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STATISTICS Introduction Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University • Lecture notes will be posted on class website – https://www.space.ntu.edu.tw/navigate/s/E2DA955C12764B48B9C04F3 6492F48D1QQY – Digital reference book: A modern introduction to probability and statistics / Dekking et al. [Electronic book] • Grades – Homeworks (40%) – Midterm (30%), Final (30%) • The R language will be used for data analysis. • A tutorial session is arranged on Thursday (6:00 – 7:00 pm). Attendance of the tutorial session is voluntary. • Class attendance rule – If you are more than 15 minutes late for the class, please do NOT enter the classroom until the next class session. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 2 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 3 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 4 • 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 5 • 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 6 Practical Applications of Statistics 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 7 Iris recognition – An Iris code consists of 2048 bits. – The iris code of the same person may change at different times and different places. Thus one has to allow for a certain percentage of mismatching bits when identifying a person. – Of the 2048 bits, 266 may be considered as uncorrelated. Hamming distance is defined as the fraction of mismatches between two iris codes. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 8 A modern introduction to probability and statistics : understanding why and how / Dekking et al. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 9 Killer Football 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 10 41 27.2 deaths, the average over the 5 days preceding and following the match. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 11 • Poisson process modeling – Occurrences of rare events 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 12 Economic Warfare Analysis During World War II – In order to obtain more reliable estimates of German war production, experts from the Economic Warfare Division of the American Embassy and the British Ministry of Economic Warfare started to analyze markings and serial numbers obtained from captured German equipment. – Each piece of enemy equipment was labeled with markings, which included all or some portion of the following information: (a) the name and location of the maker; (b) the date of manufacture; (c) a serial number; and (d) miscellaneous markings such as trademarks, mold numbers, casting numbers, etc. A modern introduction to probability and statistics : understanding why and how / Dekking et al. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 13 – The first products to be analyzed were tires taken from German aircraft shot over Britain and from supply dumps of aircraft and motor vehicle tires captured in North Africa. The marking on each tire contained the maker’s name, a serial number, and a two-letter code for the date of manufacture. – The first step in analyzing the tire markings involved breaking the two-letter date code. • It was conjectured that one letter represented the month and the other the year of manufacture, and that there should be 12 letter variations for the month code and 3 to 6 for the year code. This, indeed, turned out to be true. The following table presents examples of the 12 letter variations used by four different manufacturers. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 14 – For each month, the serial numbers could be recoded to numbers running from 1 to some unknown largest number N. – The observed (recoded) serial numbers could be seen as a subset of this. – The objective was to estimate N for each month and each manufacturer separately by means of the observed (recoded) serial numbers. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 15 – With a sample of about 1400 tires from five producers, individual monthly output figures were obtained for almost all months over a period from 1939 to mid-1943. – The following table compares the accuracy of estimates of the average monthly production of all manufacturers of the first quarter of 1943 with the statistics of the Speer Ministry that became available after the war. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 16 A modern introduction to probability and statistics : understanding why and how / Dekking et al. – The accuracy of the estimates can be appreciated even more if we compare them with the figures obtained by Allied intelligence agencies. They estimated, using other methods, the production between 900 000 and 1 200 000 per month! 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 17 The Monty Hall Problem 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 18 • Standard assumptions – The host must always open a door that was not picked by the contestant. – The host must always open a door to reveal a goat and never the car. – The host must always offer the chance to switch between the originally chosen door and the remaining closed door. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 19 • Assuming the car is worth one million NTDs and the goat 5,000 NTDs, the expected amounts of award are – 668333.33 NTDs for the choice of switching – 336666.67 NTDs for the choice of not switching. • Simulation of the Monty Hall Problem using R. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 20 Ebola Outbreak in West Africa (as of Aug. 26, 2014) 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 21 2014 West Africa Ebola • Total cases since the beginning of the 2014 outbreak 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 22 2014 West Africa Ebola • Total death counts since the beginning of the 2014 outbreak 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 23 2014 West Africa Ebola • Death rate since the beginning of the 2014 outbreak 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 24 Spatial & Temporal Rainfall Analysis 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 25 臺灣防災地圖 | Google Crisis Map http://www.google.org/crisismap/taiwan 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 26 Stochastic Modeling & Simulation • Building probability models for real world phenomena. – No matter how sophisticated a model is, it only represents our understanding of the complicated natural systems. • Generating a large number of possible realizations. • Making decisions or assessing risks based on simulation results. • Conducted by computers. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 27 • Simulation of a two-dimensional random walk Possible applications? 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 28 Exploratory Data Analysis • Features of data distributions – Histograms – Center: mean, median – Spread: variance, standard deviation, range – Shape: skewness, kurtosis – Order statistics and sample quantiles – Clusters – Extreme observations: outliers 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 29 • Histogram: frequencies and relative frequencies – A sample data set X 104.838935 22.371870 24.762863 82.708815 82.535199 115.387515 64.158533 72.895810 85.553281 102.347372 5/25/2017 265.018615 129.538575 275.440477 149.905426 150.761192 102.460651 133.663194 107.569047 96.920012 19.277535 205.279506 37.587841 70.721022 113.442704 134.931864 16.480639 139.201204 81.266071 34.202372 134.484317 146.938446 231.608794 100.717110 131.144892 174.200632 9.961515 112.180103 101.351639 45.472935 121.101643 12.577133 60.397366 33.918756 9.539663 130.360126 53.449806 105.368124 16.652365 149.996985 10.382787 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 30 • Frequency histogram 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 31 • Relative histogram 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 32 • Measures of center – Sample mean – Sample median 5/25/2017 Sample mean = 98.26067 Sample median = 101.8495 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 33 – One desirable property of the sample median is that it is resistant to extreme observations, in the sense that its value depends only the values of the middle observations, and is quite unaffected by the actual values of the outer observations in the ordered list. The same cannot be said for the sample mean. Any significant changes in the magnitude of an observation results in a corresponding change in the value of the mean. Hence, the sample mean is said to be sensitive to extreme observations. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 34 • Measures of spread – Sample variance and sample standard deviation – Range • the difference between the largest and smallest values Sample variance = 4039.931 Sample standard deviation = 63.56045 Range = 265.9008 (275.440477 – 9.539663) 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 35 • Measures of shape – Sample skewness – Sample kurtosis Sample skewness = 0.7110874 Sample kurtosis = 0.533141 (or 3.533141 in R) 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 36 • Order statistics • Sample quantiles Linear interpolation 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 37 • Box-and-whisker plot (or box plot) – A box-and-whisker plot includes two major parts – the box and the whiskers. – A parameter range determines how far the plot whiskers extend out from the box. If range is positive, the whiskers extend to the most extreme data point which is no more than range times the interquartile range (IQR) from the box. A value of zero causes the whiskers to extend to the data extremes. – Outliers are marked by points which fall beyond the whiskers. – Hinges and the five-number summary 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 38 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 39 Not “linear interpolation” – In R, a boxplot is essentially a graphical representation determined by the 5NS. The summary function in R yields a list of six numbers: 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 40 – Box-and-whisker plot of X 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 41 Seasonal variation of average monthly rainfalls in CDZ, Myanmar – Boxplots are based on average monthly rainfalls of 54 rainfall stations. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 42 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 43 • 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) – Occurrences of earthquakes • The time interval between the occurrences of two consecutive higher-than-scale 6 earthquakes. – Occurrences of typhoons • The amount of rainfalls produced by typhoons in one year (yearly typhoon rainfalls). 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 44 • 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). 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 45 Definition of Probability • Classical probability • Frequency probability • Probability model 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 46 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 . 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 47 • 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) 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 48 Remarks • The probabilities determined by the classical definition are called “a priori” probabilities since they can be derived purely by deductive reasoning. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 49 • 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 50 • Troublesome limitations in the classical definition of probability: – If the number of possible outcomes is infinite; – If possible outcomes are not equally likely. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 51 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 52 • 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: f n ( A) . P(A)= lim n 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 53 • 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 54 • 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 55 • The chain of probability definition Random experiment 5/25/2017 Sample space Event space Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University Probability space 56 Probability Model 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 57 Event and event space An 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 58 Remarks 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 59 • Probability is a mapping of sets to numbers. • Probability is not a mapping of the sample space to numbers. – The expression P( ) for is not defined. However, for a singleton event {}, P ({}) is defined. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 60 Probability space • A probability space is the triplet (, A, P[]), where is a sample space, A is an event space, and P[] is a probability function with domain A. • A probability space constitutes a complete probabilistic description of a random experiment. – The sample space defines all of the possible outcomes, the event space A 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 61 Conditional probability 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 62 Bayes’ theorem 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 63 Multiplication rule 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 64 Independent events 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 65 • 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 66 • 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. 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 67 Reading assignments • IPSUR – Chapter 2 – Chapter 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 • AMIPS – Chapter 2 – Chapter 3 5/25/2017 Lab for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University 68