Introduction STATISTICS Introduction Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.

STATISTICS IntroductionIntroduction

Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering

National Taiwan University

• Lecture notes will be posted on class website– www.rslabntu.net– Supplementary material: IRSUR by Kerns

• Grades– Homeworks (60%) [No homework copying.]– Midterm (20%), Final (20%)

• The R language will be used for data analysis.• A tutorial session is arranged on Tuesday (6:00

– 7:00 pm).• Office hour: Thursday 2:30 – 3:30 pm

04/10/23Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

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What is “statistics”?

• Statistics is a science of “reasoning” from data.

• A body of principles and methods for extracting useful information from data, for assessing the reliability of that information, for measuring and managing risk, and for making decisions in the face of uncertainty.

04/10/23 3Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

• The major difference between statistics and mathematics is that statistics always needs “observed” data, while mathematics does not.

• An important feature of statistical methods is the “uncertainty” involved in analysis.


• Statistics is the discipline concerned with the study of variability, with the study of uncertainty and with the study of decision-making in the face of uncertainty. As these are issues that are crucial throughout the sciences and engineering, statistics is an inherently interdisciplinary science.


• Extracting useful information from data• Assessing the reliability of that information– How much are we sure about our claim based on

the data?


• One of the objectives of this course is to facilitate students with a critical way of thinking.– Accuracy of weather forecasting– Accuracy of flood forecasting– Not to be fooled by the surface meaning of

statistical terminologies.


Sources of uncertainties

• Data uncertainty• Parameter uncertainty• Model structure uncertainty– An exemplar illustration


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You are given a set of (x,y) data. Apparently, Y is dependent on X.


Observed data with uncertainties (Linear model)

y = 9.9507x - 48.343

R2 = 0.9534

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Observed data with uncertainties(Power model)

y = 3.5218x1.2335

R2 = 0.8852

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The linear model fits the data better than the power model.


Theoretical model:

y = 3.5218x1.2335

R2 = 0.8852

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)60,0(~,8.2 3.1 NiidXY

Sum of squared errors (SSE) of estimates of the linear and power models (with respect to the theoretical model) are 12011.7 and 8950.08, respectively.

Theoretical model

The power model performs better than the linear model.


Key topics in statistics

• Probability• Estimation• Test of hypotheses• Regression• Forecasting• Quality control• Simulation


Deterministic vs Stochastic Models

• An abstract model is a description of the essential properties of a phenomenon that is formulated in mathematical terms. – An abstract model is used as a theoretical

approximation of reality to help us understand the world around us.

All models are wrong!


• Essentially, all models are wrong, but some are useful.

• Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful. (George E. P. Box)– Normal distribution for men’s height, grades in a

statistics class, etc.


Types of abstract models

• Deterministic model– A deterministic model describes a phenomenon

whose outcome is fixed.

• Stochastic model– A random/stochastic model describes the

unpredictable variation of the outcomes of a random experiment.


Examples

• Deterministic model– Suppose we wish to measure the area covered by

a lake that, for all practical purposes, appears to have a circular shoreline. Since we know the area A=r2, where r is the radius, we would attempt to measure the radius and substitute it in the formula.


• Stochastic model– Consider the experiment of tossing a balanced coin and

observing the upper face. It is not possible to predict with absolute accuracy what the upper face will be even if we repeat the experiment so many times. However, it is possible to predict what will happen in the long run. We can say that the probability of heads on a single toss is ½.

– P(more than 60 heads in 100 trials)


Random Experiment and Sample Space

• An experiment that can be repeated under the same (or uniform) conditions, but whose outcome cannot be predicted in advance, even when the same experiment has been performed many times, is called a random experiment.

• Can the lotto draw be considered as a random experiment?


• Examples of random experiments– The tossing of a coin. – The roll of a die. – The selection of a numbered ball (1-50) in an urn.

(selection with replacement) – The time interval between the occurrences of two

higher than scale 6 earthquakes. – The amount of rainfalls produced by typhoons in

one year (yearly typhoon rainfalls).


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• The following items are always associated with a random experiment: – Sample space. The set of all possible outcomes,

denoted by . – Outcomes. Elements of the sample space,

denoted by . These are also referred to as sample points or realizations.

– Events. Subsets of for which the probability is defined. Events are denoted by capital Latin letters (e.g., A,B,C).


Definition of Probability

• Classical probability• Frequency probability• Probability model


Classical (or a priori) probability

• If a random experiment can result in n mutually exclusive and equally likely outcomes and if nA of these outcomes have

an attribute A, then the probability of A is the fraction nA/n .


• Example 1.

Compute the probability of getting two heads if a fair coin is tossed twice. (1/4)

• Example 2.

The probability that a card drawn from an ordinary well-shuffled deck will be an ace or a spade. (16/52)


Remarks

• The probabilities determined by the classical definition are called “a priori” probabilities since they can be derived purely by deductive reasoning.


• The “equally likely” assumption requires the experiment to be carried out in such a way that the assumption is realistic; such as, using a balanced coin, using a die that is not loaded, using a well-shuffled deck of cards, using random sampling, and so forth. This assumption also requires that the sample space is appropriately defined.


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• Troublesome limitations in the classical definition of probability: – If the number of possible outcomes is infinite; – If possible outcomes are not equally likely.


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Relative frequency(or a posteriori) probability

• We observe outcomes of a random experiment which is repeated many times. We postulate a number p which is the probability of an event, and approximate p by the relative frequency f with which the repeated observations satisfy the event.


• Suppose a random experiment is repeated n times under uniform conditions, and if event A occurred nA times, then the relative frequency for which A occurs is fn(A) = nA/n. If the limit of fn(A) as n approaches infinity exists then one can assign the probability of A by:

P(A)= .)(lim Af n

n


• This method requires the existence of the limit of the relative frequencies. This property is known as statistical regularity. This property will be satisfied if the trials are independent and are performed under uniform conditions.


• Example 3

A fair coin was tossed 100 times with 54 occurrences of head. The probability of head occurrence for each toss is estimated to be 0.54.


• Example 4– Randomly draw three balls in the box at one time. • What is the sample space of the random experiment?• What is the probability of having two or more blue balls

in a draw?• What if …


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• The chain of probability definition


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Random experimen

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Sample space

Event space

Probability space

Probability Model


Event and event spaceEvent and event spaceAn event is a subset of the sample space. The class of all events associated with a given random experiment is defined to be the event space.


Remarks


• Probability is a mapping of sets to numbers. • Probability is not a mapping of the sample

space to numbers. – The expression is not defined.

However, for a singleton event , is defined.

for )(P}{ })({P


Probability space

• A probability space is the triplet (, , P[]), where is a sample space, is an event space, and P[] is a probability function with domain .

• A probability space constitutes a complete probabilistic description of a random experiment. – The sample space defines all of the possible

outcomes, the event space defines all possible things that could be observed as a result of an experiment, and the probability P defines the degree of belief or evidential support associated with the experiment.


Conditional probability


Bayes’ theorem


Multiplication rule


Independent events


• The property of independence of two events A and B and the property that A and B are mutually exclusive are distinct, though related, properties.

• If A and B are mutually exclusive events then AB=. Therefore, P(AB) = 0. Whereas, if A and B are independent events then P(AB) = P(A)P(B). Events A and B will be mutually exclusive and independent events only if P(AB)=P(A)P(B)=0, that is, at least one of A or B has zero probability.


• But if A and B are mutually exclusive events and both have nonzero probabilities then it is impossible for them to be independent events.

• Likewise, if A and B are independent events and both have nonzero probabilities then it is impossible for them to be mutually exclusive.


Summarizing data

• Qualitative data– Frequency table

Freq Relative freqRed 14 0.156

Green 16 0.178Blue 21 0.233

Yellow 9 0.100White 25 0.278Pink 5 0.056


– Bar chart

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Red Green Blue Yellow White Pink


• Quantitative data

– Histogram

35 57 43 78 77 7775 88 86 78 79 4133 88 75 72 75 7773 50 50 24 60 4060 87 59 73 83 9085 88 33 65 82 3178 95


• Boxplot





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• Dealing with outliers– Should the outliers be discarded or should they be

retained?– An example of outlier presence• Typhoon Morakot


Typhoon Morakot

• Cumulative rainfall (Aug 7, 0:00 – 24:00)






Cumulative rainfall in mm

• 2009/08/07 00:00 ~ 2009/08/09 17:00


Measures of Central Tendency

• Mean– Sum of measurements divided by the number of

measurements.• Median– Middle value when the data are sorted.

• Mode– Value or category that occurs most frequently.


Measures of Variation

• Standard Deviation - summarizes how far away from the mean the data value typically are.

• Range

n

iin xx

ns

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)1(

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minmax xxR


Reading assignment

• IPSUR (Will be covered in the tutor session)– Chapt. 2– Chapt. 3• 3.1.1, 3.1.3, 3.1.4• 3.3• 3.4.3, 3.4.4, 3.4.5, 3.4.6, 3.4.7


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Introduction STATISTICS Introduction Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University.

Documents

remote sensing hydrology

theoretical model

uncertainties linear

uncertainties power

observed data

y data

data analysis

statistics class