Logistic Regression (LR) is a measurable technique like straight relapse since LR finds a condition that predicts a result for a parallel variable, Y, from at least one reaction factors, X. Notwithstanding, not at all like straight relapse the reaction factors can be absolute or consistent, as the model doesn't carefully require nonstop information.

1. Logistic Regression
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2. Introduction
Logistic Regression is a classification algorithm. It is used to predict a binary
outcome (1 / 0, Yes / No, True / False) given a set of independent variables.
You can also think of logistic regression as a special case of linear regression
when the outcome variable is categorical, where we are using log of odds as
dependent variable.
In simple words, it predicts the probability of occurrence of an event by fitting
data to a logitfunction.
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3. Logit/Sigmoid function
The logistic function, also called the sigmoid function was developed by
statisticians to describe properties of population growth in ecology, rising
quickly and maxing out at the carrying capacity of the environment. It’s an
S-shaped curve that can take any real-valued number and map it into a value
between 0 and 1, but never exactly at those limits.
1 / (1 + e^-value)
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4. Where e is the base of the natural logarithms (Euler’s number or the EXP()
function in your spreadsheet) and value is the actual numerical value that you
want to transform. Below is a plot of the numbers between -5 and 5
transformed into the range 0 and 1 using the logistic function.
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5. Logit function
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6. Generalized Linear Model
g(E(y)) = α + βx1 +γx2
Here, g() is the linkfunction,
E(y) is the expectation oftarget variable and
α + βx1 + γx2 is the linear predictor ( α,β,γ to be predicted).
The role of link function is to ‘link’ the expectation of y to linear predictor.
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7. Problem statement
We are provided a sample of 1000 customers. We need to predict the
probability whether a customer will buy (y) a particular magazine or not.
As you can see, we’ve a categorical outcomevariable, we’ll use logistic
regression
g(y) = βo +β(Age) ----(a)
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8. Derivation
In logistic regression, we are only concerned about the probability of outcome
dependent variable ( success or failure). As described above, g() is the link
function. This function is established using two things: Probability of
Success(p) and Probability of Failure(1-p). p should meet following criteria:
1. It must always be positive (since p >= 0)
2. It must always be less than equals to 1 (since p <= 1)
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9. Since probability must always be positive, we’ll put the linear equation in
exponential form. For any value of slope and dependent variable, exponent of
this equation will never benegative.
p = exp(βo + β(Age)) = e^(βo+ β(Age)) ------- (b)
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10. To make the probability less than 1, we must divide p by a number greater than
p. This can simply be doneby:
p = exp(βo + β(Age)) / exp(βo + β(Age)) + 1 = e^(βo + β(Age)) / e^(βo + β
(Age)) + 1 ----- (c)
Using (a), (b) and (c), we can redefine the probability as:
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11. p = e^y/ 1 + e^y --- (d)
where p is the probability of success. This (d) is the Logit Function
If p is the probability of success, 1-p will be the probability of failure which can
be written as:
q = 1 - p = 1 - (e^y/ 1+ e^y) --- (e)
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12. On dividing, (d) / (e), we get,
After taking log on both side, we get,
log(p/1-p) is the link function. Logarithmic transformation on the outcome
variable allows us to model a non-linear association in a linear way. Visit: Learnbay.co

13. Final equation
After substituting value of y,we’llget:
This is the equation used in Logistic Regression. Here (p/1-p) is the odd ratio. A typical logistic model
plot is shown next. You can see probability never goes below 0 and above 1.
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14. Logit function graph
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15. Logistic regression models the probability of the default class (e.g. the first class).
For example, if we are modeling people’s Gender as male or female from their height, then the first
class could be male and the logistic regression model could be written as the probability of male given
a person’s height, or moreformally:
P(Gender=male|height)
Written another way, we are modeling the probability that an input (X) belongs to the default class
(Y=1), we can write this formallyas:
P(X) = P(Y=1|X)
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16. ln(p(X) / 1 – p(X)) = b0 + b1 * X
This equation is useful because we can see that the calculation of the output
on the right is linear again (just like linear regression), and the input on the left
is a log of the probability of the default class.
This ratio on the left is called the odds of the default class
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17. Learning Logistic model
The coefficients (Beta values b) of the logistic regression algorithm must be
estimated from your training data. This is done using maximum-likelihood
estimation.
Maximum-likelihood estimation is a common learning algorithm used by a
variety of machine learning algorithms, although it does make assumptions
about the distribution of your data
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18. The best coefficients would result in a model that would predict a value very
close to 1 (e.g. male) for the default class and a value very close to 0 (e.g.
female) for the other class. The intuition for maximum-likelihood for logistic
regression is that a search procedure seeks values for the coefficients (Beta
values) that minimize the error in the probabilities predicted by the model to
those in the data (e.g. probability of 1 if the data is the primary class).
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19. Let’s say we have a model that can predict whether a person is male or female based on
their height (completely fictitious). Given a height of 150cm is the person male or female.
We have learned the coefficients of b0 = -100 and b1 = 0.6. Using the equation above we
can calculate the probability of male given a height of 150cm or more formally
P(male|height=150). We will use EXP() for e, because that is what you can use if you type
this example into yourspreadsheet:
y = e^(b0 + b1*X) / (1 + e^(b0 + b1*X))
y = exp(-100 + 0.6*150) / (1 + EXP(-100 + 0.6*X))
y = 0.0000453978687
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20. we can snap the probabilities to a binary class value, for example:
1 if p(male) <0.5
2 if p(male) >=0.5
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