Introduction to Data Science

Dr. Irfan Yousuf

Department of Computer Science (New Campus)
UET, Lahore
(Week 3; January 29 – February 02, 2024)
Outline
• Linear Regression Analysis
• Exploratory Data Analysis
Relations Between Variables
• Functional Relation
• Statistical Relation
Functional Relation
• A functional relation between two variables is a perfect
relation where the value of the dependent variable is
uniquely determined by the value of the independent
variable.
• It is expressed by a mathematical formula.
Functional Relation
Statistical Relation
• A statistical relation between two variables is a relation
where the value of the dependent variable is NOT
uniquely determined when the level of the independent
variable is specified.
• If the values of a variable Y increase or decrease when the
values of a variable X change, there is a statistical
relationship between Variable Y and Variable X.
• It is not an exact relation.
• Examples:
• The relation between age and income
• The relation between income and expenditures
Statistical Relation
Scatter Plot
• A scatter plot (aka scatter chart, scatter graph) uses dots to
represent values for two different numeric variables.

• The position of each dot on the horizontal and vertical axis

indicates values for an individual data point.

• Scatter plots are used to observe relationships between

variables.
Scatter Plot
Regression Analysis
• Regression analysis provides a method of estimating an
average relation (often linear) between two or more variables.

• Regressor: The variable that forms the basis of estimation or

prediction (aka predictor variable or independent variable).

• Regressand: The variable whose value depend on the

independent variable is called a regressand (aka response
variable, predictand variable or dependent variable).
Regression Models
• Regression models describe the relationship between
variables by fitting a line to the observed data.

• Linear regression models use a straight line, while logistic

and nonlinear regression models use a curved line.

• Regression allows you to estimate how a dependent variable

changes as the independent variable(s) change.
Simple Linear Regression
Simple Linear Regression
- Best-fit Line
- Least Squares Method or Least Squares Regression
Simple Linear Regression
- Best-fit Line
- Least Squares Method or Least Squares Regression
Simple Linear Regression: Example
x y
14.2 215
16.4 325
11.9 185
15.2 332
18.5 406
22.1 522
19.4 412
25.1 614
23.4 544
18.1 421
22.6 445
17.2 408 b = 30.08
a = -159.4
Simple Linear Regression: Example
Residuals
- The residuals are the differences between the observed and
predicted values.
- It measures the distance from the regression line (predicted
value) and the actual observed value. In other words, it
helps us to measure error, or how well our regression line
“fits” our data.
Outliers
Simple Linear Regression: Example
Income Savings
25 2
32 3
37 5
39 2
46 10
49 7
51 15
55 14
58 8
59 15
66 24
68 10 Income = 85, Saving =?
72 12
Income = 42, Saving=?
75 15
77 13
Simple Linear Regression: Example
Income Savings X Y X*Y X^2
25 2 25 2 50 625
32 3 32 3 96 1024
37 5 37 5 185 1369
39 2 39 2 78 1521
46 10 460 2116
46 10
49 7 343 2401
49 7
51 15 765 2601
51 15
55 14 770 3025
55 14 58 8 464 3364
58 8 59 15 885 3481
a = -4.4502
59 15 66 24 1584 4356 b = 0.2741
66 24 68 10 680 4624
68 10 72 12 864 5184
72 12 75 15 1125 5625
75 15 77 13 1001 5929
77 13 809 155 9350 47245
Simple Linear Regression: Example
Income Savings
25 2
32 3
37 5
39 2
46 10
49 7
51 15
55 14
58 8
59 15
66 24
68 10
72 12
75 15
77 13
Simple Linear Regression: Example
Income Savings
25 2 Income = 85, Saving =?
32 3 Income = 42, Saving=?
37 5
39 2 a = -4.4502
46 10 b = 0.2741
49 7
51 15 Y = -4.4502 + 0.2741(X)
55 14
58 8
59 15 Income = 85, Saving = 18.84
66 24 Income = 42, Saving= 7.062
68 10
72 12
75 15
77 13
Simple Linear Regression: Example
Income Savings X Y Predicted Values
25 2 25 2 2.4023
32 3 32 3 4.321
37 5 37 5 5.6915
39 2 39 2 6.2397
46 10 46 10 8.1584
49 7 49 7 8.9807
51 15 51 15 9.5289
55 14 55 14 10.6253
58 8 58 8 11.4476
59 15 59 15 11.7217
66 24 66 24 13.6404
68 10 68 10 14.1886
72 12 72 12 15.285
75 15 75 15 16.1073
77 13 77 13 16.6555
Simple Linear Regression: Example
X Y Predicted Values
25 2 2.4023
32 3 4.321
37 5 5.6915
39 2 6.2397
46 10 8.1584
49 7 8.9807
51 15 9.5289
55 14 10.6253
58 8 11.4476
59 15 11.7217
66 24 13.6404
68 10 14.1886
72 12 15.285
75 15 16.1073
77 13 16.6555
R-squared (Goodness of Fit)
- After fitting a linear regression model, you need to
determine how well the model fits the data.
- R-squared is a goodness-of-fit measure for linear
regression models.
- This statistic indicates the percentage of the variance in
the dependent variable that the independent variable
can explain (or that is predictable from the independent
variable).
- R-squared measures the strength of the relationship
between the two variables on a convenient 0 – 100%
scale.
- It is also called Coefficient of Determination.
R-squared (Goodness of Fit)
yi = Actual Value
y-hat = Predicted Value
y-bar = Actual Mean

x y Predicted (Y)
14.2 215 267.78
16.4 325 333.97
11.9 185 198.58
15.2 332 297.87
18.5 406 397.16
22.1 522 505.47
19.4 412 424.24 Y= 30.08X – 159.4
25.1 614 595.74
23.4 544 544.59
18.1 421 385.12 R2 = 0.9168
22.6 445 520.52
17.2 408 358.04
R-squared (Goodness of Fit)
yi = Actual Value
y-hat = Predicted Value
y-bar = Actual Mean
X Y Yhat (Y - Yhat)^2 (Y -Yavg)^2
25 2 2.4023 0.1618453 69.3889
32 3 4.321 1.745041 53.7289
37 5 5.6915 0.4781723 28.4089
39 2 6.2397 17.975056 69.3889
46 10 8.1584 3.3914906 0.1089
49 7 8.9807 3.9231725 11.0889
51 15 9.5289 29.932935 21.8089
55 14 10.6253 11.3886 13.4689
58 8 11.4476 11.885946 5.4289
59 15 11.7217 10.747251 21.8089 SSRES = 241.87
66 24 13.6404 107.32131 186.869 SSTOT = 513.33
68 10 14.1886 17.54437 0.1089
72 12 15.285 10.791225 2.7889
75 15 16.1073 1.2261133 21.8089
77 13 16.6555 13.36268 7.1289
R2 = 0.5288
241.87521 513.334
Correlation

- The direction and strength of pairwise relationships

between two or more numeric variables.
- -1: Perfect negative correlation. The variables tend to
move in opposite directions (i.e., when one variable
increases, the other variable decreases).
- 0: No correlation. The variables do not have a relationship
with each other.
- 1: Perfect positive correlation. The variables tend to
move in the same direction (i.e., when one variable
increases, the other variable also increases).
Correlation
Correlation

r = 0.9575
How to Interpret Correlation?
- Strength: The greater the absolute value of the correlation
coefficient, the stronger the relationship.

- Direction: The sign of the correlation coefficient

represents the direction of the relationship.
How to Interpret Correlation?
- The correlation, denoted by r, measures the amount of
linear association between two variables.
- r is always between -1 and 1 inclusive.

- The R-squared value, denoted by R2, is the square of the

correlation. It measures the proportion of variation in
the dependent variable that can be attributed to the
independent variable.
- The R-squared value is always between 0 and 1 inclusive.
Correlation vs. Regression
- Correlation is a statistical measure that quantifies the
direction and strength of the relationship between two
numeric variables.

- Regression is a statistical technique that predicts the value

of the dependent variable Y based on the known value of
the independent variable X through an equation.
Drawbacks of Linear Regression
- Linear regression only looks at linear relationships between
dependent and independent variables.
- Linear regression looks at a relationship between the mean
of the dependent variable and the independent variables.
- Linear regression is sensitive to outliers
Exploratory Data Analysis
Data
• Data is a collection of facts, such as numbers, words,
measurements, observations or just descriptions of things.
• Qualitative data is descriptive information (it describes
something).
• Quantitative data is numerical information (numbers).
• Discrete data can only take certain values (like whole
numbers)
• Continuous data can take any value (within a range)
• Discrete data is counted, Continuous data is measured
Data Analysis
• Data analysis is the process of collecting, cleaning,
analyzing, interpreting, and visualizing data to discover
valuable insights or useful information.
Data Science
• Data science is the domain of study that deals with large
volumes of data using modern tools and techniques to find
unseen patterns, derive meaningful information, and make
business decisions.

• Data science uses complex (e.g., machine learning)

algorithms to build predictive models.
Data Analyst
• A data analyst makes sense out of existing data.
• Data analysts typically work with structured data to solve
business problems using tools like SQL, R or Python
programming languages, data visualization software, and
statistical analysis.
• Collaborating with organizational leaders to identify
informational needs
• Acquiring data from primary and secondary sources
• Cleaning and reorganizing data for analysis
• Analyzing data sets to spot trends and patterns that can be
translated into actionable insights
• Presenting findings in an easy-to-understand way to inform data-
driven decisions
Data Scientist
• A data scientist works on new ways of capturing and
analyzing data to be used by the analysts. A data scientist is
someone who creates programming code and combines it
with statistical knowledge to create insights from data.

• Gathering, cleaning, and processing raw data

• Designing predictive models and machine learning algorithms to
mine big data sets
• Developing tools and processes to monitor and analyze data
accuracy
• Building data visualization tools, dashboards, and reports
• Writing programs to automate data collection and processing
Data Engineer
• A data engineer specializes in preparing data for analytical
usage. Data Engineering involves the development of
platforms and architectures for data processing.
• Data Engineer is responsible for designing the format for
data scientists and analysts to work on.
• Build, test, and maintain dataset pipeline architectures
• Create new data validation methods and data analysis tools
• Combine raw information from different sources
• Explore ways to enhance data quality and reliability
Data Analysis
Types of Data Analysis
• Descriptive analysis looks at past data and tells what
happened. This is often used when tracking Key Performance
Indicators (KPIs), revenue, sales leads, and more.
• Exploratory analysis is an approach of analyzing data to
summarize their main characteristics, often using statistical
graphics and other data visualization methods.
• Predictive analysis predicts what is likely to happen in the
future. In this type of research, trends are derived from past
data which are then used to form predictions about the future.
• Prescriptive analysis is the most advanced form of analysis,
as it combines all your data and analytics, then outputs a
model prescription: What action to take.
Exploratory Data Analysis
• Exploratory Data Analysis (EDA) is an approach for data
analysis that employs a variety of techniques (mostly
graphical) to
• maximize insight into a data set
• uncover underlying structure
• extract important variables
• detect outliers and anomalies
• test underlying assumptions
Graphics and Exploratory Data Analysis
• Quantitative (summary)
• Graphical (plots)
Univariate Data
• Univariate data are the ones which consist of only one
variable.
• Univariate data analysis are straightforward as we are dealing
with only one variable.
• The analysis that we do in case of univariate data analysis
doesn't have to do anything with the relationships between
variables.
• The main purpose is to describe the data and find patterns
that are present.
Univariate Data
• Bar Chart
• Pie Chart
• Histogram / Frequency Distribution
Univariate Data
• Bar Chart
• Pie Chart
• Histogram / Frequency Distribution
Bivariate Data
• Bivariate data involves two different variables where we are
concerned about investigating the causes and relationship
between those 2 variables.
Bivariate Data
• Scatter Plot
• Line Plot
• Stacked Bar chart
Bivariate Data
• Scatter Plot
• Line Plot
• Stacked Bar chart
Multivariate data
• Data which involves 3 or more variables are termed as
Multivariate data. These are similar to bivariate but contains
more than one dependent variable.
• “Curse of dimension” is a trouble issue in information
visualization.
• The effectiveness of retinal visual elements (e.g. color, shape,
size) deteriorates when the number of variables increases
Multivariate data
• Enhanced Basic Plots
EDA using Python
• Matplotlib
• Seaborn
Summary
• Linear Regression
• Exploratory Data Analysis