Showing posts with label Linear Regression. Show all posts
Showing posts with label Linear Regression. Show all posts

Wednesday, August 10, 2022

Essay: My Love For Linear Regression, LASSO, And Business Education

Two years ago, during the Summer of 2020, I was enrolled in a master’s program at Georgetown University in management. It is a generalist management degree where students learn about the broader aspects of management. 


Topics like Finance, Accounting, Economics, and Supply Chain Management were covered, as well as Consulting, and Business Communication. I didn’t end up finishing the program; I withdrew due to COVID health concerns.

 

Yet, despite what by all accounts would have been a negative experience, there was one course that resonated with me. This course was Business Analytics. The subject of Business Analytics was a discipline that I have read about in Harvard Business Review, but I never had a concrete perspective on what exactly the topic was. 


The course itself, on a fundamental level, is a statistics course combined with R programming language. The more I learned in the course, the broader my perspective grew. The course is so much MORE than another math class; the skills learned in this course, can aid business managers in making high-impact decisions that are useful in organizations.

 

Now, this class was tough. Homework assignments were 3, no, 4 hours long. Students needed to assimilate the statistical material learned in class and apply them to R programming language. 


In fact, coding was the main component of the class. My instructor, Professor Jose, had one goal for the entire class. His goal was to facilitate the learning environment so that students would be able to run a linear regression model from scratch. This is a goal that is worthy of anyone learning analytics. And I am optimistic about learning even more advance methods for making predictions given any data set.

 

Since Georgetown, I have taken two sequential semesters of statistics in my home town, Miami (at FIU), to broaden my skillset, and also to acquire familiarity with the mathematical concepts that I had difficulty understanding while I was at Georgetown.


I am so much better at statistics, having done this. As a side note, I recommend continuing education at the college level during the evening, in conjunction with work during the day. You can use what you learn in the classroom immediately the next day in your work.

 

At this point, I am looking into internships, training and development programs at management consulting firms, and even research positions, that will help me acquire more analytical skills. 


I am now enrolled in a Business Analytics program at DePaul University. For the next year or so, I will be learning how to use data to make predictions about certain business questions and problems. I couldn’t be more excited.

 

Now that you are acquainted with my situation, I will get to the point of this essay—Multivariate Linear Regression.

 

Since the summer of 2021, I have been studying and learning R programming, and my experience couldn’t be more satisfying. There is something empowering about typing in a line of code and NOT seeing an error message. Those of you who have been frustrated with programming and coding will know what I am referring to.

 

At DePaul, I am learning how to code linear regression models, and there are two fundamental ways of model building that have caught my interest. 


The first way to create a linear model is to build it from the bottom up. That is, you pick a response variable, the variable that you wish to predict, and a predictor variables, the variables that impact the response. 


The method for standard linear modeling is straightforward, you begin with a response, and build the model with predictors that have a high correlation with that response. You then build out your model, adding a single predictor at a time, until you find predictors that explain the most uncertainty accounted for in an Adjusted R-Squared measurement.

 

As amazing as this method is for building multivariate linear models, there is another method that I have fallen in love with since learning of it. This method is known as a LASSO regression. This method is the opposite of the standard approach to linear regression model building in that the user can build their models from the TOP down. 


Programmers can use an algorithm that exposes ALL of the variables in a data set, which scale down their statistical correlations if they are insignificant. Analysts can then “feature select” which variables offer the highest statistical significance and use them to build an effective regression model. This regularization technique is especially useful if there are dozens, if not hundreds, of predictor variables in a dataset.

 

The LASSO regression algorithm is more in-depth, and though I have a basic understanding of it, I plan to make use of my time to run one from scratch. The more variables, the more satisfying, the more glory! 


I wish to use this brief essay to document my initial interests in regression analysis so that I can begin to acquire more advanced skills, and eventually, add value to an organization or business entity.

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Linear Regression and Regularization: The LASSO algorithm, A Brief Essay

As you all know by now, I enrolled in DePaul University’s MS in Business Analytics program this year. I am loving the program so much. I am learning new ways of solving business problems, and I am meeting such nice folks in small teams that are assigned for homework and other assignments. My professors are amazing as well!

 

At this point, I have been learning R programming for the past 7 months, since before I matriculated into the MS program. And in that time, I have studied mostly linear regression. It is simple, powerful, and relatively easy to interpret. No math wizardry needed for this!

 

For simple linear regression, there are 4 values of interest:

 

1. There is a y-variable. This is known as the “response” variable (dependent variable), and we as analysts are interested in predicting this value.

 

2. There is the y-intercept.

 

3. There is an x-variable, and this is known as the “predictor” variable (independent variable). This variable IMPACTS the response variable. Conversely, the response variable is IMPACTED by the predictor variable. In simple linear regression, there is only one predictor variable, but in multivariate regression there can be two or more.

 

4. There is a very subtle e-value, which is the approximation of all estimation errors. “E” represents the sum of ALL the differences between the actual values for y and the predicted values for y. The goal regression is to find the best values of the b coefficient and the x coefficient such that “E” is minimized.

 

To regression!

 

Running the lm() function brings me such joy, it’s a strange sort of joy. I mean, what’s not to love? You pass two variables into the function, along with the name of the data set, and BOOM, R Studio will spit out a formula with very specific numbers that are attached to the four points listed above. 


If you don’t like your R Squared score—that pesky thing just never seems to get to 1—then you can try a different predictor variable to see if your model improves.

 

This strategy seems to work well, adding and subtracting a predictor until you find a decent model worth replicating. But what if you had dozens if not hundreds of variables in your data set?  


Further, what if you could add more than one predictor variable to your model that will be able to explain more of the data that you are trying to account for? On top of that, what if you had a way of determining which variable is statistically insignificant with regards to all the other variables in your data?

 

This is where Regularization and Feature Selection comes into play.

 

To make use of Regularization (the maximization of more significant variables) and Feature Selection (the minimization of less significant variables), you can use a specific method known as LASSO regression.


In short, the LASSO “penalizes more variables in the regression,” confirmed by one of my professors at DePaul University Kellstadt Graduate School of Business. This penalization reduces the chances of our model “overfitting” the data. This is ideal as an overfitting of the data will lead to a model predicting perfectly with data that it has seen, and poorly with data that it hasn’t seen.

 

So, by eliminating statistically insignificant variables, and keeping the regression equation to 2 or 3 predictors, you can reduce your chances of overfitting, AND increase the predictive power of your model. This is all thanks to our good friend LASSO! Thank you, my good friend LASSO!

 

I am going to post a Word document with all the code to run a very basic LASSO regression. This code was sourced from a book called, “Introduction to Machine Learning with R: Rigorous Mathematical Modeling” by Scott V. Burger.

 

However, I will not be using this algorithm with the data set he uses. Instead, I will be using the bikes.csv data set that is provided in another book called “Practical Machine Learning in R,” by Notre Dame Professors Fred Nwanganga and Mike Chapple.

 

The data sets that these authors used with their demonstration of their algorithms were meticulously sourced by them, and as such, I strongly encourage you to purchase their works. I promise you will learn so much, and these books are just so rewarding to learn from.

 

Thank you so much for reading, and I hope you enjoy the LASSO process.

 

Sources for your Interest:

 

Lecture and Class Materials presented at DePaul University, Kellstadt Graduate School of Business, in a class entitled “Fundamentals of Business Analytics,” 2022, Chicago, Illinois.

 

“Practical Machine Learning in R,” by Fred Nwanganga and Mike Chapple, Professors at Notre Dame Mendoza College of Business, 2020.

 

“Introduction to Machine Learning with R: Rigorous Mathematical Modeling,” by Scott V. Burger, 2018.

 

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Regression Analysis for School Project: A Template for How Porn Stars Can Do Machine Learning Projects Independently And Efficiently

WELCOME


Hello ladies! Thank you so much for your continued support. I love writing for all of you. 


Last week, I wrote a blog post about attending college, or at least considering the prospect of school. I will link that here.  Related to that, this post details a Final Exam Project that you will likely encounter if you decide to pursue an Business Analytics master's degree. 


Originally, it was completed using Microsoft Excel, but I adapted it for you using R script. The project itself is very basic. As objective as I tried to be, there is a level of subjectivity to the analysis. I don't go into too many details with the decisions that I took to arrive at the final model, and that is a flaw of this post. I hope that you can learn through example, in this case, as opposed to explanation. 


The point of this final was to use Regression to create a prediction equation for a data set (or business problem) of interest. The scenario is fictitious, of course. But I certainly had fun with it. If you care to try it for yourself, I will post the link to the Google Drive. You can find the link under the "Data Sets" tab on the home page. 


Without further delay, I hope you enjoy this sample project.


(As always, the code that you will run in R Studio is highlighted in BLUE).


CLASS FINAL PROJECT: Stair Mart Analysis


INTRODUCTION

 

I am a business analytics consultant, and I have been asked by the General Manager of the Stair Mart Stores chain to conduct a research project to determine which variables in a supplied data set should be used for predicting annual sales for the coming year.

 

As this is the first project in my professional career as a private consultant, I feel that Regression Analysis is a great place to start. By using a specific method called OLS, or the Ordinary Least Squares Method, I can determine which variables in the supplied data set will be correlated with each other, as well as the response variable, annual net sales (ann_net_sales).

 

The OLS method is widely used, and it is considered an industry “best practice,” because it measures the distance between “predicted values and actual values” in analysis. This distance between these two points is known as the “error” term or the “residual.” The smaller this term is, the more predictive a model can be.

 

VARIABLES IN THE DATA SET

 

The data set that was given to me by the General Manager of Stair Mart Stores has six variables, which I will now describe:

 

1.         “ann_net_sales”: This variable will be my response variable, or the variable of interest that I wish to make predictions using the OLS method. This variable describes annual sales representing each observation, or store location.


2.         “sq_ft”: This variable will be one of the five predictor variables that I will use to determine in my regression model. It describes the square footage, or the size of each store.


3.         “inv”: Another independent variable that I will use to determine a prediction equation for annual net sales. This variable describes inventory of goods in each store location.


4.         “adv”: This variable represents the number of advertising dollars spent at each store location. It will also be an explanatory variable, or independent variable, in my analysis.


5.         “dist_size”: This will be the fourth independent variable that I use to determine my regression equation, and it describes the quantity of sales in a district for every 1,000 families.


6.         “comp”: This is the final explanatory variable that I will use for my regression analysis, and it describes how many competing stores (or businesses that offer similar products) are in the vicinity of that specific All Greens store location.

 

There are some important considerations in the project. Firstly, the sample size of the data set contains a total of 27 All Greens store locations. So, this will be an obstacle when determining which variables are more significant than others. 


In conducting a project on a smaller sample size, I will need to account for more error in the regression equation. Second, I do not have access to competitor data, which would be vital for running an ANOVA test. I will proceed with caution.

 

GOAL AND METHOD

 

As I previously mentioned, my goal in this project is to determine a prediction equation using the OLS method. Since all of the variables are of numerical nature, I will operate under the assumption that a regression line can be used in this instance. However, there are some steps that I need to take in order to ensure that my assumptions hold merit:

 

1.         I will use scatter plots to graph each independent variable with the response variable, ann_net_sales, to determine if there is a “linear” relationship between them.


2.         I will build out a regression with as many of the significant variables that explain as much variation in our data as possible, using p-values and the “adjusted R-Squared” score.


3.         I will eliminate the variables with lessor importance so that I can build a regression equation with two or three explanatory variables.


4.         I will use “residual” analysis to ascertain whether my assumptions regarding my analysis can be utilized with confidence.

 

Finally, I will be using R studio to conduct my analysis.

 

SCATTER PLOT ANALYSIS

 

I will begin the analysis with scatter plots of all the explanatory variables (independent variables) compared to my response variable (dependent variable). For reference, my response variable is “ann_net_sales.”

 

setwd("C:/Users/firstnamelastinitial/Documents/R")
library(tidyverse)

## -- Attaching packages --------------------------------------- tidyverse 1.3.1 --

## v ggplot2 3.3.5     v purrr   0.3.4
## v tibble  3.1.4     v dplyr   1.0.7
## v tidyr   1.1.3     v stringr 1.4.0
## v readr   2.0.1     v forcats 0.5.1

## Warning: package 'stringr' was built under R version 4.1.2

## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

 

I will bring the data set that the General Manager supplied to me in the computing environment. I will simply name the data set “stair.”

 

stair <- read.csv("stair_mart.csv", header = TRUE)

 

I will now learn about the data set.

 

glimpse(stair)

## Rows: 27
## Columns: 6
## $ ann_net_sales <int> 231000, 156000, 10000, 519000, 437000, 487000, 299000, 1~
## $ sq_ft         <int> 3000, 2200, 500, 5500, 4400, 4800, 3100, 2500, 1200, 600~
## $ inv           <int> 294000, 232000, 149000, 600000, 567000, 571000, 512000, ~
## $ adv           <int> 8200, 6900, 3000, 12000, 10600, 11800, 8100, 7700, 3300,~
## $ dist_size     <dbl> 8.2, 4.1, 4.3, 16.1, 14.1, 12.7, 10.1, 8.4, 2.1, 4.7, 12~
## $ comp          <int> 11, 12, 15, 1, 5, 4, 10, 12, 15, 8, 1, 7, 3, 14, 11, 10,~

 

This data set is small, and it only contains 27 observations (rows) and 6 variables (columns). The variables have been mentioned in an earlier section of this report, and they are numeric in nature.

 

To continue with my scatter plot analysis, I need to create x and y-axis objects to code into separate scatter plots. Here are the objects:

 

ann_net_sales <- stair$ann_net_sales
sq_ft
<- stair$sq_ft
inv
<- stair$inv
adv
<- stair$adv
dist_size
<- stair$dist_size
comp
<- stair$comp

 

Now I can analyze each object separately against my response variable. The process should look like this:

 

Here is the relationship of the square footage and sales:

 

plot(sq_ft, ann_net_sales,
    
main="Effect of Square Feet on Annual Net Sales",
    
xlab="Size", ylab="Annual Net Sales")

abline(lm(ann_net_sales~sq_ft))


Scatter Plot 1

Here is the relationship of advertising costs and sales:

 

plot(adv, ann_net_sales,
    
main="Effect of Advertising on Annual Net Sales",
    
xlab="Advertising Costs", ylab="Annual Net Sales")

abline(lm(ann_net_sales~adv))


Scatter Plot 2


Here is the relationship of the inventory and sales:

 

plot(inv, ann_net_sales,
     main="Effect of Inventory on Annual Net Sales",
     xlab="Inventory", ylab="Annual Net Sales")

abline(lm(ann_net_sales~inv))


Scatter Plot 3

Here is the relationship of the district size and sales:

 

plot(dist_size, ann_net_sales,
     main="Effect of District Size on Annual Net Sales",
     xlab="DistricT Size", ylab="Annual Net Sales")

abline(lm(ann_net_sales~dist_size))


Scatter Plot 4

finally, here is the relationship between competitor locations and sales:

 

plot(comp, ann_net_sales,
     main="Effect of Competition on Annual Net Sales",
     xlab="Competitor Locations", ylab="Annual Net Sales")

abline(lm(ann_net_sales~comp))

Scatter Plot 5

To summarize the scatter plot analysis, all of the graphs contain explanatory variables that seem to display a LINEAR relationship with the response variable. All of the relationships are POSITIVE, except the relationship between competitor locations and annual sales.

 

What does this mean?

 

Relationships that have a POSITIVE LINEAR trend:

 

As there is an increase along the x-axis, the response variable along the y-axis also increases. It is sensible to assume that the larger the store (sq_ft), the more sales are expected to be, for instance.

 

Similarly, a larger inventory (inv) can lead to more sales as customers can expect to make purchases based on more goods that the store provides.

 

One might reasonably expect that a larger advertising budget (adv) can enable stores and businesses to have more visibility, which can increase the sales of the store.

 

The size of the district (dist_size) can also impact sales. The larger the district, the larger the potential for sales.

 

Relationships that have a NEGATIVE LINEAR trend:

 

As noted on the final scatter plot, having more competitors (comp) in the vicinity of your business can impact sales negatively. Meaning, as the number of competitors increases, the amount of sales decreases. This is understandable as consumers have more options to purchase similar product offerings.

 

Finally, I can conclude that regression analysis is appropriate for this data set, as each of the scatter plot reveal seemingly linear relationships between explanatory and response variables.

 

SUMMARY STATISTICS

 

For further analysis, I can derive descriptive statistics concerning the data set:

 

summary(stair$ann_net_sales)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
##     500   98500  341000  286574  450500  570000

summary(stair$sq_ft)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
##     500    1400    3500    3326    4750    8600

summary(stair$comp)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
##   0.000   4.000   8.000   7.741  12.000  15.000

summary(stair$dist_size)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
##   1.600   4.500  11.300   9.693  14.050  16.300

summary(stair$adv)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
##    2500    4800    8100    8100   10950   17400

summary(stair$inv)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
##  102000  204000  382000  387482  551000  788000

 

I can also create a correlation plot to determine which of the explanatory variables has the largest correlation with the response variable.

 

library(corrplot)

## Warning: package 'corrplot' was built under R version 4.1.2

## corrplot 0.92 loaded

stairnumeric <- stair
stair_correlations <- cor(stairnumeric)
corrplot(stair_correlations, type = "upper")


Top Half Of Correlation Plot



For further specificity:

 

corrplot.mixed(stair_correlations)


Correlation Plot With Decimals



All of the explanatory variables show a higher linear correlation with the response variable (ann_net_sales), so a regression analysis is appropriate for this project.

 

INFERENTIAL STATISTICS

 

As I mentioned in my introduction, I will be using OLS, or Ordinary Least Squares method, to calculate the smallest distance between the actual values and predicted values, as well as the Adjusted R-Squared Score, and significance levels depicted by p-values.

 

Before running the regression, I would like to note that typically, the more variables you add to a regression equation, the higher you can expect the R-Squared value to be. 


However, in my analysis, I will seek to eliminate explanatory variables with lessor importance so as to include the most significant variables for the General Manager to focus on in her sales strategy.

 

With this in mind, there is a cost that I must incur. At the expense of selecting the most significant explanatory variables, I will have a model with a smaller R-Squared score. This is a cost that is acceptable to me, as I want my recommendation to be practical and easily implemented by the General Manager.

 

It will be easier for the General Manager to implement a practical solution to her business if she has less variables (that have the highest significance) to worry about in her upcoming sales campaign. 


In other words, more variables can be more costly to focus on, so I will recommend to her the most important ones so that she can prioritize different features in the data.

 

Here is the output of the Regression with all five explanatory variables impacting our response variable, Annual Net Sales:

 

model_all <- lm(ann_net_sales~sq_ft + adv + comp + dist_size + inv)
summary(model_all)

##
## Call:
## lm(formula = ann_net_sales ~ sq_ft + adv + comp + dist_size +
##     inv)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -26338  -9699  -4496   4040  41139
##
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)   
## (Intercept) -1.886e+04  3.015e+04  -0.626 0.538372   
## sq_ft        1.620e+01  3.544e+00   4.571 0.000166 ***
## adv          1.153e+01  2.532e+00   4.552 0.000174 ***
## comp        -5.311e+03  1.705e+03  -3.114 0.005249 **
## dist_size    1.358e+04  1.770e+03   7.671 1.61e-07 ***
## inv          1.746e-01  5.761e-02   3.032 0.006347 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 17650 on 21 degrees of freedom
## Multiple R-squared:  0.9932, Adjusted R-squared:  0.9916
## F-statistic: 611.6 on 5 and 21 DF,  p-value: < 2.2e-16

 

Knowing that more explanatory variables can be beneficial in improving an R-Square score, it is no surprise that having all of the explanatory variables in our regression equation has yielded such a high score (99.2%). In fact, at this point, I can interpret this score as such:

 

“Our multiple linear regression model can explain roughly 99.2 percent of the variability in our data.”

 

In other words, this model is strong! Further, all the p-values of each explanatory variable are statistically significant given that they are all smaller than our alpha justified by a 5% significance level. The F statistic tells us that the model is statistically significant, as well.

 

If the General Manager had unlimited resources to improve her Annual Net Sales, she might conclude that these five variables would be sufficient in a prediction equation. But there is never a case in which resources are unlimited.

 

It is true that this model has a high predictive power! Generally speaking, the higher the R Square Score, the better. Yet, there are two variables that I wish to eliminate from this model as they are variables that the General Manager cannot control in her upcoming campaign. Specifically, I wish to eliminate competition size (comp) and the size of the district (dist_size). 


The General Manager CAN'T control how many competitors are in the vicinity of Stair Mart locations. Additionally, she CAN'T control the size of the district.

 

It is important to note that our model’s predictive power will suffer in the newest iteration of our prediction equation.

 

The remaining variables will remain in the model, so our new model will look like this:

 

model_final <- lm(ann_net_sales~adv + inv + sq_ft)
summary(model_final)

##
## Call:
## lm(formula = ann_net_sales ~ adv + inv + sq_ft)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -87920 -20388   6427  23885 140000
##
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)   
## (Intercept) -9.855e+04  2.158e+04  -4.566 0.000137 ***
## adv          1.788e+01  5.725e+00   3.123 0.004776 **
## inv          3.300e-01  1.396e-01   2.364 0.026904 * 
## sq_ft        3.380e+01  8.461e+00   3.995 0.000569 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 46470 on 23 degrees of freedom
## Multiple R-squared:  0.9482, Adjusted R-squared:  0.9415
## F-statistic: 140.4 on 3 and 23 DF,  p-value: 6.308e-15

 

As expected, our Adjusted R Square score is roughly 5 percentage points lower than the previous model (94.2%). The F-statistic and the p-values associated with this model are all acceptable. As an observation, the output is telling me that inventory (inv) is of lessor significance compared to the other variables in this model.


I can interpret this new model as:


"Advertising, inventory, and square footage explains roughly 94.15% of the uncertainty in predicted annual net sales, given an significance level of 5%."


With this in mind, I am going to KEEP this model as my final recommendation for the General Manager of Stair Mart so that she can control for all of the variables that are significant in the data set. 


Further, knowing that "inventory" is a less significant variable, the General Manager can fine-tune her modifications to the create the optimal prediction equation.

 

Here is what the model looks like in numerical terms:

 

“Predicted Value of Annual Sales = -98553+ 33.80 (sq_ft) + 0.33 (inv) + 17.88 (adv) + E”

 

RESIDUAL OUTPUT

 

Now I will confirm whether my assumptions have merit with a residual plot.

 

plot(model_final)


Residual Plot 1


 


Normality Plot



Residual Analysis Plot



Residual Analysis Plot 2


In our residual vs. fitted plot, there is no observable pattern. Therefore, I can conclude that a linear regression model is appropriate for this project. In fact, in ALL of the Residual Plots, there are no patterns displayed by the data points. This is a good sign, and legitimizes the use of a regression model.

 

CONCLUSION

 

This analysis was conducted for the General Manager of the Stair Mart Franchise stores in the mid-west. The purpose was to determine whether a regression analysis would be appropriate for determining Annual Net Sales for the upcoming year based on this past year’s data. 


Since the variables were mainly numerical, and the scatter plots and residuals confirmed the appropriateness of a regression style analysis, the General Manager now has a prediction equation that she can use to plan and strategize sales campaigns for the following year.

 

Given that these variables are in her control, she can influence specific factors—size of the stores (sq_ft), inventory volume (inv), and advertisement spending (adv)—to improve sales growth for Stair Mart. Here is the prediction equation that the General Manager should use in prioritizing variables of interest:

 

“Predicted Value of Annual Sales = -98553+ 33.80 (sq_ft) + 0.33 (inv) + 17.88 (adv) + E”

 

It is vital to note that there are some flaws within this analysis. 


Firstly, it would have been more advantageous to work with more observations in the data set, specifically, more store locations. Having a larger sample size would also allow an analyst to avoid heteroscedasticity in the model production process.

 

Second, having similar corresponding competitor data would have been useful for fine-tuning the final model, and it would have given greater insight on how to manipulate the explanatory variables in the final model.

 

Finally, additional variables and data sets like customer segments (age, SES, ethnicity), climate, supply chain factors, would prove to be useful in creating a richer analysis. Although, this type of data can be very costly to obtain.



THANK YOU!


Ladies, I hope this project will serve as an appropriate learning tool for your development and transition from the Sex Work Industry. Why can't porn stars learn data analytics and machine learning, too? 


Why is it that this type of information is only available to those who are privileged? This is just not right! 


I want this blog to serve as a safe haven for all of you ladies to learn, discover and explore the wonderful world of analytics and machine learning. Are you having as much fun as I am? I hope so....


Thank you ladies! 


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