Trend and Seasonal Components


 Melissa Palmer
 1 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 informationDefinition The covariance of X and Y, denoted by cov(x, Y ) is defined by. cov(x, Y ) = E(X µ 1 )(Y µ 2 ).
Correlation Regression Bivariate Normal Suppose that X and Y are r.v. s with joint density f(x y) and suppose that the means of X and Y are respectively µ 1 µ 2 and the variances are 1 2. Definition The
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationElementary Statistics. Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination
Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination What is a Scatter Plot? A Scatter Plot is a plot of ordered pairs (x, y) where the horizontal axis is used
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 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 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 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 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 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 informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationTime Series Analysis
Time Series Analysis Time series and stochastic processes Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and GarcíaMartos
More informationChapter 6. Econometrics. 6.1 Introduction. 6.2 Univariate techniques Transforming data
Chapter 6 Econometrics 6.1 Introduction We re going to use a few tools to characterize the time series properties of macro variables. Today, we will take a relatively atheoretical approach to this task,
More informationNotes 5: More on regression and residuals ECO 231W  Undergraduate Econometrics
Notes 5: More on regression and residuals ECO 231W  Undergraduate Econometrics Prof. Carolina Caetano 1 Regression Method Let s review the method to calculate the regression line: 1. Find the point of
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 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 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 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 information2. What are the theoretical and practical consequences of autocorrelation?
Lecture 10 Serial Correlation In this lecture, you will learn the following: 1. What is the nature of autocorrelation? 2. What are the theoretical and practical consequences of autocorrelation? 3. Since
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 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 informationSec 4.1 Vector Spaces and Subspaces
Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common
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 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 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 informationAdvanced timeseries analysis
UCL DEPARTMENT OF SECURITY AND CRIME SCIENCE Advanced timeseries analysis Lisa Tompson Research Associate UCL Jill Dando Institute of Crime Science l.tompson@ucl.ac.uk Overview Fundamental principles
More informationPrentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1
STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason
More informationLesson 17: Graphing the Logarithm Function
Lesson 17 Name Date Lesson 17: Graphing the Logarithm Function Exit Ticket Graph the function () = log () without using a calculator, and identify its key features. Lesson 17: Graphing the Logarithm Function
More informationCalculus Card Matching
Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information
More informationLinearizing Data. Lesson3. United States Population
Lesson3 Linearizing Data You may have heard that the population of the United States is increasing exponentially. The table and plot below give the population of the United States in the census years 19
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationStudy & Development of Short Term Load Forecasting Models Using Stochastic Time Series Analysis
International Journal of Engineering Research and Development eissn: 2278067X, pissn: 2278800X, www.ijerd.com Volume 9, Issue 11 (February 2014), PP. 3136 Study & Development of Short Term Load Forecasting
More information7. Tests of association and Linear Regression
7. Tests of association and Linear Regression In this chapter we consider 1. Tests of Association for 2 qualitative variables. 2. Measures of the strength of linear association between 2 quantitative variables.
More informationAdvanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 28 Fourier Series (Contd.) Welcome back to the lecture on Fourier
More informationRoots of quadratic equations
CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients
More informationLecture 1: Univariate Time Series B41910: Autumn Quarter, 2008, by Mr. Ruey S. Tsay
Lecture 1: Univariate Time Series B41910: Autumn Quarter, 2008, by Mr. Ruey S. Tsay 1 Some Basic Concepts 1. Time Series: A sequence of random variables measuring certain quantity of interest over time.
More information4.3 Moving Average Process MA(q)
66 CHAPTER 4. STATIONARY TS MODELS 4.3 Moving Average Process MA(q) Definition 4.5. {X t } is a movingaverage process of order q if X t = Z t + θ 1 Z t 1 +... + θ q Z t q, (4.9) where and θ 1,...,θ q
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 informationsince by using a computer we are limited to the use of elementary arithmetic operations
> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
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 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 information3.1. Quadratic Equations and Models. Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models
3.1 Quadratic Equations and Models Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models 3.11 Polynomial Function A polynomial function of degree n, where n
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 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 informationDEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests
DEPARTMENT OF ECONOMICS Unit ECON 11 Introduction to Econometrics Notes 4 R and F tests These notes provide a summary of the lectures. They are not a complete account of the unit material. You should also
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 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 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 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 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 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 informationRegression, least squares
Regression, least squares Joe Felsenstein Department of Genome Sciences and Department of Biology Regression, least squares p.1/24 Fitting a straight line X Two distinct cases: The X values are chosen
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 informationMidterm 1. Solutions
Stony Brook University Introduction to Calculus Mathematics Department MAT 13, Fall 01 J. Viro October 17th, 01 Midterm 1. Solutions 1 (6pt). Under each picture state whether it is the graph of a function
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 informationCHAPTER 2 FOURIER SERIES
CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that =
More information55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim
Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of
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 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 informationBracken County Schools Curriculum Guide Math
Unit 1: Expressions and Equations (Ch. 13) Suggested Length: Semester Course: 4 weeks Year Course: 8 weeks Program of Studies Core Content 1. How do you use basic skills and operands to create and solve
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 informationEngineering Mathematics II
PSUT Engineering Mathematics II Fourier Series and Transforms Dr. Mohammad Sababheh 4/14/2009 11.1 Fourier Series 2 Fourier Series and Transforms Contents 11.1 Fourier Series... 3 Periodic Functions...
More informationA Multiplicative Seasonal BoxJenkins Model to Nigerian Stock Prices
A Multiplicative Seasonal BoxJenkins Model to Nigerian Stock Prices Ette Harrison Etuk Department of Mathematics/Computer Science, Rivers State University of Science and Technology, Nigeria Email: ettetuk@yahoo.com
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 information3.7 Nonautonomous linear systems of ODE. General theory
3.7 Nonautonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some
More informationChapter 1 Quadratic Equations in One Unknown (I)
Tin Ka Ping Secondary School 015016 F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) 1 1.1 Real Number System A Integers B nal Numbers C Irrational Numbers D Real
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More informationChapter 8. Quadratic Equations and Functions
Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property
More informationQuadratic Equations and Inequalities
MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose
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 informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
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 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 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 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 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 information3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes
Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.
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 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 informationIdentify examples of field properties: commutative, associative, identity, inverse, and distributive.
Topic: Expressions and Operations ALGEBRA II  STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers
More informationYiming Peng, Department of Statistics. February 12, 2013
Regression Analysis Using JMP Yiming Peng, Department of Statistics February 12, 2013 2 Presentation and Data http://www.lisa.stat.vt.edu Short Courses Regression Analysis Using JMP Download Data to Desktop
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 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 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 information12.1. VectorValued Functions. VectorValued Functions. Objectives. Space Curves and VectorValued Functions. Space Curves and VectorValued Functions
12 VectorValued Functions 12.1 VectorValued Functions Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives! Analyze and sketch a space curve given
More information3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes
Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.
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 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 informationRegression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture  2 Simple Linear Regression
Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture  2 Simple Linear Regression Hi, this is my second lecture in module one and on simple
More informationLecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)
ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process
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 informationHigh School Algebra 1 Common Core Standards & Learning Targets
High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities NQ.1. Use units as a way to understand problems
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 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 information