like this:
The "Bell Curve" is a Normal Distribution. The yellow histogram
shows some data that follows it closely, but not perfectly (which is
usual).

Many things follow a normal distribution:
 Height of People
 Size of things produced by machines
 Errors in measurements
 Blood Pressure
 Academic Test Scores
 Qunicunx (Plinko)

We say that data is normally distributed when
the histogram has:
 Mean = Median = Mode
 Symmetry around the center
 50% of the values less than the mean and
50% greater than the mean.

When we calculate the standard deviation of the mean ( σ ) we find that:
68% of values are
within 1 deviation of
the mean
95% of values are
within 2 deviations of
the mean
99.7% of values are
within 2 deviations of
the mean

 The number of standard deviations
from the mean is also called the
“standard score” or “z-score”.
 We can take any normal distribution
and convert to the standard normal
distribution.
 To convert a value to the Z-score:
 First subtract the mean.
 Then divide by the standard
deviation.
 Standardizing helps us make decisions
about the data and makes our life
easier by only having to refer to a
single table for statistical testing.

 Here is the Standard Normal Distribution with percentages for every half of a standard
deviation, and cumulative percentages:

 There are other moments about the mean, skewness and kurtosis, that can used to understand
what distribution should be used when there are 3 or more parameters to estimate.

 The Skewness statistic refers to the lopsidedness of the
distribution. If a distribution has a negative Skewness
(sometimes described as left skewed) it has a longer tail to
the left than to the right. A positively skewed distribution
(right skewed) has a longer tail to the right, and zero
skewed distributions are usually symmetric.
Interpretation:
 Skewness > 0 - Right skewed distribution - most values
are concentrated on left of the mean, with extreme values
to the right.
 Skewness < 0 - Left skewed distribution - most values are
concentrated on the right of the mean, with extreme
values to the left.
 Skewness = 0 - mean = median, the distribution is
symmetrical around the mean.

 Kurtosis is an indicator used in distribution
analysis as a sign of flattening or "peakedness"
of a distribution.
 The Kurtosis is calculated from the following
formula:
Interpretation:
 Kurtosis > 3 - Leptokurtic distribution, sharper
than a normal distribution, with values
concentrated around the mean and longer tails.
This means high probability for extreme values.
 Kurtosis < 3 - Platykurtic distribution, flatter than
a normal distribution with a wider peak. The
probability for extreme values is less than for a
normal distribution, and the values are wider
spread around the mean.
 Kurtosis = 3 - Mesokurtic distribution - normal
distribution for example.
Leptokurtic
Platykurtic

 The following table gives some examples of skewness and kurtosis for common distributions.

Here are some examples of other common distributions you may encounter:
Poisson Distribution Chi-Square Distributions

 A Boxplot is a nice way to graphically represent the data in order to communicate
the data through their quartiles.
Outlier
Right Skewed Distribution
Normal Distribution

 In statistics, a QQ plot ("Q" stands for
quantile) is a probability plot, which is a
graphical method for comparing two
probability distributions by plotting their
quantiles against each other.
 A QQ plot is used to compare the
shapes of distributions, providing a
graphical view of how properties such as
location, scale, and skewness are similar
or different in the two distributions.
 If the quantiles of the theoretical and
data distributions agree, the plotted
points fall on or near the line y = x
 This can provide an assessment of
"goodness of fit" that is graphical, rather
than reducing to a numerical summary.

 A point plot between two variables used to
understand the spread of the data.
 The spread of the data allows for us to
understand if the data has a non-linear or
linear relationship and the relative degree of
the correlation in the data.
 This technique can allow be used to detect
outliers in the data.
 A scatterplot becomes more powerful when
you incorporate categorical data as an
additional dimension.

 Scatterplot Matrices can offer a 10,000 feet view of the patterns within the data.

 When two sets of data are strongly linked
together we say they have a high correlation.
 Correlation is Positive when the values increase
together, and
 Correlation is Negative when one value
decreases as the other increases.
 Correlation can have a value ranging from:
 1 is perfect positive correlation
 0 implies that there is no correlation
 -1 is perfect negative correlation
 Correlation Does Not Imply Causation !!!!

 The creation of a correlation matrix and
correlation heat maps make it easier to have a
top level view of the data.
 Remember that data that is closer to perfect
negative or positive correlations will be stronger
for linear model building.
 We need to be careful about the assuming that
lower correlation intervals are potentially poor
variables for model building.
 Evaluation of correlation should be handled on a
case by case basis.
 Example: In behavioral sciences, it is not
uncommon to have correlations between 0.3 and
0.4 for significant variables.

 This is a powerful example on
why we need to understand the
data in order to model the
information correctly.
 Anscombe’s quartet shows how
different data spreads can
produce identical regression
models and coefficients.
 These are the same models with
identical mean, variance, and
correlation values.
 Some of these models requires
data transformations to achieve
linearity, others have outliers that
strongly influence the models
performance.

When data in a scatterplot (box plot or bell curve)
indicates that a non-linear relationship exists between
the dependent and independent variable, we can use a
data transformation to achieve linearity.
This is a vital task for models which require linearity
(regression, time series, etc…) in the variables.
Here are some types of transformations we will go over:
 Power Function
 Exponential Function
 Polynomial Function
 Logarithmic Function
 Square Root Function

Data Science - Part III - EDA & Model Selection

A power function takes the following form:
 It's important to note the location of the variable x in the power function. Later in this
activity, we will contrast the power function with the exponential function, where the
variable x is an exponent, rather than the base as in the equation.

 It could be argued that the data is somewhat linear, showing a general upward trend. However,
it is more likely that the function is nonlinear, due to the general bend.

 Let's take the logarithm of both sides of the power function:
y = 3x2
 The base of the logarithm is irrelevant; however, log x is understood to be the
natural logarithm (base e) in R. That is, log x = loge x in R.
log y = log 3x2
 The log of a product is the sum of the logs, so we can write the following.
log y = log 3 + log x2
 Another property of logs allows us to move the exponent down.
log y = log 3 + 2 log x

 The slope is 2, which is the slope indicated in log y = log 3 + 2 log x. Supposedly, the intercept
should be log 3.
 Important: If the graph of the logarithm of the response variable versus the logarithm of the
independent variable is a line, then we should suspect that the relationship between the original
variables is that of a power function.

 This box-plot is a representation of salaries for 500 employees at a University.
 What is the shape of the dataset? It depends what you are looking at…
 If you look at the middle 50% of the salaries, they look roughly symmetric.
 On the other hand, if you look at the tails (the portion of the data beyond the
quartiles), then there is clear right-skewness, and a number of high outliers. One
would expect to find a few large salaries at a university.

Original
Transformed
Power Transformation 0.3
Observations:
 If we look at the middle 50%, we don’t have symmetry (data is left skewed)
 If we look at the tails, the re-expressed data is roughly symmetric

What have we learned from this exercise?
 For some data, it can be difficult to achieve symmetry of the whole batch. Here
the power transformation was helpful in making the tails symmetric (and control
the outliers), but it does not symmetrize the middle salaries.
 In practice, one has to decide on the objective. If we are focusing on the middle
part of the salaries, then no re-expression is necessary. But if we want to look at
the whole dataset and control the outliers, then perhaps the choice of power =
0.3 is a better choice.

A exponential function takes the following form:
 In the power function, the independent variable was the base, as in y = a xb. In the
exponential function, the independent variable is now an exponent, as in y = a bx. This
is a subtle but important difference.

 Let's take the logarithm of both sides of the exponential function:
y = 3*2x
 The base of the logarithm is irrelevant; however, log x is understood to be the
natural logarithm (base e) in R. That is, log x = loge x in R.
log y = log 3(2x)
 The log of a product is the sum of the logs, so we can write the following.
log y = log 3 + log 2x
 Another property of logs allows us to move the exponent down.
log y = log 3 + x log 2

 The slope is 2, which is the slope indicated in log y = log 3 + 2 log x. Supposedly, the intercept
should be log 3.
 Important Result: If the graph of the logarithm of the response variable versus the logarithm of
the independent variable is a line, then we should suspect that the relationship between the
original variables is that of a power function.

 How did we know to use a power
function of 0.3 in the power
transformation example?
 The statisticians George Box and David
Cox developed a procedure to identify
an appropriate exponent (Lambda = l)
to use to transform data into a “normal
shape.”
 The Lambda value indicates the power
to which all data should be raised.
 In order to do this, the Box-Cox power
transformation searches from Lambda
= -5 to Lamba = +5 until the best
value is found

 Here is a table which indicates which linear transformation to apply for various
lambda values:

 A residual plot is a scatterplot of the
residuals (difference between the actual
and predicted value) against the
predicted value.
 A proper model will exhibit a random
pattern for the spread of the residuals
with no discernable shape.
 Residual plots are used extensively in
linear regression analysis for diagnostics
and assumption testing.
 For our purposes in the EDA, they can
help to identify when a transformation is
appropriate.
 If the residuals form a curvature like
shape, then we know that a
transformation will be necessary and can
explore some methods like the Box-Cox.
Random Residuals
Curved Residuals

 Transforming a data set to enhance linearity is a multi-step, trial-and-error process.
 Step 1: Conduct a standard regression analysis on the raw data.
 Step 2: Construct a residual plot.
 If the plot pattern is random, do not transform data.
 If the plot pattern is not random, continue.
 Step 3: Compute the coefficient of determination (R2).
 Step 4: Choose a transformation method.
 Step 5: Transform the independent variable, dependent variable, or both.
 Step 6: Conduct a regression analysis, using the transformed variables.
 Step 7: Compute the coefficient of determination (R2), based on the transformed variables.
 If the transformed R2 is greater than the raw-score R2, the transformation was
successful. Congratulations!
 If not, try a different transformation method.

 How do we approach a modeling task with 100’s or 1000’s of independent variables?
What if there are a significant number of variables with low correlations?

 Dimensionality reduction is the process of
reducing the number of variables under
consideration and can be represented in feature
selection and feature extraction approaches.
 Feature Selection – Methodology that attempts
to find a subset of the original variables using
some form of selection criteria. These
approaches are primarily through filtering (Ex.
information gain) and wrappers (search guided
by accuracy).
 Feature Extraction – Transforms the data in a
higher dimensional space to a space of lower
dimensions. A linear approach is called a
Principal Component Analysis (PCA).

 When attempting to describe a model containing dependent and independent
variables, it is desirable to be as accurate as possible, but at the same time to use as
few variables as possible (principal of parsimony).
 Forward Selection Procedure - Starts with no variables in the model. The term that
adds the most is identified and added to the model. The remaining variables are
analyzed and the next most “helpful” variable is added. This process continues until
none of the remaining variables will improve the model.
 Backward Selection Procedure - Starts with the full model, and proceed by
sequentially removing variables that improve the model by being deleted, until no
additional deleting will improve the model.
 Stepwise Selection Procedure – Starts like a forward selection procedure. After
adding a variable to the model, run a backward selection procedure on the existing
variable pool within the model to try and eliminate any variables. Continue until
every remaining variable is significant at the cutoff level and every excluded variable
is insignificant.

 This is an example of stepwise selection
procedure which contains a large
number of independent variables.
 After the routine has completed, note
the variable list has been reduced.
 The variables which were removed did
not meet a certain statistical confidence
level (P-Value) which was pre-defined.
 Unless we have specific subject matter
expertise and expert level knowledge of
the dataset at hand, we should set the P-
Value as 0.05 or smaller.

 The Principal Component Analysis procedure will take a number of variables and reduce
them down to a smaller number of variables called components.
 This example took the dependent variables Price, Software, Aesthetics, & Brand and created
4 Components.
 The cumulative proportion figure of 0.847 for Comp. 2 tells us that the first two components
account for 84.7% of the total variability of the data. Generally speaking, 80% accounts for
the data rather well.

 The loadings can be thought of as new variables which have been created for our
analysis.
 The way to interpret and calculate the first component variable is: Comp.1 = 0.523 *
Price + 0.177 * Software - 0.597 * Aesthetics - 0.583 * Brand

Once you have the data in the proper format,
before you perform any analysis you need to
explore the data and prepare it first.
 All of the variables are in columns and
observations in rows.
 We have all of the variables that you need.
 There is a at least one variable with a unique ID.
 Have a backup of the original dataset handy.
 We have properly addressed any data munging
tasks.

 Another important step regarding predictive model building involves splitting the
dataset into a training and testing dataset. This is performed to ensure that our models
are validated against data that was not part of the models construction.
 To accomplish this, we will randomly select observations for training dataset (70%) and
the remaining into the test dataset (30%).
Training
Dataset
Testing
Dataset

 In our previous example, we had decided to use a
split of 70% for the model building dataset or
training set and 30% for our validation dataset.
 The more data that we spend on the training set,
the better estimates that we will get. (Provided the
data is accurate.)
 Given a fixed amount of data:
 Too much spent in training wont allow us to get
a good assessment of predictive performance.
We may find a model that fits the training data
very well, but is not generalizable (over-fitting).
 Too much spent in testing wont allow us to get
a good assessment of model parameters.
 Many business consumers of these models
emphasize the need for an untouched set of
samples to evaluate performance.

 Over-fitting occurs when a model inappropriately
picks up on trends in the training set that do not
generalize to new samples.
 When this occurs, assessments of the model
based on the training set can show good
performance that does not reproduce in future
samples.
 Some predictive models have dedicated “knobs”
to control for over-fitting.
 Neighborhood size in nearest neighbor models
for example.
 The number of splits in a decision tree model.
 Often, poor choices for these parameters can
result in over-fitting. When in doubt, consult with
statisticians and data scientists on how to best
approach the tuning.

 One obvious way to detect over-fitting is the use of a test set. The predictive
performance measures that we use to evaluate the performance should be consistent
across the training and test-set.
 However, repeated “looks” at the test-set can also lead to over-fitting.
 Resampling the training samples allows us to know when we are making poor
choices for the values of these parameters (the test set is not used).
 Resampling methods try to “inject variation” into the system in order to approximate
the models performance on future samples.
 We will walk through several types of resampling methods for training set samples.

 Here, we randomly split the data into K-
distinct blocks of roughly equal size.
 We leave out the first block of data and
fit a model.
 This model is used to predict the held
out block.
 We continue this process until we’ve
predicted all K held out blocks.
 The final performance is based upon the
hold-out predictions.
 K is usually taken to be 5 or 10 and leave
one out cross validation as each sample as
a block.
K-Fold Cross Validation

 A random proportion of data (say 70%)
are used to train a model while the
remainder is used for prediction.
 The process is repeated many times and
the average performance is used.
 These splits can also be generated
using stratified sampling.
 With many iterations (20 to 100), this
procedure has smaller variance than K-
Fold CV, but is likely to be biased.
Repeated Training/Testing Splits

 Bootstrapping takes a random sample of the
data with replacement. The random sample is
the same size as the original dataset.
 Samples may be selected more than once
and each sample has approx. 63.2% chance
of showing up at least once.
 Some samples wont be selected and these
samples will be used to predict performance.
 The process is repeated multiple times (say
30-100).
 The procedure also has low variance but non
zero bias when compared to K-fold CV.
Bootstrapping

 We think that resampling will give us honest
estimates of future performance, but there is still
the issue of which model to select.
 One algorithm to select predictive models:

 There are 3 broad categories of activities that we can perform and we need to understand
how each modeling activity we are performing corresponds to each bucket.
 Predictive Models - Predictive models are models of the relation between the specific
performance of a unit in a dataset and one or more known attributes or features of the
unit. The objective of the model is to assess the likelihood that a similar unit in a different
dataset will exhibit the specific performance.
 Descriptive Models - Descriptive models quantify relationships in data in a way that is often
used to classify customers or prospects into groups. Unlike predictive models that focus on
predicting a single customer behavior (such as credit risk), descriptive models identify many
different relationships between customers or products. Descriptive modeling tools can be
utilized to develop further models that can simulate large number of individualized agents
and make predictions.
 Decision Models - Decision models describe the relationship between all the elements of a
decision — the known data (including results of predictive models), the decision, and the
forecast results of the decision — in order to predict the results of decisions involving many
variables. These models can be used in optimization, maximizing certain outcomes while
minimizing others.

 If the goal of the activity is to predict a
value in a dataset, we first need to
understand more about the dataset to
determine the correct “tool” or approach.
 Lets examine the following dataset:
 Our goal is to predict the price of a home
based upon the lot size and number of
bedrooms. Notice that there is no time
element in the dataset.
 The price variable is called the dependent
variable and the lotsize/ bedrooms are
the independent variables.
 For now, we will focus our attention on
understanding the characteristics of the
dependent variable.
Dependent
Variable
Independent
Variables

 The house price variable seems to fall into a range of values which implies that
it is continuous in nature.
 Observation: Since there is no time component for this house price in the
dataset, we can effectively consider the use of linear regression, CART, or
neural network models.

 If the dependent variable was a binary response (yes/no), there are a number
of different methods we would consider using:

 If we need to classify/cluster variables together beyond a dichotomous
variable, we can use some of the following descriptive techniques:

 There is usually an inverse relationship between
model flexibility / power and interpretability.
 In the best case, we would like a parsimonious
and interpretable model that has excellent
performance.
 Unfortunately, that is not realistic.
 A recommended strategy:
 Start with the most powerful black-box type
models.
 Get a sense of the best possible performance.
 Then fit more simplistic/ understandable
models.
 Evaluate the performance cost of using a
simpler model.

 Reside in Wayne, Illinois
 Active Semi-Professional Classical Musician
(Bassoon).
 Married my wife on 10/10/10 and been
together for 10 years.
 Pet Yorkshire Terrier / Toy Poodle named
Brunzie.
 Pet Maine Coons’ named Maximus Power and
Nemesis Gul du Cat.
 Enjoy Cooking, Hiking, Cycling, Kayaking, and
Astronomy.
 Self proclaimed Data Nerd and Technology
Lover.

 http://en.wikipedia.org/wiki/Regression_analysis
 http://www.ftpress.com/articles/article.aspx?p=2133374
 http://www.matthewrenze.com/presentations/exploratory-data-analysis-
with-r.pdf
 http://msenux.redwoods.edu/math/R/TransformingData.php
 https://exploredata.wordpress.com/2014/10/01/reexpressing-salaries/
 http://www.princeton.edu/~otorres/DataPrep101.pdf
 http://support.sas.com/documentation/cdl/en/procstat/63104/HTML/defa
ult/viewer.htm#procstat_univariate_sect040.htm
 http://www.mathsisfun.com/data/correlation.html
 http://www.mathsisfun.com/data/standard-normal-distribution.html
 http://www.edii.uclm.es/~useR-2013/Tutorials/kuhn/user_caret_2up.pdf

Data Science - Part III - EDA & Model Selection

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