First there are questions on this forum very similar to this one but trust me none matches so no duplicating please.
I have encountered two methods of linear regression using scikit's sklearn and I am failing to understand the difference between the two, especially where in first code there's a method train_test_split() called while in the other one directly fit method is called.
I am studying with multiple resources and this single issue is very confusing to me.
First which uses SVR
X = np.array(df.drop(['label'], 1))
X = preprocessing.scale(X)
y = np.array(df['label'])
X_train, X_test, y_train, y_test = cross_validation.train_test_split(X, y, test_size=0.2)
clf = svm.SVR(kernel='linear')
clf.fit(X_train, y_train)
confidence = clf.score(X_test, y_test)
And second is this one
# Split the data into training/testing sets
diabetes_X_train = diabetes_X[:-20]
diabetes_X_test = diabetes_X[-20:]
# Split the targets into training/testing sets
diabetes_y_train = diabetes.target[:-20]
diabetes_y_test = diabetes.target[-20:]
# Create linear regression object
regr = linear_model.LinearRegression()
# Train the model using the training sets
regr.fit(diabetes_X_train, diabetes_y_train)
# Make predictions using the testing set
diabetes_y_pred = regr.predict(diabetes_X_test)
So my main focus is the difference between using svr(kernel="linear") and using LinearRegression()
cross_validation.train_test_split : Splits arrays or matrices into random train and test subsets.
In second code, splitting is not random.
svm.SVR: The Support Vector Regression (SVR) uses the same principles as the SVM for classification, with only a few minor differences. First of all, because output is a real number it becomes very difficult to predict the information at hand, which has infinite possibilities. In the case of regression, a margin of tolerance (epsilon) is set in approximation to the SVM which would have already requested from the problem. But besides this fact, there is also a more complicated reason, the algorithm is more complicated therefore to be taken in consideration. However, the main idea is always the same: to minimize error, individualizing the hyperplane which maximizes the margin, keeping in mind that part of the error is tolerated.
Linear Regression: In statistics, linear regression is a linear approach for modeling the relationship between a scalar dependent variable y and one or more explanatory variables (or independent variables) denoted X. The case of one explanatory variable is called simple linear regression.
Reference:
https://cs.adelaide.edu.au/~chhshen/teaching/ML_SVR.pdf
This is what I found:
Intuitively, as all regressors it tries to fit a line to data by minimising a cost function. However, the interesting part about SVR is that you can deploy a non-linear kernel. In this case you end making non-linear regression, i.e. fitting a curve rather than a line.
This process is based on the kernel trick and the representation of the solution/model in the dual rather than in the primal. That is, the model is represented as combinations of the training points rather than a function of the features and some weights. At the same time the basic algorithm remains the same: the only real change in the process of going non-linear is the kernel function, which changes from a simple inner product to some non linear function.
So SVR allows non linear fitting problems as well while LinearRegression() is only for simple linear regression with straight line (may contain any number of features in both cases).
The main difference for these methods is in mathematics background!
We have samples X and want to predict target Y.
The Linear Regression method just minimizes the least squares error:
for one object target y = x^T * w, where w is model's weights.
Loss(w) = Sum_1_N(x_n^T * w - y_n) ^ 2 --> min(w)
As it is a convex functional the global minimum will be always found.
After taking derivative of Loss by w and transforming sums to vectors you'll get:
w = (X^T * X)^(-1)* (X^T * Y)
So, in ML (i'm sure sklearn also has the same implementation) the w is calculated according above formula.
X is train samples, when you call fit method.
In predict this weights just multiplies on X_test.
So the decision is explicit and faster (except for Big selections as finding inverse matrix in this cases is complicated task) than converging methods such as svm.
In addition: Lasso and Ridge solves the same task but have additionally the regularization on weights in their losses.
And you can calculate the weights explicit in that cases too.
The SVM.Linear does almost the same thing except it has an optimization task for maximizing the margin (i apologize but it is difficult to put it down because i didn't find out how to write in Tex format here).
So it uses gradient descent methods for finding global extremum.
Sklearn's class SVM even have attribute max_iter which is used in the converging tasks.
To sum up: Linear Regression has explicit decision and SVM finds approximate of real decision because of numerical(computational) solution.
In cs231n handout here, it says
New dataset is small and similar to original dataset. Since the data
is small, it is not a good idea to fine-tune the ConvNet due to
overfitting concerns... Hence, the best idea might be to train a
linear classifier on the CNN codes.
I'm not sure what linear classifier means. Does the linear classifier refer to the last fully connected layer? (For example, in Alexnet, there are three fully connected layers. Does the linear classifier the last fully connected layer?)
Usually when people say "linear classifier" they refer to Linear SVM (support vector machine). A linear classifier learns a weight vecotr w and a threshold (aka "bias") b such that for each example x the sign of
<w, x> + b
is positive for the "positive" class and negative for the "negative" class.
The last (usually fully connected) layer of a neural-net can be considered as a form of a linear classifier.
This is w.r.t a hybrid of ANN and logistic regression in a binary classification problem. For example in one of the papers I came across they state that "A hybrid model type is constructed by using the logistic regression model to calculate the probability of failure and then adding that value as an additional input variable into the ANN. This type of model is defined as a Plogit-ANN model".
So, for n input variables, I'm trying to understand how the additional input n+1 to a ANN is treated by the activation function (eg. a logit function) and in the summation of weights multiplied by inputs. Do we treat this probability variable n+1 as one of the standalone weights like a special type of b0 that we add in the sum of weights multiplied by inputs e.g. Summation for each Neuron = (Sum (Wi*Xi))+additional variable.
Thank you for your assistance.
According to the description provided the easiest way is to treat this as additional feature of your data. So you have a model that predicts something about your original dataset (probability of some additional thing), thus you get x -> f(x). You simply concatenate it to your feature vector so x' = [x1 x2 ... xk f(x)], and push it through the network.
However the described approach is quite naive, since you are doing these two things (training f and training neural net) completely independently), what might be more beneficial is to instead treat fitting f as an auxiliary loss and train your model jointly.
Reading course notes of Andrew NG's machine learning course it states for linear regression :
Take a training set and pass it into a learning algorithm. The algorithm outputs a function h (the hypothesis).
h takes an input and tries to output estimated value y.
It then goes on to say :
present h as : h theta(x) = theta0 + theta1x
Does this not mean the hyptohesis was not outputted by the learning algorithm, instead we just defined it as h theta(x) = theta0 + theta1x
Instead of "Take a training set and pass it into a learning algorithm. The algorithm outputs a function h (the hypothesis)." should the statement be "Take a training set and pass it into a learning algorithm. The algorithm outputs value(s) which make the hypothesis as accurate as possible" ?
In principle you are right here. A true learning algorithm as defined in learning theory is an algorithm that gets labelled instances and a whole class of possible hypotheses as input and then chooses one hypothesis as an output.
So strictly speaking, an algorithm that outputs the predictions is not a learning algorithm. But of course such an algorithm can be split into a learning algorithm - the algorithm that actually learns the parameters, here the thetas. and a prediction algorithm that transforms some input instances to our predictions which are then returned to the caller.
For the case of linear regression you want your learning algorithm to output a linear fucnction.
That is h(x) = theta0 + theta1x.
In this case the learning algorithm learns the optimal theta0 and theta1 to fit your training data.
If you wanted your learning algorithm to learn a 3rd degree polynomial the output of your learning model would be a, b, c and d
such that
h(x) = ax3 + bx2 + cx + d
But your assertion is correct, the learning algorithm chooses the best parameters to minimize the cost of an error function. Usually this is squared error + some regularization factors.
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When we have to predict the value of a categorical (or discrete) outcome we use logistic regression. I believe we use linear regression to also predict the value of an outcome given the input values.
Then, what is the difference between the two methodologies?
Linear regression output as probabilities
It's tempting to use the linear regression output as probabilities but it's a mistake because the output can be negative, and greater than 1 whereas probability can not. As regression might actually
produce probabilities that could be less than 0, or even bigger than
1, logistic regression was introduced.
Source: http://gerardnico.com/wiki/data_mining/simple_logistic_regression
Outcome
In linear regression, the outcome (dependent variable) is continuous.
It can have any one of an infinite number of possible values.
In logistic regression, the outcome (dependent variable) has only a limited number of possible values.
The dependent variable
Logistic regression is used when the response variable is categorical in nature. For instance, yes/no, true/false, red/green/blue,
1st/2nd/3rd/4th, etc.
Linear regression is used when your response variable is continuous. For instance, weight, height, number of hours, etc.
Equation
Linear regression gives an equation which is of the form Y = mX + C,
means equation with degree 1.
However, logistic regression gives an equation which is of the form
Y = eX + e-X
Coefficient interpretation
In linear regression, the coefficient interpretation of independent variables are quite straightforward (i.e. holding all other variables constant, with a unit increase in this variable, the dependent variable is expected to increase/decrease by xxx).
However, in logistic regression, depends on the family (binomial, Poisson,
etc.) and link (log, logit, inverse-log, etc.) you use, the interpretation is different.
Error minimization technique
Linear regression uses ordinary least squares method to minimise the
errors and arrive at a best possible fit, while logistic regression
uses maximum likelihood method to arrive at the solution.
Linear regression is usually solved by minimizing the least squares error of the model to the data, therefore large errors are penalized quadratically.
Logistic regression is just the opposite. Using the logistic loss function causes large errors to be penalized to an asymptotically constant.
Consider linear regression on categorical {0, 1} outcomes to see why this is a problem. If your model predicts the outcome is 38, when the truth is 1, you've lost nothing. Linear regression would try to reduce that 38, logistic wouldn't (as much)2.
In linear regression, the outcome (dependent variable) is continuous. It can have any one of an infinite number of possible values. In logistic regression, the outcome (dependent variable) has only a limited number of possible values.
For instance, if X contains the area in square feet of houses, and Y contains the corresponding sale price of those houses, you could use linear regression to predict selling price as a function of house size. While the possible selling price may not actually be any, there are so many possible values that a linear regression model would be chosen.
If, instead, you wanted to predict, based on size, whether a house would sell for more than $200K, you would use logistic regression. The possible outputs are either Yes, the house will sell for more than $200K, or No, the house will not.
Just to add on the previous answers.
Linear regression
Is meant to resolve the problem of predicting/estimating the output value for a given element X (say f(x)). The result of the prediction is a continuous function where the values may be positive or negative. In this case you normally have an input dataset with lots of examples and the output value for each one of them. The goal is to be able to fit a model to this data set so you are able to predict that output for new different/never seen elements. Following is the classical example of fitting a line to set of points, but in general linear regression could be used to fit more complex models (using higher polynomial degrees):
Resolving the problem
Linear regression can be solved in two different ways:
Normal equation (direct way to solve the problem)
Gradient descent (Iterative approach)
Logistic regression
Is meant to resolve classification problems where given an element you have to classify the same in N categories. Typical examples are, for example, given a mail to classify it as spam or not, or given a vehicle find to which category it belongs (car, truck, van, etc ..). That's basically the output is a finite set of discrete values.
Resolving the problem
Logistic regression problems could be resolved only by using Gradient descent. The formulation in general is very similar to linear regression the only difference is the usage of different hypothesis function. In linear regression the hypothesis has the form:
h(x) = theta_0 + theta_1*x_1 + theta_2*x_2 ..
where theta is the model we are trying to fit and [1, x_1, x_2, ..] is the input vector. In logistic regression the hypothesis function is different:
g(x) = 1 / (1 + e^-x)
This function has a nice property, basically it maps any value to the range [0,1] which is appropiate to handle propababilities during the classificatin. For example in case of a binary classification g(X) could be interpreted as the probability to belong to the positive class. In this case normally you have different classes that are separated with a decision boundary which basically a curve that decides the separation between the different classes. Following is an example of dataset separated in two classes.
You can also use the below code to generate the linear regression
curve
q_df = details_df
# q_df = pd.get_dummies(q_df)
q_df = pd.get_dummies(q_df, columns=[
"1",
"2",
"3",
"4",
"5",
"6",
"7",
"8",
"9"
])
q_1_df = q_df["1"]
q_df = q_df.drop(["2", "3", "4", "5"], axis=1)
(import statsmodels.api as sm)
x = sm.add_constant(q_df)
train_x, test_x, train_y, test_y = sklearn.model_selection.train_test_split(
x, q3_rechange_delay_df, test_size=0.2, random_state=123 )
lmod = sm.OLS(train_y, train_x).fit() lmod.summary()
lmod.predict()[:10]
lmod.get_prediction().summary_frame()[:10]
sm.qqplot(lmod.resid,line="q") plt.title("Q-Q plot of Standardized
Residuals") plt.show()
Simply put, linear regression is a regression algorithm, which outpus a possible continous and infinite value; logistic regression is considered as a binary classifier algorithm, which outputs the 'probability' of the input belonging to a label (0 or 1).
The basic difference :
Linear regression is basically a regression model which means its will give a non discreet/continuous output of a function. So this approach gives the value. For example : given x what is f(x)
For example given a training set of different factors and the price of a property after training we can provide the required factors to determine what will be the property price.
Logistic regression is basically a binary classification algorithm which means that here there will be discreet valued output for the function . For example : for a given x if f(x)>threshold classify it to be 1 else classify it to be 0.
For example given a set of brain tumour size as training data we can use the size as input to determine whether its a benine or malignant tumour. Therefore here the output is discreet either 0 or 1.
*here the function is basically the hypothesis function
They are both quite similar in solving for the solution, but as others have said, one (Logistic Regression) is for predicting a category "fit" (Y/N or 1/0), and the other (Linear Regression) is for predicting a value.
So if you want to predict if you have cancer Y/N (or a probability) - use logistic. If you want to know how many years you will live to - use Linear Regression !
Regression means continuous variable, Linear means there is linear relation between y and x.
Ex= You are trying to predict salary from no of years of experience. So here salary is independent variable(y) and yrs of experience is dependent variable(x).
y=b0+ b1*x1
We are trying to find optimum value of constant b0 and b1 which will give us best fitting line for your observation data.
It is a equation of line which gives continuous value from x=0 to very large value.
This line is called Linear regression model.
Logistic regression is type of classification technique. Dnt be misled by term regression. Here we predict whether y=0 or 1.
Here we first need to find p(y=1) (wprobability of y=1) given x from formuale below.
Probaibility p is related to y by below formuale
Ex=we can make classification of tumour having more than 50% chance of having cancer as 1 and tumour having less than 50% chance of having cancer as 0.
Here red point will be predicted as 0 whereas green point will be predicted as 1.
Cannot agree more with the above comments.
Above that, there are some more differences like
In Linear Regression, residuals are assumed to be normally distributed.
In Logistic Regression, residuals need to be independent but not normally distributed.
Linear Regression assumes that a constant change in the value of the explanatory variable results in constant change in the response variable.
This assumption does not hold if the value of the response variable represents a probability (in Logistic Regression)
GLM(Generalized linear models) does not assume a linear relationship between dependent and independent variables. However, it assumes a linear relationship between link function and independent variables in logit model.
| Basis | Linear | Logistic |
|-----------------------------------------------------------------|--------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------|
| Basic | The data is modelled using a straight line. | The probability of some obtained event is represented as a linear function of a combination of predictor variables. |
| Linear relationship between dependent and independent variables | Is required | Not required |
| The independent variable | Could be correlated with each other. (Specially in multiple linear regression) | Should not be correlated with each other (no multicollinearity exist). |
In short:
Linear Regression gives continuous output. i.e. any value between a range of values.
Logistic Regression gives discrete output. i.e. Yes/No, 0/1 kind of outputs.
To put it simply, if in linear regression model more test cases arrive which are far away from the threshold(say =0.5)for a prediction of y=1 and y=0. Then in that case the hypothesis will change and become worse.Therefore linear regression model is not used for classification problem.
Another Problem is that if the classification is y=0 and y=1, h(x) can be > 1 or < 0.So we use Logistic regression were 0<=h(x)<=1.
Logistic Regression is used in predicting categorical outputs like Yes/No, Low/Medium/High etc. You have basically 2 types of logistic regression Binary Logistic Regression (Yes/No, Approved/Disapproved) or Multi-class Logistic regression (Low/Medium/High, digits from 0-9 etc)
On the other hand, linear regression is if your dependent variable (y) is continuous.
y = mx + c is a simple linear regression equation (m = slope and c is the y-intercept). Multilinear regression has more than 1 independent variable (x1,x2,x3 ... etc)
In linear regression the outcome is continuous whereas in logistic regression, the outcome has only a limited number of possible values(discrete).
example:
In a scenario,the given value of x is size of a plot in square feet then predicting y ie rate of the plot comes under linear regression.
If, instead, you wanted to predict, based on size, whether the plot would sell for more than 300000 Rs, you would use logistic regression. The possible outputs are either Yes, the plot will sell for more than 300000 Rs, or No.
In case of Linear Regression the outcome is continuous while in case of Logistic Regression outcome is discrete (not continuous)
To perform Linear regression we require a linear relationship between the dependent and independent variables. But to perform Logistic regression we do not require a linear relationship between the dependent and independent variables.
Linear Regression is all about fitting a straight line in the data while Logistic Regression is about fitting a curve to the data.
Linear Regression is a regression algorithm for Machine Learning while Logistic Regression is a classification Algorithm for machine learning.
Linear regression assumes gaussian (or normal) distribution of dependent variable. Logistic regression assumes binomial distribution of dependent variable.
The basic difference between Linear Regression and Logistic Regression is :
Linear Regression is used to predict a continuous or numerical value but when we are looking for predicting a value that is categorical Logistic Regression come into picture.
Logistic Regression is used for binary classification.