Trend and Seasonal Components


 Melissa Palmer
 3 years ago
 Views:
Transcription
1 Chapter 2 Trend and Seasonal Components If the plot of a TS reveals an increase of the seasonal and noise fluctuations with the level of the process then some transformation may be necessary before doing any further analysis of the TS. For example, the fluctuations in the UK unemployment data and the sales of an industrial heater data are considerably reduced (evened) by log transformation; compare Figures 1.2 and 2.1 and Figures 1.7 and 2.2. Our aim is to estimate and eliminate the deterministic components m(t) and s(t) in such a way that the random noise Y t is a stationary process, in the sense that its fluctuations are stable and it has no trend. A formal definition of stationarity will be given in Chapter 4. Such a process can be modelled and analyzed using the well developed theory of stationary TS. All these components are parts of the original TS and their overall analysis leads to the analysis of X t. 2.1 Elimination of Trend in the Absence of Seasonality When there is no seasonality, the model (1.1) simplifies to X t = m t + Y t, t = 0, 1,...,n, (2.1) where E(X t ) = m t, that is, we assume that E(Y t ) = Least Square Estimation of Trend We assume that m(t) = m(t,β) is a function of time t, such as a polynomial of degree k, depending on some unknown parameters β = (β 0,β 1,...,β k ) T. We 11
2 12 CHAPTER 2. TREND AND SEASONAL COMPONENTS Log(Unemployment) Year Figure 2.1: Natural logarithm of the UK unemployment data, compare Fig Log(Sales) Year Figure 2.2: Natural logarithm of the data: Sales of an industrial heater.
3 2.1. ELIMINATION OF TREND IN THE ABSENCE OF SEASONALITY 13 estimate the parameters by fitting the function to the data x t, that is by minimizing the sum of squares of differences x t m(t,β). To find a general form of the estimator we work with the rvs X t rather than their realizations x t. We find the estimator β of β by minimizing the function: S(β) = (X t m(t,β)) 2. (2.2) For example, consider a linear trend m(t,β 0,β 1 ) = β 0 + β 1 t. Then S(β) = (X t β 0 β 1 t) 2. To minimize S(β) with respect to the parameters β = (β 0,β 1 ) T we need to calculate the derivatives of S(β) with respect to β i, i = 0, 1, compare them to zero and solve the resulting equations, i.e., ds(β) dβ 0 ds(β) dβ 1 = 2 = 2 (X t β 0 β 1 t) = 0 (X t β 0 β 1 t)t = 0. This can be written as X t nβ 0 β 1 X t t β 0 t = 0 t β 1 t 2 = 0. (2.3) Solving the first equation for β 0 we obtain β 0 = 1 n X t β 1 1 n t = X t β 1 t. Hence, equations (2.3) can be written as { β0 = X t β 1 t nx t t β 0 n t β 1 nt 2 = 0.
4 14 CHAPTER 2. TREND AND SEASONAL COMPONENTS 30 Residuals Year Figure 2.3: Residuals of a linear fit to the UK consumption data; compare Fig. 1.8 Then substituting β 0 to the second equation we can calculate the formula for β 1 and obtain the estimators of both parameters β 0 = X t β 1 t β 1 = X tt X t t t 2 ( t). 2 We need to distinguish an estimator as a function of rvs from an estimate (i.e., a value of the function for a given set of data). This should be obvious from the context. Fitting a linear trend to the UK consumption data (see Figure 1.8) we obtain the following estimates Hence the estimate of the trend is The random noise is then estimated as β 0 = , β1 = m(t) = t. ŷ t = x t m t = x t ( t). The plot (see Figure 2.3) of the residuals does not show any clear trend. However it indicates correlations of the random noise. In this course we will be modelling
5 2.1. ELIMINATION OF TREND IN THE ABSENCE OF SEASONALITY LS_RegressResids LS_Residuals_Lag LS_Residuals_Lag LS_Residuals_Lag Figure 2.4: Correlation matrix for the residuals of the Least Squares Regression fit; UK consumption data. this kind of random variables to be able to predict future values of the TS, for example next year values of consumption in the UK. The matrix plot shows that indeed the residuals are dependent, at least up to two neighboring values.
6 16 CHAPTER 2. TREND AND SEASONAL COMPONENTS Smoothing by a Moving Average Let q be a positive integer. We call W t = 1 2q + 1 q X t+j (2.4) a symmetric twosided moving average of the process {X t }. Simple example Take the following realization of {X t }: {x t } = { } For q = 1 we have the following realizations of W t : w 1 =? w 2 = 1 10 ( ) = 3 3 w 3 = 1 3 ( ) = 9 3 w 4 = 1 11 ( ) = 3 3. w 9 = 1 20 ( ) = 3 3 w 10 =? Equation (2.4) defines W t for q + 1 t n q. Problem: How to calculate values of W t for t q and for t > n q? One possibility is to define X t = X 1 for t < 1 and X t = X n for t > n. Then in the simple example we would get x 0 = x 1 = 5 and x 11 = x 10 = 9 and w 1 = 1 3 ( ) = 13 3 w 10 = 1 3 ( ) = What can we achieve calculating moving average? If X t = m t + Y t, then W t = 1 2q + 1 q m t+j + 1 2q + 1 q Y t+j.
7 2.1. ELIMINATION OF TREND IN THE ABSENCE OF SEASONALITY 17 Changes in Temperature and Moving Average Series Year Figure 2.5: Surface air temperature change for the globe. The TS data for years [Degrees Celsius] and a symmetric five elements moving average. If we can assume that the sum of noise values is close to zero, then W t evaluates the trend m t. It is a good evaluation of m t if, over [t q,t + q], the trend is approximately linear. It is used as an estimator of trend and we write m t = W t = 1 2q + 1 Then, the estimator of the noise is q X t+j for q + 1 t n q. Ŷ t = X t W t. The moving average values w t for q = 2 for the surface air temperature change for the globe data are shown in Figure 2.5 and the residuals y t = x t w t in Figure 2.6. There is no apparent trend in the residuals. The process {W t } is a linear function (linear filter) of random variables {X t }. In general, it may be written as W t = a j X t+j, (2.5) where j= 1 for q j q, a j = 2q for j > q,
8 18 CHAPTER 2. TREND AND SEASONAL COMPONENTS Residuals Year Figure 2.6: Residuals after removing trend calculated as a symmetric five elements moving average. Surface air temperature change for the globe data, if W t is a symmetric twosided ma. Graphically, it could be presented as in Figure 2.7. It smoothes the data by averaging, that is by removing rapid fluctuations (bursts). It attenuates the noise. The set of coefficients {a j },...,q is also called a linear filter. If q m t = a j X t+j then we say that the filter {a j } passes through without distortion. The weights {a j } neither have to be all equal nor symmetric. By applying appropriate weights it is possible to include a wide choice of trend functions. Suppose that the trend follows a polynomial of order three and we want to find a seven points filter {a j } j= 3,...,3, that is a moving average W t = a j X t+j. j= 3
9 2.1. ELIMINATION OF TREND IN THE ABSENCE OF SEASONALITY 19 {x t } {w Linear Filter Σa j x t } t+j If the trend is Figure 2.7: Graphical representation of a linear filter m t = β 0 + β 1 t + β 2 t 2 + β 3 t 3, then a filter which passes through without distortion is such that m t = W t and so a j X t+j = β 0 + β 1 t + β 2 t 2 + β 3 t 3. j= 3 We are interested in the middle point, i.e., for j = 0. It is convenient to take values of t such that the middle one is zero. For a seven point average they will be t = 3, 2, 1, 0, 1, 2, 3. Then the trend at the middle point, and so the value of the moving average, is m(t = 0) = β 0. To find β 0 we may apply the Least Squares method for the seven points. Differentiating the function ( S(β) = Xt (β 0 + β 1 t + β 2 t 2 + β 3 t 3 ) ) 2 with respect to vector β gives the following four equations S(β) ( = 2 Xt (β 0 + β 1 t + β 2 t 2 + β 3 t 3 ) ) = 0 β 0 S(β) β 1 = 2 S(β) β 2 = 2 S(β) β 3 = 2 ( Xt (β 0 + β 1 t + β 2 t 2 + β 3 t 3 ) ) t = 0 ( Xt (β 0 + β 1 t + β 2 t 2 + β 3 t 3 ) ) t 2 = 0 ( Xt (β 0 + β 1 t + β 2 t 2 + β 3 t 3 ) ) t 3 = 0.
10 20 CHAPTER 2. TREND AND SEASONAL COMPONENTS After simple manipulation we obtain ( X t = β0 + β 1 t + β 2 t 2 + β 3 t 3) X t t = X t t 2 = X t t 3 = ( β0 + β 1 t + β 2 t 2 + β 3 t 3) t ( β0 + β 1 t + β 2 t 2 + β 3 t 3) t 2 ( β0 + β 1 t + β 2 t 2 + β 3 t 3) t 3. However, odd powers of t sum to zero, that is t = t 3 = t 5 = 0, what simplifies the equations to X t = 7β β 2 X t t = 28β β 3 X t t 2 = 28β β 2 X t t 3 = 196β β 3 of which only equations 1 and 3 involve β 0, so we can ignore the other two equations. Solving equations 1 and 3 for β 0 then gives β 0 = 1 21 ( 2X 3 + 3X 2 + 6X 1 + 7X 0 + 6X 1 + 3X 2 2X 3 ). The trend value for any point is then the weighted average of the seven points of which that point is in the centre, m t = W t = 1 21 ( 2X t 3 + 3X t 2 + 6X t 1 + 7X t + 6X t+1 + 3X t+2 2X t+3 ).
11 2.1. ELIMINATION OF TREND IN THE ABSENCE OF SEASONALITY 21 The weights are ( 2 21, 3 21, 6 21, 7 21, 7 21, 3 21, 2 ) = 1 ( 2, 3, 6, 7, 6, 3, 2), which due to the symmetry can be written in a short way, such as 1 ( 2, 3, 6, 7), 21 the bold typeface indicating the middle weight. In general, if we want to find a linear filter {a j } appropriate for fitting a polynomial trend of order k, we have to minimize S(β) = q ( Xt (β 0 + β 1 t β k t k ) ) 2 t= q and find the solution for β 0 in terms of X t, t = q,...,q. The following filters are often applied: {a j } being successive terms in the expansion ( )2q. For q = 1 we get ( 1 {a j } j= 1,0,1 = 4, 1 2, 1 ). 4 For large q the weights approximate to a normal curve. So called Spencer s 15point moving average (used for smoothing mortality statistics to obtain life tables). Here q = 7 and the weights are {a j } j= 7,...,7 = 1 ( 3, 6, 5, 3, 21, 46, 67,74). 320 The Henderson s moving average, which aims to follow a cubic polynomial trend. For example, for q = 4 the Henderson s m.a. is {a j } j= 4,...,4 = ( 0.041, 0.010, 0.119, 0.267,0.330). Having estimated the trend we can calculate the residuals, as in the previous case, i.e., q q ŷ t = x t w t = x t a j x t+j = b j x t+j,
12 22 CHAPTER 2. TREND AND SEASONAL COMPONENTS x t Σaj x t+j w t Σb k w t+k z t where Figure 2.8: Graphical representation of a double linear filter and if a j = 1 then b j = 0. { aj, for j = q,...,q,j 0 b j = 1 a 0, for j = 0 Sometimes it may be necessary to apply filters more than once to obtain a stationary process. Two filters might be represented as in Figure 2.8. The resulting time series {Z t } can be written as s s q Z t = b k W t+k = a j X t+k+j = k= s k= s b k For example, for s = q = 1 we obtain 1 1 Z t = a j X t+k+j = k= 1 b k j= 1 s+q i= (s+q) c i X t+i. b 1 a 1 X t 2 + (b 1 a 0 + b 0 a 1 )X t 1 + (b 1 a 1 + b 0 a 0 + b 1 a 1 )X t + 2 (b 0 a 1 + b 1 a 0 )X t+1 + b 1 a 1 X t+2 = c i X t+i. i= 2 The weights {c i } can be obtained by so called convolution of the weights {a j } and {b k }, the operation denoted by, {c i } = {a j } {b k }, where c i is a sum of all products a j b k for which j + k = i. Using this operation in the above example gives {c i } = (a 1,a 0,a 1 ) (b 1,b 0,b 1 ) = (a 1 b 1, a 0 b 1 + a 1 b 0, a 1 b 1 + a 0 b 0 + a 1 b 1, a 0 b 1 + a 1 b 0, a 1 b 1 ) = (c 2,c 1,c 0,c 1,c 2 ).
13 2.1. ELIMINATION OF TREND IN THE ABSENCE OF SEASONALITY 23 If one of the moving averages is not symmetric, we calculate the convoluted weights in the same way, see the example below. (a 1,a 0,a 1 ) (b 0,b 1 ) = (a 1 b 0, a 0 b 0 + a 1 b 1, a 0 b 1 + a 1 b 0, a 1 b 1 ) = (c 1,c 0,c 1,c 2 ). With {a j } = ( 1, 1, ) and {bk } = ( 1 {c i } = ( 1 3, 1 3, 1 ) 3, 1 2 2) we get ( 1 2, 1 ) = 2 and the new series Z t is the following combination of X t ( 1 6, 1 3, 1 3, 1 ). 6 Z t = 1 6 X t X t X t X t Differencing Differencing is a trend removing operation by using special kind of a linear filter with weights ( 1, 1). This procedure is often repeated until a stationary series is obtained. We denote the first (lag 1) difference operator by, that is X t = X t X t 1. Here another operator comes into play, it is called the backward shift operator, usually denoted by B, such that BX t = X t 1. Hence, Note that In general Similarly, X t = X t BX t = (1 B)X t. X t 2 = BX t 1 = B(BX t ) = B 2 X t. B j X t = X t j. 2 X t = ( X t ) = (1 B)(1 B)X t = (1 2B + B 2 )X t = X t 2X t 1 + X t 2.
14 24 CHAPTER 2. TREND AND SEASONAL COMPONENTS x t  x t Year Figure 2.9: The differenced data: UK Consumption, compare Figures 1.1 and 1.8 By the way, this is what we would get by convolution of two linear filters with weights ( 1, 1) (a 1,a 0 ) (b 1,b 0 ) = ( 1, 1) ( 1, 1) = (( 1) ( 1), ( 1) ( 1), 1 1) = (1, 2, 1) = (c 2, c 1, c 0 ). In general, we may write j X t = ( j 1 X t ), for j 1, 0 X t = X t. Assume that a TS model is X t = m t + Y t, where the trend m t is a polynomial of degree k. Then for k = 1 we have X t = X t X t 1 = m t + Y t (m t 1 + Y t 1 ) = β 0 + β 1 t [β 0 + β 1 (t 1)] + Y t = β 1 + Y t.
15 2.1. ELIMINATION OF TREND IN THE ABSENCE OF SEASONALITY 25 Similarly, for k = 2 we obtain 2 X t = X t 2X t 1 + X t 2 = m t + Y t 2(m t 1 + Y t 1 ) + m t 2 + Y t 2 = β 0 + β 1 t + β 2 t 2 2[β 0 + β 1 (t 1) + β 2 (t 1) 2 ] + β 0 + β 1 (t 2) + β 2 (t 2) Y t = 2β Y t. For a polynomial trend of degree k we have k X t = k!β k + k Y t. It means that if the noise fluctuates about zero, then kth differencing of a TS with a polynomial trend of degree k should give a stationary process with mean about k!β k. Example UK consumption data. The data show a linear trend. The least squares methods gives the following estimates ˆm(t) = t and we can see from the graph of the differenced data (Figure 2.9) that the mean is close to β 1 and that there is no trend.
2.2 Elimination of Trend and Seasonality
26 CHAPTER 2. TREND AND SEASONAL COMPONENTS 2.2 Elimination of Trend and Seasonality Here we assume that the TS model is additive and there exist both trend and seasonal components, that is X t = m t +
More informationTime series Forecasting using HoltWinters Exponential Smoothing
Time series Forecasting using HoltWinters Exponential Smoothing Prajakta S. Kalekar(04329008) Kanwal Rekhi School of Information Technology Under the guidance of Prof. Bernard December 6, 2004 Abstract
More information3. Regression & Exponential Smoothing
3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a
More informationMoving averages. Rob J Hyndman. November 8, 2009
Moving averages Rob J Hyndman November 8, 009 A moving average is a time series constructed by taking averages of several sequential values of another time series. It is a type of mathematical convolution.
More informationIntroduction to time series analysis
Introduction to time series analysis Margherita Gerolimetto November 3, 2010 1 What is a time series? A time series is a collection of observations ordered following a parameter that for us is time. Examples
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationLINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL
Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables
More informationCURVE FITTING LEAST SQUARES APPROXIMATION
CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationSection A. Index. Section A. Planning, Budgeting and Forecasting Section A.2 Forecasting techniques... 1. Page 1 of 11. EduPristine CMA  Part I
Index Section A. Planning, Budgeting and Forecasting Section A.2 Forecasting techniques... 1 EduPristine CMA  Part I Page 1 of 11 Section A. Planning, Budgeting and Forecasting Section A.2 Forecasting
More informationNovember 16, 2015. Interpolation, Extrapolation & Polynomial Approximation
Interpolation, Extrapolation & Polynomial Approximation November 16, 2015 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationTime Series and Forecasting
Chapter 22 Page 1 Time Series and Forecasting A time series is a sequence of observations of a random variable. Hence, it is a stochastic process. Examples include the monthly demand for a product, the
More information8. Time Series and Prediction
8. Time Series and Prediction Definition: A time series is given by a sequence of the values of a variable observed at sequential points in time. e.g. daily maximum temperature, end of day share prices,
More informationChapter 4: Vector Autoregressive Models
Chapter 4: Vector Autoregressive Models 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)...
More informationRob J Hyndman. Forecasting using. 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1
Rob J Hyndman Forecasting using 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1 Outline 1 Regression with ARIMA errors 2 Example: Japanese cars 3 Using Fourier terms for seasonality 4
More informationDemand Forecasting When a product is produced for a market, the demand occurs in the future. The production planning cannot be accomplished unless
Demand Forecasting When a product is produced for a market, the demand occurs in the future. The production planning cannot be accomplished unless the volume of the demand known. The success of the business
More informationDERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Realvalued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we
More information(Refer Slide Time: 1:42)
Introduction to Computer Graphics Dr. Prem Kalra Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture  10 Curves So today we are going to have a new topic. So far
More informationTIME SERIES ANALYSIS
TIME SERIES ANALYSIS L.M. BHAR AND V.K.SHARMA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi0 02 lmb@iasri.res.in. Introduction Time series (TS) data refers to observations
More information5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
More informationUsing R for Linear Regression
Using R for Linear Regression In the following handout words and symbols in bold are R functions and words and symbols in italics are entries supplied by the user; underlined words and symbols are optional
More informationLINEAR EQUATIONS IN TWO VARIABLES
66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that
More informationMULTIPLE REGRESSION WITH CATEGORICAL DATA
DEPARTMENT OF POLITICAL SCIENCE AND INTERNATIONAL RELATIONS Posc/Uapp 86 MULTIPLE REGRESSION WITH CATEGORICAL DATA I. AGENDA: A. Multiple regression with categorical variables. Coding schemes. Interpreting
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.
More informationLinear Equations and Inequalities
Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109  Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................
More informationIntroduction to the Finite Element Method (FEM)
Introduction to the Finite Element Method (FEM) ecture First and Second Order One Dimensional Shape Functions Dr. J. Dean Discretisation Consider the temperature distribution along the onedimensional
More information2013 MBA Jump Start Program
2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of
More informationTime Series Analysis
Time Series Analysis Forecasting with ARIMA models Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and GarcíaMartos (UC3MUPM)
More informationPart 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable
More informationTime Series Analysis
Time Series Analysis Identifying possible ARIMA models Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and GarcíaMartos
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationStatistics in Retail Finance. Chapter 6: Behavioural models
Statistics in Retail Finance 1 Overview > So far we have focussed mainly on application scorecards. In this chapter we shall look at behavioural models. We shall cover the following topics: Behavioural
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More informationthe points are called control points approximating curve
Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
More informationIrrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers.
Irrational Numbers A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Definition: Rational Number A rational number is a number that
More informationA characterization of trace zero symmetric nonnegative 5x5 matrices
A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationSales and operations planning (SOP) Demand forecasting
ing, introduction Sales and operations planning (SOP) forecasting To balance supply with demand and synchronize all operational plans Capture demand data forecasting Balancing of supply, demand, and budgets.
More information7 Time series analysis
7 Time series analysis In Chapters 16, 17, 33 36 in Zuur, Ieno and Smith (2007), various time series techniques are discussed. Applying these methods in Brodgar is straightforward, and most choices are
More informationFunctions: Piecewise, Even and Odd.
Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 2122, 2015 Tutorials, Online Homework.
More informationState Space Time Series Analysis
State Space Time Series Analysis p. 1 State Space Time Series Analysis Siem Jan Koopman http://staff.feweb.vu.nl/koopman Department of Econometrics VU University Amsterdam Tinbergen Institute 2011 State
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More information4. Simple regression. QBUS6840 Predictive Analytics. https://www.otexts.org/fpp/4
4. Simple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/4 Outline The simple linear model Least squares estimation Forecasting with regression Nonlinear functional forms Regression
More informationSoftware Review: ITSM 2000 Professional Version 6.0.
Lee, J. & Strazicich, M.C. (2002). Software Review: ITSM 2000 Professional Version 6.0. International Journal of Forecasting, 18(3): 455459 (June 2002). Published by Elsevier (ISSN: 01692070). http://0
More informationProcedure for Graphing Polynomial Functions
Procedure for Graphing Polynomial Functions P(x) = a n x n + a n1 x n1 + + a 1 x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine
More informationThe Method of Least Squares. Lectures INF2320 p. 1/80
The Method of Least Squares Lectures INF2320 p. 1/80 Lectures INF2320 p. 2/80 The method of least squares We study the following problem: Given n points (t i,y i ) for i = 1,...,n in the (t,y)plane. How
More informationMGT 267 PROJECT. Forecasting the United States Retail Sales of the Pharmacies and Drug Stores. Done by: Shunwei Wang & Mohammad Zainal
MGT 267 PROJECT Forecasting the United States Retail Sales of the Pharmacies and Drug Stores Done by: Shunwei Wang & Mohammad Zainal Dec. 2002 The retail sale (Million) ABSTRACT The present study aims
More informationANALYSIS OF TREND CHAPTER 5
ANALYSIS OF TREND CHAPTER 5 ERSH 8310 Lecture 7 September 13, 2007 Today s Class Analysis of trends Using contrasts to do something a bit more practical. Linear trends. Quadratic trends. Trends in SPSS.
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationCOMP6053 lecture: Time series analysis, autocorrelation. jn2@ecs.soton.ac.uk
COMP6053 lecture: Time series analysis, autocorrelation jn2@ecs.soton.ac.uk Time series analysis The basic idea of time series analysis is simple: given an observed sequence, how can we build a model that
More informationSales forecasting # 2
Sales forecasting # 2 Arthur Charpentier arthur.charpentier@univrennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting
More informationForecasting in supply chains
1 Forecasting in supply chains Role of demand forecasting Effective transportation system or supply chain design is predicated on the availability of accurate inputs to the modeling process. One of the
More informationCorrelation key concepts:
CORRELATION Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d)
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationLecture 5 Principal Minors and the Hessian
Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and
More informationSimple linear regression
Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationWEB APPENDIX. Calculating Beta Coefficients. b Beta Rise Run Y 7.1 1 8.92 X 10.0 0.0 16.0 10.0 1.6
WEB APPENDIX 8A Calculating Beta Coefficients The CAPM is an ex ante model, which means that all of the variables represent beforethefact, expected values. In particular, the beta coefficient used in
More informationy t by left multiplication with 1 (L) as y t = 1 (L) t =ª(L) t 2.5 Variance decomposition and innovation accounting Consider the VAR(p) model where
. Variance decomposition and innovation accounting Consider the VAR(p) model where (L)y t = t, (L) =I m L L p L p is the lag polynomial of order p with m m coe±cient matrices i, i =,...p. Provided that
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationPromotional Forecast Demonstration
Exhibit 2: Promotional Forecast Demonstration Consider the problem of forecasting for a proposed promotion that will start in December 1997 and continues beyond the forecast horizon. Assume that the promotion
More information17. SIMPLE LINEAR REGRESSION II
17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.
More informationCubic Functions: Global Analysis
Chapter 14 Cubic Functions: Global Analysis The Essential Question, 231 Concavitysign, 232 Slopesign, 234 Extremum, 235 Heightsign, 236 0Concavity Location, 237 0Slope Location, 239 Extremum Location,
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More informationTIME SERIES ANALYSIS
TIME SERIES ANALYSIS Ramasubramanian V. I.A.S.R.I., Library Avenue, New Delhi 110 012 ram_stat@yahoo.co.in 1. Introduction A Time Series (TS) is a sequence of observations ordered in time. Mostly these
More informationA RegimeSwitching Model for Electricity Spot Prices. Gero Schindlmayr EnBW Trading GmbH g.schindlmayr@enbw.com
A RegimeSwitching Model for Electricity Spot Prices Gero Schindlmayr EnBW Trading GmbH g.schindlmayr@enbw.com May 31, 25 A RegimeSwitching Model for Electricity Spot Prices Abstract Electricity markets
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationPiecewise Cubic Splines
280 CHAP. 5 CURVE FITTING Piecewise Cubic Splines The fitting of a polynomial curve to a set of data points has applications in CAD (computerassisted design), CAM (computerassisted manufacturing), and
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Spring 2012 Homework # 9, due Wednesday, April 11 8.1.5 How many ways are there to pay a bill of 17 pesos using a currency with coins of values of 1 peso, 2 pesos,
More informationProbability and Random Variables. Generation of random variables (r.v.)
Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationThe PageRank Citation Ranking: Bring Order to the Web
The PageRank Citation Ranking: Bring Order to the Web presented by: Xiaoxi Pang 25.Nov 2010 1 / 20 Outline Introduction A ranking for every page on the Web Implementation Convergence Properties Personalized
More information1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each xcoordinate was matched with only one ycoordinate. We spent most
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationPredict the Popularity of YouTube Videos Using Early View Data
000 001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 041 042 043 044 045 046 047 048 049 050
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationCHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression
Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationTesting for Granger causality between stock prices and economic growth
MPRA Munich Personal RePEc Archive Testing for Granger causality between stock prices and economic growth Pasquale Foresti 2006 Online at http://mpra.ub.unimuenchen.de/2962/ MPRA Paper No. 2962, posted
More informationTime Series Analysis and Forecasting
Time Series Analysis and Forecasting Math 667 Al Nosedal Department of Mathematics Indiana University of Pennsylvania Time Series Analysis and Forecasting p. 1/11 Introduction Many decisionmaking applications
More informationObjectives of Chapters 7,8
Objectives of Chapters 7,8 Planning Demand and Supply in a SC: (Ch7, 8, 9) Ch7 Describes methodologies that can be used to forecast future demand based on historical data. Ch8 Describes the aggregate planning
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationCHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises
CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =
More informationGetting Correct Results from PROC REG
Getting Correct Results from PROC REG Nathaniel Derby, Statis Pro Data Analytics, Seattle, WA ABSTRACT PROC REG, SAS s implementation of linear regression, is often used to fit a line without checking
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationTime Series Analysis. 1) smoothing/trend assessment
Time Series Analysis This (not surprisingly) concerns the analysis of data collected over time... weekly values, monthly values, quarterly values, yearly values, etc. Usually the intent is to discern whether
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationRegression III: Advanced Methods
Lecture 16: Generalized Additive Models Regression III: Advanced Methods Bill Jacoby Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Goals of the Lecture Introduce Additive Models
More information