Test: Regression Analysis - Civil Engineering (CE) MCQ

Concept:

Learning algorithms: These algorithms are used in machine learning to help technology in human learning process. It is used to process data to extract patterns appropriate for application in a new system.

Explanation:

From above given options, only a) and b) are learning algorithms.

a) Logistic regression: It is used for binary classification problems. It is used to examine and describe the relationship between a binary variable and set of predicator variables.  The primary objective of logistic regression is to model the mean of the response variables, given a set or predicator variables.

b) Back propagation: It is the essence of neural net training. It is the method of fine tuning the weights of a neural net based on the error rate obtained in the previous iteration. It is a standard method of training artificial neural networks. 

Concept:

Learning algorithms: These algorithms are used in machine learning to help technology in human learning process. It is used to process data to extract patterns appropriate for application in a new system.

Explanation:

From above given options, only a) and b) are learning algorithms.

a) Logistic regression: It is used for binary classification problems. It is used to examine and describe the relationship between a binary variable and set of predicator variables.  The primary objective of logistic regression is to model the mean of the response variables, given a set or predicator variables.

b) Back propagation: It is the essence of neural net training. It is the method of fine tuning the weights of a neural net based on the error rate obtained in the previous iteration. It is a standard method of training artificial neural networks. 

Formula

The regression line of y on x is given as

y - y̅ = r × σ_y(x - x̅)/σ_x

Calculation

According to the question

y - 2 = 0.8 × 9(x - 1)/3

⇒ y - 2 = 2.4(x - 1)

∴ y = 2 + 2.4(x - 1)

The regression coefficient are independent of the change of the origin. But , they are not independent of the change of the scale. It means there will be no effect on the regression coefficient if any constant is subtracted from the values of x and y

∴ After subtracting constant 60 from each value of X and Y, the regression coefficient is not changed.

Given

X̅ = mean of sales = 40

Y̅ = mean of advertisement cost = 6

Formula

Regression equation of X and Y = X - X̅ = r(x/y)(Y - Y̅)

Calculation

⇒ X - 40 = 0.09(10/1.5)(Y - 6)

⇒ X - 40 = (9/1.5)(Y - 6)

⇒ X - 40 = 6(Y - 6)

⇒ X - 40 = 6Y - 36

⇒ X = 6Y + 4

⇒ X = 6(10) + 4

∴ The likely sales for a proposed advertisement expenditure of Rs. 10 crore is 64 crore.

• If a quantitative variable in regression analysis has "m" categories, one can add "m-1" dummy variables to the model. Dummy coding or indicator variable coding are terms used to describe this method.
• To represent categorical data in a regression model, utilize dummy coding. One category of the original variable is identified as the reference or baseline category, and one binary (dummy) variable is created for each of the other categories. Usually, the category with the lowest or most frequent value is the reference category.

Hence, In Regression Analysis, if a quantitative variable has 'm' categories, one can introduce Only m -1 dummy variables.

 The correct answer is Options A, C, and D only.

• Option A: Dimension reduction methods are used to reduce the number of predictor components. This is done by identifying the underlying patterns in the data and then representing the data in a lower-dimensional space.
• Option C: Dimension reduction methods can provide a framework for interpretability of the results. This is because it can help to simplify the data and make it easier to understand the relationships between the variables.
• Option D: Dimension reduction methods can help to ensure that these components are independent. This is because the goal of dimension reduction is to identify the underlying patterns in the data, and independent components do not share any common patterns.
• Option B is incorrect because dimension reduction methods do not necessarily ensure that the components are dependent. In fact, the goal of dimension reduction is to identify the underlying patterns in the data, and independent components do not share any common patterns.
• Option E is incorrect because dimension reduction methods are used to reduce the number of predictor components, not increase them

Given

Regression lines

x + 2y - 5 = 0

2x + 3y - 8 = 0

var(x)= σ_x = 12

Calculation

x + 2y - 5 = 0    ------(i)

Let y = - x/2 + 5/2 be the regression line of y on x [ from equation 1]

2x + 3y - 8

x = -(3/2)y + 8/2 be the regressiopn line of x on y

⇒ b_xy = -1/2 and b_yx = -3/2

b_xy = Regression line of y on x

b_yx = Regression line of x on y

We know that regression coefficient = r = √(b_yx × b_xy)

⇒ r = √(-1/2 × -3/2)

∴ r = √3/2 < 1

σ_x = 12 = 2√3

We know that b_yx = r (σ_y/σ_x)

⇒ -1/2 = √3/2 (σ_y/2√3)

⇒ σ_y = - 2

∴ var(y) = variance of y =(-2)² = 4

The given regression equations are:

y = 0.516x + 33.73 ⇒ 0.516 x – y = -33.73

x = 0.512 y + 32.52 ⇒ x – 0.512 y = 32.52

By solving the above two equations, we get

x = 67.6, y = 68.6

Important Points:

The line of regression of y on x is

The line of regression of x on y is

 is called the regression co-efficient of y on x and is denoted by b_yx

 is called the regression co-efficient of x on y and is denoted by b_xy

If the line is in the form of y₁ = m₁ x₁ + c₁, then the regression co-efficient byx = m₁

If the line is in the form of x₂ = y₂ m₂ + c₃, then the regression co-efficient bxy = m₂

Correlation co-efficient 

If x and y are uncorrelated variables then there can not be any line of regression between them drawn and therefore there is no line of regression relationship between them

∴ Option a is correct

Important Points

The line of regression = The line  y = a + bx  which is fitted  to a set of n points (x_i, y_i) by the method of least square called line of regression of y on x. similarly if regression x = a + by is fitted to (x_i, y_i) is called line of regression of x on y

The coefficient of correlation is the GM between the regression coefficient.

Y = a + bX

Where, a and b are constants

b = it is called the regression coefficient of Y on X and is denoted by b_yx. It measures the change in Y corresponding to a unit change in X.

Thus, b_yx = Slope of the line of regression of Y on X = b

X = c + dY

where, c and d are constants

d = it is called the regression coefficient of X on Y and is denoted by b_xy. It measures the change in X corresponding to a unit change in Y.

Thus, b_xy = Slope of the line of regression of X on Y = d

Correlation coefficient = √b_yx b_xy

⇒ √bd