Mastering Multiple Linear Regression with Python for Artificial Intelligence

Welcome to our comprehensive guide on mastering Multiple Linear Regression (MLR) for Artificial Intelligence (AI) using Python. If you're eager to unlock the power of data-driven decision-making and build cutting-edge AI models, you're in the right place!

In today's digital age, AI is transforming industries, revolutionizing processes, and reshaping the way we approach problem-solving. At the heart of many AI applications lies MLR, a fundamental statistical technique that enables us to analyze relationships between multiple independent variables and predict outcomes with remarkable accuracy.

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But before we dive into the intricacies of MLR and its applications in AI, let's take a step back and understand why mastering this technique is crucial for success in the ever-evolving landscape of artificial intelligence.

Why Master MLR for Artificial Intelligence?

MLR serves as the backbone of many Artificial Intelligence algorithms, providing a solid foundation for understanding complex datasets, identifying patterns, and making informed predictions. Whether you're a seasoned data scientist, an aspiring AI enthusiast, or a business leader looking to leverage AI for strategic decision-making, proficiency in MLR is indispensable.

By mastering MLR, you gain the ability to:

• By mastering MLR, you gain the ability to:

  1. Extract valuable insights from diverse datasets: MLR equips you with the tools to uncover hidden relationships between variables, enabling you to extract actionable insights and drive data-informed decisions.

  2. Build predictive models with precision: With Multiple Linear Regression, you can develop sophisticated predictive models that accurately forecast future outcomes, anticipate trends, and optimize business processes.

  3. Enhance Artificial Intelligence capabilities across industries: From owners-operated HVAC service to transportation businesses, MLR empowers organizations to harness the full potential of AI, revolutionizing industries and driving innovation at scale.

  4. Optimize fleet management and customer behavior: Leveraging MLR insights, transportation businesses can analyze customer behavior, optimize routes, and enhance fleet management, leading to improved operational efficiency and customer satisfaction.

  5. Engage subscribers with tailored content: For non-profit organizations, MLR enables the analysis of subscribers' interests and preferences, facilitating the creation of personalized content and events that resonate with their audience, driving engagement and loyalty.

  6. Maximize college marketing efforts: In the realm of college marketing, MLR offers a strategic approach to understanding potential student reach, identifying target demographics, and optimizing marketing campaigns to attract prospective students effectively.

What to Expect in This Guide

In this comprehensive guide, we'll walk you through everything you need to know to master MLR for AI. From understanding the fundamentals of MLR and implementing Python code to handling multicollinearity issues and optimizing model performance, we've got you covered.

Each section is carefully crafted to provide practical insights, actionable strategies, and real-world examples, ensuring that you not only grasp the concepts but also gain the confidence to apply them in your AI projects.

So, whether you're embarking on your AI journey or looking to elevate your existing skills, join us as we unravel the mysteries of MLR and embark on a transformative journey towards AI mastery.

Ready to unlock the secrets of MLR and revolutionize your approach to artificial intelligence? Let's dive in!

The Power of Multiple Linear Regression (MLR) for Artificial Intelligence

Welcome to the gateway of AI mastery! In this section, we'll embark on an enlightening journey into the world of Multiple Linear Regression (MLR) and its profound implications for artificial intelligence. Get ready to demystify complex datasets, uncover hidden patterns, and unleash the predictive prowess of MLR like never before.

Understanding the Essence of MLR

At its core, MLR represents the cornerstone of data-driven decision-making, offering a systematic approach to understanding relationships between multiple variables and predicting outcomes with precision. Unlike its predecessor, Simple Linear Regression which One-to-One relation (also called BiVariate), Multiple Linear Regression (MLR) is One-to-Many (also called MultiVariate), allows us to navigate the intricate web of interconnected variables, making it a vital tool in the arsenal of Artificial Intelligence practitioners.

The Art of Data Preparation

Before diving headfirst into MLR, it's imperative to lay the groundwork with meticulous data preparation. From defining independent and dependent variables to conducting relevance checks and addressing missing values, each step in the data preparation process plays a pivotal role in shaping the success of our MLR endeavors.

1. Define the Independent Variable IV (Y)
2. Generate a list of potential Dependent Variables DVs (X)
3. Prepare Dummy Variable (more about this later)
4. Inspect the dataset for missing values (if necessary remove records with missing data)
5. Ideally, run a relevance check, the relationship between each IV and DVs using scatter plots or correlations
6. Check the relationships between the independent variables (Colinearity check or Multicollinearity - critical)
7. Remove Redundant DVs
8. Find and Remove Outliers
9. If Necessary Segment your Multiple Linear Regression into Sub_MLRs

Navigating the Dataset: A Practical Example

Let's embark on a practical journey into the heart of MLR by exploring a rich housing dataset, brimming with insights just waiting to be unearthed. Our dataset, sourced from OpenML.org, offers a treasure trove of information, with 81 columns and 1460 rows waiting to reveal their secrets.

In our quest to predict the sale value of residential properties with unparalleled accuracy, we'll scrutinize a myriad of factors, ranging from square footage and bedroom counts to bathroom amenities and beyond. However, to ensure our analysis remains focused and relevant, we've narrowed our scope to properties with a "Normal Sale" condition, excluding outliers and irrelevant data points that may skew our results.

As we delve deeper into the dataset, we encounter a diverse array of variables, including both numerical values and categorical variables. To effectively handle this heterogeneity, we'll leverage the power of dummy variables, a potent technique for encoding categorical data into a format suitable for MLR analysis.

But our journey doesn't end here. With the dataset in hand, we'll embark on a voyage of exploration and discovery, breaking the data into training and testing sets to facilitate model development and validation. Through meticulous preparation and strategic analysis, we'll unlock the latent potential of our dataset, paving the way for groundbreaking insights and transformative AI applications.

So, strap in and prepare for an exhilarating journey into the heart of data-driven decision-making. The road ahead may be challenging, but with perseverance and ingenuity, we'll unravel the mysteries of MLR and unlock the keys to AI innovation. Welcome aboard, fellow adventurer—the adventure begins soon.

Scaling the MLR Summit

As we ascend the MLR summit, scalability emerges as a guiding principle in our quest for AI mastery. By embracing an iterative approach to model building and optimization, we equip ourselves with the agility to tackle real-life challenges and forge ahead on the path to AI enlightenment.

Your Multiple Linear Regression Journey Begins Here

With the groundwork laid and your tools at the ready, it's time to embark on your MLR odyssey. Armed with insights and strategies from our previous discussions, you're now well-equipped to navigate the intricate world of Multiple Linear Regression (MLR) and unlock its transformative potential for artificial intelligence.

Stay tuned as we delve deeper into the nuances of MLR, unraveling its mysteries and unlocking the gates to AI innovation. The future of AI awaits, and your journey begins now.

Code and Analysis:

import pandas
import numpy
from sklearn import linear_model

df = pandas.read_csv("housing_openml_v1.csv")

X = df[['GrLivArea', 'BedroomAbvGr','FullBath']]
y = df['SalePrice']

X = X.iloc[:,:].values # Independent variable (X) 
y = y.iloc[:].values   # Dependent variable (Y)

regr = linear_model.LinearRegression()
regr.fit(X, y)

sale_Price =154000 # Actual price $154,000
predicted_Price = regr.predict([[1060, 3, 1]])

numpy.set_printoptions(precision=0)
Err = predicted_Price - sale_Price
print('Predicted Price: $', predicted_Price, "Error: $", Err, "    Error Percent:  %", 100*Err/sale_Price)

CoeffDet = regr.score(X,y)*100
print('Coeff Det R^2: {:.1f}'.format(CoeffDet),'%')

Assessing Predictive Accuracy

The results are in, and they reveal a predicted sale price of $111,917, a considerable deviation from the actual price of $154,000. This discrepancy translates to an underprediction of $42,083, representing a significant -27% error. At first glance, such a high error rate may raise questions and concerns about the reliability of our model.

However, it's essential to contextualize these findings within the broader framework of MLR analysis. While the -27% error pertains to a single data point, it may not accurately reflect the overall performance of our algorithm. To gain a more comprehensive understanding, we turn to the Coefficient of Determination, or R-squared, a powerful metric that evaluates the model's ability to explain the variability in the dataset.

Unveiling the Coefficient of Determination

At 62.7%, the Coefficient of Determination paints a more nuanced picture of our model's performance. This metric indicates that approximately two-thirds of the variation in housing prices can be attributed to the variables we've selected for our MLR analysis. While not perfect, this level of explanatory power suggests that our model captures a significant portion of the underlying patterns in the data.

This visualizes the journey into MLR, providing insights into both the challenges and the potential of this powerful technique for AI innovation.

Predicted Price:     $111,917     |     Error:     -$42,083     |     Error Percent:      -27%
Coeff Det R^2:      62.7 %

Navigating Challenges: Managing Data Complexity

As we delve deeper into our analysis, questions arise:

• Did the buyer overpay?
• Is the algorithm flawed?
• Can we optimize its performance?

Evaluating the effectiveness of the algorithm demands scrutiny, yet with 81 rows in our dataset, tackling multicollinearity tests and scatter matrix visualizations presents a formidable task.

Strategic Data Inspection

In confronting this data deluge, strategic data inspection emerges as our beacon of clarity. Leveraging Python's versatile tools, we employ cool functions like 'describe' to swiftly glean insights into critical dataset characteristics. This approach streamlines our analysis, enabling us to prioritize key data facets while maximizing efficiency.

Harnessing the Power of 'describe'

By harnessing the 'describe' function, we gain a panoramic view of our dataset's statistical landscape, from mean values to quartile ranges. This holistic perspective empowers us to detect anomalies and inconsistencies, informing decisions on data preprocessing and model refinement.

#typicall it is df.describe(include='all')
#since the dataset is to big we use the print() or we out the result to a separate file 
print(df.describe(include='all'))
temp = df.describe( include='all')
temp.to_csv('describe_output.csv')

Unveiling Insights from the 'describe' function output

Upon scrutinizing the 'describe_output.csv', a revealing discrepancy emerges: the standard deviation for 'GrLivArea' exceeds expectations, suggesting potential outliers or data irregularities. Swiftly pivoting to 'LotArea' as a more reliable predictor yields improved results, enhancing the robustness of our MLR model.

Key Takeaways: Insights and Learnings

Our journey through data exploration underscores the pivotal role of tools like 'describe' in uncovering hidden insights and anomalies. By detecting outliers early in the process, we refine our feature selection and bolster the accuracy of our MLR model.

In essence, the 'describe' function serves as our compass, guiding us towards data-driven decision-making and empowering us to extract meaningful insights. As we continue our voyage through MLR and AI, let us carry forward the lessons learned from our 'describe' journey—lessons that will shape our data analysis and model development endeavors for years to come.

Visualizing Data Relationships: Unveiling Insights through Scatter Matrix

As we advance in our exploration of MLR, visualizing the intricate relationships between variables emerges as a crucial step in unraveling hidden patterns within our dataset. Among the arsenal of analytical tools at our disposal, the scatter matrix stands out, offering a panoramic view of pairwise correlations that can significantly enhance our model development process.

Harnessing the Power of Data Visualization

Before immersing ourselves in the scatter matrix, it's imperative to ensure our dataset undergoes meticulous refinement to optimize it for analysis. Leveraging the 'describe' function, we gain invaluable statistical insights into key dataset features, empowering us to pinpoint outliers and inconsistencies that might impede the accuracy of our MLR model.

import pandas as pd
import numpy
from sklearn import linear_model

df = pandas.read_csv("housing_openml_v1.csv")

pd.options.display.float_format = '{:,.1f}'.format
print(df.describe(include='all'))
temp = df.describe( include='all')
temp.to_csv('out.csv')

X = df[['LotArea', 'BedroomAbvGr','FullBath']]
y = df['SalePrice']

X = X.iloc[:,:].values # Independent variable (X) 
y = y.iloc[:].values   # Dependent variable (Y)

regr = linear_model.LinearRegression()
regr.fit(X, y)

sale_price =154000 #$154,000
predicted_Price = regr.predict([[8246, 3, 1]])

numpy.set_printoptions(precision=0)
print('Predicted Price: $', predicted_Price)

Err = predicted_Price - sale_price
print("Error: $", Err, "    Error Percent:  %", 100*Err/sale_price)

Interpreting Prediction Results

Upon scrutinizing the prediction outcomes, we initially observe an apparent enhancement in accuracy, evidenced by a -13% error rate. However, a deeper analysis unveils a substantial decline in the Coefficient of Determination, plummeting from 62.7% to 36.9%. This variance suggests that while our model excels in predicting individual data points, its efficacy diminishes when applied holistically to the entire dataset. Notably, the discrepancy appears to be driven by the disparity between LotArea and GrLivArea, indicating that GrLivArea serves as a superior predictor for the dataset as a whole.

We intentionally spotlighted this unit to underscore the distinction between analyzing individual units versus the dataset as a whole. It's crucial to bear in mind that our ultimate goal is to develop a model that accurately encapsulates the broader dataset dynamics.

Predicted Price:     $133,709     |     Error:     -$20,291     |     Error Percent:      -13%
Coeff Det R^2:      36.9 %

Assessing Model Robustness: Impact of Independent Variables

In our journey through Multiple Linear Regression (MLR), we encounter pivotal crossroads where the selection of independent variables wields immense influence over the predictive prowess of our model. Let's embark on a journey to unravel how the inclusion or exclusion of specific variables can sculpt the performance landscape of our model.

Analyzing Variable Impact

Initially, our MLR model incorporated Square footage, bedroom count, and Full Bath amenities as independent variables. To gauge the significance of each variable, 

At the genesis of our MLR odyssey, our model embraced "Square footage", "bedroom count", and "Full Bath" amenities as its independent variables. To gauge the significance of each variable, we systematically assess their impact on prediction accuracy.

Let's check what happens when we remove one independent variable, the column labeled "Full Bath"!

import pandas as pd
import numpy
from sklearn import linear_model

df = pandas.read_csv("housing_openml_v1.csv")

X = df[['LotArea', 'BedroomAbvGr']]
y = df['SalePrice']

X = X.iloc[:,:].values # Independent variable (X) 
y = y.iloc[:].values   # Dependent variable (Y)

regr = linear_model.LinearRegression()
regr.fit(X, y)

sale_price =154000 #$154,000
predicted_Price = regr.predict([[8246, 3]])
numpy.set_printoptions(precision=0)
Err = predicted_Price - sale_price
print('Predicted Price: $', predicted_Price, "Error: $", Err, "    Error Percent:  %", 100*Err/sale_price)
CoeffDet = regr.score(X,y)*100
print('Coeff Det R^2: {:.1f}'.format(CoeffDet),'%')

Result

Predicted Price:     $173,177     |     Error:     -$19,177     |     Error Percent:      -12%
Coeff Det R^2:      12.0 %

Evaluating Model Performance

Upon meticulous analysis, we observe a noteworthy decrease in the Coefficient of Determination plummeting from 62.7% to 12%, indicating a substantial decline in predictive accuracy when excluding the 'Full Bath' variable. While assessing individual data points may yield insights, it's imperative to evaluate the model's overall performance across the dataset.

Refining Multiple Linear Regression Model Evaluation for Artificial Intelligence

In our quest for mastery over Multiple Linear Regression (MLR) models, a crucial step lies in refining our evaluation techniques to glean deeper insights into their performance within the realm of Artificial Intelligence. Let's embark on a journey of refinement, where we enhance our model evaluation process to unlock its full potential.

Comprehensive Model Evaluation

To embark on a comprehensive evaluation of our MLR model, we pivot our strategy to incorporate both training and testing datasets. By partitioning the data into a 70% allocation for training and 30% for testing, we gain a panoramic view of our model's predictive prowess. This strategic maneuver not only showcases the performance of our code but also provides a robust assessment of its predictive capabilities across diverse datasets. We start the code with only 2 independent variables but we will add more independent variables as we progress through.

import pandas as pd
import numpy as np
from sklearn import linear_model

df = pandas.read_csv("housing_openml_v1.csv")

X = df[['LotArea', 'BedroomAbvGr']]
y = df['SalePrice']

X = X.iloc[:,:].values # Independent variable (X) 
y = y.iloc[:].values   # Dependent variable (Y)

#Split the data with 70% for training and 30% for testing
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state = 100)

regr = linear_model.LinearRegression()
regr.fit(X, y)

y_pred_Price = regr.predict(X_test)

#Model Evaluation
from sklearn import metrics
MeanAbEr = metrics.mean_absolute_error(y_test, y_pred_Price)
RMSqEr = np.sqrt(metrics.mean_squared_error(y_test, y_pred_Price))
MeanSqEr = metrics.mean_squared_error(y_test, y_pred_Price)
Scor = regr.score(X,y)*100

print('Mean Absolute Error: {:,.0f}'.format(MeanAbEr))
print('Mean Square Error: {:,.0f}'.format(MeanSqEr))
print('Root Mean Square Error: {:,.0f}' .format(RMSqEr))
print('Coeff Det R^2: {:.1f}'.format(Scor),'%')

Interpreting Evaluation Results

Upon analyzing the evaluation results, we observe that while the coefficient of determination remains unchanged, our model delivers a Mean Absolute Error of $47,664, a Mean Square Error of $4,185,186,139, and a Root Mean Square Error of $64,693. These metrics offer valuable insights into the accuracy and precision of our model, providing a robust foundation for further refinement and optimization. All we did is we re-wrote the code for a better understanding of the dataset. 

Mean Absolute Error: 47,664 
Mean Square Error: 4,185,186,139
Root Mean Square Error: 64,693
Coeff Det R^2: 12.05 %

Of critical importance before developing a Multilinear regression model, as stated in the data preparation list, is to check for multicollinearity. While analysts could go about making a pairwise multicollinearity analysis, also known as a correlation matrix, a technique known as Variation Inflation Factor VIF, is what we use.

Mitigating Multicollinearity

A pivotal consideration before delving into the development of a Multilinear Regression model is the detection and mitigation of multicollinearity. While conventional approaches may involve pairwise multicollinearity analyses, we employ a sophisticated technique known as Variation Inflation Factor (VIF) to unveil hidden correlations among independent variables. Variance inflation factor measures how much the behavior (variance) of an independent variable is influenced, or inflated, by its interaction/correlation with the other independent variables. This strategic approach ensures the integrity and reliability of our MLR model, paving the way for enhanced predictive capabilities and informed decision-making in the realm of Artificial Intelligence.

Detecting Multicollinearity: Variance Inflation Factor using Python

In our relentless pursuit of perfecting the Multiple Linear Regression (MLR) model, we plunge into the depths of data intricacies, focusing our attention on unraveling the mysteries of multicollinearity through the powerful tool of Variance Inflation Factor (VIF) analysis. By scrutinizing the interplay among independent variables, we strive to unearth potential correlations that could sway the accuracy of our model within the realm of Artificial Intelligence.

Conducting VIF Analysis

To commence our voyage into VIF analysis, we meticulously assemble a comprehensive array of independent variables from our dataset, ranging from structural dimensions to temporal attributes. Leveraging the robust capabilities of Python and the sophisticated functionalities of the Statsmodels library, we embark on our journey of discovery.

import pandas as pd
import numpy as np

df = pd.read_csv("housing_openml_v1.csv")

''' 
All the indpendent variables for VIF analysis

X = df[['MSSubClass', 'LotArea', 'YearBuilt', 'YearRemodAdd',  'BsmtFinSF1', 'BsmtFinSF2', 'BsmtUnfSF',
        'TotalBsmtSF', '1stFlrSF', '2ndFlrSF', 'LowQualFinSF', 'GrLivArea', 'BsmtFullBath', 'BsmtHalfBath', 'FullBath',
        'HalfBath', 'BedroomAbvGr', 'KitchenAbvGr', 'TotRmsAbvGrd',  'Fireplaces', 'GarageCars', 'GarageArea', 'WoodDeckSF',
        'OpenPorchSF', 'EnclosedPorch', '3SsnPorch', 'ScreenPorch', 'PoolArea', 'MiscVal','MoSold', 'YrSold']]

Note: we removed 'TotalBsmtSF'  'LowQualFinSF' as they induced error in the code
'''

X = df[['MSSubClass','LotArea','YearBuilt','YearRemodAdd','BsmtUnfSF','BsmtFinSF1','BsmtFinSF2','1stFlrSF',
         '2ndFlrSF','GarageCars','GrLivArea','BsmtFullBath','BsmtHalfBath','FullBath','HalfBath','BedroomAbvGr',
         'KitchenAbvGr','TotRmsAbvGrd','Fireplaces','GarageArea','WoodDeckSF','OpenPorchSF','EnclosedPorch', 
         '3SsnPorch','ScreenPorch','PoolArea','MiscVal','MoSold','YrSold']]

import statsmodels.api as sm
X = sm.add_constant(X) # You will have a headache if you do not add this line

from statsmodels.stats.outliers_influence import variance_inflation_factor
vif_df = pd.DataFrame()

vif_df["feature"] = X.columns
vif_df["VIF"] = [variance_inflation_factor(X.values, i) for i in range(len(X.columns))]

print(vif_df.round(1))

 Interpreting Variance Inflation Factor (VIF) Analysis & Results:

Upon the culmination of our VIF analysis, a treasure trove of insights emerges, shedding light on the intricate correlations among independent variables:

          feature        VIF
0           const  2453429.5
1      MSSubClass        1.5
2         LotArea        1.2
3       YearBuilt        2.8
4    YearRemodAdd        1.7
5       BsmtUnfSF        3.5
6      BsmtFinSF1        4.5
7      BsmtFinSF2        1.5
8        1stFlrSF       65.7
9        2ndFlrSF       93.4
10     GarageCars        5.2
11      GrLivArea      124.0
12   BsmtFullBath        2.2
13   BsmtHalfBath        1.2
14       FullBath        2.9
15       HalfBath        2.2
16   BedroomAbvGr        2.3
17   KitchenAbvGr        1.5
18   TotRmsAbvGrd        4.9
19     Fireplaces        1.5
20     GarageArea        4.8
21     WoodDeckSF        1.2
22    OpenPorchSF        1.2
23  EnclosedPorch        1.2
24      3SsnPorch        1.0
25    ScreenPorch        1.1
26       PoolArea        1.0
27        MiscVal        1.0
28         MoSold        1.0
29         YrSold        1.1

Interpreting VIF Results:

• The VIF results illuminate the degree of multicollinearity prevalent among our independent variables.
• Variables exhibiting VIF values exceeding the threshold of 5 denote substantial correlation with other variables, necessitating meticulous scrutiny.
• To enhance the robustness of our model, we adhere to the prescribed cutoffs, electing to eliminate variables surpassing the VIF threshold of 5, namely: "1stFlrSF", "2ndFlrSF", "GrLivArea", and "GarageCars".

Key Takeaways:

• VIF analysis unveils hidden correlations, offering invaluable insights into the intricacies of our dataset.
• Rigorous assessment and strategic variable selection are imperative to fortify the predictive capabilities of our MLR model within the domain of Artificial Intelligence. 

Refining Multiple Linear Regression the Better Method

In the relentless pursuit of perfection within the realm of Multiple Linear Regression (MLR), we embark on a journey of continuous refinement, aiming to elevate our predictive model to unparalleled heights of accuracy and reliability. Through a systematic approach and the integration of advanced methodologies like Variance Inflation Factor (VIF), we unravel the latent potential embedded within our dataset, ensuring that every adjustment propels us closer to predictive excellence.

X = df[['MSSubClass', 'LotArea', 'YearBuilt', 'YearRemodAdd', 'BsmtFinSF1', 'BsmtFinSF2', 'BsmtUnfSF',
        'TotalBsmtSF', 'LowQualFinSF', 'GrLivArea', 'BsmtFullBath', 'BsmtHalfBath', 'FullBath','HalfBath', 
        'BedroomAbvGr', 'KitchenAbvGr', 'TotRmsAbvGrd', 'Fireplaces', 'GarageArea', 'WoodDeckSF',
        'OpenPorchSF', 'EnclosedPorch', '3SsnPorch', 'ScreenPorch', 'PoolArea', 'MiscVal','MoSold', 'YrSold']]

Enhanced Model Performance

With a refined arsenal of independent variables meticulously curated through rigorous analysis, our Multiple Linear Regression MLR model emerges as a beacon of predictive prowess. Following the judicious removal of redundant variables identified through VIF analysis, our model undergoes a transformative evolution, as evidenced by the surge in the coefficient of determination (R-squared) from a modest 12.05% to a commanding 84.4%. Moreover, significant reductions in Mean Absolute Error (MAE), Mean Square Error (MSE), and Root Mean Square Error (RMSE) underscore the enhanced precision and reliability of our predictions. This monumental leap signifies that a substantial portion of variability in sale prices is accurately encapsulated by our model, heralding a new era of predictive precision.

Mean Absolute Error: 19,895
Mean Square Error: 748,060,970
Root Mean Square Error: 27,351
Coeff Det R^2: 84.4 %

In our unwavering quest for precision, we stand poised to introduce the concept of dummy variables and embark on yet another iteration of VIF analysis. By embracing an iterative approach to model refinement, we push the boundaries of predictive accuracy, unraveling deeper insights into the intricate dynamics governing residential property sales. Stay tuned for the forthcoming chapter of our MLR odyssey, where every adjustment brings us closer to unlocking the full potential of our predictive model.