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STATISTICS Linear Statistical Models Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

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The Method of Least Squares Consider the data shown in the following table and figure. We are interested in fitting a straight line to the points in order to obtain a simple mathematical relationship for runoff and rainfall. 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 2

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 3

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Intuitively, we want that, for each observed value of rainfall, the corresponding value of runoff will be as close as possible to the observed value. It is equivalent to say that we want the vertical deviations to be as small as possible. 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 4

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One method of constructing such a straight line to fit the observed data is called the method of least squares. It requires the sum of the squares of the vertical deviations of all the points from the fitted line to be a minimum. Let the rainfall and runoff data in the above figure be respectively represented by x and y. The fitted line is expressed by 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 5

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 6

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 7

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 8

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 9

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Remarks 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 10

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 11

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 12 Given a value of x, what dose the predicted value of y really represent?

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– It is unlikely that the predicted value will be the same as the observed value at all times. – It may even be possible that the predicted value is the same as the observed value only in very few cases. – In some cases, the predicted values are far different from observed values. We are sure that the linear model may overpredict or underpredict the observed values. 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 13

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Linear statistical model 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 14 Random component We are not able to predict y without errors due to existence of the random component. If a phenomenon is stochastic in nature, it cannot be predicted without errors.

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 15

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Coefficient of determination How well does the least squares line explain the variation in the data? The coefficient of determination represents the proportion of data variation that can be explained by the linear regression model. 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 16

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Estimating the variance of Y | x 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 19 Note: The variance of Y|x is NOT the same as the variance of Y. RSS (Residual sum of squares) = SSE (sum of squared errors)

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 20

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Unbiasedness of the least squares estimators 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 21

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Confidence intervals of the regression coefficients Pivotal quantities 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 22

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Hypothesis tests for regression coefficients 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 23

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 25

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 26

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 27

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 28

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2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 29

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Simple linear regression using R Useful material – Chapter 11 of Introduction to Probability and Statistics Using R (G. J. Kerns) is highly recommended. – /ac /Class8/Using%20R%20for%20linear %20regression.pdf /ac /Class8/Using%20R%20for%20linear %20regression.pdf 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 30

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Defining linear regression models 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 31

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Conducting regression lm(y~model) 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 32

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Other useful commands 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 33

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– For prediction (x values not observed) 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 34

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Graphing the Confidence and Prediction Bands 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 35 You may want to change it. For example, data.frame(x=seq(20,30,by=0.5))

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Confidence and prediction intervals 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 36 Line of prediction. It represents the estimated conditional expectation of y given x.

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Multiple regression – The following slides are provided for your reference only. Due to the time constraint, they will not be covered in this class. 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 38

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Now let s consider fitting a linear function of several variables. Suppose that we have the following data set: 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 39

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The Linear Regression Model 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 44

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Covariance and Correlation Coefficient Suppose we have observed the following data. We wish to measure both the direction and the strength of the relationship between Y and X. 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 49

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The Analysis of Variance (ANOVA) 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 57

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Given X, Y s are independent normal random variables, i.e., The residual sum of squares (or sum of squared errors, SSE) is expressed by 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 58

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The total sum of squares corrected for the mean is referred to as the total variation. This total variation is split up in two parts: – the regression part (SSR m ) explained by the model, and – the residual part (SSE). 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 64

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The ratio is known as the coefficient of determination. If the coefficient of determination is large then the model provides a good fit to the data. It also represents the part of the total variation which is explained by the model. 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 65

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Properties of the Estimators 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 69

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Confidence Intervals 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 73

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The 100(1 – )% confidence interval of 2 is 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 74

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However, the true value of is unknown, the above equation can not be used to establish the confidence interval of. We then use s to substitute and it is known that has a t-distribution with (n – p) degree of freedom. 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 76

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The 100(1 – )% confidence interval of is 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 77

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Example 1 A scientist carries out an experiment on the relationship between the yield Y of a crop and the amount of irrigation water X. It is believed that the relationship between expected yield and amount of irrigation water (ignore the units) can be described adequately as 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 80

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The data shown in the following table were collected in the field. 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 81

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Example 2 Data in the following table are rainfall (x) and runoff (y) measured during the rainy season in a study area. A regression model is postulated for the above data 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 86

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Test of Hypotheses 2014/1/31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 92

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