#linear regression

Predicting Wine Quality by comparing Linear Regression with Machine Learning techniques. Comparing Linear Regression with kNN, Decision Tree and Random Forest with Bayesian Inference to Predict Wine Quality in Python. We use python and Jupyter Notebook to download, extract, transform and analyze data about the physicochemical properties which make up wine, and use them to predict…

Today’s Book of the Day is Mathematics for Machine Learning, written by Marc Peter Deisenroth, Aldo Faisal, and Cheng Soon Ong in 2020 and published by Cambridge University Press. Marc Peter Deisenroth is DeepMind Chair in Artificial Intelligence at the Department of Computer Science, University College London. His research areas include data-efficient learning, probabilistic modeling, and…

Introduction:

Multiple regression is a branch of linear regression which can be used to analyse more than two variables. In multiple regression there is one response and more than one predictor variables whereas in linear regression where one response variable and one predictor variable. The predicator variables are the dependent variables and the response variable are the independent variables. Considering the equation for multiple regression,

Y=mx1+mx2+mx3=b

Where Y is the response variable

m1, m2, m3 are predictor variables

Let us discuss two problems regarding multiple regression

Analysis using R:

Multiple regression using R is one of the widely and often used method which is easy to use and handle.

DATA SET USED:

· https://github.com/grantgasser/Complete-Multiple-Regression

Using this dataset, we study of the relation between degree of brand liking (Y) and moisture content (X1) and sweetness (X2) of the product, the following results were obtained from the experiment based on a completely randomized design.

Some of the steps which we has to be followed are

1. Load and view the dataset

2. Identifying the data linearity in R

3. Plotting the graph

4. Implementation of Multiple Regression

5. Prediction and Interpretation

Brand Preference:

In a small-scale experimental study of the relation between degree of the brand liking (Y) and moisture content (XI) and sweetness (X2) of the product, the following results were obtained from the experiment based on a completely randomized design (data are coded:)

Analyzation of the data:

Scatter plot:

The diagnostic aids show that firstly, there are no outliers and the distribution for each variable is normal. Additionally, looking at the correlation matrix, Y and X1 have significant positive correlation, Y and X2 are positively correlated, but less so than Y and X1 and there’s no correlation between X1 and X2.

Correlation Matrix:

The correlation matrix of the variables is plotted to check the correlation between the variables.

Summary:

The value of multiple R- squared is 0.9521 and the adjusted R- squared value is 0.9447. When the variable X2, is added to X1 we get a p-value of about 2.01e-05. F- statistic variable is larger than 1. Y= 37.65 + 4.425X1 + 4.375X2 is the result of the regression model. Holding the other variables constant, increasing one unit of X1 results in a 4.425 rise in brand liking degree, while increasing one unit of X2 results in a 4.375 increase in brand liking degree. Because the P values for each variable are less than 0.05, both X1 and X2 are significant.

QQ plot:

In this QQ- plot the points plotted all fall in the same line which clearly determines that the residuals follow normal distribution. There are no outliers and errors .

Shapiro test:

Shapiro Wilk test is a statistic normality test for a random data set. It can be used to analyse if the data set is normally distributed. By analysing the values, we get,

Model validation:

Regression vs Residual Plot:

The above given plot is a residual plot which indicates a pattern in the residuals and the fitted plot. Although the distribution appears to be pretty normal, there are outliers on both sides of the median, with more outliers to the right.

Breusch-Pagan test:

This is test can be used to determine whether the heteroscedasticity is present in a regression analysis

Prediction and Confidence level:

newX = data. frame (X1=newX1, X2 = newX2)

#Confidence interval (95%)

predict (fit, newX, interval="confidence")

#Prediction Interval (95%)

predict (fit, newX, interval="prediction"

Output:

Interpretation:

Linear Regression

The relation between a dependent and an independent variable can be seen or predicted using linear regression models. When two or more independent variables are employed in a regression analysis, the model is referred to as a multiple regression model rather than a linear model.

Simple linear regression is a technique for predicting the value of one variable from the value of another. With linear regression, the relationship between the two variables is represented as a straight line.

In multiple regression, a dependent variable has a linear relationship with two or more independent variables. The dependent and independent variables may not follow a straight line if the relationship is non-linear.

When two or more variables are used to track a response, linear and non-linear regression are used. The non-linear regression is based on trial-and-error assumptions and is comparatively difficult to implement.

 Multiple Linear Regression

Multiple linear regression is a statistical analysis technique that uses two or more factors to predict a variable's outcome. It is also known as multiple regression and is an extension of linear regression. The dependent variable is the one that has to be predicted, and the factors that are used to forecast the value of the dependent variable are called independent or explanatory variables.

 Analysts can use multiple linear regression to determine the model's variance and the relative contribution of each independent variable. There are two types of multiple regression: linear and non-linear regression.

 Multiple Regression Equation

The equation for multiple regression with three predictor variables (x) predicting variable y is as follows:

 Y = B0 + B1 * X1 + B2 * X2 +B3 * X3

The beta coefficients are represented by the "B" values, which are the regression weights. They are the correlations between the predictor and the result variables.

Yi is predictable variable or dependent variable

B0 is the Y Intercept

B1 and B2 the regression coefficients represent the change in y as a function of a one-unit change in x1 and x2, respectively.

 Multiple Linear Regression Assumptions

        I.            The Independent Variables are not Much Correlated

Multicollinearity, which occurs when the independent variables are highly correlated, should not be present in the data. This will make finding the specific variable that contributes to the variance in the dependent variable difficult.

    II.            Relationship Between Dependent and Independent Variables

Each independent variable has a linear relationship with the dependent variable. A scatterplot is constructed and checked for linearity to check the linear relationships. If the scatterplot relationship is non-linear, the data is transferred using statistical software or a non-linear regression is done.

 III.            Observation Independence

The observations should be of each other, and the residual values should be independent. The Durbin Watson statistic works best for this.

 Multiple Linear Regression in R

Analyzing the linear relationship between Stock Index Prices and Unemployment rate in the Economy.

Multiple linear regression can be done in a variety of methods, although statistical software is the most popular. R, a free, powerful, and easily accessible piece of software, is one of the most widely used. We'll start by learning how to perform regression with R, then look at an example to make sure we understand everything.

 Steps to Apply Multiple Linear Regression in R

Step 1: Data Collection

The data needed for the forecast is gathered and collected. The purpose is to use two independents

·         Unemployment Rate

·         Interest Rate

to forecast the stock index price (the dependent variable) of a fictional economy.

Step 2: Capturing the Data in R

Data Capturing using the code in R and Importing Excel file from save folder.

Step 3: Checking Data Linearity with R

It is critical to ensure that the dependent and independent variables have a linear relationship. Scatter plots or R code can be used to do this. Scatter plots are a quick technique to check for linearity. We need to make sure that various assumptions are met before using linear regression models. Most importantly, you must ensure that the dependent variable and the independent variable(s) have a linear relationship.

In this we'll check the Linear Relationship exist between:

·         Stock Index Price (Dependent Variable) and Interest Rate (Independent Variable)

·         Stock Index Price (Dependent Variable) and Unemployment Rate (Independent Variable)

Below is the code that is used in R to plot the relations between the dependent variable that is Stock Index Price and Interest Rate.

From the Graph we can notice that there is Indeed a Linear relationship exist between the dependent variable Stock Index Price and Independent variable Interest Rate.

In can be specifically noted that When interest rates rise, the price of the stock index rises as well.

In the second scenario, we can plot the link between the Stock Index Price and the Unemployment Rate using the code below:

As you can see, the Stock Index Price and the Unemployment Rate have a linear relationship: when unemployment rates rise, the stock index price drops. we still have a linear correlation, although with a negative slope.

Step 4: Perform Multiple Linear Regression In R

To generate a set of coefficients, use code to conduct multiple linear regression in R. Template to perform Multiple Linear Regression in R is as below:

M1 <- lm (Dependent Variable ~ First Independent Variable + Second Independent Variable, Data= X)

Summary (M1)

Using the Template, the Code follows:

We will get the following summary if you execute the code in R:

The model's residuals ('Residuals'). The model fulfils heteroscedasticity assumptions if the residuals are roughly centered around zero and have similar spread on both sides (median -6.248, and min and max -158.2 and 118.8). The model's regression coefficients ('Coefficients').

To construct the multiple linear regression equation, utilize the coefficients from the summary as follows:

Stock Index Price = (Intercept) + (Interest Rate) X1* (Unemployment Rate) X2

Once you've entered the numbers from the summary: 

Stock Index Price = (1798.4) + (345.5) X1 * (-250.1) X2

Adjusted R-squared: Measures the model's fit, with a higher number indicating a better fit.

The p-value is Pr(>|t|):  Statistical significance is defined as a p-value of less than 0.05.

Step 5: Make Predictions

 To predict the Stock Index Price from the collected Data it is noted that,

X1 <= Interest Rate = 1.5

X2 <= Unemployment Rate = 5.8

And when this data in equated into the regression Equation we obtain:

Stock Index Price = (1798.4) + (345.5) * (1.5) + (-250.1) * (5.8)

Stock Index Price = 866.066666

 The Final Predicted data for the stock Index Price using Multiple Linear Regression is 866.07.

Conclusion

The stock market and our economy's relationship frequently converge and diverges from one another. The gross domestic product, unemployment, inflation, and a slew of other indicators all represent the state of the economy. These trends are expected to show the economy and markets moving in lockstep in the long run.

When the unemployment rate is high, the Monetary Policy lowers the interest rate, which causes stock market prices to rise.

11/17/2020

Howdy guys! Sphelon here back from elsewhere

Time for another art post...well I have nothing else to show in the meantime cause I’m doing the remaing early FANFIC fanart in a while **BATTLE REHIME

please refer to my child account @sphelon8565​ tumblr page:

https://sphelon8565.tumblr.com/

------------------------------------Anyways back to this art post!

meet Harumi Yoshihara ~ A fresh college graduate from Japan in searching for an acting career to the WestCoast. 1st appearing in my ARTWORK since 2017 over here; I expanded her role as a protagonist in her own story yet to be developed:

https://www.deviantart.com/sphelon8565/art/Original-Character-AS-0004-676536771

Her story is that of a hard working average optimistic to down-to-earth lovely girl, yet is NOT without human errors though she could appear to be that of an average +A to +B College Student, she herself knows that she is sentimental, a person who values moral values and determinations alike and often stands up to her feet after certain things not getting in her own accord. A Well-rounded person, she sets herself a goal to become a well-known celebrity but doesn’t know where to start after botching up her theater audition in a casting role, thus begins her journey into the stars.

Her development sketches can be found here as early as this year and I may change her story over time.

https://amazingsphelon.tumblr.com/post/616411132081700864/original-character-sketches-i-did-last-year-2019

More details to come after this post soon....I have alot of sketches about her that I haven’t uploaded yet so in the meantime here’s a short showcase about her.