Linear logisticregression

Linear and Logistic Regression
Let us refresh our memory about two important classes of data
science problems; Regression is where you explain an unknown
numeric variable as a function of several known variables. In
classification, an unknown categorical variable is computed as a
function of known variables. So, the goal in classification is
binning the data.
Typically, the unknown variable is the most difficult for the
business to compute but has a lot of value (like how much
revenue the customer is likely to give in the next 12 months,
which of the customers are likely to buy etc.). The known
variables on the other hand can be easily measured (like age,
gender of the customers).
The business measures both known and unknown variables for
some time and then as a data scientist you build models that
help them guess accurately the unknown variables.
We will in this class study two of the most popular data
science models namely linear regression and logistic regression.
Linear regression as the name suggests is a regression model.
Logistic regression, however, is a classification problem. It is
just named that way.
Let us learn three different aspects of both these techniques.
 In scientific aspects, let us gain mathematical intuition of
these techniques. As an engineer, you may never code
these models yourself. But, the intuition is very helpful.
 In engineering aspects, let us get hands-on and build
some models on real data sets.
 In consulting aspects, let us learn to interpret the outcome
in a systematic manner.

Linear regression
An agricultural scientist is measuring how the profit of a
specific seed varies as a function of ground height from the
sea level.
Yield (Profits/acre) as a function of ground level from sea
(meters)
Now, as a data scientist, you need to build a model to
compute the difficult to measure, unknown variable yield as a
function of easy to measure known variable, ground level.
We are working with only one known variable in this case. It
is called univariate regression. While in most real life
situations you will have more than one known variable (in this
case, the rain fall, type of sand etc. will also most likely
influence the outcome), the univariate case is easy to plot and
understand. So, let us learn this and then move to
multivariate cases.
The goal is to plot a line that fits best through this data.
Yield (profit/acre) of a seed as a function of altitude

Linear model through the data
How does one fit this line? As our goal is to predict the
yield as a function of height, we want to fit a line that gives
the closest possible yields given heights.
A possible equation is
𝑦 = 𝜃0 + 𝜃1 𝑥
For each possible value of 𝜃0and 𝜃1, we get a different line
as shown below
𝐿𝑒𝑓𝑡: 𝜃1 = 0 𝐶𝑒𝑛𝑡𝑒𝑟: 𝜃0 = 0 𝑎𝑛𝑑 𝑅𝑖𝑔ℎ𝑡: 𝐵𝑜𝑡ℎ 𝑎𝑟𝑒 𝑛𝑜𝑛 𝑧𝑒𝑟𝑜
The goal is then to pick those two parameters such that for
every data point, we make the best possible prediction.

In machine, learning each possible combination of 𝜃0 and 𝜃1 is
called a hypothesis. So, there is an infinite space of possible
hypotheses. We use a learning algorithm like linear regression
to identify the best hypothesis.
The learning algorithm in this case should help us choose the
right set of 𝜃0 and 𝜃1that minimizes the discrepancy between
the predicted and actual values. Let us define a function
called cost function which is the difference between the
predicted and original values
Cost function,
𝐽 =
1
2𝑚
∑((𝜃0 + 𝜃1 𝑥𝑖) − 𝑦𝑖)2
𝑚
1
We square the term to ensure that positives and negatives do
not get cancelled. So, the cost function is the average of the
squares of differences of predicted and real values of all
points. 2 in the denominator makes the mathematics easier.
Let us get more intuition with a single parameter hypothesis
and cost functions.
ℎ = 𝜃1 𝑥
𝐽 =
1
2𝑚
∑((𝜃1 𝑥𝑖) − 𝑦𝑖)2
𝑚
1

Note that while hypothesis function is dependent on attributes,
cost function is dependent on parameters. Hypothesis function
is linear and cost function is quadratic.
For two parameters, the cost function will be something like
The algorithm should choose the parameters to minimize the
cost function.

For the linear regression, there is an analytical way to
minimize the cost function.Where you take the partial derivative
of cost functions with respect to 𝜃0 and 𝜃1 and solve them
simultaneously.
However, defining a cost function and minimizing it is a much
more general problem in machine learning and for most of the
cost functions, there are no analytical solutions. A very
standard algorithm used for finding minimum in such cases is
gradient descent. Gradient means slope. Hence, gradient
descent is moving in the direction of maximum slope.
But, depending on where we start, we can take different paths.
Mathematically, what we are repeating until convergence

𝜃𝑗
𝑛+1
= 𝜃𝑗
𝑛
− 𝛼
𝜕
𝜕𝜃𝑗
𝐽(𝜃0
𝑛
, 𝜃1
𝑛)𝑓𝑜𝑟 𝑗 = 0 𝑎𝑛𝑑 1
We update 𝜃0 and 𝜃1 simultaneously.𝛼is called the learning rate.
It tells how big a step we must take at each point. It is
called the learning rate. Those of you familiar with calculus
can verify that for linear regression, this becomes
For the linear regression, the cost function will be bow shaped
and hence we will always reach the same minimum if we use
gradient descent. But, in most general cases, gradient descent
will find only a local minimum.
Let us understand gradient descent a bit more again with a
single parameter example.
Clearly, gradient descent always moves you towards minima.
At minima, as derivative is zero, it keeps you unmoved.
A low learning rate means baby steps. A high learning rate
means non-convergence as shown below. However, once you

finalize on a learning rate, you need not change it as the
slope reduces as you go near the minimum value.
This is in fact a disadvantage of gradient descent that you
need to experiment and find the best learning rate. But, if
you normalize data and choose a good learning rate through
experimentation, you converge fairly well.
There are more advanced and complex algorithms for finding
the minima of a cost function. But, gradient descent is the
most simplest and popular.
One advantage is that if you have multiple features, gradient
descent can be easily scaled up. Let us say, in our original
problem, we added rain fall, type of soil also in our analysis.
The hypothesis now takes the following form
ℎ = 𝜃0 + 𝜃1 𝑥1 + 𝜃2 𝑥2 + 𝜃3 𝑥3
To simplify this, let us use the matrix notation. Let us
introduce a term 𝑥0which is always equal to 1. Then, we
introduce two column vectors X and 𝜃.
So, the hypothesis for a multivariate regression can be written
as 𝜃 𝑇
𝑋. The gradient descent equation will then be

Actually, for linear regression, the cost function can be
minimized analytically by taking derivatives with respect to each
of the parameters and simultaneously solving them. Let us
say, we are working on the following data
Using linear algebra, we can show that the analytical solution
is
However, gradient descent works best for very large dimensional
problem (attributes more than 10,000). It also is used
extensively for other cost functions (logistic regression, neural
networks etc.).
When there are categorical variable, they are directly
implemented if they are binary. For variables with more bins,
dummy variables are created.
Great! That is all the science you need to know about linear
regression. Let us do some R. Over to code!

Analyzing linear regression.
In the car.test.frame data, let us plot Disp. as a function of
HP and Weight
Firstly, the R square is a good indication of how good the fit
is. It says here that, we explain 73.7% of the variance in
the data. So, the model reduces the variance in the data by
73.7% compared to the simple mean.
The equation is
Disp. = -88.79 + 0.85*HP+ 0.04*Weight
So, both HP and Weight have a proportional relation.
If one of them is negative, that meant it had an inverse
relation.
As the coefficient of HP is larger than that of Weight, change
in HP has a higher impact that change in Weight on the
displacement. When you have multiple variables, analyzing
coefficients is a good way to see which attribute impacts the
dependent attribute most. For such analyses, it is better to
normalize all the dependent attributes.

The standard error is the SD of each of the predicted values.
The Pr indicates how confident the model is about the sign of
the coefficients and intercept. So, low Prindicates good and
confident fit. T value indicates how many SDs away the
value is from zero. A high t value is better.
Logistic regression
As mentioned before, logistic regression is a classification
problem.
Logistic regression extends the ideas of linear regression to the
situation where the dependent variable Y, is categorical. We
can think of categorical variable as dividing the observations
into classes.
For example, if Y denotes whether a particular customer is
likely to purchases a product (1) or not likely to purchase (0)
we have a categorical variable with 2 categories or classes (0
and 1).
Let us say, we want to model it as a function of income
level. One thing we can do is to model using linear
regression.

As linear regression out puts a numeric value and not 0 and
1, we can identify a threshold value of y and define such that
above that it is 1 and below that is 0. As you can see, it
works fine in the above example. But, imagine a slightly
different data set. As you can see linear regression and the
same threshold give a poorer result.
In addition, linear regression hypothesis can be much larger
than 1 or much smaller than zero and hence thresholding
becomes difficult.
In logistic regression we take two steps:
1. Find the estimates of the probabilities of belonging to
each class. Case when Y = 0 or 1, the probability of
belonging to class 1, P(Y=1)

2. Use a cut-off value on these probabilities in order to
classify each case in one of the classes. For example in
binary case, a cut-off of 0.5 means that the cases with
an estimated probability of P(Y=1) > 0.5 are classified as
belonging to class 1, whereas cases with P(Y=0) < 0.5
are classified as belonging to class 0. The cut-off need
not be set at 0.5. When the event in the question is a
low-probability event, a higher than average cut-off value,
although below 0.5 may be sufficient to classify. Deciding
a cut-off value is an ‘Art’ rather than science.
Model
The hypothesis in linear regression is
h =𝜃 𝑇
𝑋
However, this can go from -∞ to ∞.So, we modify it into a
sigmoid or logistic function as shown below
h =
1
1+𝑒−𝜃 𝑇 𝑋
(This is called the logistic response function)
When x= -∞, yi = 0; When x= ∞, yi = 1; When x= 0, y=
0.5. Clearly, y always takes a value between 0 and 1 as x
varies from -∞to ∞.

One interprets the value as the probability of y belonging to
class 1. So, if the output of logistic regression is 0.7, then
the probability that the class is 1 is 0.7
From the above graph, it is evident that if 𝜃 𝑇
𝑋>0, then y>0.5.In a
binary classification, it means that the probability that class is 1
is more than 0.5.
Let us look at the decision boundaries of logistic regression on
the data set where the crosses are represented as 1 and
circles are 0.
The hypothesis is of the form
ℎ = 𝑔(𝜃0 + 𝜃1 𝑥1 + 𝜃2 𝑥2)
g here is the logistic function. Let us say, we compute the
parameters of hypothesis and they turn out to be -3, 1, 1.
Then as we learned above, we can conclude that the record
is 1 (or cross) when -3+x1+x2> 0 or x1+x2>3. This is shown
in the following figure

For more complex data sets, the decision boundaries have to
be more complex.
We need to use a quadratic hypothesis space to describe it.
ℎ = 𝑔(𝜃0 + 𝜃1 𝑥1 + 𝜃2 𝑥2 + 𝜃3 𝑥1
2
+ 𝜃4 𝑥2
2
)
Again, let us say, somehow we find that the correct values of
the parameters are -1,0,0,1,1.
Then, we get the hypothesis that class is 1 (or cross) if 𝑥1
2
+
𝑥2
2
− 1>0.

So, by going higher order polynomials, we can construct more
complex decision boundaries.
Cost function of a logistic regression
The cost function of linear regression would become very
complex if used directly in logistic regression as the hypothesis
is of the form of a sigmoid.
Gradient descent would lead to local minimum points and
hence cannot be used. We use a different cost function

This looks complex. But, if we plot the cost functions, the
intuition becomes clear.
If y=1, and the prediction is 1, then cost =0 (from left hand
side). Similarly, cost becomes zero whenever a correct
prediction is made and it becomes very high when incorrect
predictions are made.
The above cost function can be simplified as
𝐶𝑜𝑠𝑡 = −𝑦𝑙𝑜𝑔(ℎ 𝜃) − ((1 − 𝑦)log(1 − ℎ 𝜃))
Clearly, this equation reduces to the above cost function based
on the value of y.
We use gradient descent to then find the parameters
thatminimize the cost function.
Analyzing the logistic regression
Let us go back and write our logistic regression equation.
𝑝𝑖 =
1
1 + 𝑒−(𝜃0+𝜃1 𝑥)
1 + 𝑒−(𝜃0+𝜃1 𝑥)
=
1
𝑝𝑖
𝑒−(𝜃0+𝜃1 𝑥)
=
1 − 𝑝𝑖
𝑝𝑖
𝑒(𝜃0+𝜃1 𝑥)
=
𝑝𝑖
1−𝑝𝑖
(this is called the odds of success)Eq(1)

log (
𝑝𝑖
1−𝑝𝑖)
) = 𝜃0 + 𝜃1 𝑥Eq(2)
From Eq(2), the log of odds has a linear relationship with the
independent variables. From Eq(1), a few things become clear
Odds = 𝑒 𝜃0. 𝑒 𝜃1 𝑥
…
Hence, in logistic regression, the terms are multiplicative and
not additive like linear regression. If x increases by 1 and
everything remains the same, the odds increase by 𝑒 𝜃1.Another
way of saying is that the log of odds increase by 𝜃1.
Here is some intuition
 Logistic regression models are multiplicative in their inputs.
 The exponent of each coefficient tells you how a unit
change in that input variable affects the odds ratio of the
response being true.
 Overly large coefficient magnitudes, overly large error bars
on the coefficient estimates, and the wrong sign on a
coefficient could be indications of correlated inputs.
 Coefficients that tend to infinity could be a sign that an
input is perfectly correlated with a subset of your
responses. Or put another way, it could be a sign that
this input is only really useful on a subset of your data,
so perhaps it is time to segment the data.
Case 5.3: Logistic regression example
We will take the example given in
http://www.ats.ucla.edu/stat/r/dae/logit.htm. OK! over to R code.
The summary of the model is given as

In the output above, the first thing we see is the call, this is
R reminding us what the model we ran was, what options we
specified, etc.
Next we see the deviance residuals, which are a measure of
model fit. This part of output shows the distribution of the
deviance residuals for individual cases used in the model.
Below we discuss how to use summaries of the deviance
statistic to assess model fit.
The next part of the output shows the coefficients, their
standard errors, the z-statistic (sometimes called a Wald z-
statistic), and the associated p-values. Both gre and gpa are
statistically significant, as are the three terms for rank. The
logistic regression coefficients give the change in the log odds
of the outcome for a one unit increase in the predictor
variable.

 For every one unit change in gre, the log odds of
admission (versus non-admission) increases by 0.002.
 For a one unit increase ingpa, the log odds of being
admitted to graduate school increases by 0.804.
 The indicator variables for rank have a slightly different
interpretation. For example, having attended an
undergraduate institution with rank of 2, versus an
institution with a rank of 1, decreases the log odds of
admission by -0.675.

Linear logisticregression

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Linear logisticregression