1. Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling Introduction STATISTICS Introduction Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

3. Lab for Remote Sensing Hydrology and Spatial Modeling Dept 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.

4. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Extracting useful information from data Assessing the reliability of that information How much are we sure about our claim based on the data?

6. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University One of the objectives of this course is to facilitate students with a critical and precise way of thinking. Accuracy of weather forecasting Accuracy of landuse classification using remote sensing images Accuracy of flood forecasting

7. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Key topics in statistics Probability Estimation Test of hypotheses Regression Forecasting Quality control Simulation

8. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

9. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

10. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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=  r 2, where r is the radius, we would attempt to measure the radius and substitute it in the formula.

11. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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)

12. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University The Random Experiment and Sample Space An experiment that can be repeated under similar conditions, but whose outcome cannot be predicted in advance, even when the same experiment has been performed many times is called a random experiment.

13. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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).

14. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Definition of Probability Classical probability Frequency probability Probability model

15. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Classical (or a priori) probability If a random experiment can result in n mutually exclusive and equally likely outcomes and if n A of these outcomes have an attribute A, then the probability of A is the fraction n A / n.

16. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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)

17. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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. Troublesome limitations in the classical definition of probability: If the number of possible outcomes is infinite; If possible outcomes are not equally likely.

18. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

19. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Suppose a random experiment is repeated n times under uniform conditions, and if event A occurred n A times, then the relative frequency for which A occurs is f n (A) = n A /n. If the limit of f n (A) as n approaches infinity exists then one can assign the probability of A by: P(A)=.

20. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

21. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

22. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Probability Model

23. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

24. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Remarks

25. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

26. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

27. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Conditional probability

28. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Bayes ’ theorem

29. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Multiplication rule

30. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Independent events

31. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

32. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

33. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Summarizing data Qualitative data Frequency table

34. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Bar chart

35. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Quantitative data Histogram

36. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Boxplot

37. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

38. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

39. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

40. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Measures of Variation Standard Deviation - summarizes how far away from the mean the data value typically are. Range