Gradient Decent in Linear Regression.pptx
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1) Gradient descent is an algorithm used to find the best fit linear regression line for a dataset by minimizing the cost function. 2) The cost function measures the error between predicted and actual values, using mean squared error. 3) Gradient descent works by iteratively calculating the partial derivatives of the cost function with respect to the slope and intercept and adjusting those values in the anti-gradient direction with each iteration until the cost is minimized.
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Linear regression has many practical uses. Most applications fall into one of the following two broad categories:
If the goal is error reduction in prediction or forecasting, linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables. After developing such a model, if additional values of the explanatory variables are collected without an accompanying response value, the fitted model can be used to make a prediction of the response.
If the goal is to explain variation in the response variable that can be attributed to variation in the explanatory variables, linear regression analysis can be applied to quantify the strength of the relationship between the response and the explanatory variables, and in particular to determine whether some explanatory variables may have no linear relationship with the response at all, or to identify which subsets of explanatory variables may contain redundant information about the response.
CHAPTER TENObjectiveA brief introduction of the basic concept
CHAPTER TEN
Objective:
A brief introduction of the basic concepts of Forecasting Tools like Moving average, Weighted Moving Average, Exponential Smoothing will be used to develop projection models.
Chapter Content:
Forecasting techniques:
Into our class we will use a simple product to manufacturer. A plush eraser will be our product (Note: Don’t blame my drawing, only look at and enjoy it.).
The first technique will be Moving Average (MA). This forecasting technique consists in the estimate of a average value from historical data that move as the new present value it’s know. This average is determinated by a series of terms or established periods (n). The quantity of periods (n) will be based in the variation that it exists between the historical data. If there is large variation, the value (n) must be greater to reflect the variation. If there is small variation, the value (n) can be smaller.
Let us suppose that the following table shown the eraser’s demand for first six months of production.
Period (Month)
Demand
1
1250
2
1590
3
1340
4
1510
5
1486
6
1440
Using the Moving Average equation:
(
)
n
i
t
A
t
MA
å
-
=
)
(
Where: MA(t) is the forecasting for period t
A(t-i) is the present for period t-i
(n) is the number of periods to average
If we looking for the forecasting for the fifth period, using n=2 and n=3, which would be the answer?
N=2
N=3
A(4) = 1510
A(4) = 1510
A(3) = 1340
A(3) = 1340
---------------
A(2) = 1590
Σ = 2850
Σ = 4440
n = 2
n = 3
MA(5) = 1425
MA(5) = 1480
The average changes of period when calculating the next forecasting. When forecast the sixth period, the terms to be used for the average change according to the following example:
N=2
N=3
A(5) = 1486
A(5) = 1486
A(4) = 1510
A(4) = 1510
---------------
A(3) = 1340
Σ = 2996
Σ = 4336
n = 2
n = 3
MA(5) = 1498
MA(5) = 1445.3
The next technique known like Weighted Moving Average, this technique to difference of regular moving average, each period have a weight assigned as output probability. The Moving average to divide the periods sum between the value (n), indirectly,
it’s giving the same probability o weight to each period to determine the forecasting.
(
)
(
)
å
-
-
=
i
t
xW
i
t
A
t
WMA
)
(
Where t is the hoped period and i value run from 1 to n.
Example, determining the sixth forecasting, with (n) = 2
Mov. Average Reg
Weighted MA
A(5) = 1486 x (50%)
A(5) = 1486 x (75%)
A(4) = 1510 x (50%)
A(4) = 1510 x (25%)
MA(6) = 1498
WMA(6) = 1492
This technique allow to assign a weight or probability according to expect behavior from marketing influences. I.e. to assign greater weight to the period value most recent a cause of a promotion. The quantity of periods or term to be used for estimate the forecast depends of the variation that exists between the historical data. That means, follow the same concept of moving average.
The third technique is the Exponential Smoothing. This forecasting technique allows a ...
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The document discusses linear regression, an algorithm used in machine learning and statistics. Linear regression is a simple and well-understood algorithm that finds a linear relationship between input variables (x) and an output variable (y). It assumes the dependence of y on x1, x2, etc. is linear. The document provides examples of using linear regression to predict house prices based on characteristics or advertising sales based on budgets. It also discusses calculating the mean squared error to evaluate a regression model's performance and the gradient descent algorithm used to minimize error and learn the model's weights.
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5) The black line is the regression line because it get closest to the sample points (minimizes error between the points and the line). The red line has a larger error; that is, larger total distance from points to the line.
6) Because it is reasonable to suppose that costs are dependent on production volume (since units are produced, directly resulting in costs), then regression is more appropriate for this data since regression is appropriate when an cause-and-effect relationship is assumed.
7) C
8) a) r = 0.8;
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c) p = 0.117
d) There is no evidence of a significant correlation between X and Y in the population because we did not reject the null of H0: = 0.
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10) Yes, there appears to be a straight line relationship between the variables. Linear regression appears to be appropriate. The regression output is:
11) a) T = -0.09, p = 0.929, do not reject Ho, conclude there is no evidence of a relationship
b) R2 = 0.002 = 0.2%, No because value is very close to zero
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d) Regression line is Y^ = 1.26 – 0.035X.
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CHAPTER TENObjectiveA brief introduction of the basic concept
CHAPTER TEN
Objective:
A brief introduction of the basic concepts of Forecasting Tools like Moving average, Weighted Moving Average, Exponential Smoothing will be used to develop projection models.
Chapter Content:
Forecasting techniques:
Into our class we will use a simple product to manufacturer. A plush eraser will be our product (Note: Don’t blame my drawing, only look at and enjoy it.).
The first technique will be Moving Average (MA). This forecasting technique consists in the estimate of a average value from historical data that move as the new present value it’s know. This average is determinated by a series of terms or established periods (n). The quantity of periods (n) will be based in the variation that it exists between the historical data. If there is large variation, the value (n) must be greater to reflect the variation. If there is small variation, the value (n) can be smaller.
Let us suppose that the following table shown the eraser’s demand for first six months of production.
Period (Month)
Demand
1
1250
2
1590
3
1340
4
1510
5
1486
6
1440
Using the Moving Average equation:
(
)
n
i
t
A
t
MA
å
-
=
)
(
Where: MA(t) is the forecasting for period t
A(t-i) is the present for period t-i
(n) is the number of periods to average
If we looking for the forecasting for the fifth period, using n=2 and n=3, which would be the answer?
N=2
N=3
A(4) = 1510
A(4) = 1510
A(3) = 1340
A(3) = 1340
---------------
A(2) = 1590
Σ = 2850
Σ = 4440
n = 2
n = 3
MA(5) = 1425
MA(5) = 1480
The average changes of period when calculating the next forecasting. When forecast the sixth period, the terms to be used for the average change according to the following example:
N=2
N=3
A(5) = 1486
A(5) = 1486
A(4) = 1510
A(4) = 1510
---------------
A(3) = 1340
Σ = 2996
Σ = 4336
n = 2
n = 3
MA(5) = 1498
MA(5) = 1445.3
The next technique known like Weighted Moving Average, this technique to difference of regular moving average, each period have a weight assigned as output probability. The Moving average to divide the periods sum between the value (n), indirectly,
it’s giving the same probability o weight to each period to determine the forecasting.
(
)
(
)
å
-
-
=
i
t
xW
i
t
A
t
WMA
)
(
Where t is the hoped period and i value run from 1 to n.
Example, determining the sixth forecasting, with (n) = 2
Mov. Average Reg
Weighted MA
A(5) = 1486 x (50%)
A(5) = 1486 x (75%)
A(4) = 1510 x (50%)
A(4) = 1510 x (25%)
MA(6) = 1498
WMA(6) = 1492
This technique allow to assign a weight or probability according to expect behavior from marketing influences. I.e. to assign greater weight to the period value most recent a cause of a promotion. The quantity of periods or term to be used for estimate the forecast depends of the variation that exists between the historical data. That means, follow the same concept of moving average.
The third technique is the Exponential Smoothing. This forecasting technique allows a ...
Linear_Regression
The document discusses linear regression, an algorithm used in machine learning and statistics. Linear regression is a simple and well-understood algorithm that finds a linear relationship between input variables (x) and an output variable (y). It assumes the dependence of y on x1, x2, etc. is linear. The document provides examples of using linear regression to predict house prices based on characteristics or advertising sales based on budgets. It also discusses calculating the mean squared error to evaluate a regression model's performance and the gradient descent algorithm used to minimize error and learn the model's weights.
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This document provides an introduction to machine learning concepts including supervised and unsupervised learning. It discusses linear regression with one variable and multiple variables. For linear regression with one variable, it describes the hypothesis function, cost function, gradient descent algorithm, and makes predictions using a housing dataset. For multiple variables, it introduces feature normalization and applies the concepts to predict housing prices based on size, bedrooms and price in a real estate dataset. The document provides code examples to implement the algorithms.
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The document discusses various regression techniques including ridge regression, lasso regression, and elastic net regression. It begins with an overview of advancements in regression analysis since the late 1800s/early 1900s enabled by increased computing power. Modern high-dimensional data often has many independent variables, requiring improved regression methods. The document then provides technical explanations and formulas for ordinary least squares regression, ridge regression, lasso regression, and their properties such as bias-variance tradeoffs. It explains how ridge and lasso regression address limitations of OLS through regularization that shrinks coefficients.
BUS 305 SOLUTIONS TOPRACTICE PROBLEMS EXAM 21) B2) B3.docx
BUS 305: SOLUTIONS TO
PRACTICE PROBLEMS EXAM 2
1) B
2) B
3) No, fan pattern (heteroscedasticity)
4) No, nonlinear relationship between X and Y
5) The black line is the regression line because it get closest to the sample points (minimizes error between the points and the line). The red line has a larger error; that is, larger total distance from points to the line.
6) Because it is reasonable to suppose that costs are dependent on production volume (since units are produced, directly resulting in costs), then regression is more appropriate for this data since regression is appropriate when an cause-and-effect relationship is assumed.
7) C
8) a) r = 0.8;
b) T = 1.31;
c) p = 0.117
d) There is no evidence of a significant correlation between X and Y in the population because we did not reject the null of H0: = 0.
9) Note: the following are not complete answers to Question 11; they are just enough for you to know whether your short answer addressed the correct things.
a) 1 = population slope, b1 = sample slope. On exam, would also want to address what you know (or don’t know) about each of these and how each is found.
b) An outlier can “drag” the regression line toward it. On the exam, also think about how this would affect the quality of your regression model and the predictions.
10) Yes, there appears to be a straight line relationship between the variables. Linear regression appears to be appropriate. The regression output is:
11) a) T = -0.09, p = 0.929, do not reject Ho, conclude there is no evidence of a relationship
b) R2 = 0.002 = 0.2%, No because value is very close to zero
c) Correlation = r = -0.0421. No, there is not a strong relationship between these variables. The correlation is nearly 0.
d) Regression line is Y^ = 1.26 – 0.035X.
Y^ = 1.26 – 0.035(100) = 1.26 – 3.5 = -2.24. No this does not make sense because you cannot have a negative number of near misses. It is not wise to predict with this model. The R-squared value is extremely low (essentially 0%), which means that there is no relationship at all between near misses and flights in this data. Therefore, predicting misses from flights is meaningless.
e) b1 = -0.035. As Number of flights increases by 1, we expect number of near misses to go down by 0.035. Or, put another way, as flights increases by 1000, we expect number of near misses to go down by 35. No, this does not make sense. We would assume that as flights increase, so would near misses.
12) a. Multiple regression is a direct extension of simple regression, except that now we have more than one independent (X) variable.
b. Note: the following is not a complete answer; it is just enough for you to know whether your short answer addressed the correct things: Multicollinearity is when the independent variables are highly correlated with one another. On the exam, also indicate how this affects the model, how one can identify if it is present, and what can be done to correct it.
c. Dummy variables are us ...
Direct methods
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficients with one column replaced by constants.
3) Gaussian elimination transforms the matrix of coefficients into upper triangular form using elementary row operations, then back substitution can be used to solve for the unknowns.
Direct Methods to Solve Linear Equations Systems
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2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
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Direct methods
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2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
Linear regression
Machine learning is a type of artificial intelligence that allows systems to learn from data and improve automatically without being explicitly programmed. There are several types of machine learning algorithms, including supervised learning which uses labeled training data to predict outcomes, unsupervised learning which finds patterns in unlabeled data, and reinforcement learning which interacts with its environment to discover rewards or errors. Linear regression is an example machine learning model that fits a linear equation to describe the relationship between a dependent variable and one or more independent variables. It works by minimizing the residual sum of squares to find the coefficients that produce the best fitting line.
Applied Numerical Methods Curve Fitting: Least Squares Regression, Interpolation
Correction with the misspelled langrange.
and credits to the owners of the pictures (Fantasmagoria01, eugene-kukulka, vooga, and etc.) . I do not own all of the pictures used as background sorry to those who aren't tagged.
The presentation contains topics from Applied Numerical Methods with MATHLAB for Engineers and Scientist 6th and International Edition.
Chapter05
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Machine Learning Interview Question and Answer
These slides will help you to crack interviews for product-based companies if you are planning your career in Data Science, Artificial Intelligence, etc.
Ann a Algorithms notes
Gradient descent is an optimization algorithm used to minimize a cost function by iteratively adjusting parameter values in the direction of the steepest descent. It works by calculating the derivative of the cost function to determine which direction leads to lower cost, then taking a step in that direction. This process repeats until reaching a minimum. Gradient descent is simple but requires knowing the gradient of the cost function. Backpropagation extends gradient descent to neural networks by propagating error backwards from the output to calculate gradients to update weights.
Linear algebra application in linear programming
Linear programming is used to maximize or minimize quantities subject to constraints. It can be applied to problems with any number of variables and constraints, as long as the relationships are linear. Key aspects include defining an objective function to optimize, determining the feasible region where all constraints are satisfied, and finding extreme points where the objective function may be maximized or minimized. An example problem involves determining how to allocate candy mixtures to maximize revenue given constraints on available ingredients. The optimal solution is found at an extreme point within the bounded feasible region.
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This document provides an overview of kernel machines and the kernel trick in machine learning. It discusses how the kernel trick allows projecting data into a higher dimensional space to make it linearly separable. It describes using kernels like polynomial kernels in the dual formulation to calculate dot products without explicitly performing the projection. The kernel trick avoids having to compute in the higher dimensional space, improving computational efficiency.
Course Assignment : Skip gram
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- 1. Gradient Decent in Linear Regression
- 2. A linear regression model attempts to explain the relationship between a dependent (output variables) variable and one or more independent (predictor variable) variables using a straight line. This straight line is represented using the following formula:
- 3. Where, y: Dependent variable x: Independent variable m: Slope of the line (For a unit increase in the quantity of X, Y increases by m.1 = m units.) c: y intercept (The value of Y is c when value of X is 0)
- 4. The first step in finding a linear regression equation is to determine if there is a relationship between the two variables. We can do this by using Correlation coefficient and scatter plot. When a correlation coefficient shows that data is likely to be able to predict future outcomes and a scatter plot of the data appears to form a straight line, we can use simple linear regression to find a predictive function. Let us consider an example.
- 5. From the scatter plot we can see there is a linear relationship between Sales and marketing spent. Next step is to find a straight line between Sales and Marketing that explain the relationship between them. But there can be multiple lines that can pass through these points.
- 6. So how do we know which of these lines is the best fit line? That’s the problem that we will solve in this article. For this we will first look at the cost function.
- 7. Cost Function Cost Function The cost is the error in our predicted value. We will use Mean Squared Error function to calculate the cost.
- 8. Our goal is to minimize the cost as much as possible in order to find the best fit line. We are not going to try all the permutation and combination of m and c (inefficient way) to find best fit line. For that we will use Gradient Decent Algorithm.
- 9. Gradient Decent Algorithm Gradient Decent is an algorithm that finds best fit line for a given training dataset in less number of iteration. If we plot m and c against MSE, it will acquire a bowl shape (As shown in the diagram below)
- 10. For some combination of m and c we will get the least Error (MSE). That combination of m and c will give us our best fit line. The algorithm starts with some value of m and c (usually starts with m=0, c=0). We calculate MSE (cost) at point m=0, c=0. Let say the MSE (cost) at m=0, c=0 is 100. Then we reduce the value of m and c by some amount (Learning Step). We will notice decrease in MSE (cost). We will continue doing the same until our loss function is a very small value or ideally 0 (which means 0 error or 100% accuracy). Step By Step Algorithm Let m = 0 and c = 0. Let L be our learning rate. It could be a small value like 0.01 for good accuracy.
- 11. Learning rate gives the rate of speed where the gradient moves during gradient descent. Setting it too high would make your path instable, too low would make convergence slow. Put it to zero means your model isn’t learning anything from the gradients. 2. Calculate the partial derivative of the Cost function with respect to m. Let partial derivative of the Cost function with respect to m be Dm (With little change in m how much Cost function changes)
- 12. Similarly let’s find the partial derivative with respect to c. Let partial derivative of the Cost function with respect to c be Dc (With little change in c how much Cost function changes).
- 13. 3. Now update the current values of m and c using the following equation: 4. We will repeat this process until our Cost function is a very small (ideally 0). Gradient Decent Algorithm gives optimum values of m and c. With these values of m and c we will get the equation of the best-fit line and ready to make predictions.