Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

5: Time Series and Forecasting

Last updated

Apr 22, 2025
Save as PDF
- 4.7: Quantitative Problems
- 5.0: Introduction

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

5.0: Introduction
This page discusses time series analysis, which studies data collected over time to uncover trends and make predictions. It highlights its applications in various fields like finance, sports, and politics, as well as its ability to address future uncertainties. The chapter emphasizes the importance of tools for analyzing time series data, aiming to improve forecasting accuracy and quantify uncertainties in predictions.
5.1: Introduction to Time Series Analysis
This page provides an overview of time series data, highlighting its significance in forecasting across fields like business and healthcare. It covers the analysis process, related forecasting methods from naïve to linear regression, and the importance of validating model accuracy. It also details the visualization of S&P 500 time series data in Python using Matplotlib, emphasizing clarity with appropriate formatting and grid lines while referencing Excel for additional context.
5.2: Components of Time Series Analysis
This page outlines key components of time series data, including trend, cyclic and seasonal variations, and noise. It discusses the decomposition of time series through various methods to analyze these components. Trends indicate long-term directions, while seasonal variations occur at regular intervals, illustrated through quarterly kayak rental revenue data showing high summer and low winter values.
5.3: Time Series Forecasting Methods
This page covers time series analysis techniques, including decomposition into trends, seasonal and cyclic variations, and noise. It elaborates on methods like Simple Moving Averages (SMA), Exponential Moving Averages (EMA), and ARIMA models for forecasting and illustrates their applications using Python and Excel, particularly on datasets like the S&P 500 Index and U.S. coal consumption.
5.4: Forecast Evaluation Methods
This page covers time series forecasting, focusing on error measurement techniques like MAE, RMSE, MAPE, and sMAPE. It highlights the importance of prediction intervals in assessing future value ranges and discusses margin of error and confidence intervals. Using Python's `statsmodels.tsa.arima.model` library, it illustrates how to create prediction intervals, setting an 80% confidence level and visualizing original and forecasted values, showing that uncertainty increases with longer forecasts.
5.5: Key Terms
This page offers a detailed overview of time series analysis, presenting definitions and explanations of essential concepts like decomposition methods, stationarity tests, predictive models, and error metrics. It highlights the significance of components such as trend, seasonality, and cycles for effective forecasting. Key terms such as autocorrelation and differencing are also defined, ensuring a thorough understanding of the subject matter for analyzing time-dependent data.
5.6: Group Project
This page outlines a project comprising two key analyses of financial time series data. Project A aims to predict S&P 500 Index trends using smoothing techniques like simple moving averages (SMA) and exponential moving averages (EMA), while exploring the effects of varying window sizes and alpha values.
5.7: Critical Thinking
This page discusses time series data characteristics with examples like monthly expenses and weather measurements. It encourages trend analysis of a graph and the identification of a moving average trendline from provided data. Additionally, it introduces error measures such as MAE, RMSE, and MAPE, urging the identification of the most and least accurate models based on these metrics, while also exploring the reasons for their accuracy.
5.8: Quantitative Problems
This page outlines analytical tasks for time series data, focusing on seasonal variations, trend calculations, and seasonality adjustments in sales and COVID-19 data. It details methods for smoothing data using centered simple moving averages (SMA) and exponential moving averages (EMA), as well as analyzing seasonal components through ACF plots and STL decomposition. The content also covers evaluating model accuracy with error measures such as MAE, RMSE, MAPE, and sMAPE.

Support Center

How can we help?