Introduction to Machine Learning (Part-7)

 Linear Regression : In Linear regression, we are bound with the straight line which is our decision boundary. The value of y can be same for different values of x. Can be represented as : 

According to Gaussian :
 
P(y|x=x1) = (1/(√2π)*σ)* e-(y-mean)2/2σ2

Through linear regression equation we define yi' = β1x1 + β0

We are predicting value of y as   yi' = β1x1 + β0+ e  where e is the error or noise.

Let us assume  σ for every value of x is same and  β1 = 0 then,

                              β0*argmin C = Σni=1(yi -β0)2

                                By differentiating on both sides

                               dC/dβ0 = Σni=1(0 + 2β0 - 2yi) = 0   

                                     2Σni=1β0 - 2Σni=1yi = 0

                                         nβ0 - Σyi = 0

                                         β0 = Σyi/n

Here it comes out to be the best value for   β0 which is the mean average of y values observed in the data.

Now comes the question , How can we determine the Goodfit of our linear regression Model??

We can check the good fit of the linear regression model by the R2 matrix which generalize Linear regression.

                       R2  = 1 -Σni=1(yi - (Σβjxj + β0))2 / Σni=1(yi - mean)2

Where βjxj  is independent variable. 

The good fit can be done by matrices. The more the value of R close to 1 , it means independent variable has sizable impact on predictions of output. We don't want  βj  to be dependent on the range or domain of the variable.For the solution of the same we do normalization of variables beforehand. Normalization can be done for this using Min-Max Sealing . For that we define variable Z.

                                   Z = x - xmin/xmax - xmin

Now for the actual value of yi, 

                                        yi = β0 + Σmi=1βjxj + e

                                  Therefore  yi = yi' + e

                                Cost function (C) = Σni=1(yi - (Σβjxj + β0))2

                                                          =   yi - (Σβjxj + β0) - e



Comments

Post a Comment

Popular posts from this blog

Supervised Learning(Part-5)

Image
  Decision Trees (Continued) Attribute Selection Measures If the dataset consists of N attributes then deciding which attribute to place at the root or at different levels of the tree as internal nodes is a complicated step. By just randomly selecting any node to be the root can’t solve the issue. If we follow a random approach, it may give us bad results with low accuracy. For solving this attribute selection problem, researchers worked and devised some solutions. They suggested using some criteria like : Entropy, Information gain, Gini index, Gain Ratio, Reduction in Variance Chi-Square These criteria will calculate values for every attribute. The values are sorted, and attributes are placed in the tree by following the order i.e, the attribute with a high value(in case of information gain) is placed at the root. While using Information Gain as a criterion, we assume attributes to be categorical, and for the Gini index, attributes are assumed to be continuous. Entropy Entropy is a me
Read more

Supervised Learning(Part-2)

Image
  Support Vector Machine A Support Vector Machine (SVM) is a discriminative classifier formally defined by a separating hyperplane. In other words, given labeled training data, the algorithm outputs an optimal hyperplane which categorizes new examples.  In two dimensional space this hyperplane is a line dividing a plane in two parts where in each class lay in either side. Suppose given plot of two label classes on graph as shown in image. You might have come up with something similar to following image (image B). It fairly separates the two classes. Any point that is left of line falls into black circle class and on right falls into blue square   In real world application, finding perfect class for millions of training data set takes lot of time. This is called regularization parameter.  Two terms regularization parameter and gamma are tuning parameters in SVM classifier. Varying those we can achieve considerable non linear classification line with more accuracy in reasonable amount of
Read more

Convolutional Neural Networks(Part-2)

Image
  Cross- Correlation Cross-correlation  is a  measure of similarity  of two series as a function of the displacement of one relative to the other. This is also known as a  sliding  dot product  or  sliding inner-product . It is commonly used for searching a long signal for a shorter, known feature. It has applications in  pattern recognition ,  single particle analysis ,  electron tomography ,  averaging ,  cryptanalysis , and  neurophysiology .  Steps followed in Cross- Correlation Take two matrices with same dimensions. Multiply them one by one, element by element (i.e , not the dot product, just the simple multiplication). Sum the elements together. VGG16 VGG16 is a convolutional neural network model proposed by K. Simonyan and A. Zisserman from the University of Oxford in the paper “Very Deep Convolutional Networks for Large-Scale Image Recognition”.  The model achieves 92.7% top-5 test accuracy in ImageNet,  which is a dataset of over 14 million images belonging to 1000 classes. A
Read more