In each linear regression equation, there will be errors or deviations. The Least-Squares technique mentions the solution for minimising errors or squares of deviations. Usually, this method is implemented in data fitting. The use of linear regression Google Sheets helps you to determine the Least Squares error accurately.
The optimal result intends to decrease the residuals or sum of squared errors showing the differences between the experimental value and the equivalent fitted value stated in the model.
To find least squares, first, we define a linear relationship between the independent variable (X) and dependent variable (Y). It is vital to come up with the formula to determine the sum of errors' squares. Ultimately, this formula helps to find out the variation in observed data.
The linear relationship between these variables is:
Y = c + mX
The aim is to find the values of c and m to determine the minimum error for the specified dataset.
When employing the Least Squares method, we aim to minimise the error. So, now we must proceed with calculating the error. The loss function in machine learning indicates the difference between the actual value and the predicted value.
Let’s use the Quadratic Loss Function to measure the error. Its formula is:
c = y’ –mx’
In the equation of m, x’ shows the mean of all the values in the input X. y’ shows the mean of all the values in output variable Y. Users can retrieve further predictions by implementing corresponding linear regression programs in Python.
Using Python for further analysis of the Least Square method may not yield high accuracy as we simply take a straight line and force it to fit into the specified data optimally. However, it can be helpful to gauge the magnitude of the real value. It serves as an excellent first step for novices in Machine Learning. The Google Sheets linear regression can help measure the least square error.
When we fit a set of points to a regression line, it is assumed that some linear relationship exists between X and Y. The regression line lets you predict the target variable Y for an input value of X.
The following equation corresponds to this:
µY|X = α0 + α1X1
However, for any particular observation, there can be a deviation between the actual value of Y and the predicted value. These deviations are known as errors or residuals. The more efficiently the line fits the data, the smaller the error will be.
But how to find the regression line that best fits these data? Does it help calculate slope and intercept values for the particular regression line?
Finding a line capable of minimising model errors to fit data to the line manually is necessary. However, when you include data to the line, specific errors will be positive, while others will be negative. This means that few actual values would be greater than their predicted values. On the other hand, some of the actual values would also be lower than the predicted values.
When we add all the errors, the sum comes out to zero. So, the challenge is how to determine the overall error? The answer is squaring the errors and finding a line that minimises the sum of the squared errors.
∑e2=∑(Yt−Y’t)2
Here, e = error
Yt - Y’t = deviation between the actual and predicted value of the target variable.
With the above equation, the Least Squares method determines the values of slope and intercept coefficient. These coefficients will minimise the total squared errors. This method makes the sum of the square of the errors as tiny as possible. Hence, the total is the least likely value when all errors are squared and added.
In linear regression, the regression coefficients let you predict an unknown variable's value with a known variable's help. The variables in a regression equation get multiplied by some magnitudes. These magnitudes are regression coefficients. Based on the regression coefficients, the linear regression plots the best-fitted line.
This section helps you thoroughly learn regression coefficients, their formula, and their interpretation.
Regression coefficients are approximations of specific unknown parameters to determine the relationship between a predictor variable and the actual variable. These coefficients help predict the value of an unknown variable with the help of a known variable.
Linear regression analysis measures how a change in an independent variable affects the dependent variable using the best-fitted straight line.
Before finding the values of regression coefficients, you must check whether the variables adhere to a linear relationship or not. You can use the correlation coefficient and interpret the equivalent value to check this.
Linear regression aims to find the straight line equation that establishes the relationship between two or multiple variables. Suppose we have a simple regression equation: y = 5x + 3. Here, 5 is the coefficient, x is the predictor, and 3 is the constant term.
According to the equation of the best-fitted line Y = aX + b, the formulas for the regression coefficients are:
Use this equation to find the coefficient of X:
n is the number of data points in the specified data sets; its formula is:
Now insert the values of regression coefficients in Y= n + mX
Understanding the nature of the regression coefficient assists you in predicting the unknown variable. It gives an idea of the amount the dependent variable changes with a unit change in an independent variable.
If the sign of regression coefficients is positive, there is a direct relationship between these variables. So, if the independent variable increases, the dependent variable increases, and vice versa.
If the sign of regression coefficients is negative, there is an indirect relationship between these variables. So, if the independent variable increases, the dependent variable decreases, and vice versa.
Using regression Google Sheets can provide the exact interpretation of regression coefficients.
Linear regression is a statistical technique to comprehend the relationship between variables x and y. Before conducting linear regression, make sure the below assumptions are met:
If there is a violation of assumptions, linear regression results can be unreliable.
Every assumption discussed below explains how to determine linear regression if it's met and steps to perform if the assumption violates.
It assumes the existence of a linear relationship between the dependent variable (y) and the independent variable (x).
The easiest way to determine assumption fulfilment is to prepare a scatter plot of x vs. y. It helps you visually know the linear relationship between these variables. If the plot shows points falling across a straight line, there is some kind of linear relationship between them, and this assumption is fulfiled.
Solutions to try if this assumption is violated:
When you prepare a scatter plot of x and y values, notice no linear relationship exists between them, you have the following options:
i. Implement a non-linear transformation to the independent or dependent variables. You can implement non-linear transformation using log, square root, or reciprocal of the independent or dependent variables.
ii. Add an independent variable to the model.
In this assumption, the residuals are independent. There is zero correlation between successive residuals in the time series data. It implies residuals do not steadily grow more prominent with time.
Observing a residual time series plot is the easiest way to check this assumption fulfilment, showing a graph of residuals vs. time. Most of the residual autocorrelations must be inside the 95% confidence bands close to zero. These are present at approx. +/- 2 above the square root of n (where n is the sample size). The Durbin-Watson test also helps you check the fulfilment of this assumption.
Solutions to try if this assumption is violated:
Here are a few solutions you can try based on how this assumption is violated:
If the serial correlation is positive, add lags of the dependent or independent variable to a particular model.
For the serial correlation to be negative, ensure no variables are over-differenced.
For periodic correlation, add periodic dummy variables to the model.
In this assumption, the residuals bear constant variance at each level of x. The existence of heteroscedasticity in a regression analysis makes it difficult to rely on the analysis results. Particularly, heteroscedasticity enlarges the difference in the regression coefficient estimates. There are high odds for the regression model to state that a term in the model is statistically substantial, although it’s not.
The easiest way to recognise heteroscedasticity is to create a fitted value vs. residual plot. After fitting a regression line to a data set, you can prepare a scatterplot representing the model’s fitted values vs. residuals of corresponding values. With the increase in the fitted values, the residuals spread out more, and the cone shape shows the existence of heteroscedasticity.
Solutions to try if this assumption is violated:
i. Transformation of the dependent variable:
The common way of transforming the dependent variable is to take its log. For example, suppose we use population size to predict the total number of fruit shops in a town. Here, the population size is the independent variable, and the number of fruit shops is a dependent variable. We can use the log of the dependent variable (population size) instead of the dependent variable itself to predict the number of fruit shops. Following this approach usually eliminates heteroscedasticity.
ii. Weighted regression:
This form of regression allocates weight to every data point depending on the variance of its fitted value. It provides small weights to those data points bearing higher variances, decreasing their squared residuals' value. Overall, proper weights can discard heteroscedasticity.
iii. Redefine the dependent variable:
A typical method to redefine the dependent variable is using a rate instead of the raw value. Let’s consider the example discussed in solution-i. Rather than using the population size to predict the number of fruit shops in a town, use population size to indicate the number of fruit shops per capita.
In many cases, this approach decreases the variability between more significant populations as we measure the number of fruit shops per person instead of the absolute amount of fruit shops.
Assumption-4: Normality
The model’s residuals are normally distributed.
How to determine Normality assumption:
i. Visual testing using Q-Q plots.
A Q-Q plot (quantile-quantile plot) is helpful to know whether a model’s residuals obey a normal distribution. The normality assumption is fulfiled when the points on the plot coarsely create a straight diagonal line.
ii. Using formal statistical tests:
This solution checks the normality assumption using formal statistical tests such as Shapiro-Wilk, Jarque-Barre, Kolmogorov-Smirnov, or D’Agostino-Pearson. These tests are sensitive to huge sample sizes. They usually determine that the residuals are not normal when the sample size is big. Therefore, using graphical methods like the Q-Q plot to test this assumption is better.
Solutions to try if this assumption is violated:
Firstly, make sure any outliers don’t lay an immense influence on the distribution. If outliers are present, you need to confirm their real values; no data entry errors are allowed.
You can implement a non-linear transformation to the dependent or independent variable. For example, you can apply the dependent or independent variable's square root, log, or reciprocal.
A simple method to plot a graph of X and Y is to use the Google Sheets linear regression. In Google Sheets, the linear regression line plots the data with the help of a scatter plot. You need to choose the data range to plot (including headers). Next, open the Insert menu, and choose the Chart option A. It will insert a new chart. The Chart Editor sidebar will be shown.
The linear regression equation depicts the linear relationship between X and Y variables. It is identical to the slope formula.
Linear Regression Formula:
Y= nX + m
The least-squares is the greatest technique to fit a regression line in an XY plot. It determines the best-fitting line for a specific data set. This is because it decreases the sum of squares of the vertical variance from every data point to the line.
If a point accurately rests on the fitted line, the value of its upright variance is 0. The reason is the variations are first squared and added. Hence, their negative and positive values will not be annulled. The corresponding straight line is the least-squares regression line (LSRL).
Let’s assume X as an independent variable and Y as a dependent variable. The equation of the population regression line:
Y = α0 + α1X
here, α0: constant
α1: regression coefficient
If a random sample of observations is considered, the equation of the regression line is:
ŷ = α0+ α1x
here x: independent variable
ŷ: the predicted value of the dependent variable
α0: constant
α1: regression coefficient
A linear regression line equation is:
Y = m + nX
Here the X (independent) variable is plotted on the X-axis and the Y (dependent) variable on the Y-axis. The m is the intercept (value of y when x = 0), and n is the slope of the line.
Multiple Regression Line Formula:
y= m +n1x1 +n2x2 + n3x3 +…+ ntxt + u
Examples to solve the linear regression equation:
Let’s find a linear regression equation for the two sets of data:
x1 = 2, x2 = 4, x3 = 6, x4 = 8
y1 = 3, y2 = 5, y3 = 7, y4 = 9
Find the value of Σx, Σy, Σx2, and Σxy.
Based on the given data set,
x12 = 4, x22 = 16, x32 = 36, x42 = 64
x1y1 = 6, x2y2 = 20, x3y3 = 42, x4y4 = 72
Σx = x1+ x2 + x3 + x4 = 20
Σy = y1+ y2 + y3 + y4 = 24
Σx2 = x12+ x22 + x32 + x42 = 120
Σxy = x1y1 + x2y2 + x3y3 + x4y4 = 140
Using the formula of the linear equation y=m+nx, calculate the values of m and n.
Using the formula, find the value of a and b
here s = number of datasets = 4
Now put all the calculated values in the above equations of m and n.
m = ((24×120) − (20×140))/((4×120)−400)
m = 1
n = 4((140) – (20×24))/((4×120)−(400))
n = -17
Hence, m = 1 and n = -17
So, the linear equation Y = m + nx is now Y = 1 - 17x
When you pursue one of the best linear regression courses, you become familiar with all aspects of linear regression, including how to find the equation of the regression line. The corresponding linear regression training aims to acquaint students with how to plot the regression line from the derived equation.
The standard form of the linear regression line equation is y = mx + c. It denotes an equation of a line with a y-intercept of c and a gradient of m. It needs the y-intercept ‘c’ of the line and the slope value ‘m’. Another name for this equation is the slope-intercept form. In machine learning and artificial intelligence, this equation helps predict the values depending on the values of the input variable.
Here is the description of the slope and intercept:
Slope: The alphabet m denotes the gradient or slope of the line. Its value can be positive, negative, or zero. Furthermore, one can calculate from the tangent of the inclination angle of this line concerning the X-axis or a line parallel to the X-axis.
Intercept: The alphabet c denotes the intercept of the line. This intercept measures the length at which the line crosses the y-axis from the origin. The intercept is also indicated through the point (0, c) on the Y-axis. At this point, the line is passing. The point (0, c) is c units far from the origin.
How to derive y = mx + c equation:
You can derive this equation from other significant forms of equations of a line. A few of these forms of equations are as below:
Deriving the Slope Formula:
The slope formula helps to derive the y = mx + c equation. It takes (0, c) point on the Y-axis and an arbitrary point (x, y) over the line. Using these two points, find the slope ‘m’. It first calculates the variance of the y coordinates of these two points and then divides it by the difference of the x coordinates among them.
m = (y - c)/(x - 0)
m = (y - c)/(x)
mx = y - c
So, y = mx + c
Hence, the Slope Formula helps derive the line equation's slope-intercept form.
Point Slope Form:
To derive this form of the line equation, you need the slope of the line and a point. Suppose the slope of the line is m, and the point is (0,c). These two values help find the point-slope form equation as below:
(y - c) = m(x - 0)
y - c = mx
So, y = mx + c
Hence, the Point-Slope form helps derive the line's y = mx + c equation.
To calculate the correlation coefficient, firstly, you need to determine the variables' covariance. The covariance value is then divided by the multiplication of standard deviations of the given variables.
Here is the equation to find the correlation coefficient:
ρxy = Cov(x,y) /σxσy
here, ρxy: Pearson's product-moment correlation coefficient
Cov(x,y): Covariance of variables x and y
σx: Standard deviation of x
σy: Standard deviation of y
In the above formula, the term Cov(x,y) is covariance. It provides the joint relationship among two random variables. Its formula is:
here,
n = Total number of values of x or y
x,y = random variables
xi = data value of x
yi = data value of y
x’= mean of all the values of x
y’= mean of all the values of y
The formula for the correlation coefficient is:
Here,
n = Total number of values of x or y
Σx = Total of all values of the first variable
Σy = Total of all values of the second variable
Σxy = Sum of products of x and y values
Σx2 = Sum of squares of the first variable
If one independent variable is considered, it is known as simple linear regression. It is known as multiple linear regression if numerous independent variables are used. Simple linear regression helps you estimate the relationship between two quantitative variables. You can use the simple linear regression Google Sheets to analyse simple regression further accurately.
Simple linear regression is practical when you want to know:
The value of a dependent variable at a specific value of the independent variable (for example, the number of sales at a specific festive season)
How powerful the relationship between two variables is (for example, the relationship between the number of sales and the festive season)
Assumption of Simple Linear regression:
It assumes a linear relationship between the independent and dependent variables. The best fit line across the data points is a straight line.
Objectives of Simple Linear regression algorithm:
The examples include the relationship between salary and experience, investment and sales, income and expenditure, etc., to model the relationship between the two variables. Forecasting observations like weather forecasting based on temperature, a company's income based on the investment in a year, etc.
The equation of the Simple Linear Regression model:
y= α0+ α1x+ ε
here,
α0= the intercept of the regression line (obtained with x = 0)
α1= the slope of the regression line (denotes whether the line is increasing or decreasing)
ε = the error term.
Let’s understand Simple Linear Regression with a practical example:
For example, a social researcher wants to establish the relationship between salary and happiness. Suppose there are 200 people surveyed whose salary ranges from 20k to 70k, and they are asked to rank their happiness on a scale of 1 to 5.
When you detect the linear relationship between the dependent and independent variable, the linear regression algorithm is more suitable than other machine learning algorithms due to its ease. Moreover, the equations of linear regression are easy to interpret and easy to master.
Linear regression perfectly fits linearly separable datasets and is frequently used to determine the nature of the relationship among variables. Obtaining a linear regression certification implies comprehending the less complex linear regression models.
Although linear regression is prone to over-fitting, one can prevent it with the help of certain dimensionality, reduction techniques, cross-validation, and regularisation (L1 and L2) techniques. The regularisation technique is easy to implement and can competently reduce the complexity of a linear regression function. Hence, it decreases the risk of overfitting.
i. Assumes independence between variables:
The linear regression algorithm assumes a linear relationship between dependent and independent variables. It assumes a straight-line relationship and predicts the independence between attributes.
ii. Incomplete description of relationships among variables:
Linear regression focuses on the relationship between the mean of the independent variables and the dependent variables. The mean is not a comprehensive description of a single variable. Hence, linear regression is not a complete description of relationships between variables.
iii. Susceptible to underfitting:
Underfitting occurs when a machine learning model cannot capture the data. Typically, this situation is observed when the hypothesis function can’t correctly fit the data. Linear regression undertakes a linear relationship between input and output variables. Hence, it can’t correctly fit the complex data.
iv. Low accuracy:
In most real-life scenarios, there doesn’t exist a linear relationship among the dataset's variables. Therefore, a straight line doesn't correctly fit the data. For such cases, a more complex function can effectively capture the data leading linear regression models to show low accuracy.
A dataset’s outliers are variances or extreme values that diverge from the other data points of a particular distribution. These data outliers can significantly degrade the performance of a machine learning model. Hence, they usually lead to low accuracy of the models.
Practical Application-1:
Businesses frequently use linear regression to determine the relationship between income and advertising expenditure. We can understand this application from the perspective of linear regression.
For example, businesses may use a simple linear regression model where advertising expenditure is considered the predictor variable and income is the response variable.
So, the equation of the regression model becomes:
Income = β0 + β1 (ad expenditure)
Coefficient β0 shows the total expected income when ad expenditure is zero.
Coefficient β1 shows the average change in the total income when ad expenditure increases by one unit (for example, one dollar).
If the β1 value is negative, there is more ad expenditure and less income.
If the β1 value is close to zero, the ad expenditure has little impact on income.
If β1 is positive, more ad expenditure is linked with more income.
Based on the value of β1, an organisation can decide whether to increase or decrease its ad spending.
Practical Application-2:
Medical researchers frequently use the linear regression algorithm to determine the relationship between patients' blood pressure and drug dosage.
Suppose the researchers observe different drug dosages in patients and notice the change in their blood pressure. This application can be mapped as a simple linear regression model, with dosage as the predictor variable and blood pressure as the response variable.
So, the equation of the linear regression model will be:
Blood pressure = β0 + β1 (drug dosage)
Coefficient β0 shows the expected blood pressure when drug dosage is zero.
Coefficient β1 shows the average change in patients' blood pressures when drug dosage increases by one unit.
If the β1 value is negative, the drug dosage increases with a decrease in blood pressure.
If the β1 value is close to zero, the drug dosage increases without changing blood pressure.
If β1 is positive, the drug dosage increases with an increase in blood pressure.
Based on the value of β1, medical researchers may alter the drug dosage for the patient.
Practical Application-3:
Data scientists for professional dance teams usually use the linear regression model to determine the effect of various dance training programs on dancers’ performance.
For example, suppose data scientists want to examine how weekly cardio sessions and Zumba sessions influence the total points a dancer scores. In this case, make a multiple linear regression model with the help of cardio sessions and Zumba sessions as the predictor variables and total score points as the response variable.
The equation of the linear regression model will be:
Total points scored = β0 + β1 (cardio sessions) + β2 (zumba sessions)
Coefficient β0 shows the predictable points a dancer scores who don’t participate in cardio and Zumba sessions.
Coefficient β1 shows the average change in total points scored when weekly cardio sessions increase by one. Here, the assumption is the number of weekly Zumba sessions stays unchanged.
Coefficient β2 shows the average change in total points scored when weekly Zumba sessions increase by one. Here, the assumption is the number of weekly cardio sessions stays unchanged.
Based on the β1 and β2 values, the data scientists recommend each dancer in which session they must participate to maximise their points.
The following section highlights major advantages of pursuing an online Tableau course over an offline one:
Training from industry experts:
The superlative aspect of pursuing a linear regression online course is that candidates will not suffer political limitations when learning. You can apply for any available Tableau online courses depending on your previous knowledge of this domain, budget, and schedule.
Plenty of offline Tableau courses may not have qualified and experienced industry experts or may include only a few industry experts. Conversely, any linear regression online course is fully equipped with sufficient numbers of trained and experienced industry experts. Therefore, the students attain enough guidance and can solve their doubts in a user-friendly manner.
In any Tableau course, the instructors hold the practical experience and accurately know how to teach each Tableau and linear regression concept. However, offline mentors might lack this aspect.
Leading online Tableau and linear regression courses derive inspiration for projects from practical scenarios. It assists the learners in perceiving the existing market condition and understanding patterns existing in the data visualisation. Moreover, it opens exciting opportunities to learners hoping to get into reputable organisations.
Additional benefits of online Tableau courses compared to offline Tableau courses:
Good choice for freshers, students, job seekers, and professionals
Imparts Tableau teaching with Live projects and Demo projects
Thorough guidance is provided with the backup classes and video-recorded classes
Provides remote assistance for all-inclusive support to students
Each online batch has a limited number of students
Overview and uses of Tableau
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The factors influencing the Tableau industry growth in the period 2022-23 are increasing execution of cloud computing services, increasing demand for data generation, high demand for Internet penetration, enhanced infrastructural development, and the upward trend of Bring-Your-Own-Device (BYOD).
The Tableau industry’s market value in the period 2022-23 is$1,016.5 Mn
This industry provides the services likely to accelerate with the continual developments in business intelligence technologies.
Tableau is the simplest visualisation tool for data handling. Certainly, automation requires a huge volume of data analyses, and Tableau serves crucial data for that. Furthermore, Tableau is also producing its personal space in domains, for example, business intelligence and data analytics. From this aspect, we can understand the importance of linear regression in data analytics.
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Candidates completing the Tableau course can effectively deal with the assigned duties in any job role like data analysts, data professionals, business analysts, and more. It requires professional intelligence to deal with sensitive data. As a result, it creates a huge demand for skilled and certified Tableau professionals. Furthermore, the massive demand for linear regression models in machine learning applications increases the need for pursuing Tableau courses in India.
If you have made up your mind to work in an MNC, the likelihood of getting recruited in any of the below job profiles can be increased after you finish the Tableau course in India:
Consultant in Tableau
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Dell
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In India, the average salary offered to Tableau Specialists is approx. INR 14 lac per year. Experienced Tableau Specialists can receive a salary up to 20 lac per year.
The salary of a Tableau Specialist can differ based on several factors. Here we outline a few factors:
Recruiter | Average Salary (per annum) |
TEKsystems | INR 20,00,000 - INR 21,00,000 |
Cognizant Technology Solutions | INR 6,13,815 |
TCS | INR 5,22,782 |
Accenture | INR 7,87,484 |
Infosys | INR 5,64,972 |
Beatinfo Consulting | INR 7,32,949 |
Tech Mahindra | INR 5,01,134 |
Job locations | Average Salary (per annum) |
Bangalore | INR 10,00,000 |
Chennai | INR 9,97,000 |
Hyderabad | INR 9,75,000 |
Pune | INR 8,50,000 |
Gurgaon | INR 7,80,000 |
Job titles | Average Salary (per annum) |
Data Analyst | INR 12,30,000 |
Business Objects Developer | INR 5,43,000 |
Microstrategy Developer | INR 5,15,000 |
Business Intelligence Developer | INR 3,60,000 |
A Tableau Specialist's salary abroad can vary based on many factors. Here we outline a few factors:
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Scotiabank | $83K - $89K |
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Operations Analyst | $97,817 |
Senior Operations Analyst | $101,045 |
Senior Business Analyst | $124,000 |
Senior Information Security Analyst | $160,425 |
Senior Reporting Analyst | $162,000 |
Technical Consultant | $131,625 |
Over and above the general knowledge of computer applications, candidates equipped with other pertinent skills can obtain higher-paying jobs in different countries. Some of these skills are the cutting-edge business intelligence technologies such as Microsoft Power BI and Oracle BI.
The knowledge and working experience with SQL Server Data Tools, namely SQL Server Integration Services (SSIS), SQL Server Analytics Services (SSAS), and SQL Server Reporting Services (SSRS), can assure admirable paying jobs for the Tableau Specialists abroad.
The know-how of ETL Tools (including Talend and Informatica) can enhance your chances of grabbing an exceptional job as a Tableau Specialist Abroad.
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Analyse movie data from the past 100 years and find out various insights to determine what makes a movie do well.
Solve a real industry problem through the concepts learnt in exploratory data analysis
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This is an ETL project which will cover the topics like Apache Sqoop, Apache Spark and Apache Redshift
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One of the most widespread methods to enhance the accuracy of a linear regression model is “The Outlier Treatment.” This method is quite useful for boosting accuracy because the regression is quite sensitive to outliers. Hence, it becomes crucial to treat outliers with proper values.
The linear regression graph shows that it fits a straight line, minimising the inconsistencies between the predicted and the original output values. The relationship between the variables is linear. Francis Galton first used the term ‘regression’ in his 1866 paper entitled ‘Regression towards mediocrity in hereditary stature’. He only utilised the word in the perspective of regression toward the mean. Subsequently, the term was used by others to indicate linearity, and therefore, linear regression is called linear.
Linear regression analysis helps predict a variable's value depending on another variable's value. The variable whose value you want to predict is the dependent variable. The variable used to indicate the value of another variable is known as the independent variable. Linear regression analysis helps to know which predictors in a model are statistically essential and which are not. Moreover, this analysis can provide a confidence interval for every regression coefficient it estimates.
Linear regression predicts a dependent variable value (b) depending on the given independent variable (a). It models the linear relationship between one or more variables. Every observation comprises two values. One is for the dependent variable, and another is for the independent variable. Linear regression works such that it allows the model to predict outputs for inputs it has never observed before.
A regression line can feature a Positive or Negative Linear Relationship. If the dependent variable progresses on the Y-axis and the independent variable progresses on the X-axis, it is called a Positive linear relationship. Conversely, if the dependent variable’s value reduces on the Y-axis and the independent variable’s value increases on the X-axis, it is called a Negative linear relationship.
The Cost function optimises the regression coefficients and measures the performance of a linear regression model. It helps find the accuracy of the mapping function, which maps the input variable to the output variable. Moreover, it enables you to determine the optimal values for a0 and a1 that offers the best fit line for given data points. The alternate name of this mapping function is the Hypothesis function.
Multiple regression analysis is a statistical method for examining the relationship between a single dependent variable and multiple independent variables. Its key objective is to use those independent variables whose values can forecast the value of the single dependent variable.
Yes, Ordinary least squares (OLS) is a linear least squares method for assessing the unknown parameters within a linear regression model. It is the technique used to determine the simple linear regression of a given data set. OLS estimates the relationship between the variables by minimising the sum of the squares in the variance between the observed and predicted values of the dependent variable aligned as a straight line.
Linear Regression deals with regression problems, whereas Logistic regression deals with classification problems. Linear regression offers a continuous output, while Logistic regression offers discrete output. Linear Regression aims to find the best-fitted line, but Logistic regression fits the line values to the sigmoid curve. The method to calculate loss function in linear regression is mean square error but its maximum likelihood estimation in the logistic regression.
You get a negative value of the linear regression coefficient when the value of the independent variable increases with the decrease in the value of the dependent variable. The negative coefficient value indicates how much the mean of the dependent variable differs when there is a one-unit change in the independent variable. In this evaluation, the values of other variables stay constant.
In linear regression, overfitting takes place when the model is very complex. Usually, this situation arises when there are more parameters than the number of observations. A linear regression model with overfitting will not perfectly generalise to new data. So, it will perform efficiently on training data but poorly on test data. Factors responsible for overfitting are –(i) outliers in the train data and (ii) train and test data belonging to different distributions.
When using a linear regression model, the widespread way to fit curves for the data is to incorporate polynomial terms like cubed or squared predictors. Commonly, you need to select the model order based on the number of bends you require in your line. Every increment in the exponent generates one more bend into the curved fitted line.
One of the key assumptions of linear regression is that the residues aren’t correlated. This is often not the case with the time series data. In case there are autocorrelated residues, linear regression cannot capture all the trends within the data. Therefore, linear regression is generally not used for time series.
Including outliers and influential cases can significantly alter the magnitude of regression coefficients. They can also change the coefficient signs, i.e. from negative to positive or vice versa. Their empirical results can be erroneous when abnormal observations are ignored, specifically concerning dependent variables.