Characteristics and statistics of digital remote sensing imagery

Size: px
Start display at page:

Download "Characteristics and statistics of digital remote sensing imagery"

Transcription

1 Characteristics and statistics of digital remote sensing imagery There are two fundamental ways to obtain digital imagery: Acquire remotely sensed imagery in an analog format (often referred to as hard-copy) and then convert it to a digital format through the process of digitization, and Acquire remotely sensed imagery already in a digital format, such as that obtained by a Landsat sensor system. 1

2 Image Digitization: Analog to Digital (A/D) Conversion Using Linear Array Scanners Image digitization is the process of turning a hard-copy analog imagery into a digital image. (e.g., a des-top scanner)

3 RGB = True Color. Any addition array that collect spectral information beyond visible light will be able to generate pseudo color image. Additional array? 3

4 Panchromatic Blac & White Infrared True-color vs. Pseudo (false)-color 4

5 With all images are digital, nowledge and sills in digital image processing are necessary. Digital Image With raster data structure, each image is treated as an array of values of the pixels. Image data is organized as rows and columns (or lines and pixels) start from the upper left corner of the image. Each pixel (picture element) is treated as a separate unite. 5

6 Statistics of Digital Images: Looing at the frequency of occurrence of individual brightness values in the image displayed in a histogram Viewing individual pixel brightness values at specific locations or within a geographic area; Computing univariate descriptive statistics to determine if there are unusual anomalies in the image data; and Computing multivariate statistics to determine the amount of between-band correlation (e.g., to identify redundancy). Statistics of Digital Images It is useful to calculate fundamental univariate and multivariate statistics of the multispectral remote sensor data. This normally involved computing of maximum and minimum value for each band of imagery, the range, mean, standard deviation, between-band variance-covariance matrix, correlation matrix, and frequencies of brightness values for each band, which are used to produce histograms. Such statistics provide valuable information necessary for processing and analyzing remote sensing data. 6

7 Digital Images: A population is an infinite or finite set of elements. A sample is a subset of the elements taen from a population used to mae inferences about certain characteristics of the population. (e.g., training signatures) Large samples drawn randomly from natural populations usually produce a symmetrical frequency distribution. Most values are clustered around some central value, and the frequency of occurrence declines away from this central point. A graph of the distribution appears bell shaped is called a normal distribution. 7

8 Histogram and Its Significance to Digital Remote Sensing Image Processing Many statistical tests used in the analysis of remotely sensed data assume that the brightness values recorded in a scene are normally distributed. Unfortunately, remotely sensed data may not be normally distributed. In such instance, nonparametric statistical theory may be preferred. Forest Water 8

9 Histogram and Its Significance to Digital Remote Sensing Image Processing The histogram is a useful graphic representation of the information content of a remote sensing image, which provide readers with an appreciation of the quality of the original data, e.g. whether it is low in contrast, high in contrast, or multimodal in nature. Univariate Descriptive Image Statistics Measures of Central Tendency in Remote Sensor Data Mode: is the value that occurs most frequently in a distribution and is usually the highest point on the curve. Multiple modes are common in image dataset. Median: is the value midway in the frequency distribution. Mean: is the arithmetic average and if defined as the sum of all observations divided by the number of observations. 9

10 Measures of Central Tendency in Remote Sensor Data The mean is the arithmetic average and is defined as the sum of all brightness value observations divided by the number of observations. It is the most commonly used measure of central tendency. The mean ( ) of a single band of imagery composed of n brightness values (BV i ) is computed using the formula: Sample mean is a poor measure of central tendency when the set of observations is sewed or contains an outlier. 9 Further measurements are needed. Outlier Mean = 7/9 = 3 (Does not represent the dataset well) Sewed Mean = 9/9 = 1 Mean = 9/9 = 1 10

11 Measures of Dispersion Measures of the dispersion about the mean of a distribution provide valuable information about the image. For example, the range of a band of imagery (range ) is computed as the difference between the maximum (max ) and minimum (min ) values: Measures of Dispersion Unfortunately, when the minimum or maximum values are extreme or unusual observations, the range could be a misleading measure of dispersion. Such extreme values are not uncommon because the remote sensor data are often collected by detector systems with delicate electronics that can experience spies in voltage and other unfortunate malfunctions. When unusual values are not encountered, the range is a very important statistic often used in image enhancement functions such as min max contrast stretching. 11

12 Measures of Dispersion Variance: is the average squared deviation of all possible observations from the sample mean. The variance of a band of imagery, var, is computed using the equation: BV 11 BV 1 BV 13 BV 1 BV BV 3 BV 31 BV 3 BV 33 var n ( i 1 BV n i 1 ) BV 11 BV 1 BV 13 BV 1 BV BV 3 var n ( i 1 BV i n 1 ) BV 31 BV 3 BV 33 var 9 i 1 (1 1 ) var ( n i 1 54 ( BV i ) n 1 (1 (1 (18 (1 8 (13 ( (14 (1 1

13 Standard Deviation (s ): is the positive square root of the variance. s var A small s. suggests that observations clustered tightly around a central value. A large s indicates that values are scattered widely about the mean. The sample having the largest variance or standard deviation has the greater spread among the values of the observations. Mean Standard Deviations Standard Deviation (s ): is the positive square root of the variance. s var var ( n i 1 54 ( BV i n 1 (1 ) (1 (18 (1 8 (13 (1 (14 (1 s var

14 Measures of Distribution (Histogram) Asymmetry and Pea Sharpness Sewness is a measure of the asymmetry of a histogram and is computed using the formula: BV 11 BV 1 BV 13 BV 1 BV BV 3 BV 31 BV 3 BV 33 A perfectly symmetric histogram has a sewness value of zero. Histogram of A Single Band of Landsat Thematic Mapper Data Max. = 10 Min. = 6 Mean = 7 Median = 5 Mode = 9 14

15 Histogram of Thermal Infrared Imagery of a Thermal Plume in the Savannah River Max. = 188 Min. = 38 Mean = 73 Median = 68 Mode = 75 Remote Sensing Multivariate Statistics Remote sensing is often concerned with the measurement of how much radiant flux is reflected or emitted from an object in more than one band (e.g., in red and near-infrared bands). It is useful to compute multivariate statistical measures such as covariance and correlation among the several bands to determine how the measurements covary. 15

16 Remote Sensing Multivariate Statistics The different remote-sensing-derived spectral measurements for each pixel often change together in some predictable fashion. Landsat-7 ETM+ Band 1 (Blue band) Landsat-7 ETM+ Band (Green band) Remote Sensing Multivariate Statistics If there is no relationship between the brightness value in one band and that of another for a given pixel, the values are mutually independent; i.e., an increase or decrease in one band s brightness value is not accompanied by a predictable change in another band s brightness value. Landsat-7 ETM+ Band 3 (Red band) Landsat-7 ETM+ Band 4 (Near IR band) 16

17 Correlation between Multiple Bands of Remotely Sensed Data To estimate the degree of interrelation between variables in a manner not influenced by measurement units, the correlation coefficient, r, is commonly used. The correlation between two bands of remotely sensed data, r l, is the ratio of their covariance (cov l ) to the product of their standard deviations (s s l ); thus: r l cov s s l l Correlation Coefficient: A correlation coefficient of +1 indicates a positive, perfect relationship between the brightness values of the two bands. A correlation coefficient of -1 indicates that the two bands are inversely related. A correlation coefficient of zero suggests that there is no linear relationship between the two bands of data. 17

18 Remote Sensing Multivariate Statistics Because spectral measurements of individual pixels may not be independent, some measure of their mutual interaction is needed. This measure, called the covariance, is the joint variation of two variables about their common mean. To calculate covariance, we first compute the corrected sum of products (SP) defined by the equation: 18

19 Remote Sensing Multivariate Statistics It is computationally more efficient to use the following formula to arrive at the same result: This quantity is called the uncorrected sum of products. Remote Sensing Multivariate Statistics Covariance is calculated by dividing SP by (n 1). Therefore, the covariance between brightness values in bands and l, cov l, is equal to: 19

20 Covariance: is the joint variation of two variables about their common mean. SP l is the corrected Sum of Products between bands and l. SP cov l l n 1 SP l n ( i 1 BV i )( BV il l ) Band = Band l l = 1 SP l = (19-9)x(1-1)+(1-9)x(1-1)+(1-9)x(1-1)+(18-9)x(1-1)+(1-9)x(1-1)+(13-9)x(1-1)+ (1-9)x(1-1)+(14-9)x(1-1)+(1-9)x(1-1) = 0 Cov l = 0 Covariance: A more efficient formula is: n SP l ( BV i BV il ) i 1 n BV i 1 n i BV i 1 n il Band Band l Total = Total = 9 SP l = [(19x1)+(1x1)+(1x1)+(18x1)+(1x1)+(13x1)+(1x1)+(14x1)+(1x1)] [91x9/9] = 0 Cov l = 0 The sums of products (SP) and sums of squares (SS) can be computed for all possible band combinations. 0

21 Format of a Variance-Covariance Matrix Band 1 (green) Band (red) Band 3 (nearinfrared) Band 4 (nearinfrared) Band 1 SS 1 cov 1, cov 1,3 cov 1,4 Band cov,1 SS cov,3 cov,4 Band 3 cov 3,1 cov 3, SS 3 cov 3,4 Band 4 cov 4,1 cov 4, cov 4,3 SS 4 1

22 Remote Sensing Multivariate Statistics Variance covariance and correlation matrices are the ey for principal components analysis (PCA), feature selection, classification and accuracy assessment.

Data Mining Part 2. Data Understanding and Preparation 2.1 Data Understanding Spring 2010

Data Mining Part 2. Data Understanding and Preparation 2.1 Data Understanding Spring 2010 Data Mining Part 2. and Preparation 2.1 Spring 2010 Instructor: Dr. Masoud Yaghini Introduction Outline Introduction Measuring the Central Tendency Measuring the Dispersion of Data Graphic Displays References

More information

SAMPLE MIDTERM QUESTIONS

SAMPLE MIDTERM QUESTIONS Geography 309 Sample MidTerm Questions Page 1 SAMPLE MIDTERM QUESTIONS Textbook Questions Chapter 1 Questions 4, 5, 6, Chapter 2 Questions 4, 7, 10 Chapter 4 Questions 8, 9 Chapter 10 Questions 1, 4, 7

More information

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

More information

10-3 Measures of Central Tendency and Variation

10-3 Measures of Central Tendency and Variation 10-3 Measures of Central Tendency and Variation So far, we have discussed some graphical methods of data description. Now, we will investigate how statements of central tendency and variation can be used.

More information

Content DESCRIPTIVE STATISTICS. Data & Statistic. Statistics. Example: DATA VS. STATISTIC VS. STATISTICS

Content DESCRIPTIVE STATISTICS. Data & Statistic. Statistics. Example: DATA VS. STATISTIC VS. STATISTICS Content DESCRIPTIVE STATISTICS Dr Najib Majdi bin Yaacob MD, MPH, DrPH (Epidemiology) USM Unit of Biostatistics & Research Methodology School of Medical Sciences Universiti Sains Malaysia. Introduction

More information

Resolutions of Remote Sensing

Resolutions of Remote Sensing Resolutions of Remote Sensing 1. Spatial (what area and how detailed) 2. Spectral (what colors bands) 3. Temporal (time of day/season/year) 4. Radiometric (color depth) Spatial Resolution describes how

More information

Data Exploration Data Visualization

Data Exploration Data Visualization Data Exploration Data Visualization What is data exploration? A preliminary exploration of the data to better understand its characteristics. Key motivations of data exploration include Helping to select

More information

STATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI

STATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI STATS8: Introduction to Biostatistics Data Exploration Babak Shahbaba Department of Statistics, UCI Introduction After clearly defining the scientific problem, selecting a set of representative members

More information

Chapter 2: Systems of Linear Equations and Matrices:

Chapter 2: Systems of Linear Equations and Matrices: At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,

More information

DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.

DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE STATISTICS The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize,

More information

Module 3: Correlation and Covariance

Module 3: Correlation and Covariance Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis

More information

Data. ECON 251 Research Methods. 1. Data and Descriptive Statistics (Review) Cross-Sectional and Time-Series Data. Population vs.

Data. ECON 251 Research Methods. 1. Data and Descriptive Statistics (Review) Cross-Sectional and Time-Series Data. Population vs. ECO 51 Research Methods 1. Data and Descriptive Statistics (Review) Data A variable - a characteristic of population or sample that is of interest for us. Data - the actual values of variables Quantitative

More information

A Introduction to Matrix Algebra and Principal Components Analysis

A Introduction to Matrix Algebra and Principal Components Analysis A Introduction to Matrix Algebra and Principal Components Analysis Multivariate Methods in Education ERSH 8350 Lecture #2 August 24, 2011 ERSH 8350: Lecture 2 Today s Class An introduction to matrix algebra

More information

Chapter 3: Central Tendency

Chapter 3: Central Tendency Chapter 3: Central Tendency Central Tendency In general terms, central tendency is a statistical measure that determines a single value that accurately describes the center of the distribution and represents

More information

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that

More information

Engineering Problem Solving and Excel. EGN 1006 Introduction to Engineering

Engineering Problem Solving and Excel. EGN 1006 Introduction to Engineering Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques

More information

Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement

Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.

More information

GCSE HIGHER Statistics Key Facts

GCSE HIGHER Statistics Key Facts GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information

More information

( ) ( ) Central Tendency. Central Tendency

( ) ( ) Central Tendency. Central Tendency 1 Central Tendency CENTRAL TENDENCY: A statistical measure that identifies a single score that is most typical or representative of the entire group Usually, a value that reflects the middle of the distribution

More information

PIXEL-LEVEL IMAGE FUSION USING BROVEY TRANSFORME AND WAVELET TRANSFORM

PIXEL-LEVEL IMAGE FUSION USING BROVEY TRANSFORME AND WAVELET TRANSFORM PIXEL-LEVEL IMAGE FUSION USING BROVEY TRANSFORME AND WAVELET TRANSFORM Rohan Ashok Mandhare 1, Pragati Upadhyay 2,Sudha Gupta 3 ME Student, K.J.SOMIYA College of Engineering, Vidyavihar, Mumbai, Maharashtra,

More information

In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing.

In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing. 3.0 - Chapter Introduction In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing. Categories of Statistics. Statistics is

More information

03 The full syllabus. 03 The full syllabus continued. For more information visit www.cimaglobal.com PAPER C03 FUNDAMENTALS OF BUSINESS MATHEMATICS

03 The full syllabus. 03 The full syllabus continued. For more information visit www.cimaglobal.com PAPER C03 FUNDAMENTALS OF BUSINESS MATHEMATICS 0 The full syllabus 0 The full syllabus continued PAPER C0 FUNDAMENTALS OF BUSINESS MATHEMATICS Syllabus overview This paper primarily deals with the tools and techniques to understand the mathematics

More information

Inferential Statistics

Inferential Statistics Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

More information

MTH304: Honors Algebra II

MTH304: Honors Algebra II MTH304: Honors Algebra II This course builds upon algebraic concepts covered in Algebra. Students extend their knowledge and understanding by solving open-ended problems and thinking critically. Topics

More information

Remote Sensing Image Processing

Remote Sensing Image Processing Remote Sensing Image Processing -Pre-processing -Geometric Correction -Atmospheric correction -Image enhancement -Image classification Division of Spatial Information Science Graduate School Life and Environment

More information

StatTools Assignment #1, Winter 2007 This assignment has three parts.

StatTools Assignment #1, Winter 2007 This assignment has three parts. StatTools Assignment #1, Winter 2007 This assignment has three parts. Before beginning this assignment, be sure to carefully read the General Instructions document that is located on the StatTools Assignments

More information

Mathematics. Probability and Statistics Curriculum Guide. Revised 2010

Mathematics. Probability and Statistics Curriculum Guide. Revised 2010 Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction

More information

Descriptive Statistics

Descriptive Statistics Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

More information

Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo

Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo Readings: Ha and Ha Textbook - Chapters 1 8 Appendix D & E (online) Plous - Chapters 10, 11, 12 and 14 Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability

More information

Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion

Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion Statistics Basics Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion Part 1: Sampling, Frequency Distributions, and Graphs The method of collecting, organizing,

More information

Frequency distributions, central tendency & variability. Displaying data

Frequency distributions, central tendency & variability. Displaying data Frequency distributions, central tendency & variability Displaying data Software SPSS Excel/Numbers/Google sheets Social Science Statistics website (socscistatistics.com) Creating and SPSS file Open the

More information

Dimensionality Reduction: Principal Components Analysis

Dimensionality Reduction: Principal Components Analysis Dimensionality Reduction: Principal Components Analysis In data mining one often encounters situations where there are a large number of variables in the database. In such situations it is very likely

More information

business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar

business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel

More information

13.2 Measures of Central Tendency

13.2 Measures of Central Tendency 13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers

More information

Bracken County Schools Curriculum Guide Math

Bracken County Schools Curriculum Guide Math Unit 1: Expressions and Equations (Ch. 1-3) Suggested Length: Semester Course: 4 weeks Year Course: 8 weeks Program of Studies Core Content 1. How do you use basic skills and operands to create and solve

More information

Selecting the appropriate band combination for an RGB image using Landsat imagery

Selecting the appropriate band combination for an RGB image using Landsat imagery Selecting the appropriate band combination for an RGB image using Landsat imagery Ned Horning Version: 1.0 Creation Date: 2004-01-01 Revision Date: 2004-01-01 License: This document is licensed under a

More information

An introduction to Value-at-Risk Learning Curve September 2003

An introduction to Value-at-Risk Learning Curve September 2003 An introduction to Value-at-Risk Learning Curve September 2003 Value-at-Risk The introduction of Value-at-Risk (VaR) as an accepted methodology for quantifying market risk is part of the evolution of risk

More information

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

More information

Data Analysis: Describing Data - Descriptive Statistics

Data Analysis: Describing Data - Descriptive Statistics WHAT IT IS Return to Table of ontents Descriptive statistics include the numbers, tables, charts, and graphs used to describe, organize, summarize, and present raw data. Descriptive statistics are most

More information

UNDERSTANDING MULTIPLE REGRESSION

UNDERSTANDING MULTIPLE REGRESSION UNDERSTANDING Multiple regression analysis (MRA) is any of several related statistical methods for evaluating the effects of more than one independent (or predictor) variable on a dependent (or outcome)

More information

Exercise 1.12 (Pg. 22-23)

Exercise 1.12 (Pg. 22-23) Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

More information

Numerical Summarization of Data OPRE 6301

Numerical Summarization of Data OPRE 6301 Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting

More information

WATER BODY EXTRACTION FROM MULTI SPECTRAL IMAGE BY SPECTRAL PATTERN ANALYSIS

WATER BODY EXTRACTION FROM MULTI SPECTRAL IMAGE BY SPECTRAL PATTERN ANALYSIS WATER BODY EXTRACTION FROM MULTI SPECTRAL IMAGE BY SPECTRAL PATTERN ANALYSIS Nguyen Dinh Duong Department of Environmental Information Study and Analysis, Institute of Geography, 18 Hoang Quoc Viet Rd.,

More information

Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

More information

Statistical Concepts and Market Return

Statistical Concepts and Market Return Statistical Concepts and Market Return 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 2 2. Some Fundamental Concepts... 2 3. Summarizing Data Using Frequency Distributions...

More information

High School Algebra 1 Common Core Standards & Learning Targets

High School Algebra 1 Common Core Standards & Learning Targets High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities N-Q.1. Use units as a way to understand problems

More information

The Role of SPOT Satellite Images in Mapping Air Pollution Caused by Cement Factories

The Role of SPOT Satellite Images in Mapping Air Pollution Caused by Cement Factories The Role of SPOT Satellite Images in Mapping Air Pollution Caused by Cement Factories Dr. Farrag Ali FARRAG Assistant Prof. at Civil Engineering Dept. Faculty of Engineering Assiut University Assiut, Egypt.

More information

Lecture 7 Cogsci 109. Thurs. Oct. 12, 2006

Lecture 7 Cogsci 109. Thurs. Oct. 12, 2006 Lecture 7 Cogsci 109 Thurs. Oct. 12, 2006 Announcements Homework 2 is posted. Office hours Midterm is coming up somewhere in the next couple of weeks Anything lectured on, presented in section, in the

More information

Chapter 3 Descriptive Statistics: Numerical Measures. Learning objectives

Chapter 3 Descriptive Statistics: Numerical Measures. Learning objectives Chapter 3 Descriptive Statistics: Numerical Measures Slide 1 Learning objectives 1. Single variable Part I (Basic) 1.1. How to calculate and use the measures of location 1.. How to calculate and use the

More information

Chapter 15 Multiple Choice Questions (The answers are provided after the last question.)

Chapter 15 Multiple Choice Questions (The answers are provided after the last question.) Chapter 15 Multiple Choice Questions (The answers are provided after the last question.) 1. What is the median of the following set of scores? 18, 6, 12, 10, 14? a. 10 b. 14 c. 18 d. 12 2. Approximately

More information

Environmental Remote Sensing GEOG 2021

Environmental Remote Sensing GEOG 2021 Environmental Remote Sensing GEOG 2021 Lecture 4 Image classification 2 Purpose categorising data data abstraction / simplification data interpretation mapping for land cover mapping use land cover class

More information

4.1 Exploratory Analysis: Once the data is collected and entered, the first question is: "What do the data look like?"

4.1 Exploratory Analysis: Once the data is collected and entered, the first question is: What do the data look like? Data Analysis Plan The appropriate methods of data analysis are determined by your data types and variables of interest, the actual distribution of the variables, and the number of cases. Different analyses

More information

CHINHOYI UNIVERSITY OF TECHNOLOGY

CHINHOYI UNIVERSITY OF TECHNOLOGY CHINHOYI UNIVERSITY OF TECHNOLOGY SCHOOL OF NATURAL SCIENCES AND MATHEMATICS DEPARTMENT OF MATHEMATICS MEASURES OF CENTRAL TENDENCY AND DISPERSION INTRODUCTION From the previous unit, the Graphical displays

More information

How Landsat Images are Made

How Landsat Images are Made How Landsat Images are Made Presentation by: NASA s Landsat Education and Public Outreach team June 2006 1 More than just a pretty picture Landsat makes pretty weird looking maps, and it isn t always easy

More information

INTRODUCTION REMOTE SENSING

INTRODUCTION REMOTE SENSING INTRODUCTION REMOTE SENSING dr.ir. Jan Clevers Centre for Geo-Information Dept. Environmental Sciences Wageningen UR Remote Sensing --> RS Sensor at a distance EARTH OBSERVATION EM energy Earth RS is a

More information

Stats Review Chapters 3-4

Stats Review Chapters 3-4 Stats Review Chapters 3-4 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

More information

Chapter 3: Data Description Numerical Methods

Chapter 3: Data Description Numerical Methods Chapter 3: Data Description Numerical Methods Learning Objectives Upon successful completion of Chapter 3, you will be able to: Summarize data using measures of central tendency, such as the mean, median,

More information

MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

Descriptive Statistics. Understanding Data: Categorical Variables. Descriptive Statistics. Dataset: Shellfish Contamination

Descriptive Statistics. Understanding Data: Categorical Variables. Descriptive Statistics. Dataset: Shellfish Contamination Descriptive Statistics Understanding Data: Dataset: Shellfish Contamination Location Year Species Species2 Method Metals Cadmium (mg kg - ) Chromium (mg kg - ) Copper (mg kg - ) Lead (mg kg - ) Mercury

More information

Homework 3. Part 1. Name: Score: / null

Homework 3. Part 1. Name: Score: / null Name: Score: / Homework 3 Part 1 null 1 For the following sample of scores, the standard deviation is. Scores: 7, 2, 4, 6, 4, 7, 3, 7 Answer Key: 2 2 For any set of data, the sum of the deviation scores

More information

F. Farrokhyar, MPhil, PhD, PDoc

F. Farrokhyar, MPhil, PhD, PDoc Learning objectives Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc To recognize different types of variables To learn how to appropriately explore your data How to display data using graphs How

More information

Chapter 10. Key Ideas Correlation, Correlation Coefficient (r),

Chapter 10. Key Ideas Correlation, Correlation Coefficient (r), Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0-: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables

More information

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions. Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

More information

DATA INTERPRETATION AND STATISTICS

DATA INTERPRETATION AND STATISTICS PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE

More information

Table 2-1. Sucrose concentration (% fresh wt.) of 100 sugar beet roots. Beet No. % Sucrose. Beet No.

Table 2-1. Sucrose concentration (% fresh wt.) of 100 sugar beet roots. Beet No. % Sucrose. Beet No. Chapter 2. DATA EXPLORATION AND SUMMARIZATION 2.1 Frequency Distributions Commonly, people refer to a population as the number of individuals in a city or county, for example, all the people in California.

More information

Factor Analysis. Chapter 420. Introduction

Factor Analysis. Chapter 420. Introduction Chapter 420 Introduction (FA) is an exploratory technique applied to a set of observed variables that seeks to find underlying factors (subsets of variables) from which the observed variables were generated.

More information

Remote Sensing of Global Climate Change

Remote Sensing of Global Climate Change of Global Climate Change 2009 TECHNICAL STUFF... WHERE ON THE EARTH IS THAT??? Students should understand various geographic coordinate systems used to locate images, and how to translate these into distances

More information

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

More information

LEARNING OBJECTIVES SCALES OF MEASUREMENT: A REVIEW SCALES OF MEASUREMENT: A REVIEW DESCRIBING RESULTS DESCRIBING RESULTS 8/14/2016

LEARNING OBJECTIVES SCALES OF MEASUREMENT: A REVIEW SCALES OF MEASUREMENT: A REVIEW DESCRIBING RESULTS DESCRIBING RESULTS 8/14/2016 UNDERSTANDING RESEARCH RESULTS: DESCRIPTION AND CORRELATION LEARNING OBJECTIVES Contrast three ways of describing results: Comparing group percentages Correlating scores Comparing group means Describe

More information

Introduction to Quantitative Methods

Introduction to Quantitative Methods Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................

More information

Lean Six Sigma Training/Certification Book: Volume 1

Lean Six Sigma Training/Certification Book: Volume 1 Lean Six Sigma Training/Certification Book: Volume 1 Six Sigma Quality: Concepts & Cases Volume I (Statistical Tools in Six Sigma DMAIC process with MINITAB Applications Chapter 1 Introduction to Six Sigma,

More information

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( ) Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

More information

Calculation of Minimum Distances. Minimum Distance to Means. Σi i = 1

Calculation of Minimum Distances. Minimum Distance to Means. Σi i = 1 Minimum Distance to Means Similar to Parallelepiped classifier, but instead of bounding areas, the user supplies spectral class means in n-dimensional space and the algorithm calculates the distance between

More information

Big Ideas in Mathematics

Big Ideas in Mathematics Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards

More information

Describing Data. We find the position of the central observation using the formula: position number =

Describing Data. We find the position of the central observation using the formula: position number = HOSP 1207 (Business Stats) Learning Centre Describing Data This worksheet focuses on describing data through measuring its central tendency and variability. These measurements will give us an idea of what

More information

Concepts in Investments Risks and Returns (Relevant to PBE Paper II Management Accounting and Finance)

Concepts in Investments Risks and Returns (Relevant to PBE Paper II Management Accounting and Finance) Concepts in Investments Risks and Returns (Relevant to PBE Paper II Management Accounting and Finance) Mr. Eric Y.W. Leung, CUHK Business School, The Chinese University of Hong Kong In PBE Paper II, students

More information

Digital image processing

Digital image processing 746A27 Remote Sensing and GIS Lecture 4 Digital image processing Chandan Roy Guest Lecturer Department of Computer and Information Science Linköping University Digital Image Processing Most of the common

More information

Module 4: Data Exploration

Module 4: Data Exploration Module 4: Data Exploration Now that you have your data downloaded from the Streams Project database, the detective work can begin! Before computing any advanced statistics, we will first use descriptive

More information

not to be republished NCERT Measures of Central Tendency

not to be republished NCERT Measures of Central Tendency You have learnt in previous chapter that organising and presenting data makes them comprehensible. It facilitates data processing. A number of statistical techniques are used to analyse the data. In this

More information

Proceedings - AutoCarto Columbus, Ohio, USA - September 16-18, Rajinder S. Nagi

Proceedings - AutoCarto Columbus, Ohio, USA - September 16-18, Rajinder S. Nagi Maintaining detail and color definition when integrating color and grayscale rasters using No Alteration of Grayscale or Intensity (NAGI) fusion method Rajinder S. Nagi ABSTRACT: Integrating color raster

More information

An introduction to using Microsoft Excel for quantitative data analysis

An introduction to using Microsoft Excel for quantitative data analysis Contents An introduction to using Microsoft Excel for quantitative data analysis 1 Introduction... 1 2 Why use Excel?... 2 3 Quantitative data analysis tools in Excel... 3 4 Entering your data... 6 5 Preparing

More information

EXPLORING SPATIAL PATTERNS IN YOUR DATA

EXPLORING SPATIAL PATTERNS IN YOUR DATA EXPLORING SPATIAL PATTERNS IN YOUR DATA OBJECTIVES Learn how to examine your data using the Geostatistical Analysis tools in ArcMap. Learn how to use descriptive statistics in ArcMap and Geoda to analyze

More information

Descriptive statistics parameters: Measures of centrality

Descriptive statistics parameters: Measures of centrality Descriptive statistics parameters: Measures of centrality Contents Definitions... 3 Classification of descriptive statistics parameters... 4 More about central tendency estimators... 5 Relationship between

More information

The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

More information

Summarizing and Displaying Categorical Data

Summarizing and Displaying Categorical Data Summarizing and Displaying Categorical Data Categorical data can be summarized in a frequency distribution which counts the number of cases, or frequency, that fall into each category, or a relative frequency

More information

Dr. Peter Tröger Hasso Plattner Institute, University of Potsdam. Software Profiling Seminar, Statistics 101

Dr. Peter Tröger Hasso Plattner Institute, University of Potsdam. Software Profiling Seminar, Statistics 101 Dr. Peter Tröger Hasso Plattner Institute, University of Potsdam Software Profiling Seminar, 2013 Statistics 101 Descriptive Statistics Population Object Object Object Sample numerical description Object

More information

AP STATISTICS REVIEW (YMS Chapters 1-8)

AP STATISTICS REVIEW (YMS Chapters 1-8) AP STATISTICS REVIEW (YMS Chapters 1-8) Exploring Data (Chapter 1) Categorical Data nominal scale, names e.g. male/female or eye color or breeds of dogs Quantitative Data rational scale (can +,,, with

More information

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics. Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

More information

Lecture 2: Descriptive Statistics and Exploratory Data Analysis

Lecture 2: Descriptive Statistics and Exploratory Data Analysis Lecture 2: Descriptive Statistics and Exploratory Data Analysis Further Thoughts on Experimental Design 16 Individuals (8 each from two populations) with replicates Pop 1 Pop 2 Randomly sample 4 individuals

More information

Descriptive Data Summarization

Descriptive Data Summarization Descriptive Data Summarization (Understanding Data) First: Some data preprocessing problems... 1 Missing Values The approach of the problem of missing values adopted in SQL is based on nulls and three-valued

More information

Multiple regression - Matrices

Multiple regression - Matrices Multiple regression - Matrices This handout will present various matrices which are substantively interesting and/or provide useful means of summarizing the data for analytical purposes. As we will see,

More information

2. Simple Linear Regression

2. Simple Linear Regression Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according

More information

Data Mining: Exploring Data. Lecture Notes for Chapter 3. Introduction to Data Mining

Data Mining: Exploring Data. Lecture Notes for Chapter 3. Introduction to Data Mining Data Mining: Exploring Data Lecture Notes for Chapter 3 Introduction to Data Mining by Tan, Steinbach, Kumar What is data exploration? A preliminary exploration of the data to better understand its characteristics.

More information

Diagrams and Graphs of Statistical Data

Diagrams and Graphs of Statistical Data Diagrams and Graphs of Statistical Data One of the most effective and interesting alternative way in which a statistical data may be presented is through diagrams and graphs. There are several ways in

More information

GRADES 7, 8, AND 9 BIG IDEAS

GRADES 7, 8, AND 9 BIG IDEAS Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for

More information

Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012

Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012 Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization GENOME 560, Spring 2012 Data are interesting because they help us understand the world Genomics: Massive Amounts

More information

CHAPTER THREE COMMON DESCRIPTIVE STATISTICS COMMON DESCRIPTIVE STATISTICS / 13

CHAPTER THREE COMMON DESCRIPTIVE STATISTICS COMMON DESCRIPTIVE STATISTICS / 13 COMMON DESCRIPTIVE STATISTICS / 13 CHAPTER THREE COMMON DESCRIPTIVE STATISTICS The analysis of data begins with descriptive statistics such as the mean, median, mode, range, standard deviation, variance,

More information

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

More information

, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (

, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients ( Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we

More information