UNIVARIENT & BIVARIENT GEO-STATISTICAL ANALYSIS

Course: Special Topics in Remote Sensing & GIS

Mirza Muhammad WaqarContact:

mirza.waqar@ist.edu.pk+92-21-34650765-79 EXT:2257

RG712

What is Statistics About?

Statistics is the science of collecting, organizing, analyzing and interpreting data in order to make decisions

Statistics is the science of data-based decision making in the case of uncertainty

PlanStatistical Cycle

Problem

DataAnalysis

Conclusion

Statistical Analysis

1. Problem

"I wonder if there are differences between...“

What information will you need to answer the question?

Identify two or more sub-groups of the population to compare.

What variables are likely to show differences?

2. Plan

If collecting data you will need to plan a survey of questionnaire.

Using available data sets is recommended If using a data set decide what sub-groups of

data are needed and choose from the available variables (choose carefully so you can answer the problem

3. Data

Collect data by making a survey or questionnaire, OR take a sample from large data set. (at least 30 values)

For example, Census data Clean the data set before continuing

4. Analysis

Analyze the data to find similarities and differences.

You will need measures of central tendency (mean, median, mode) AND measures of spread (range, inter quartile range, standard deviation)

Use technology to calculate the statistics: calculator, or EXCEL (using excel)

5. Conclusion

Remember that you are analysing and comparing data from a SAMPLE from a population

Is there a difference between the subgroups? Comparisons made from a Box-and-Whisker

graph Comparisons bases on measures of central

tendency Comparisons made from measures of spread

Role of Statistics in GIS

To describe and summarize spatial data. To make generalizations concerning complex

spatial patterns. To use samples of geographic data to infer

characteristics for a larger set of geographic data. To determine if the magnitude or frequency of

some phenomenon differs from one location to another.

To learn whether an actual spatial pattern matches some expected pattern.

What is Geostatistics?

Applies the theories of statistical inference to geographic phenomena.

Methods of geostatistics are used in petroleum geology, hydrogeology, hydrology, meteorology, oceanography, geochemistry

A way of describing the spatial continuity as an essential feature of natural phenomena.

Recognized to have emerged in the early 1980’s as a hybrid of mathematics, statistics, and mining engineering.

Some Useful Definitions

Data – information coming from observations, counts, measurements or responses. The data you will be analyzing will almost always be

a sample form a population. Population – the collection of all outcomes,

responses, measurements or counts that are of interest.

Sample – a subset of a population. We will almost always be dealing with samples and

hopping to make inference about the population.

Some Useful Definitions

Parameter – numerical description of a characteristic of the population.

Statistic – a description of a characteristic of the sample. We will often wish to make inferences about

parameter based on statistics.

Some Useful Definitions

Descriptive Statistics – relate to organizing, summarizing and displaying data.

Inferential Statistics – relate to using a sample to draw conclusions about a population. Inferential statistics involves drawing a conclusion

from some data.

Inferences vs. Descriptive

Consider: Average length of females and males: 90cm and

100cm respectively. Descriptive statistics: the values. Inference: males are (in general) taller than

females.

Descriptive Statistics

3 categories of descriptive statics in geostatistics

Univariate Descriptive Statistics Use to describe and summarize single data/variable

Bivariate Descriptive Statistics Use to describe relationship between two data/variable

Spatial Descriptive Statistics Describe data in term of space and time

Univariate Description

Describe and summarize single variable Graphical methods

Histogram Cumulative Frequency

Numerical methods divides in three categories Measurement of location Measurement of spread Measurement of shape

Univariate Description

Measurement of location Measurement of center location

Mean Median Mode

Measurement of other part Qunatile Quartile percentile

Measurement of spread (variability) Variance Standard Deviation Inter-Quartile range

Measurement of shape (symmetry & length) Coefficient of skewness Coefficient of Variation

Frequency Table and Histogram

Histogram – is a bar graph that plots the frequency of distribution of dataset. The horizontal scale is representing classes/bin The vertical scale measures the frequencies of the

classes. Consecutive boundaries much touch

Ideal Histogram for Image Analysis

Water

Vegetation

Soil

Urban Area

Band A

Freq

uenc

y (f

)

Actual Histogram from Image Analysis

Water

Vegetation

Soil

Urban Area

Band A

Freq

uenc

y (f

)

Histogram from Image Analysis

Very informative tool for analysis. Histogram define the contrast of satellite image.

More the BV’s range, more the contrast.

Low Contrast Histogram High Contrast Histogram

Histogram from Image Analysis

We can also identify the largest land cover in satellite image by histogram.

Rough quantification of landcovers can be made using histogram. This rough quantification leads to correct

quantification. Using histogram, range of a particular landcover

can be identified in aspect of BV.

Frequency Table

To develop a histogram a frequency table is used.

Frequency table: records how often observed values fall within certain intervals or classes.

Constructing a Frequency Distribution

Decide on the number of classes to include in the frequency distribution.

Find the class width as follows: Determine the range of the data Divide the range by the number of classes and round up to the next

convenient number Find the class limits:

Start with the lowest value as the lower limit of the first class, add the class width to this to obtain the lower limit for the second class, etc.

Place a mark in the row for the class corresponding to each data point

Count the number of marks in each class.

Frequency Table

Cumulative Frequency Table and Histogram

Cumulative frequency of a class is the sum of the frequency of that class and all previous classes.

The cumulative frequency for the last class is always n.

Cumulative Frequency Tables

Cumulative Histogram

Measure of Location

It provide us the information about where various part (information) of data lies

Center of data can be find by Mean Median Mode

Location of other parts of the data are given by the quantiles

Mean Median Mode

Mean – average of all the data points in the data/distribution Unique and unbiased Based on every data point in the dataset Can be sensitive to outlaying observations

Median – middle value in an ordered array of number. Unaffected by extremely large and extremely small values.

Mode – the most frequently occurring value in a dataset. Unlike the mean and median, the mode is not always uniquely

defined. Bimodal – two values having same number of instances in the data Multimodal – three or more values having same number of occurrences

Univarient Statistics for Image Analysis

The histogram of satellite image can not be the uni-mode data. Number of mode represents how many land covers

exists in the satellite image. We can’t make decision about transition zone

using histogram.

Univarient Statistics for Image Analysis

Water

Vegetation

Soil

Urban Area

Band A

Freq

uenc

y (f

)

Water

Vegetation

Soil

Urban Area

Band A

Freq

uenc

y (f

)

Which Measure is Best?

No clear answer to this question. The mean can be influenced by outliers while the

mode may not be particularly “typical central value”.

Statistical inference based on the median and the mode is difficult.

Percentiles

Divide a group of data into 100 parts At least n% od data live below the nth percentile, and

most (100-n)% of the data lie above the nth percentile.

Example – 90th percentile indicates that at least 90% of the data lie below it, and at most 10% of the data live above it.

The median and the 50% percentile have the same value.

Percentiles (i): Computational Procedure

Organize the data into an ascending ordered array.

Calculated percentile location i Determine the percentile’s location and its

value. If i is a whole number, the percentile is the

average of the value at the i and (i+1) positions. If i is not a whole number, the percentile is at

(i+1) position in the order array.

Percentiles: Example

Raw Data: 14, 12, 19, 23, 5, 13, 28, 17 Order Array: 5, 12, 13, 14, 17, 19, 23, 28 Location of 30th percentile i = = 2.4

The location index, i, is not a whole number; i+1=2.4+1=3.4; the whole number portion is 3; the 30th percentile is at the 30th location of the array; the 30th percentile is 13.

Quartiles

Formulae in EXCEL

Calculating Means: Average(data) Calculating Median: Median(data) Calculating Mode: Mode(data) Calculating Minimum: min(data) Calculating Maximum: max(data) Calculating Quartile: QUARTILE(data,quart) Calculating Percentile: PERCENTILE(array,k)

Measure of Spread/Variation

Measure of variability describe the spread or the dispersion of a dataset.

Common measures of variability Range Interquartile Range Mean Absolute Deviation Variance Standard Deviation Coefficient of Variation

Range

The difference between the largest and the smallest values in a set od data

Simple to compute Ignore all data points except two extremes

Range = Maximum – Minimum

Range tells us about the spread of data. Some time range provides us very biased

information when outliers exists in data

Interquartile Range

Range of values between the first and third quartiles

Less influenced by extremes Interquartile Range = Q3 – Q1

Deviation, Variance and Standard Deviation

The deviation of a data entry x in a population data set is the difference between x and population mean µ, i.e.

Deviation of x = x - µ

The sum of the deviation over entries is zero.

Mean Absolute Deviation

Average of the absolute deviation from the mean

X X - µ |X - µ|

5 -8 8

9 -4 4

16 3 3

17 4 4

18 5 5

0 24

M.A.D. =

M.A.D. = = 4.8

Variance

The population variance is the sum of squared deviation over all entries:

Population Variance = σ2 =

Population Variance

Average of squared deviation from the arithmetic mean

X X - µ (X - µ)2

5 -8 64

9 -4 16

16 3 9

17 4 16

18 5 25

0 130

σ2 =

M.A.D. = = 26.0

Sample Variance

S2 =

Variance for Image Analysis

For variance analysis, we go for comparative analysis.

By comparing variance of all bands we come to know that which band has more dispersion.

Band # Variance

Band 1 572

Band 2 634

Band 3 93

Band 4 224

Band 5 336

Band 7 325

Variance for Image Analysis

Less the variance, it depicts that the homogeneity of the data is high.

Outlier can disturb the variance.

Standard Deviation

The population standard deviation is the square root of the population variance i.e.

= = σ

Standard Deviation

Square root of the variance

X X - µ (X - µ)2

5 -8 64

9 -4 16

16 3 9

17 4 16

18 5 25

0 130

σ =

σ = =

Standard Deviation of Sampleσ =

Empirical Rules

Data are normally distributed (or approximately normally distributed)

Distance from the mean % of values falling within distance

µ ± 1σ 68

µ ± 2σ 95

µ ± 3σ 99.7

Shape of Distribution - Systematic

A frequency distribution is systematic when a vertical line can be drawn through the middle of a graph of distribution and the resulting halves are mirror images.

Shape of Distribution - Uniform

A frequency distribution is uniform when the number of entries in each class is equal.

Shape of Distribution - Skewed

A frequency distribution is skewed right (or positively skewed) if its tail extends to the right (mode < median < mean)

Shape of Distribution - Skewed

A frequency distribution is skewed left (of negatively skewed) if its tail extends to the left (mode > median > mean)

Measure of Shape

Shape of the distribution is described by Coefficient of skewness Coefficient of kurtosis

Coefficient of Skewness

Sknewness Absence of symmetry Extreme values in one side of distribution

Symmetry measure for skewness = Where E is Expected value (mean) If S<0, distribution is negatively skewed (skewed to

the left) If S=0, distribution is symmetric (not skewed) If S>0, distribution is positively skewed (skewed to

the right)

Skewness

Skewness

Kurtosis

Describes the shape of the curve about the mean Kurtosis is based on the size of distribution’s tail A measure of weather the curve of distribution

is: Bell Shaped – normal distribution Peaked – large tail (Leptokurtic) Flat – small tail (Platykurtic)

Kurtosis & Skewness

Coefficient of Kurtosis

The following formula can be used to calculate kurtosis:

Kurtosis = - 3

Kurtosis can be expressed as a number or value A value of kurtosis = 0 indicates symmetrical or no

kurtosis Positive value = leptokurtic Negative value = platykurtic

1. Covariance

2. Correlation

Multivariate Statistical Parameter

Covariance

How the two variables are varying with respect to each other.

Bands having same information content has high covariance and vice versa.

Optimum index factor (OIF) can be used to identify those bands which contain distinct information content.

Correlation

It is the measurement of linear relationship between the variables.

Correlation-Covariance Matrix

* Band 1 Band 2 Band 3 Band 4 Band 5 Band 7

Band 1 1 0.5 0.7 0.2 -0.4 0.9

Band 2 0.5 1 0.25 0.15 0.75 0.65

Band 3 0.7 0.25 1 0.29 -0.45 -0.1

Band 4 0.2 0.15 0.29 1 0.12 -0.25

Band 5 -0.4 0.75 -0.45 0.12 1 0.19

Band 7 0.9 0.65 -0.1 -0.25 0.19 1

Correlation Coefficient: +1 Direct RelationshipCorrelation Coefficient: 0 No RelationshipCorrelation Coefficient: -1 Indirect Relationship

Questions & Discussion