DOCSLIB.ORG

9 Polynomial Regression

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
9 Polynomial Regression MAS/MASOR/MASSM - Statistical Analysis - Autumn Term 9 Polynomial regression 9.1 Example An experiment in chemical engineering was carried out to obtain information about the e®ect of temperature on a glass laminate. The following table gives values of a quantity known as \log damping decrement" (y) for ten controlled temperatures (x) in degrees Celsius. x y -30 0.053 -20 0.057 -10 0.061 0 0.068 10 0.072 20 0.081 30 0.093 40 0.105 50 0.115 60 0.130 The following output shows the construction of the data frame that contains the data. The plot of the data suggests that there is a curvilinear relationship between y and x, which could be modelled by a polynomial curve. We begin by ¯tting a straight line to the data. Note that the function coef provides us with estimated regression coe±cients to more decimal places than does the function summary. The ¯tted line, y = 0:070745455 + 0:000850303x; explains almost 96% of the variation in the response variable, which looks very impres- sive, and the F -statistic for the regression is highly signi¯cant. However, to check the adequacy of the model, we should inspect the residuals, which we obtain using the function residuals and here store as the vector res1. A plot of the residuals against x exhibits a very strong pattern, which shows that the straight line model is inadequate and indicates that we might ¯t a quadratic to the data. y 0.06 0.08 0.10 0.12 -20 0 20 40 60 x Figure 1: Plot of y vs. x 1 > x <- seq(-30,60,by=10) > y <- c(0.053,0.057,0.061,0.068,0.072,0.081,0.093,0.105,0.115,0.130) > glass <- data.frame(y,x) > plot(x,y) > glass1.lm <- lm(y~x, data = glass) > summary(glass1.lm) Call: lm(formula = y ~ x, data = glass) Residuals: Min 1Q Median 3Q Max -0.007248 -0.003127 -0.0005 0.00288 0.008236 Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 0.0707 0.0020 34.8099 0.0000 x 0.0009 0.0001 13.5573 0.0000 Residual standard error: 0.005697 on 8 degrees of freedom Multiple R-Squared: 0.9583 F-statistic: 183.8 on 1 and 8 degrees of freedom, the p-value is 8.418e-007 Correlation of Coefficients: (Intercept) x -0.4629 > coef(glass1.lm) (Intercept) x 0.07074545 0.000850303 > res1 <- residuals(glass1.lm) > plot(x,res1) res1 -0.005 0.0 0.005 -20 0 20 40 60 x Figure 2: Plot of res1 vs. x The general situation which we consider is the one in which there are n observed pairs of values (xi; yi); i=1; : : : ; n, of the variables x; y, and we wish to ¯t a k-th order polynomial relationship of the form 2 k yi = ¯0 + ¯1xi + ¯2xi + ::: + ¯kxi + ²i i=1; : : : ; n; (1) 2 2 for some k with k < n, where the errors ²i are assumed to be NID(0; σ ). This is a special case of the multiple linear regression model in which the regressor variables are x; x2; : : : ; xk. ² Although we are ¯tting a non-linear relationship, the model (1) is linear in the parameters ¯0; ¯1; ¯2; : : : ; ¯k. The design matrix X corresponding to the above polynomial regression is a special case of the one for multiple linear regression, with 0 2 k 1 1 x1 x1 : : : x1 B C B 1 x x2 : : : xk C B 2 2 2 C X = B . C : @ . A 2 k 1 xn xn : : : xn In S-PLUS, given the variables x and y, we can de¯ne new variables x2; : : : ; xk and then carry out the multiple regression of y on x; x2; : : : ; xm for values of m with m · k to ¯t a polynomial of order k or less. What degree of polynomial should we ¯t? In the following output for our example, the variables x2, x3 and x4 are constructed and the data frame is extended to include the newly constructed variables. Next we consider a multiple linear regression of y on x and x2. > x2 <- x*x > x3 <- x2*x > x4 <- x3*x > glass <- data.frame(y,x,x2,x3,x4) > rm(x,x2,x3,x4,y) > attach(glass) > glass2.lm <- lm(y~x+x2, data = glass) > summary(glass2.lm) Call: lm(formula = y ~ x + x2, data = glass) Residuals: Min 1Q Median 3Q Max -0.001764 -0.0008682 0.00006894 0.0007739 0.001614 Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 0.0666 0.0006 117.9214 0.0000 x 0.0006 0.0000 29.5321 0.0000 x2 0.0000 0.0000 12.3261 0.0000 Residual standard error: 0.001278 on 7 degrees of freedom Multiple R-Squared: 0.9982 F-statistic: 1902 on 2 and 7 degrees of freedom, the p-value is 2.657e-010 Correlation of Coefficients: (Intercept) x x 0.2107 x2 -0.5906 -0.7645 3 > coef(glass2.lm) (Intercept) x x2 0.06663182 0.0006446212 6.856061e-006 > res2 <- residuals(glass2.lm) > plot(x,res2) res2 -0.001 0.0 0.001 -20 0 20 40 60 x Figure 3: Plot of res2 vs. x Note that the values of b0 and b1 have changed from what they were for the simple linear regression. We now obtain the ¯tted equation y = 0:06663182 + 0:0006446212x + 0:000006856061x2: The introduction of the variable x2 gives a signi¯cant improvement in the ¯t. Note ¯rstly the increase from the previous regression in the value of R2 and the dramatic increase in the value of the F-statistic in the ANOVA. However, more fundamentally, the fact that the t-statistic t = 12:3261 for x2 is highly signi¯cant shows that a highly signi¯cant improvement in ¯t is achieved by the inclusion of the variable x2. An apparently random scatter of points in the plot of the residuals (res2) against x indicates that we now have an adequate model. Nevertheless, we try a further regression, introducing a cubic term x3, to see if we can improve even further on the ¯t of the model. 4 > glass3.lm <- lm(y~x+x2+x3, data = glass) > summary(glass3.lm) Call: lm(formula = y ~ x + x2 + x3, data = glass) Residuals: Min 1Q Median 3Q Max -0.001762 -0.0008706 0.00007343 0.0007716 0.00161 Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 0.0666 0.0008 87.0416 0.0000 x 0.0006 0.0000 21.0199 0.0000 x2 0.0000 0.0000 5.4096 0.0016 x3 0.0000 0.0000 -0.0078 0.9940 Residual standard error: 0.001381 on 6 degrees of freedom Multiple R-Squared: 0.9982 F-statistic: 1087 on 3 and 6 degrees of freedom, the p-value is 1.355e-008 Correlation of Coefficients: (Intercept) x x2 x -0.2570 x2 -0.7546 0.2853 x3 0.6036 -0.6397 -0.8808 The introduction of the additional regressor variable x3 does not result in a signi¯cant im- provement in the ¯t. The value of the t-statistic for x3 is t = ¡0:0078, which is certainly 2 not signi¯cant (p = 0.9940). In fact the value of s = MSR has increased (s = 0:001381), so that the introduction of the x3 term into the model has been counter-productive. In looking for an appropriate degree of polynomial to ¯t we go through a stepwise pro- cedure of a special kind, where at each stage only the next power term is considered. If we include the regressor variable xm in the model then it is natural to include all lower powers of x as well. From the following output, which uses the function cor for correlation, we see that the regressor variables x2, x3 and x4 are particularly highly correlated with each other. This illustrates that problems of multicollinearity will arise and become ever more serious as further powers of x are introduced into the regression. > cor(glass) y x x2 x3 x4 y 1.0000000 0.9789228 0.8770847 0.9288534 0.8378634 x 0.9789228 1.0000000 0.7644708 0.8560000 0.7312607 x2 0.8770847 0.7644708 1.0000000 0.9479438 0.9589104 x3 0.9288534 0.8560000 0.9479438 1.0000000 0.9732258 x4 0.8378634 0.7312607 0.9589104 0.9732258 1.0000000 5 9.2 Orthogonal polynomials A way of dealing with problems of multicollinearity in polynomial regression is to work with what are known as orthogonal polynomials. Consider an alternative version of the model (1), yi = ®0 + ®1P1(xi) + ®2P2(xi) + ::: + ®kPk(xi) + ²i i=1; : : : ; n: (2) The regressor variables are functions P1(x);P2(x);:::;Pk(x) of the variable x, where, for r =1; : : : ; k, Pr(x) is a polynomial of order r, to be speci¯ed more precisely later. We thus have a linear model y = X® + ²; with design matrix X given by 0 1 1 P1(x1) P2(x1) :::Pk(x1) B C B 1 P (x ) P (x ) :::P (x ) C B 1 2 2 2 k 2 C X = B . C : @ . A 1 P1(xn) P2(xn) :::Pk(xn) We choose the polynomial functions P1(x);P2(x);:::;Pk(x) in such a way that the design matrix X is orthogonal, i.e., the columns of X are orthogonal to each other. We thus require that Xn Pr(xi) = 0; r =1; : : : ; k (3) i=1 and Xn Pr(xi)Ps(xi) = 0; r =2; : : : ; k; s=1; : : : ; r¡1: (4) i=1 A set of polynomials P1(x);P2(x);:::;Pk(x) which satis¯es the conditions of Equations (3) and (4) is said to be orthogonal.
Load more

Recommended publications
  • Simple Linear Regression with Least Square Estimation: an Overview
    Aditya N More et al, / (IJCSIT) International Journal of Computer Science and Information Technologies, Vol. 7 (6) , 2016, 2394-2396 Simple Linear Regression with Least Square Estimation: An Overview Aditya N More#1, Puneet S Kohli*2, Kshitija H Kulkarni#3 #1-2Information Technology Department,#3 Electronics and Communication Department College of Engineering Pune Shivajinagar, Pune – 411005, Maharashtra, India Abstract— Linear Regression involves modelling a relationship amongst dependent and independent variables in the form of a (2.1) linear equation. Least Square Estimation is a method to determine the constants in a Linear model in the most accurate way without much complexity of solving. Metrics where such as Coefficient of Determination and Mean Square Error is the ith value of the sample data point determine how good the estimation is. Statistical Packages is the ith value of y on the predicted regression such as R and Microsoft Excel have built in tools to perform Least Square Estimation over a given data set. line The above equation can be geometrically depicted by Keywords— Linear Regression, Machine Learning, Least Squares Estimation, R programming figure 2.1. If we draw a square at each point whose length is equal to the absolute difference between the sample data point and the predicted value as shown, each of the square would then represent the residual error in placing the I. INTRODUCTION regression line. The aim of the least square method would Linear Regression involves establishing linear be to place the regression line so as to minimize the sum of relationships between dependent and independent variables.
    [Show full text]
  • Application of Polynomial Regression Models for Prediction of Stress State in Structural Elements
    Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 4 (2016), pp. 3187-3199 © Research India Publications http://www.ripublication.com/gjpam.htm Application of polynomial regression models for prediction of stress state in structural elements E. Ostertagová, P. Frankovský and O. Ostertag Assistant Professor, Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering, Technical University of Košice, Slovakia Associate Professors, Department of Applied Mechanics and Mechanical Engineering Faculty of Mechanical Engineering, Technical University of Košice, Slovakia Abstract This paper presents the application of linear regression model for processing of stress state data which were collected through drilling into a structural element. The experiment was carried out by means of reflection photoelasticity. The harmonic star method (HSM) established by the authors was used for the collection of final data. The non-commercial software based on the harmonic star method enables us to automate the process of measurement for direct collection of experiment data. Such software enabled us to measure stresses in a certain point of the examined surface and, at the same time, separate these stresses, i.e. determine the magnitude of individual stresses. A data transfer medium, i.e. a camera, was used to transfer the picture of isochromatic fringes directly to a computer. Keywords: principal normal stresses, harmonic star method, simple linear regression, root mean squared error, mean absolute percentage error, R-squared, adjusted R-squared, MATLAB. Introduction Residual stresses are stresses which occur in a material even if the object is not loaded by external forces. The analysis of residual stresses is very important when determining actual stress state of structural elements.
    [Show full text]
  • Generalized Linear Models
    Generalized Linear Models Advanced Methods for Data Analysis (36-402/36-608) Spring 2014 1 Generalized linear models 1.1 Introduction: two regressions • So far we've seen two canonical settings for regression. Let X 2 Rp be a vector of predictors. In linear regression, we observe Y 2 R, and assume a linear model: T E(Y jX) = β X; for some coefficients β 2 Rp. In logistic regression, we observe Y 2 f0; 1g, and we assume a logistic model (Y = 1jX) log P = βT X: 1 − P(Y = 1jX) • What's the similarity here? Note that in the logistic regression setting, P(Y = 1jX) = E(Y jX). Therefore, in both settings, we are assuming that a transformation of the conditional expec- tation E(Y jX) is a linear function of X, i.e., T g E(Y jX) = β X; for some function g. In linear regression, this transformation was the identity transformation g(u) = u; in logistic regression, it was the logit transformation g(u) = log(u=(1 − u)) • Different transformations might be appropriate for different types of data. E.g., the identity transformation g(u) = u is not really appropriate for logistic regression (why?), and the logit transformation g(u) = log(u=(1 − u)) not appropriate for linear regression (why?), but each is appropriate in their own intended domain • For a third data type, it is entirely possible that transformation neither is really appropriate. What to do then? We think of another transformation g that is in fact appropriate, and this is the basic idea behind a generalized linear model 1.2 Generalized linear models • Given predictors X 2 Rp and an outcome Y , a generalized linear model is defined by three components: a random component, that specifies a distribution for Y jX; a systematic compo- nent, that relates a parameter η to the predictors X; and a link function, that connects the random and systematic components • The random component specifies a distribution for the outcome variable (conditional on X).
    [Show full text]
  • Lecture 18: Regression in Practice
    Regression in Practice Regression in Practice ● Regression Errors ● Regression Diagnostics ● Data Transformations Regression Errors Ice Cream Sales vs. Temperature Image source Linear Regression in R > summary(lm(sales ~ temp)) Call: lm(formula = sales ~ temp) Residuals: Min 1Q Median 3Q Max -74.467 -17.359 3.085 23.180 42.040 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -122.988 54.761 -2.246 0.0513 . temp 28.427 2.816 10.096 3.31e-06 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 35.07 on 9 degrees of freedom Multiple R-squared: 0.9189, Adjusted R-squared: 0.9098 F-statistic: 101.9 on 1 and 9 DF, p-value: 3.306e-06 Some Goodness-of-fit Statistics ● Residual standard error ● R2 and adjusted R2 ● F statistic Anatomy of Regression Errors Image Source Residual Standard Error ● A residual is a difference between a fitted value and an observed value. ● The total residual error (RSS) is the sum of the squared residuals. ○ Intuitively, RSS is the error that the model does not explain. ● It is a measure of how far the data are from the regression line (i.e., the model), on average, expressed in the units of the dependent variable. ● The standard error of the residuals is roughly the square root of the average residual error (RSS / n). ○ Technically, it’s not √(RSS / n), it’s √(RSS / (n - 2)); it’s adjusted by degrees of freedom. R2: Coefficient of Determination ● R2 = ESS / TSS ● Interpretations: ○ The proportion of the variance in the dependent variable that the model explains.
    [Show full text]
  • Chapter 2 Simple Linear Regression Analysis the Simple
    Chapter 2 Simple Linear Regression Analysis The simple linear regression model We consider the modelling between the dependent and one independent variable. When there is only one independent variable in the linear regression model, the model is generally termed as a simple linear regression model. When there are more than one independent variables in the model, then the linear model is termed as the multiple linear regression model. The linear model Consider a simple linear regression model yX01 where y is termed as the dependent or study variable and X is termed as the independent or explanatory variable. The terms 0 and 1 are the parameters of the model. The parameter 0 is termed as an intercept term, and the parameter 1 is termed as the slope parameter. These parameters are usually called as regression coefficients. The unobservable error component accounts for the failure of data to lie on the straight line and represents the difference between the true and observed realization of y . There can be several reasons for such difference, e.g., the effect of all deleted variables in the model, variables may be qualitative, inherent randomness in the observations etc. We assume that is observed as independent and identically distributed random variable with mean zero and constant variance 2 . Later, we will additionally assume that is normally distributed. The independent variables are viewed as controlled by the experimenter, so it is considered as non-stochastic whereas y is viewed as a random variable with Ey()01 X and Var() y 2 . Sometimes X can also be a random variable.
    [Show full text]
  • 1 Simple Linear Regression I – Least Squares Estimation
    1 Simple Linear Regression I – Least Squares Estimation Textbook Sections: 18.1–18.3 Previously, we have worked with a random variable x that comes from a population that is normally distributed with mean µ and variance σ2. We have seen that we can write x in terms of µ and a random error component ε, that is, x = µ + ε. For the time being, we are going to change our notation for our random variable from x to y. So, we now write y = µ + ε. We will now find it useful to call the random variable y a dependent or response variable. Many times, the response variable of interest may be related to the value(s) of one or more known or controllable independent or predictor variables. Consider the following situations: LR1 A college recruiter would like to be able to predict a potential incoming student’s first–year GPA (y) based on known information concerning high school GPA (x1) and college entrance examination score (x2). She feels that the student’s first–year GPA will be related to the values of these two known variables. LR2 A marketer is interested in the effect of changing shelf height (x1) and shelf width (x2)on the weekly sales (y) of her brand of laundry detergent in a grocery store. LR3 A psychologist is interested in testing whether the amount of time to become proficient in a foreign language (y) is related to the child’s age (x). In each case we have at least one variable that is known (in some cases it is controllable), and a response variable that is a random variable.
    [Show full text]
  • Mixed-Effects Polynomial Regression Models Chapter 5
    Mixed-effects Polynomial Regression Models chapter 5 1 Figure 5.1 Various curvilinear models: (a) decelerating positive slope; (b) accelerating positive slope; (c) decelerating negative slope; (d) accelerating negative slope 2 Figure 5.2 More curvilinear models: (a) positive to negative slope (β0 = 2, β1 = 8, β2 = −1.2); (b) inverted U-shaped slope (β0 = 2, β1 = 11, β2 = −2.2); (c) negative to positive slope (β0 = 14, β1 = −8, β2 = 1.2); (d) U-shaped slope (β0 = 14, β1 = −11, β2 = 2.2) 3 Expressing Time with Orthogonal Polynomials Instead of 1 1 1 1 1 1 0 0 X = Z = 0 1 2 3 4 5 0 1 4 9 16 25 use √ 1 1 1 1 1 1 / 6 √ 0 0 − − − X = Z = 5 3 1 1 3 5 / 70 √ 5 −1 −4 −4 −1 5 / 84 4 Figure 4.5 Intercept variance changes with coding of time 5 Top 10 reasons to use Orthogonal Polynomials 10 They look complicated, so it seems like you know what you’re doing 9 With a name like orthogonal polynomials they have to be good 8 Good for practicing lessons learned from “hooked on phonics” 7 Decompose more quickly than (orthogonal?) polymers 6 Sound better than lame old releases from Polydor records 5 Less painful than a visit to the orthodonist 4 Unlike ortho, won’t kill your lawn weeds 3 Might help you with a polygraph test 2 Great conversation topic for getting rid of unwanted “friends” 1 Because your instructor will give you an F otherwise 6 “Real” reasons to use Orthogonal Polynomials • for balanced data, and CS structure, estimates of polynomial fixed effects β (e.g., constant and linear) won’t change when higher-order
    [Show full text]
  • Linear Regression Using Stata (V.6.3)
    Linear Regression using Stata (v.6.3) Oscar Torres-Reyna [email protected] December 2007 http://dss.princeton.edu/training/ Regression: a practical approach (overview) We use regression to estimate the unknown effect of changing one variable over another (Stock and Watson, 2003, ch. 4) When running a regression we are making two assumptions, 1) there is a linear relationship between two variables (i.e. X and Y) and 2) this relationship is additive (i.e. Y= x1 + x2 + …+xN). Technically, linear regression estimates how much Y changes when X changes one unit. In Stata use the command regress, type: regress [dependent variable] [independent variable(s)] regress y x In a multivariate setting we type: regress y x1 x2 x3 … Before running a regression it is recommended to have a clear idea of what you are trying to estimate (i.e. which are your outcome and predictor variables). A regression makes sense only if there is a sound theory behind it. 2 PU/DSS/OTR Regression: a practical approach (setting) Example: Are SAT scores higher in states that spend more money on education controlling by other factors?* – Outcome (Y) variable – SAT scores, variable csat in dataset – Predictor (X) variables • Per pupil expenditures primary & secondary (expense) • % HS graduates taking SAT (percent) • Median household income (income) • % adults with HS diploma (high) • % adults with college degree (college) • Region (region) *Source: Data and examples come from the book Statistics with Stata (updated for version 9) by Lawrence C. Hamilton (chapter 6). Click here to download the data or search for it at http://www.duxbury.com/highered/.
    [Show full text]
  • Linear Regression in Matrix Form
    Statistics 512: Applied Linear Models Topic 3 Topic Overview This topic will cover • thinking in terms of matrices • regression on multiple predictor variables • case study: CS majors • Text Example (KNNL 236) Chapter 5: Linear Regression in Matrix Form The SLR Model in Scalar Form iid 2 Yi = β0 + β1Xi + i where i ∼ N(0,σ ) Consider now writing an equation for each observation: Y1 = β0 + β1X1 + 1 Y2 = β0 + β1X2 + 2 . Yn = β0 + β1Xn + n TheSLRModelinMatrixForm Y1 β0 + β1X1 1 Y2 β0 + β1X2 2 . = . + . . . . Yn β0 + β1Xn n Y X 1 1 1 1 Y X 2 1 2 β0 2 . = . + . . . β1 . Yn 1 Xn n (I will try to use bold symbols for matrices. At first, I will also indicate the dimensions as a subscript to the symbol.) 1 • X is called the design matrix. • β is the vector of parameters • is the error vector • Y is the response vector The Design Matrix 1 X1 1 X2 Xn×2 = . . 1 Xn Vector of Parameters β0 β2×1 = β1 Vector of Error Terms 1 2 n×1 = . . n Vector of Responses Y1 Y2 Yn×1 = . . Yn Thus, Y = Xβ + Yn×1 = Xn×2β2×1 + n×1 2 Variance-Covariance Matrix In general, for any set of variables U1,U2,... ,Un,theirvariance-covariance matrix is defined to be 2 σ {U1} σ{U1,U2} ··· σ{U1,Un} . σ{U ,U } σ2{U } ... σ2{ } 2 1 2 . U = . . .. .. σ{Un−1,Un} 2 σ{Un,U1} ··· σ{Un,Un−1} σ {Un} 2 where σ {Ui} is the variance of Ui,andσ{Ui,Uj} is the covariance of Ui and Uj.
    [Show full text]
  • Reading 25: Linear Regression
    Linear regression Class 25, 18.05 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Be able to use the method of least squares to fit a line to bivariate data. 2. Be able to give a formula for the total squared error when fitting any type of curve to data. 3. Be able to say the words homoscedasticity and heteroscedasticity. 2 Introduction Suppose we have collected bivariate data (xi; yi), i = 1; : : : ; n. The goal of linear regression is to model the relationship between x and y by finding a function y = f(x) that is a close fit to the data. The modeling assumptions we will use are that xi is not random and that yi is a function of xi plus some random noise. With these assumptions x is called the independent or predictor variable and y is called the dependent or response variable. Example 1. The cost of a first class stamp in cents over time is given in the following list. .05 (1963) .06 (1968) .08 (1971) .10 (1974) .13 (1975) .15 (1978) .20 (1981) .22 (1985) .25 (1988) .29 (1991) .32 (1995) .33 (1999) .34 (2001) .37 (2002) .39 (2006) .41 (2007) .42 (2008) .44 (2009) .45 (2012) .46 (2013) .49 (2014) Using the R function lm we found the ‘least squares fit’ for a line to this data is y = −0:06558 + 0:87574x; where x is the number of years since 1960 and y is in cents. Using this result we ‘predict’ that in 2016 (x = 56) the cost of a stamp will be 49 cents (since −0:06558 + 0:87574x · 56 = 48:98).
    [Show full text]
  • Design and Analysis of Ecological Data Landscape of Statistical Methods: Part 1
    Design and Analysis of Ecological Data Landscape of Statistical Methods: Part 1 1. The landscape of statistical methods. 2 2. General linear models.. 4 3. Nonlinearity. 11 4. Nonlinearity and nonnormal errors (generalized linear models). 16 5. Heterogeneous errors. 18 *Much of the material in this section is taken from Bolker (2008) and Zur et al. (2009) Landscape of statistical methods: part 1 2 1. The landscape of statistical methods The field of ecological modeling has grown amazingly complex over the years. There are now methods for tackling just about any problem. One of the greatest challenges in learning statistics is figuring out how the various methods relate to each other and determining which method is most appropriate for any particular problem. Unfortunately, the plethora of statistical methods defy simple classification. Instead of trying to fit methods into clearly defined boxes, it is easier and more meaningful to think about the factors that help distinguish among methods. In this final section, we will briefly review these factors with the aim of portraying the “landscape” of statistical methods in ecological modeling. Importantly, this treatment is not meant to be an exhaustive survey of statistical methods, as there are many other methods that we will not consider here because they are not commonly employed in ecology. In the end, the choice of a particular method and its interpretation will depend heavily on whether the purpose of the analysis is descriptive or inferential, the number and types of variables (i.e., dependent, independent, or interdependent) and the type of data (e.g., continuous, count, proportion, binary, time at death, time series, circular).
    [Show full text]
  • Lecture 2: Linear Regression
    Lecture 2: Linear regression Roger Grosse 1 Introduction Let's jump right in and look at our first machine learning algorithm, linear regression. In regression, we are interested in predicting a scalar-valued target, such as the price of a stock. By linear, we mean that the target must be predicted as a linear function of the inputs. This is a kind of supervised learning algorithm; recall that, in supervised learning, we have a collection of training examples labeled with the correct outputs. Regression is an important problem in its own right. But today's dis- cussion will also highlight a number of themes which will recur throughout the course: • Formulating a machine learning task mathematically as an optimiza- tion problem. • Thinking about the data points and the model parameters as vectors. • Solving the optimization problem using two different strategies: deriv- ing a closed-form solution, and applying gradient descent. These two strategies are how we will derive nearly all of the learning algorithms in this course. • Writing the algorithm in terms of linear algebra, so that we can think about it more easily and implement it efficiently in a high-level pro- gramming language. • Making a linear algorithm more powerful using basis functions, or features. • Analyzing the generalization performance of an algorithm, and in par- ticular the problems of overfitting and underfitting. 1.1 Learning goals • Know what objective function is used in linear regression, and how it is motivated. • Derive both the closed-form solution and the gradient descent updates for linear regression. • Write both solutions in terms of matrix and vector operations.
    [Show full text]