Probability & Statistical Inference Lecture 1

MSc in Computing (Data Analytics)

Lecture Outline Introduction

General Info Questionnaire

Introduction to Statistics Statistics at work The Analytics Process Descriptive Statistics & Distributions Graphs and Visualisation

Introduction Name : Aoife D’Arcy Email: aoife@theanalyticsstore.com Bio: Managing Director and Chief Consultant at the

Analytics Store, has degrees in statistics, computer science, and financial & industrial mathematics. With over 10 years of experience in analytics consultancy with major national and international companies in banking, finance, insurance, manufacturing and gaming; Aoife has developed particular expertise in risk analytics, fraud analytics, and customer insight analytics.

Lecture Notes: Will be available online on

www.comp.dit.ie/bmacnamee and later on webcourses

TMP-1Data Mining

TMP-5Research Writing & Scientific Literature

TMP-0Probability & Statistical Inference

TMP-2Data & Database Design for Data

Analytics

SPEC9260Geographic Information Systems

SPEC 9270Machine Learning

TECH9250Complex and Adaptive Agent Based

Computation

INTC 9141 Enterprise Systems Integration

TECH9280Security

TECH9290Ubiquitous Computing

BUS9290Legal Issues for Knowledge

Management

SENG X01Software Project Management

TMP-6Research Methods & Proposal Writing

TMP-4Case Studies in Computing

INTC9221Strategic Issues in IT

INTC9231Internet Systems

SPEC9290Universal Design for Knowledge

Management

TMP-7Research Project & Dissertaion

Core Module

Option Module

Pre-requisite

TMP-3Data Management

SPEC 9160Problem Solving Communication &

Innovation

MATH 4807Financial Mathematics - I

MATH 4810Queuing Theory & Markov Processes

MATH 4814Decision Theory & Games

MATH 4821Industrial & Commercial Statistics

MATH 4809Linear Programming

MATH 4818 Financial Mathematics - II

TMP-9Language Technology

TMP-10Designing and Building Semantic Web

Applications

Programme Overview

Course OutlineWeek Topic

1 Introduction to Statistics2 & 3 Probability Theory4 Introduction to SAS Enterprise Guide5 Probability Distributions6 Confidence Intervals7 & 8 Hypothesis testing9 Assignment10 - 12 Regression Analysis13 Revision

Exam & Assignment

Exam The end of term exam accounts for 60% of

the overall mark

Assignment The assignment is worth 40% of the overall

mark. The assignment will be handed out in week 5 Week 9’s class will be dedicated to working

on the assignment.

Software SAS Enterprise Guide will be the software that

will be used during the course.

Applied Statistics and Probability for EngineersJohn Wiley & SonsDouglas C. Montgomery

Probability and Statistics for Engineers and ScientistsPearson EducationR.E. Walpole, R.H. Myers, S.L. Myers, K. Ye

Modelling Binary DataChapman & HallDavid Collett

Probability and Random Processes Oxford University PressG. Grimmett & D. Stirzaker

Statistical InferenceBrooks/ColeGeorge Casella

Recommended Reading

Questionnaire

Section 1: Statistics at work

Statistics in Everyday Life With the increase in the amount of data

available and advancement`s in the power of computers, statistics are being used more and more frequently. We are constantly reading about surveys done where 3 out 5 people prefer brand X or research has shown that having tomatoes in your diet can reduce the risk of dieses Y.

Is it good that statistics are used so much and what happens

when statistics are misused?

Statistics can be misleading An ad claimed:

“9 Out of 10 Dentists prefer Colgate” What is wrong with this statement?

During the Obama presidential election the follow was stated:“According to the Advertising Project, one out of three McCain ads has been negative, criticizing Obama. Nine out of 10 Obama ads have been positive, stressing his own background and ideas.” What is wrong with this statement?

Misinterpreted Statistics can be Devastating In 1999 Sally Clarke was wrongly convicted of

the murder of two of her sons. The case was widely criticised because of the way statistical evidence was misrepresented in the original trial, particularly by paediatrician Professor Sir Roy Meadow.

He claimed that, for an affluent non-smoking family like the Clarks, the probability of a single cot death was 1 in 8,543, so the probability of two cot deaths in the same family was around "1 in 73 million" (8543 × 8543).

What is wrong with this assumption?

Video

Challenges As an Analytics practitioner you will face a

number of challenges:

Create insight from data Interpret statistic correctly Communicate statistically driven insight in a way

that is clearly understood

The Analytics Process & Statistics

Section Overview Statistics and Analytics Introduction to CRISP

Predictive Analytics Is Multidisciplinary

Databases

StatisticsPatternRecognition

KDD

MachineLearning AI

Neurocomputing

Predictive Analytics

Data Warehousing

CRISP-DM Evolution Over 200 members of the CRISP-DM SIG

worldwide DM Vendors: SPSS, NCR, IBM, SAS, SGI, Data

Distilleries, Syllogic, etc System Suppliers/Consultants: Cap Gemini, ICL

Retail, Deloitte & Touche, etc End Users: BT, ABB, Lloyds Bank, AirTouch,

Experian, etc Crisp-DM 2.0 is due…

Complete information on CRISP-DM is available at: http://www.crisp-dm.org/

CRISP-DM Features of CRISP-DM:

Non-proprietary Application/Industry neutral Tool neutral Focus on business issues

As well as technical analysis Framework for guidance Experience base

Templates for Analysis

Data

Business Understandin

g

Data Understandin

g

Data Preparation

Modelling

Evaluation

Deployment

Phases & Generic TasksBusiness

Understanding Data

Understanding Data

Preparation Modeling Deployment Evaluation

Determine Business Objectives

AssessSituation

DetermineData Mining

Goals

ProduceProject Plan

Business Understanding

This initial phase focuses on understanding the project objectives and requirements from a business perspective, then converting this knowledge into a data mining problem definition and a preliminary plan designed to achieve the objectives

Phases & Generic TasksBusiness

Understanding Data

Understanding Data

Preparation Modeling Deployment Evaluation

CollectInitialData

DescribeData

ExploreData

VerifyData

Quality

Data Understanding

The data understanding phase starts with an initial data collection and proceeds with activities in order to get familiar with the data, to identify data quality problems, to discover first insights into the data or to detect interesting subsets to form hypotheses for hidden information.

Phases & Generic TasksBusiness

Understanding Data

Understanding Data

Preparation Modeling Deployment Evaluation

SelectData

CleanData

ConstructData

IntegrateData

FormatData

Data Preparation

The data preparation phase covers all activities to construct the data that will be fed into the modelling tools from the initial raw data. Data preparation tasks are likely to be performed multiple times and not in any prescribed order. Tasks include table, record and attribute selection as well as transformation and cleaning of data for modelling tools.

Phases & Generic TasksBusiness

Understanding Data

Understanding Data

Preparation Modeling Deployment Evaluation

SelectModelingTechnique

GenerateTest Design

BuildModel

AssessModel

Modelling

In this phase, various modelling techniques are selected and applied and their parameters are calibrated to optimal values. Typically, there are several techniques for the same data mining problem type. Some techniques have specific requirements on the form of data. Therefore, stepping back to the data preparation phase is often necessary.

Phases & Generic TasksBusiness

Understanding Data

Understanding Data

Preparation Modeling Deployment Evaluation

EvaluateResults

ReviewProcess

DetermineNext Steps

Evaluation

Before proceeding to final deployment of a model, it is important to thoroughly evaluate it and review the steps executed to construct it to be certain it properly achieves the business objectives. A key objective is to determine if there is some important business issue that has not been sufficiently considered. At the end of this phase, a decision on the use of the data mining results should be reached.

Phases & Generic TasksBusiness

Understanding Data

Understanding Data

Preparation Modeling Deployment Evaluation

Plan Deployment

Plan Monitoring &

Maintenance

ProduceFinal

Report

ReviewProject

Deployment

Creation of a model is generally not the end of the project. Even if the purpose of the model is to increase knowledge of the data, the knowledge gained will need to be organized and presented in a way that the customer can use it. Depending on the requirements, the deployment phase can be as simple as generating a report or as complex as implementing a repeatable data mining process across the enterprise.

Crisp - DM

Business Understanding

Data Understanding

Data Preparation

Modelling

Evaluation

Deployment.

Crisp – DM – Areas covered in this course

Business Understanding

Data Understanding

Data Preparation

Modelling

Evaluation

Deployment

Section 2: Descriptive Statistics & Distributions

Topics1. Introduction to Statistics2. The Basics 3. Measures of location: Mean, Median & Mode.4. Measures of location & Skew.5. Measures of dispersion: range, standard

deviation (variance) & interquartile range.

Introduction to Statistics According to The Random House College

Dictionary, statistics is “the science that deals with the collection, classification, analysis and interpretation of numerical facts or data.” In short, statistics is the science of data.

There are two main branches of Statistics: The branch of statistics devoted to the

organisation, summarization and the description of data sets is called Descriptive Statistics.

The branch of statistics concerned with using sample data to make an inference about a large set of data is called Inferential Statistics.

Process of Data Analysis

Population

Representative Sample

Sample Statistic

A Statistical population is a data set that is our target of interest.

A sample is a subset of data selected from the target population.

If your sample is not representative then it is referred to as being bias

Describe

Make

Inference

Types of Data There are a number of data types that we will

be considering.  These can be split into hierarchy of 4 levels of

measurement.1. Categorical

a) Nominalb) Ordinal

2. Intervala) Discreteb) Continuous

Describing Distributions

Describing Distributions

Measures of Location (Central Tendency)

Numbers that attempt to express the location of data on the number line

Variable(s) are said to be distributed over the number line - so we talk of distributions of numbers

•Want a measure of the location of this data on the number line.

•There is 'symmetry' around this point in this particular data – hence the term central tendency

Arithmetic Mean (average) The mean of a data set is one of the most

commonly used statistics. It is a measure of the central tendency of the data set.

The mean of a sample is denoted by (pronounced x bar) and the mean of a population is denoted by µ (pronounced mew).

They are both ( and µ ) computed using the same formula.

X___

X___

n

n

iX 1

Arithmetic Mean - Example

Example: Ages of Students in 1st year history of Art degree course

18, 18, 18, 18, 19, 19, 20, 20, 58

Mean of ages here is 23.11 – but this is not a ‘typical value or a value around which the observed values cluster.

The same thing tends to happen with values that are strictly positive: average salaries, house prices etc.

We say that the mean is sensitive to extreme values

Median The middle value of the ordered set of

values, i.e. 50% higher and 50% lower.

Example: The class age data again 18, 18, 18, 18, 19, 19, 20, 20, 58  The data is ordered, and n = 9, so the

middle number is (n+1)/2 = (9+1)/2 = 5th value = 19

  => median = 19 years

Median• Robust with regard to extreme values

• Often a real value in the distribution or close to 2 real values - in that sense tends to be more typical of actually observed values

Mode The most commonly occurring value in a distribution

Example: The class age data again 18, 18, 18, 18, 19, 19, 20, 20, 58

 The mode is 18 years as it occurs more than any other

Tends to show where the data is concentrated

Mode: 18 Mean: 23.11 Median: 19 

Skew – The Shape of a DistributionThere are a number of ways of describing the shape of a distribution.

We will consider only one – skew.

Skew is a measure of how asymmetric a distribution is.

Symmetric Distributions  = skew is zero

There are few very large data points which create a 'tail' going to the right (i.e. up the number line)

Note: No axis of symmetry here - skew > 0 (i.e. it is positive)

Example: Lifetime of people, house prices

Positive Skew

There are few very small data points which create a 'tail' going to the left (i.e. down the number line)

Note: No axis of symmetry here - skew < 0 (i.e. it is negative)

Examples: Examination Scores, reaction times for drivers

Negative Skew

Mean, Median & Mode are the same and are found in the middle

66

5 6 74 5 6 7 8

3 4 5 6 7 8 9

Mean = 102/17 = 6Median = 6Mode = 6

Skew & Measures of Location - Symmetry

Mode

MedianMean

6

6

5 6 7

5 6 7 8 9

5 6 7 8 9 10 11

Mean = 121/17 = 7.12Median = 7Mode = 6

In general: Mode < Median < Mean

Positive Skew

Mode

MedianMean

Mean = 83/17 = 4.89Median = 5Mode = 6

In general: Mode > Median > Mean

66

5 6 73 4 5 6 7

1 2 3 4 5 6 7

Negative Skew

• The Mean, Mode and Median all 250 for both companies

• But not the same - look at the difference in ‘spread’ of bills

• Need a measure of spread (dispersion) as well as location to describe a distribution

Measures of Spread (Dispersion)

Location

Spread

Range Simplest measure of spread = largest -

smallest

Example for data in histograms:  Esat: Largest = €335 Smallest = €180

Range = €335 - €180 = €155

Meteor: Largest = €295 Smallest = €210Range = €295 - €210 = €85

Very simple to compute

Easy to interpret

Does not use all the data

Subject to effect by extreme values

Range Example: The class age data again

18, 18, 18, 18, 19, 19, 20, 20, 58 

Range: 58-18 = 40 years

 Is this really indicative of the spread of ages?

 => if the mature student was not there, range would be 2 years - so just 1 extreme value has huge effect on range

Typical Deviation – Average Deviation Consider the following data:

OBS Data Mean Deviation

1 3 5 -2

2 4 5 -1

3 8 5 3

Sum 15 15 0

Mean 5 5 0

Typical Deviation – Average Squared Deviation (Variance) Consider the following data:

OBS Data Deviation (Deviation)2

1 3 -2 4

2 4 -1 1

3 8 3 9

Sum 15 0 14

Mean 5 0 14/3

Variance – the formula 1. Square the deviations around the mean before

summing. NB. quantities will be in squared units e.g. cm2 > not original scale:

2. Divide by n-1 (?) to get the average of (deviations )2

2 xx

2

2

1

n

xxs

Standard Deviation

2

1

n

xxs

Take the square root of the variance . The value is in the original unit

Quantiles The nth quantile is a value that has a

proportion n of the sample taking values smaller than it and 1-n taking values larger than it.

For Example: if your grade in an industrial engineering class was located at the 84th percentile, then 84% of the grades were lower than your grade and 16% were higher.

The median is the 50th percentile. The 25th percentile and the 75th percentile are called the lower (1st) quartile and upper (3rd) quartile respectively.

The difference between the lower and upper quartile is called the inter-quartile range.

ExampleClass Age data: 18, 18, 18, 18, 19, 19, 20, 20, 58Order No: 1 2 3 4 5 6 7 8 9 1st Quartile = 1+(n-1)/4 =1+ (8)/4 = 3rd score => 18  Median = 1+2×(n-1)/4 = 1+2×(8)/4 = 5th score =>

19 3rd Quartile = 1+3×(n-1)/4 = 1+3×(8)/4 = 7th score =>

20

Interquartile Range : 20 - 18 = 2 years

Coefficient of Variation A problem with s is that it is was scale specific - i.e.

comparison of s calculated on difference scales is hard to do.

Example:  Distribution A: 8, 9, 10, 11, 12, 13, 14 Distribution B: 1008, 1009, 1010, 1011, 1012, 1013,

1014 Use two of the measures of spread we have

RangeRange for A: 14 - 8 = 6Range for B: 1014 - 1008 = 6

  Standard Deviation (s)

s for A:= 2.16 s for B:= 2.16

Why are the standard deviations the same ?s measures average deviation around the mean.

A A-Mean

B B-Mean

8 -3 1008 -3 9 -2 1009 -2 10 -1 1010 -1 11 0 1011 0 12 1 1012 1 13 2 1013 2 14 3 1014 3

Mean = 11 Mean =1011

Coefficient of variation

%100. X

sVC

• C.V. is unit-less (i.e. scale-less) • Can compare difference measurement systems

and standardise for differences in scale E.G. for data above; A: C.V. = ( 2.16 / 11 ) 100% => 19.6 %B: C.V. = ( 2.16 / 1011 ) 100% => 0.2 %

Section 3: Graphs and Visualisation

A way of letting people get a 'picture' of relationships in the data set.

The simpler the better should be a rule in graphical display.

People can remember pictures better.

A good graph should show something that is not easy to ‘see’ using tables.

Graphical Displays

Bar Charts Used to display categorical data or

discrete data with a modest number of values.

A Bar is drawn to represent each category. The Bar height represents the frequency

or % in each category. Allows for visual comparison of relative

frequencies. Need to draw up a frequency distribution

table first.

Table 1.- Counts in each exercise categoryExercise FrequencyV. High 32High 30Medium 52Low 32None 36

Frequency of Exercise Levels from Exercise Data Set

0

10

20

30

40

50

60

V. High High Medium Low None

Freq

uenc

y

So, 5 categories 5 bars heights of bar are the frequencies Clear to see hierarchy in frequency, and can

make a guess at relative percentages between categories

E.G. ‘Low’ looks about 60% of ‘Medium’ Actual = ( 32/52 ) 100% = 61.53 %

Note appropriate title and axis labels

0

10

20

30

40

50

60

V. High HighMedium Low None

Fre

qu

en

cy

Frequency of Exercise Levels from Exercise Data Set

Do NOT use 3D effects –  The angling loses information

Also colouring effects can distract

Can use more than one set of bars to subdivide groups e.g. same data – subdivided by gender

Table 2. Exercise Level By Sex

Gender

Exercise Female

Male

V. High 13 19High 12 18Medium 22 30Low 16 16None 8 28

0

5

10

15

20

25

30

35

Freq

uenc

y

Exercise level by Gender

Female Male

 Another way to subdivide groups in the barsDivide segments in each bar to represent the frequency ( or % ) of each group in that category

0

10

20

30

40

50

60

Freq

uen

cy

Exercise level by Gender

MaleFemale

Component bar charts

Histogram Histograms are among the most widely

used method for displaying continuous data

Has similarities with bar chart – but definitely not the same!

A rectangle is drawn to represent the frequency in a grouped frequency distribution table

Components; 2 axes, x and y

Histogram x axis: Grouped intervals are chosen to

appropriately display data.

y axis: Heights are chosen to represent frequency, or some form of relative frequency, % or density.

Example: From the Exercise data set - want to look at height variable for all people. 

E.G. Construct grouped frequency distribution table of peoples heights from the Exercise dataset.

Table 3. Heights cm

Frequency

>150 - 155 3

>155-160 10

>160-165 29

>165-170 37

>170-175 44

>175-180 34

>180-185 19

>185-190 6

Total 182

• Choose intervals to reflect meaningful groupings,

OR

• Choose largest number of intervals that avoids jaggedness.

Be careful with choice of intervals as shape can change.

40455055606570758085

140 150 160 170 180 190 200

Wei

ght

Height

Scatter Plot of Weight by Height

Scatterplots

Used for plotting data over timeX-axis is a time lineY-axis is the value changing over time

Profit - £0,000'sQuarter 1992 1993 1994

1 114 116 1282 142 150 1583 155 153 1694 136 140 159

80

100

120

140

160

180

1 2 3 4 1 2 3 4 1 2 3 4

1992 1993 1994Shows ‘Trend’ & ‘Seasonality’

Time Series Plots