Time series analysis involves detecting patterns in variables measured over time to aid in forecasting future values. A time series can consist of four components: long-term trend, cyclical effect, seasonal effect, and random variation. Smoothing techniques like moving averages and exponential smoothing are used to remove random variation and identify other components. Seasonal indexes express the degree to which different seasons vary from the average time series value and can be computed using a multiplicative model.
This document discusses quantitative forecasting methods. Quantitative forecasting depends on data and analytical techniques to predict future demand based on past demand information. Some common quantitative forecasting methods discussed include time series analysis, causal models, and simulation. Time series methods like simple moving averages, weighted moving averages, and exponential smoothing are explained as techniques to forecast future demand based on historical data trends. Linear regression models are also mentioned as a way to establish relationships between demand and other factors. Key factors that influence the selection of a forecasting method include data availability, required time horizon, accuracy needs, and available resources.
This document provides an overview of operations management forecasting models and their applications. It defines forecasting and lists its common uses. The key components of a forecast and the forecasting process are described. Both qualitative and quantitative forecasting approaches are discussed, along with their advantages and disadvantages. Specific forecasting techniques covered include time series methods, regression methods, moving averages, exponential smoothing, and naive forecasts. Examples are provided to illustrate weighted moving averages and exponential smoothing.
This powerpoint presentation was done as part of the course STAT 591 titled Mater's Seminar during Third semester of MSc. Agricultural Statistics at Agricultural College, Bapatla under ANGRAU, Andhra Pradesh.
The document provides an overview of time series analysis. It defines a time series as numerical data obtained at regular time intervals that can be analyzed to describe patterns, fit models, and make forecasts. Time series are different from other data because observations are not independent. The document discusses the key components of time series, including trend, seasonal variation, cyclical variation, and irregular variation. It also covers techniques for smoothing time series data, such as moving averages, and measuring seasonal effects through seasonal indices. The overall goal of time series analysis is to understand and separate out the different variations in a time series to better predict future trends.
This document provides an overview of forecasting in the aviation industry. It defines forecasting as predicting future demand based on past data to aid planning, analysis, and control. The document outlines several forecasting methods, including causal, trend analysis, and judgmental. Causal forecasts use statistical relationships between variables, trend analysis extrapolates past trends, and judgmental forecasts rely on expert opinions. The document emphasizes that forecasting is important for strategic planning, budgeting, marketing, production, and comparing actual performance to predictions.
The document discusses various quantitative forecasting techniques including time series methods like moving averages and exponential smoothing. It provides examples of how to calculate 3-period moving averages and exponential smoothing forecasts using sample sales data. Exponential smoothing places more weight on recent observations compared to moving averages. The smoothing constant determines how quickly older data is discounted.
This document discusses various forecasting methods including qualitative methods like panel consensus and quantitative methods like time series analysis. It explains moving averages, weighted moving averages, and exponential smoothing for time series forecasting. Moving averages are simple to calculate but do not respond well to trends while exponential smoothing accounts for trends using smoothing constants. Linear regression can also be used to explore relationships between dependent and independent variables for forecasting. Overall the key points are that forecasting predicts future demand based on past data, different quantitative methods are suited to different situations, and accuracy depends on how well past patterns predict the future.
Forecasting is essential for business operations and involves estimating future events and trends. There are two main types of forecasting: quantitative and qualitative. Quantitative forecasting uses historical data and mathematical models, while qualitative forecasting relies on expert opinions. Common quantitative forecasting methods include moving averages, exponential smoothing, and time series models. Moving averages calculate the average demand over a set time period to smooth out fluctuations. Exponential smoothing places more emphasis on recent data by applying weighting factors. Qualitative methods include jury of executive opinion, Delphi method, and consumer surveys. Forecasting allows businesses to better plan operations and prepare for the future.
This document discusses quantitative forecasting methods. Quantitative forecasting depends on data and analytical techniques to predict future demand based on past demand information. Some common quantitative forecasting methods discussed include time series analysis, causal models, and simulation. Time series methods like simple moving averages, weighted moving averages, and exponential smoothing are explained as techniques to forecast future demand based on historical data trends. Linear regression models are also mentioned as a way to establish relationships between demand and other factors. Key factors that influence the selection of a forecasting method include data availability, required time horizon, accuracy needs, and available resources.
This document provides an overview of operations management forecasting models and their applications. It defines forecasting and lists its common uses. The key components of a forecast and the forecasting process are described. Both qualitative and quantitative forecasting approaches are discussed, along with their advantages and disadvantages. Specific forecasting techniques covered include time series methods, regression methods, moving averages, exponential smoothing, and naive forecasts. Examples are provided to illustrate weighted moving averages and exponential smoothing.
This powerpoint presentation was done as part of the course STAT 591 titled Mater's Seminar during Third semester of MSc. Agricultural Statistics at Agricultural College, Bapatla under ANGRAU, Andhra Pradesh.
The document provides an overview of time series analysis. It defines a time series as numerical data obtained at regular time intervals that can be analyzed to describe patterns, fit models, and make forecasts. Time series are different from other data because observations are not independent. The document discusses the key components of time series, including trend, seasonal variation, cyclical variation, and irregular variation. It also covers techniques for smoothing time series data, such as moving averages, and measuring seasonal effects through seasonal indices. The overall goal of time series analysis is to understand and separate out the different variations in a time series to better predict future trends.
This document provides an overview of forecasting in the aviation industry. It defines forecasting as predicting future demand based on past data to aid planning, analysis, and control. The document outlines several forecasting methods, including causal, trend analysis, and judgmental. Causal forecasts use statistical relationships between variables, trend analysis extrapolates past trends, and judgmental forecasts rely on expert opinions. The document emphasizes that forecasting is important for strategic planning, budgeting, marketing, production, and comparing actual performance to predictions.
The document discusses various quantitative forecasting techniques including time series methods like moving averages and exponential smoothing. It provides examples of how to calculate 3-period moving averages and exponential smoothing forecasts using sample sales data. Exponential smoothing places more weight on recent observations compared to moving averages. The smoothing constant determines how quickly older data is discounted.
This document discusses various forecasting methods including qualitative methods like panel consensus and quantitative methods like time series analysis. It explains moving averages, weighted moving averages, and exponential smoothing for time series forecasting. Moving averages are simple to calculate but do not respond well to trends while exponential smoothing accounts for trends using smoothing constants. Linear regression can also be used to explore relationships between dependent and independent variables for forecasting. Overall the key points are that forecasting predicts future demand based on past data, different quantitative methods are suited to different situations, and accuracy depends on how well past patterns predict the future.
Forecasting is essential for business operations and involves estimating future events and trends. There are two main types of forecasting: quantitative and qualitative. Quantitative forecasting uses historical data and mathematical models, while qualitative forecasting relies on expert opinions. Common quantitative forecasting methods include moving averages, exponential smoothing, and time series models. Moving averages calculate the average demand over a set time period to smooth out fluctuations. Exponential smoothing places more emphasis on recent data by applying weighting factors. Qualitative methods include jury of executive opinion, Delphi method, and consumer surveys. Forecasting allows businesses to better plan operations and prepare for the future.
Exponential smoothing uses all past time series values to generate forecasts, with less weight given to older values. It works by calculating a smoothed level (Lt) at each period t as a weighted average of the current value (yt) and the previous smoothed level (Lt-1). The forecast for the next period (Ft+1) is then set equal to the current smoothed level (Lt). Choosing the smoothing constant (α) determines how responsive the smoothed level is to recent changes in the data.
This document discusses Hoshin management, which is a Japanese strategic planning process used to align organizational activities with key goals. It involves the following components:
1. Annual planning cycles where managers collaboratively set goals and metrics to measure progress.
2. Deploying goals and means for achieving them throughout the organization using a "catchball" process of discussion and analysis.
3. Monitoring metrics regularly to ensure goals are on track and make corrections if needed.
4. Checking in at the end of the cycle to evaluate weaknesses and inform planning for the next year.
The purpose is to focus all employees and tasks on critical priorities and enable rapid response to changing conditions. It is compared to management
- Forecasting helps reduce risk and uncertainty in decision making by predicting future outcomes.
- There are three main types of forecasting methods: qualitative, extrapolative/time series, and causal/explanatory.
- Time series forecasting uses historical data patterns to predict future values, accounting for trends, seasonality, cycles, and randomness. Common time series forecasting techniques include moving averages, weighted moving averages, and exponential smoothing.
The document discusses various forecasting techniques used to predict future values based on historical data patterns. It describes time series models like moving averages, exponential smoothing and trend projections that rely solely on past values to forecast. It also covers decomposition of time series data into trend, seasonality, cycles and random components. The document provides examples of scatter plots to visualize relationships in time series data and defines accuracy measures like MAD, MSE and MAPE to evaluate forecast errors. Overall it provides an overview of quantitative forecasting methods and how to implement them.
This document discusses time series analysis and forecasting methods. It covers several key topics:
1. Time series decomposition which involves separating a time series into seasonal, trend, cyclical, and irregular components. Seasonal and trend components are then modeled and forecasts are made by recomposing these components.
2. Common forecasting techniques including exponential smoothing to reduce random variation, modeling seasonality using seasonal indices, and incorporating trends and cycles.
3. The process of time series forecasting which involves decomposing historical data, modeling each component, and recomposing forecasts by applying the component models to future periods. Accuracy and sources of error in forecasts are also discussed.
This document discusses exponential smoothing techniques for time series forecasting. It introduces simple, double, and triple exponential smoothing. Simple exponential smoothing works for stationary time series, double exponential smoothing adds a trend component for trending time series, and triple exponential smoothing (the Holt-Winters method) further adds seasonal components to handle seasonality. The document discusses parameters, components, extensions, and evaluation metrics for exponential smoothing models.
Forecasting techniques, time series analysisSATISH KUMAR
This document discusses forecasting techniques and time series analysis. It defines forecasting as the estimation or prediction of future outcomes, trends, or behavior through the use of statistics. The document outlines several key points:
- It describes the meaning, definition, features, process, importance, advantages, and limitations of forecasting.
- It discusses various qualitative and quantitative forecasting methods including regression analysis, business barometers, input/output analysis, surveys, and time series analysis.
- It explains the components of time series analysis including secular trends, seasonal variations, cyclical variations, and irregular variations.
- It provides examples of each type of variation and discusses their importance for time series forecasting.
This document discusses demand forecasting techniques used by product managers. It defines demand forecasting as using statistical data and market determinants to predict future demand. There are two types of forecasts: passive, which assume no changes to company actions, and active, which account for likely changes. Short term forecasts relate to periods under a year and are used for production, sales, pricing and target policies. Long term forecasts cover multiple years and are used for business, workforce and financial planning. The document outlines various demand forecasting techniques including consumer and opinion polls, market experiments, and analytical methods.
Forecasting Quantitative - Time Series.pptbookworm65
The document discusses various quantitative time series forecasting models including causal models and time series models. It describes stationary time series models including the naïve model, moving average models, and exponential smoothing. It explains that moving average models reduce random variation by averaging past data, and that exponential smoothing requires less data storage than moving averages as it applies a smoothing constant to weight the most recent period.
- Forecasting involves making predictions about future market conditions and demand. It is an important part of business planning but forecasts will always be imperfect.
- Market size refers to the number of potential buyers and sellers in a market. Understanding market size is important for launching new products or services. Qualitative and quantitative models can be used to forecast market size.
- Qualitative models include expert opinion methods like the Delphi method and jury of executive opinion. Quantitative time series models analyze historical demand patterns using techniques like moving averages, exponential smoothing, and regression analysis. These techniques help minimize forecast errors.
This document discusses various quantitative forecasting techniques including time series models. It provides an overview of moving averages, exponential smoothing, trend projections, and decomposition models. Examples are given to illustrate computing forecasts using a three-month simple moving average and a three-month weighted moving average. Exponential smoothing is also introduced as a type of moving average that requires less data to compute forecasts.
This document discusses quantitative approaches to forecasting, including time series analysis and forecasting techniques. It covers the components of a time series, including trends, cycles, seasonality, and irregular components. Specific quantitative forecasting approaches covered include smoothing methods like moving averages, weighted moving averages, and exponential smoothing. Examples are provided to demonstrate how to perform moving averages and exponential smoothing on time series data for sales of headache medicine. The document aims to teach readers how to analyze time series data and select appropriate forecasting techniques.
Learning material on Measurement of Seasonal variations prepared in accordance to VTU I Sem MBA syllabus for the subject Business Statistics & Analytics
Interventions required to meet business objectives - from Forecasting Methods,
Forecast Accuracy / Error Reduction,
Integrate – Sales Forecast / Production to undertaking a CPFR
Time series decomposition involves breaking down a time series into various components: trend, seasonality, and error/noise. There are different decomposition models such as additive and multiplicative. Smoothing methods like moving averages are used to estimate the trend-cycle component by reducing random variation. Box-Jenkins models combine autoregressive (AR) and moving average (MA) terms to model time series, and involve identification, estimation, and diagnostic stages.
Basic Concepts, Components of time series. The trend, Fitting of trend by least square method and moving average method, uses of time series in business.
This document provides an overview of time series analysis and forecasting techniques. It discusses key concepts such as stationary and non-stationary time series, additive and multiplicative models, smoothing methods like moving averages and exponential smoothing, autoregressive (AR), moving average (MA) and autoregressive integrated moving average (ARIMA) models. The document uses examples to illustrate how to identify patterns in time series data and select appropriate models for description, explanation and forecasting of time series.
The document provides an overview of a time series analysis and forecasting course. It discusses key topics that will be covered including descriptive statistics, correlation, regression, hypothesis testing, clustering, time series analysis and forecasting techniques like TCSI and ARIMA models. It notes that the presentation serves as class notes and contains informal high-level summaries intended to aid the author, and encourages readers to check the website for updated versions of the document.
1) This document provides an introduction to time series analysis, beginning with a review of basic probability concepts like sample spaces, random variables, and distributions.
2) It defines stochastic processes as collections of time-indexed random variables and discusses their properties like stationarity and dependence structure. Examples like Brownian motion are provided.
3) Key concepts in time series analysis are introduced, including the autocovariance and autocorrelation functions, which describe the dependence between observations in a time series as a function of the time lag between them.
This document discusses demand forecasting methods. It categorizes forecasting by level (macro, industry, product) and product type (non-durable goods, durable goods, capital goods, new products). Common forecasting approaches include survey methods, market experiments, Delphi method, experts' opinions, statistical methods, trend projection, barometric techniques, and econometric methods. Advantages of demand forecasting are predicting the future, keeping customers happy, remaining competitive, reducing inventory costs, and preparing for changes in sales. Limitations include changes in fashion, consumer psychology, lack of past data, and lack of experienced experts.
Exponential smoothing uses all past time series values to generate forecasts, with less weight given to older values. It works by calculating a smoothed level (Lt) at each period t as a weighted average of the current value (yt) and the previous smoothed level (Lt-1). The forecast for the next period (Ft+1) is then set equal to the current smoothed level (Lt). Choosing the smoothing constant (α) determines how responsive the smoothed level is to recent changes in the data.
This document discusses Hoshin management, which is a Japanese strategic planning process used to align organizational activities with key goals. It involves the following components:
1. Annual planning cycles where managers collaboratively set goals and metrics to measure progress.
2. Deploying goals and means for achieving them throughout the organization using a "catchball" process of discussion and analysis.
3. Monitoring metrics regularly to ensure goals are on track and make corrections if needed.
4. Checking in at the end of the cycle to evaluate weaknesses and inform planning for the next year.
The purpose is to focus all employees and tasks on critical priorities and enable rapid response to changing conditions. It is compared to management
- Forecasting helps reduce risk and uncertainty in decision making by predicting future outcomes.
- There are three main types of forecasting methods: qualitative, extrapolative/time series, and causal/explanatory.
- Time series forecasting uses historical data patterns to predict future values, accounting for trends, seasonality, cycles, and randomness. Common time series forecasting techniques include moving averages, weighted moving averages, and exponential smoothing.
The document discusses various forecasting techniques used to predict future values based on historical data patterns. It describes time series models like moving averages, exponential smoothing and trend projections that rely solely on past values to forecast. It also covers decomposition of time series data into trend, seasonality, cycles and random components. The document provides examples of scatter plots to visualize relationships in time series data and defines accuracy measures like MAD, MSE and MAPE to evaluate forecast errors. Overall it provides an overview of quantitative forecasting methods and how to implement them.
This document discusses time series analysis and forecasting methods. It covers several key topics:
1. Time series decomposition which involves separating a time series into seasonal, trend, cyclical, and irregular components. Seasonal and trend components are then modeled and forecasts are made by recomposing these components.
2. Common forecasting techniques including exponential smoothing to reduce random variation, modeling seasonality using seasonal indices, and incorporating trends and cycles.
3. The process of time series forecasting which involves decomposing historical data, modeling each component, and recomposing forecasts by applying the component models to future periods. Accuracy and sources of error in forecasts are also discussed.
This document discusses exponential smoothing techniques for time series forecasting. It introduces simple, double, and triple exponential smoothing. Simple exponential smoothing works for stationary time series, double exponential smoothing adds a trend component for trending time series, and triple exponential smoothing (the Holt-Winters method) further adds seasonal components to handle seasonality. The document discusses parameters, components, extensions, and evaluation metrics for exponential smoothing models.
Forecasting techniques, time series analysisSATISH KUMAR
This document discusses forecasting techniques and time series analysis. It defines forecasting as the estimation or prediction of future outcomes, trends, or behavior through the use of statistics. The document outlines several key points:
- It describes the meaning, definition, features, process, importance, advantages, and limitations of forecasting.
- It discusses various qualitative and quantitative forecasting methods including regression analysis, business barometers, input/output analysis, surveys, and time series analysis.
- It explains the components of time series analysis including secular trends, seasonal variations, cyclical variations, and irregular variations.
- It provides examples of each type of variation and discusses their importance for time series forecasting.
This document discusses demand forecasting techniques used by product managers. It defines demand forecasting as using statistical data and market determinants to predict future demand. There are two types of forecasts: passive, which assume no changes to company actions, and active, which account for likely changes. Short term forecasts relate to periods under a year and are used for production, sales, pricing and target policies. Long term forecasts cover multiple years and are used for business, workforce and financial planning. The document outlines various demand forecasting techniques including consumer and opinion polls, market experiments, and analytical methods.
Forecasting Quantitative - Time Series.pptbookworm65
The document discusses various quantitative time series forecasting models including causal models and time series models. It describes stationary time series models including the naïve model, moving average models, and exponential smoothing. It explains that moving average models reduce random variation by averaging past data, and that exponential smoothing requires less data storage than moving averages as it applies a smoothing constant to weight the most recent period.
- Forecasting involves making predictions about future market conditions and demand. It is an important part of business planning but forecasts will always be imperfect.
- Market size refers to the number of potential buyers and sellers in a market. Understanding market size is important for launching new products or services. Qualitative and quantitative models can be used to forecast market size.
- Qualitative models include expert opinion methods like the Delphi method and jury of executive opinion. Quantitative time series models analyze historical demand patterns using techniques like moving averages, exponential smoothing, and regression analysis. These techniques help minimize forecast errors.
This document discusses various quantitative forecasting techniques including time series models. It provides an overview of moving averages, exponential smoothing, trend projections, and decomposition models. Examples are given to illustrate computing forecasts using a three-month simple moving average and a three-month weighted moving average. Exponential smoothing is also introduced as a type of moving average that requires less data to compute forecasts.
This document discusses quantitative approaches to forecasting, including time series analysis and forecasting techniques. It covers the components of a time series, including trends, cycles, seasonality, and irregular components. Specific quantitative forecasting approaches covered include smoothing methods like moving averages, weighted moving averages, and exponential smoothing. Examples are provided to demonstrate how to perform moving averages and exponential smoothing on time series data for sales of headache medicine. The document aims to teach readers how to analyze time series data and select appropriate forecasting techniques.
Learning material on Measurement of Seasonal variations prepared in accordance to VTU I Sem MBA syllabus for the subject Business Statistics & Analytics
Interventions required to meet business objectives - from Forecasting Methods,
Forecast Accuracy / Error Reduction,
Integrate – Sales Forecast / Production to undertaking a CPFR
Time series decomposition involves breaking down a time series into various components: trend, seasonality, and error/noise. There are different decomposition models such as additive and multiplicative. Smoothing methods like moving averages are used to estimate the trend-cycle component by reducing random variation. Box-Jenkins models combine autoregressive (AR) and moving average (MA) terms to model time series, and involve identification, estimation, and diagnostic stages.
Basic Concepts, Components of time series. The trend, Fitting of trend by least square method and moving average method, uses of time series in business.
This document provides an overview of time series analysis and forecasting techniques. It discusses key concepts such as stationary and non-stationary time series, additive and multiplicative models, smoothing methods like moving averages and exponential smoothing, autoregressive (AR), moving average (MA) and autoregressive integrated moving average (ARIMA) models. The document uses examples to illustrate how to identify patterns in time series data and select appropriate models for description, explanation and forecasting of time series.
The document provides an overview of a time series analysis and forecasting course. It discusses key topics that will be covered including descriptive statistics, correlation, regression, hypothesis testing, clustering, time series analysis and forecasting techniques like TCSI and ARIMA models. It notes that the presentation serves as class notes and contains informal high-level summaries intended to aid the author, and encourages readers to check the website for updated versions of the document.
1) This document provides an introduction to time series analysis, beginning with a review of basic probability concepts like sample spaces, random variables, and distributions.
2) It defines stochastic processes as collections of time-indexed random variables and discusses their properties like stationarity and dependence structure. Examples like Brownian motion are provided.
3) Key concepts in time series analysis are introduced, including the autocovariance and autocorrelation functions, which describe the dependence between observations in a time series as a function of the time lag between them.
This document discusses demand forecasting methods. It categorizes forecasting by level (macro, industry, product) and product type (non-durable goods, durable goods, capital goods, new products). Common forecasting approaches include survey methods, market experiments, Delphi method, experts' opinions, statistical methods, trend projection, barometric techniques, and econometric methods. Advantages of demand forecasting are predicting the future, keeping customers happy, remaining competitive, reducing inventory costs, and preparing for changes in sales. Limitations include changes in fashion, consumer psychology, lack of past data, and lack of experienced experts.
This chapter discusses time-series forecasting and index numbers. It aims to develop basic forecasting models using smoothing methods like moving averages and exponential smoothing. It also covers trend-based forecasting using linear and nonlinear regression models. Time-series data contains trend, seasonal, cyclical, and irregular components that must be accounted for. Forecasting future values involves identifying patterns in historical data and extending those patterns into the future.
The document presents an overview of demand forecasting techniques, including opinion polling and statistical methods. Opinion polling involves directly surveying consumers, salespeople, and experts for their projections. Statistical techniques include trend projection, barometric analysis, and regression. Trend projection analyzes historical sales data to identify patterns, barometric links economic indicators, and regression correlates demand to independent variables like income, price. The document discusses various approaches under each technique like consumer surveys, expert opinions, time series analysis, leading/lagging indicators to estimate future demand for production planning.
Time series analysis involves analyzing data collected over time. A time series is a set of data points indexed in time order. The key components of a time series are trends, seasonality, cycles, and irregular variations. Trend refers to the long-term movement of a time series over time. Seasonality refers to periodic fluctuations that occur each year, such as higher sales in winter. Cyclical variations are longer term fluctuations in business cycles. Irregular variations are random, unpredictable fluctuations. Time series analysis is important for forecasting, economic analysis, and business planning. Common methods for analyzing time series components include moving averages, least squares regression, decomposition models, and harmonic analysis.
This document discusses sales budgeting, forecasting, and control. It covers developing sales budgets to plan and coordinate sales, types of budgets including sales, selling expense, and administrative budgets. Forecasting methods like macro, micro, qualitative, and quantitative are described. Sales forecasting is used for production scheduling, pricing, promotion, and financial planning. Control involves setting standards, evaluating performance, and correcting deviations to optimize sales, profits, and revenue.
There are three main types of forecasting methods: qualitative, extrapolative, and causal. Qualitative methods rely on expert opinions and are useful for medium to long range forecasting. Extrapolative methods use past historical demand data to identify patterns and extrapolate them into the future. Causal methods use statistical models based on historical demand data and other variables that influence demand. Some specific forecasting techniques mentioned include the Delphi technique, market surveys, scenario writing, moving averages, weighted moving averages, exponential smoothing, regression analysis, and econometric methods.
Demand forecasting involves determining what products are needed, where, when, and in what quantities. It is a customer-focused activity that supports logistics planning like capacity, inventory, and business planning. Demand forecasting techniques can be qualitative like surveys or quantitative like time series analysis and regression models. The choice of technique depends on factors like the time period, data availability, and purpose of the forecast. Effective demand forecasting provides benefits like reduced uncertainties and improved operations.
This document discusses various forecasting techniques. It covers qualitative and quantitative methods as well as different time horizons for forecasting. Specific quantitative techniques discussed include moving averages, exponential smoothing, regression analysis, and double exponential smoothing. Moving averages and exponential smoothing are described as methods for forecasting stationary time series. Exponential smoothing provides a weighted average of past observations with more weight given to recent observations. Double exponential smoothing accounts for trends by smoothing changes in the intercept and slope over time.
This document provides an overview of time series analysis and its key components. It discusses that a time series is a set of data measured at successive times joined together by time order. The main components of a time series are trends, seasonal variations, cyclical variations, and irregular variations. Time series analysis is important for business forecasting, understanding past behavior, and facilitating comparison. There are two main mathematical models used - the additive model which assumes data is the sum of its components, and the multiplicative model which assumes data is the product of its components. Decomposition of a time series involves discovering, measuring, and isolating these different components.
This document discusses various methods for classifying and forecasting demand. It categorizes demand based on whether goods are for consumers or producers, whether they are perishable or durable, and whether demand is derived, autonomous, for a firm or industry, or for total markets versus market segments. It then discusses demand forecasting and different quantitative and qualitative techniques for forecasting, including expert opinion methods, complete/sample consumer enumeration surveys, sales force opinion surveys, and consumer end use surveys. Each technique is described along with its advantages and disadvantages.
The document provides an overview of time-series forecasting methods, including:
1) It discusses trend analysis, seasonality, cyclical behavior, and various forecasting techniques such as the ratio-to-moving average method and exponential smoothing.
2) Exponential smoothing is described as a forecasting method that gives the largest weight to present observations and smaller, geometrically declining weights to past observations.
3) An example demonstrates exponential smoothing on a time series using weighting factors of 0.4 and 0.8, showing the smoothed series for each weight.
This document provides an overview of time series data mining. It begins with an introduction to time series data and examples of time series similarity search tasks. It then discusses major time series mining tasks like indexing, clustering, classification, prediction and anomaly detection. Distance measures for time series similarity search are explained, including Dynamic Time Warping which allows for nonlinear time alignments. Dimensionality reduction techniques like Fourier analysis and discretization using Symbolic Aggregate Approximation are also summarized. The document is presented as an introduction to key concepts and techniques in time series data mining.
The document discusses time series decomposition methods. It explains that decomposition separates a time series into various components including trend, seasonality, and irregular components. Additive and multiplicative decomposition models are described. The process of computing seasonal indices using ratios is demonstrated with an example of hotel occupancy rates. Graphs of the original, smoothed, and deseasonalized time series are shown. SPSS procedures for conducting time series decomposition are also summarized.
Statistical Quality Control (SQC) is used to evaluate organizational quality through statistical tools. SQC can be classified into descriptive statistics, statistical process control, and acceptance sampling. Descriptive statistics describe quality characteristics and relationships through measures like the mean and standard deviation. Statistical process control uses random sampling to determine if a process is producing products within a predetermined range. Acceptance sampling involves random inspection of samples to determine if an entire lot should be accepted or rejected. Control charts graphically show whether sample data falls within the normal variation limits.
The document discusses three methods for trend projection in demand forecasting: 1) graphic method of fitting a trend line visually to data, 2) least squares method of fitting a trend line algebraically, and 3) smoothing methods such as moving averages that predict values based on averages of past data. It provides examples of applying a three-period and five-period moving average to sample time series data and calculating the root-mean-square error to evaluate forecast accuracy. Finally, it introduces exponential smoothing, which forecasts the next period as a weighted average of the actual current value and previous forecast.
This document provides an overview of time-series analysis techniques. It defines the components of a time series as trend, seasonal variation, and cyclical variation. Smoothing techniques like moving averages and exponential smoothing are described to reduce random fluctuations and expose patterns. Methods for separating out the trend, seasonal, and cyclical components are outlined, including linear regression for trends and calculating seasonal factors. The goal of time-series analysis is described as describing historical patterns and making forecasts by projecting patterns into the future.
The document discusses time series data that contain deterministic trend and seasonal components in addition to irregular components. It describes how to decompose a time series into these components using either additive or multiplicative models. The key steps involve estimating the trend using a moving average, estimating the seasonal factors by averaging residuals from the trend, and identifying the irregular component as the residual.
This document discusses time series analysis and various time series models. It introduces fundamental concepts like stationarity and summarizes common time series models including white noise, random walks, moving average (MA) models, autoregressive (AR) models, and autoregressive integrated moving average (ARIMA) models. Examples of generating and analyzing each type of time series are demonstrated in R.
This document provides an overview of time series forecasting techniques. It discusses the components of time series data including trends, cycles, seasonality and irregular fluctuations. It also covers stationary and non-stationary time series. Forecasting techniques covered include naive methods, smoothing techniques like moving averages and exponential smoothing, and decomposition methods. Regression models for trend analysis and measuring forecast accuracy are also discussed.
This document discusses quantitative forecasting methods, including time series and causal models. It covers key time series components like trend, seasonality, and cycles. Three main time series methods are described: smoothing, trend projection, and trend projection adjusted for seasonal influence. Moving averages and exponential smoothing are explained as common techniques for forecasting stationary time series. The document also covers decomposing a time series into trend, seasonal, and irregular components. Regression methods are mentioned as another approach when a trend is present in the data.
Time series analysis examines patterns in data over time. It relies on identifying trends, measuring past patterns to forecast the future, and decomposing time series into four main components: secular trends, cyclical movements, seasonal variations, and irregular variations. Secular trends represent long-term direction, while cyclical and seasonal variations have recurring patterns over different time scales. Various techniques can depict trends and identify variations, including freehand drawing, semi-averages, moving averages, least squares, and exponential smoothing.
This document defines and explains time series analysis. It contains:
- A definition of time series as observations taken at specified time intervals, usually equal.
- Time series establish relationships between an independent variable (time) and dependent variable (data).
- Components of time series include trends, seasonal variations, cyclical variations, and irregular variations.
- Methods for measuring trends include free hand curve, semi-average, moving average, and least squares.
- Time series analysis has various applications including forecasting, budgeting, stock analysis, and economic forecasting.
The multiple linear regression model aims to predict water cases produced from four predictor variables: run time, downtime, setup time, and efficiency. Preliminary analysis found run time has the highest correlation to water cases. Residual analysis showed non-constant variance, so a square root transformation of water cases was tested but did not improve the model. Further analysis is needed to develop the best-fitting multiple linear regression model.
This article is the part taken from the draft version of the Book: Scientific Guide to Price Action and Pattern Trading (Wisdom of Trend, Cycle, and Fractal Wave). Full version of the book can be found from the link below:
https://algotrading-investment.com/portfolio-item/scientific-guide-to-price-action-and-pattern-trading/
Fractal Pattern Scanner provides the way to measure the fractal cycles in forex and stock market data. In addition, the turning Point Probability and trend Probability can be measured with Fractal Pattern Scanner at every price action in your chart. For the simplest usage, Fractal Pattern Scanner can turn the simple support and resistance to the powerful killer strategy. Below is the landing page to Fractal Pattern Scanner in MetaTrader 4 and MetaTrader 5.
https://www.mql5.com/en/market/product/49170
https://www.mql5.com/en/market/product/49169
https://algotrading-investment.com/portfolio-item/fractal-pattern-scanner/
Below is the landing page for Optimum Chart (Standalone Charting and Analytical Platform).
https://algotrading-investment.com/2019/07/23/optimum-chart/
When fitting loss data (insurance) to a distribution, often the parameters that provide a good overall fit will understate the density in the tail.
This method allows one to split the distribution into 2 portions, and use a Pareto distribution to fit the tail.
Presented at the CAS Spring Meeting in Seattle, May 2016.
This document provides examples and formulas for calculating mean absolute deviation and coefficient of dispersion. Mean absolute deviation is a measure of how far data points deviate from the mean. It is calculated by taking the average of the absolute differences between each data point and the mean. Coefficient of dispersion is calculated by dividing the mean absolute deviation by the mean and multiplying by 100 to express it as a percentage. Two examples are provided to demonstrate calculating mean absolute deviation and coefficient of dispersion using data sets and frequency distributions.
Introduction. Data presentation
Frequency distribution. Distribution center indicators. RMS. Covariance. Effects of
diversification. Choice of the weighing method.
More: https://ek.biem.sumdu.edu.ua/
Moving averages, weighted moving averages, and exponential smoothing are three forecasting methods appropriate for time series data with a horizontal pattern. A moving average is calculated by averaging successive groups of observations, deleting the oldest and adding the newest at each step. Weighted moving averages assign greater weight to recent data. Exponential smoothing similarly gives most weight to the most recent observation, with weights exponentially decreasing for older data.
A Study on the Short Run Relationship b/w Major Economic Indicators of US Eco...aurkoiitk
The objective of this study
was to develop an economic indicator system for the US
economy that will help to forecast the turning points in the
aggregate level of economic activity. Our primary concern
is to study the short run relationship between the major
economic indicators of US economy (eg: GDP, Money
Supply, Unemployment Rate, Inflation rate, Federal Fund
Rate, Exchange Rate, Government Expenditure &
Receipt, Crude Oil Price, Net Import & Export).
The Vasicek model is one of the earliest stochastic models for modeling the term structure of interest rates. It represents the movement of interest rates as a function of market risk, time, and the equilibrium value the rate tends to revert to. This document discusses parameter estimation techniques for the Vasicek one-factor model using least squares regression and maximum likelihood estimation on historical interest rate data. It also covers simulating the term structure and pricing zero-coupon bonds under the Vasicek model. The two-factor Vasicek model is introduced as an extension of the one-factor model.
3. Introduction
Any variable that is measured over time in
sequential order is called a time series.
We analyze time series to detect patterns.
The patterns help in forecasting future
values of the time series.
Predicted value
The time series exhibit a
downward trend pattern.
3
4. 20.1 Components of a Time
Series
A time series can consist of four
components.
Long - term trend (T).
Cyclical effect (C).
Seasonal effect (S).
Random variation (R).
A trend is a long term relatively
smooth pattern or direction, that
persists usually for more than one year.
4
5. Components of a Time Series
A time series can consists of four
components.
Long - term trend (T)
Cyclical variation (C)
Seasonal variation (S)
Random variation (R)
6/90 6/93 6/96 6/99 6/02
A cycle is a wavelike pattern describing
Cycles are seldom regular, and
a long term behavior (for more than one
often appear in combination with
year).
other components.
5
6. Components of a Time Series
A time series can consists of four
components.
Long - term trend (T).
Cyclical effect (C).
Seasonal effect (S).
Random variation (R).
The seasonal component of the time series 6/97 12/97 6/98 12/98 6/99
exhibits a short term (less than one year)
calendar repetitive behavior.
6
7. Components of a Time Series
A time series can consists of four
components.
Long - term trend (T).
Cyclical effect (C).
Seasonal effect (S).
Random variation (R).
We try to remove random variation
Random variation comprises the irregular thereby, identify the other components.
unpredictable changes in the time series.
It tends to hide the other (more predictable)
components.
7
8. 20.2 Smoothing
Techniques
To produce a better forecast we need to
determine which components are present
in a time series.
To identify the components present in the
time series, we need first to remove the
random variation.
This can be done by smoothing
techniques.
8
9. Moving Averages
A k-period moving average for time
period t is the arithmetic average of the
time series values around period t.
– For example: A 3-period moving average at period t
is calculated by (yt-1 + yt + yt+1)/3
9
10. Moving Average: Example
Example 20.1
To forecast future gasoline sales, the last four
years quarterly sales were recorded.
Calculate the three-quarter and five-quarter
moving average. Show the relevant graphs.
Xm20-01 Period Year/Quarter Gas Sales
1 1 1 39
2 2 37
3 3 61
4 4 58
5 2 1 18
6 2 56 10
12. Moving Average: Smoothing
effect
Moving Average
3-period moving average
100
Value
50
0 Comment: We can identify
1 2 3 4 5 6 7 8 9 10 11 12a 13 14component .
trend 15 16
Data Point
Notice how the averaging process removes some of
the random variation.
12
13. Moving Average: Smoothing
effect
3-period moving average
100
80
60
40
The 5-period 20
0
moving average
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
removes more
variation than the 5-period moving average
3-period moving 100
80
average. 60
40
20
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 13
14. Centered Moving Average
With an even number of observations
included in the moving average, the average
is placed between the two periods in the
middle.
To place the moving average in an actual
time period, we need to center it.
Two consecutive moving averages are
centered by taking their average, and placing
it in the middle between them. (This is used to
graph the points.)
14
15. Centered Moving Average:
Example
Calculate the 4-period moving average and
center it, for the data given below:
Period Time series Moving Avg. Centerd
Mov.Avg.
(2.5)1 15 19.0
(3.5)
2 27 20.25
21.5
(4.5)3 20
17.5
19.50
4 14
5 25
15
6 11
16. Exponential Smoothing
The exponential smoothing – When smoothing the
method provides smoothed time series at time t,
values for all the time > the exponential
periods observed. smoothing method
considers all the data
available at t
– The moving average
method does not the(yt, yt-1,…)
moving average method
considers only the
provide smoothed
observations included in the
values (moving calculation of the average
average values) for value, and “forgets” the
the first and last set of rest.
periods. 16
17. Exponentially Smoothed Time
Series
S = wy + (1-w)St-1
Stt = wytt + (1-w)St-1
St = exponentially smoothed time series at
time t.
yt = time series at time t.
St-1 = exponentially smoothed time series at
time t-1.
w = smoothing constant, where 0 ≤ w ≤ 1.
17
18. The Exponentially Smoothed
Process
Example 20.2 (Xm20-01)
Calculate the gasoline sale smoothed time series
using exponential smoothing with w = .2, and w = .7.
Period Gas
Sales
Set S1 = y1 1 39 S1 = 39
S2 = wy2 + (1-w)S1 2 37 S2 = (.2)(37) + (1-.2)(39) = 38.6
S3 = wy3 + (1-w)S2 3 61 S3 = (.2)(61) + (1-.2)(38.6) = 43.1
4 58
5 18
6 56
. .
. .
18
19. The Exponentially Smoothed
Process
Example 20.2-continued
Exponential Smoothing, w=.2 Small ‘w’ provides
100 a lot of smoothing
Value
50 Neither value of ‘w’
0 reveals the seasonality.
1
3
5
7
9
11
13
15
Exponential Smoothing , w=.7
100
Value
50
0
1 3 5 7 9 11 13 15
19
20. The Exponentially Smoothed
Process
We will let Minitab do our exponential
smoothing
Stat>Time Series > Single Exp Smoothing
This will give us the smoothed graph
20
21. 20.3 Trend and Seasonal Effects
Trend Analysis
The trend component of a time series
can be linear or non-linear.
It is easy to isolate the trend component
using linear regression.
For linear trend use the model y = β0 + β1t +
ε.
For non-linear trend with one (major) change
in slope use the quadratic model y = β0 + β1t
+ β2t2 + ε 21
22. Seasonal Analysis
Seasonal variation may occur within a
year or within a shorter period (month,
week)
To measure the seasonal effects we
construct seasonal indexes.
Seasonal indexes express the degree to
which the seasons differ from the average
time series value across all seasons.
22
23. Computing Seasonal Indexes
Remove the effects of the seasonal and
random variations by regression analysis
yt = b 0 + b 1 t
ˆ
• For each time period compute the ratio
yt/yt > This is based on the Multiplicative Model.
which removes most of the trend variation
>
• For each season calculate the average of yt/yt
which provides the measure of seasonality.
• Adjust the average above so that the sum of averages
of all seasons is 1 (if necessary)
23
24. The additive and multiplicative models
There are two mostly used general models to
describe the composition of a time series
(Assuming no cyclical effects).
The multiplicative model y t = Tt × S t × R t
The additive model yt = Tt + St + Rt
In the seasonal index analysis performed here we use
the multiplicative model because it is mathematically
easier to handle.
24
25. Seasonal indexes analysis
The multiplicative model
y t Tt × S t × R t
= = St × R t
ˆ
yt Tt
The regression line represents trend.
yt
• We see that the ratio represents the seasonality and
ˆ
yt
the random effects.
• Averaging these ratios for each season type removes
the random effects and leaves the seasonality.
25
26. Seasonal indexes analysis
The multiplicative model
y t Tt × S t × R t
= = St × R t
ˆ
yt Tt
Rate/Predicted rate
1.5
Note how no trend is observed, but
1 seasonality and randomness
0.5
still exist.
0
1 3 5 7 9 11 13 15 17 19
yt
• Averaging the ratios for each season type removes
ˆ
yt
the random effects and leaves the seasonality. 26
27. Computing Seasonal Indexes
Example 20.3 (Xm20-03)
Calculate the quarterly seasonal indexes
for hotel occupancy rate in order to
measure seasonal variation.
Data:
Year Quarter Rate Year Quarter Rate Year Quarter Rate
1996 1 0.561 1998 1 0.594 2000 1 0.665
2 0.702 2 0.738 2 0.835
3 0.8 3 0.729 3 0.873
4 0.568 4 0.6 4 0.67
1997 1 0.575 1999 1 0.622
2 0.738 2 0.708
3 0.868 3 0.806
4 0.605 4 0.632
27
28. Rationale for seasonal indexes
Computing Seasonal Indexes
Perform regression analysis for the model
y = β0 + β1t + ε where t represents the time,
and y represents the occupancy rate.
Time (t) Rate ˆ
y = .639368 + .005246t
1 0.561
2 0.702
3 0.800
4 0.568
Rate
5 0.575
6 0.738
7 0.868
8 0.605 0 5 10 15 20 25
. . The regression line represents trend.
t
. .
28
29. yt / yt
>
The Ratios
t yt ˆ
yt Ratio
1 .561 .645 .561/.645=.870
2 .702 .650 .702/.650=1.08
3 ………………………………………………….
=.639368+.005245(1)
No trend is observed, but
yt seasonality and randomness
Rate/Predicted rate
ˆt
y still exist.
1.5
1
0.5
0
13
17
1
3
5
7
9
11
15
19
29
30. The Average Ratios by Seasons
Rate/Predicted rate
0.870
1.080
• To remove most of the random variation
1.221 but leave the seasonal effects,average
0.860 ˆ
the terms yt / yt for each season.
0.864
1.100
1.284 Rate/Predicted rate
0.888
0.865 1.5
1.067
1.046 1
0.854
0.5
0.879
0.993 0
1.122
1 3 5 7 9 11 13 15 17 19
0.874 Average ratio for quarter 1: (.870 + .864 + .865 + .879 + .913)/5 = .878
0.913
1.138
Average ratio for quarter 2: (1.080+1.100+1.067+.993+1.138)/5 = 1.076
1.181 Average ratio for quarter 3: (1.221+1.284+1.046+1.122+1.181)/5 = 1.171
0.900 30
Average ratio for quarter 4: (.860 +.888 + .854 + .874 + .900)/ 5 = .875
31. Adjusting the Average
Ratios
In this example the sum of all the averaged ratios
must be 4, such that the average ratio per season is
equal to 1.
If the sum of all the ratios is not 4, we need to adjust
them proportionately.
Suppose the(Seasonal averaged ratio) (number of seasons)
sum of ratios is equal to 4.1. Then each
Seasonal index be multipliedSum of averaged ratios
ratio will = by 4/4.1.
In our problem the sum of all the averaged ratios is equal to 4:
.878 + 1.076 + 1.171 + .875 = 4.0.
No normalization is needed. These ratios become the
seasonal indexes.
31
32. Interpreting the Seasonal
Indexes
The seasonal indexes tell us what is the ratio
between the time series value at a certain
season, and the overall seasonal average.
17.1% above the
In our problem: annual average
117.1%
7.6% above the 12.5% below the
annual average annual average
Annual average
occupancy (100%) 107.6%
87.8% 87.5%
12.2% below the
annual average
Quarter 1 Quarter 2 Quarter 3 Quarter 4 Quarter 1 Quarter 2 Quarter 3 Quarter 4
32
33. The Smoothed Time Series
The trend component and the seasonality
component are recomposed using the
y = ˆ ˆ t
ˆ
multiplicative model. T × S = (.639 + .0052t ) S
ˆ t t t
In period #1 ( quarter 1): ˆ ˆ ˆ
y1 = T1 × S1 = (.639 + .0052(1))(.878) = .566
ˆ ˆ ˆ
y 2 = T2 × S 2 = (.639 + .0052(2))(1.076) = .699
In period #2 ( quarter 2):
0.9
Actual series Smoothed series
0.8
0.7
0.6
0.5 The linear trend (regression) line
1 3 5 7 9 11 13 15 17 19
33
34. Deseasonalized Time Series
Seasonally adjusted time series = Actual time series
Seasonal index
By removing the seasonality, we can
identify changes in the other components of
the time series, that might have occurred
over time.
34
35. Deseasonalized Time Series
In period #1 ( quarter 1): y 1 / SI1 = .561/ .870 = .639
In period #2 ( quarter 2): y 2 / SI2 = .708 1.076 = .652
In period #5 ( quarter 1): y 5 / SI1 = .575 .870 = .661
There was a gradual increase in occupancy rate
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25
35
36. 20.4 Introduction to
Forecasting
There are many forecasting models available
o o o
*
*
? *
o o
*
o * o o
*
* Model 1 * Model 2
Which model
performs better?
36
37. 20.4 Introduction to
Forecasting
A forecasting method can be selected
by evaluating its forecast accuracy
using the actual time series.
The two most commonly used measures of forecast accuracy
are: n
Mean Absolute Deviation ∑ y t − Ft
MAD = t =1
Sum of Squares for Forecast Error n
n
SSE = ∑ ( y t − Ft )2
t =1 37
38. Measures of Forecast
Accuracy
Choose SSE if it is important to avoid
(even a few) large errors. Otherwise,
use MAD.
A useful procedure for model selection.
Use some of the observations to develop several
competing forecasting models.
Run the models on the rest of the observations.
Calculate the accuracy of each model (by comparing to
actual data at a later time period).
Select the model with the best (lowest) accuracy measure.
38
39. Selecting a Forecasting
Model
Annual data from 1970 to 1996 were used to
develop three forecasting models.
Use MAD and SSE to determine which model
performed best for 1997, 1998, 1999, and
2000.
Forecast value
Year Actual y Model 1 Model 2 Model 3
1997 129 136 118 130
1998 142 148 141 146
1999 156 150 158 170
2000 183 175 163 180
39
40. Solution
For model
Actual y Forecast for y
1 in 1991 in 1991
129 −136 + 142 −148 + 156 −150 + 183 −175
MAD = = 6.75
4
SSE = (129 − 136) + (142 − 148) + (156 − 160) + (183 − 175) 2 = 185
2 2 2
Summary of results
Model 1 Model 2 Model 3
MAD 6.75 8.5 5.5
SSE 185 526 222
40
41. Selecting a Forecasting
Model
In this case, Model 2 is inferior to both Models 1
and 3
Using MAD Model 3 is best
Using SSE Model 1 is most accurate
Choice – prefer model that consistently
produces moderately accurate forecasts (Model
3) or one whose forecasts come quite close
usually but misses badly in a small number of
time periods (Model 1)
41
42. 20.5 Forecasting Models
The choice of a forecasting technique
depends on the components identified in
the time series.
The techniques discussed next are:
Exponential smoothing
Seasonal indexes
Autoregressive models (a brief discussion)
42
43. Forecasting with Exponential
Smoothing
The exponential smoothing model can
be used to produce forecasts when the
time series…
exhibits gradual(not a sharp) trend
no cyclical effects
no seasonal effects
Forecast for period t+k is computed by
Ft+k = St
t is the current period; St = ωyt + (1-ω)St-1 43
44. Forecasting with Exponential
Smoothing - Example
The quarterly sales (in $ millions) of a
department store chain were recorded for
the years 1997 – 2000
44
46. Forecasting with Exponential
Smoothing - Example
Si ngl e Ex ponent i al Smoot hi ng Pl ot f or Sal es
55
Y16 = 45
Variable
Actual
50 Fits
Smoothing Constant
45 Alpha 0.4
40
Accuracy Measures
MAPE 22.4486
S16 =
MAD 7.2311
41.8
Sales
MSD 69.2590
35
30
25
20
2 4 6 8 10 12 14 16
I ndex
46
47. Forecasting with Exponential
Smoothing - Example
Forecast for period t+k is computed by
(t = 16) Ft+k = St
t is the current period; St = ωyt + (1-ω)St-1
F17= S16 = 41.8
F18= S16 = 41.8
F19= S16 = 41.8
F20= S16 = 41.8
47
48. Computing Seasonal Indexes
Example 20.3 (Xm20-03)
Calculate the quarterly seasonal indexes
for hotel occupancy rate in order to
measure seasonal variation.
Data:
ˆ ˆ ˆ
yt = Tt × S t = (.639 + .0052t ) S t
ˆ
Year Quarter Rate Year Quarter Rate Year Quarter Rate
1996 1 0.561 1998 1 0.594 2000 1 0.665
2 0.702 2 0.738 2 0.835
3 0.8 3 0.729 3 0.873
4 0.568 4 0.6 4 0.67
1997 1 0.575 1999 1 0.622
2 0.738 2 0.708
3 0.868 3 0.806
4 0.605 4 0.632
48
49. Forecasting with Seasonal
Indexes; Example
Example 20.5
Use the seasonal indexes calculated in
Example 21.3 along with the trend line, to
forecast each quarter occupancy rate in 2001.
The solution procedure
Use simple linear regression to find the trend line.
Use the trend line to calculate the trend values.
To calculate Ft multiply the trend value for period t by the
seasonal index of period t.
49
50. Forecasting with Seasonal
Indexes; Example
The solution procedure
1. Use simple linear regression to find the trend line.
From Minitab (Using decomposition )
Yt = .639 + .00525t
50
51. Forecasting with Seasonal
Indexes; Example
The solution procedure:
2. Use the trend line to calculate the trend values
Yt = .639 + .00525t
We previously used 20 time periods
Quarter 1: 0.639 + 0.00525(21) = 0.749
Quarter 2: 0.639 + 0.00525(22) = 0.755
Quarter 3: 0.639 + 0.00525(23) = 0.760
Quarter 4: 0.639 + 0.00525(24) = 0.765
51
52. Forecasting with Seasonal
Indexes; Example
The solution procedure
3. To calculate Ft multiply the trend value for period t by
the seasonal index of period t.
Quarter 1: 0.749 X 0.878 = 0.658
Quarter 2: 0.755 X 1.076 = 0.812
Quarter 3: 0.760 X 1.171 = 0.890
Quarter 4: 0.765 X 0.875 = 0.670
We forecast that the quarterly occupancy rates for
the next year will be 65.8%, 81.2%, 89% and
67%.
52
53. Forecasting with Seasonal
Indexes: Example (Summary)
Solution
The trend line was obtained from the
regression analysis. y t = .639 + .00525t
ˆ
– For the year 2001 we have:
y21 = .639 + .00525 (21) = .749
ˆ F21 = .749(.878)
t Trend value Quarter SI Forecast
21 .749 1 .878 .658
22 .755 2 1.076 .812
23 .760 3 1.171 .890
24 .765 4 .875 .670
53
54. Problem 20-47
The revenues (in $millions) of a chain of
ice-cream stores are listed for each
quarter during the previous 5 years.
Use the handout with the plots etc. to
answer the questions.
54
56. 20-47
1. Plot the time series. Is there a
pattern? Does there appear to be
A long term trend
A cyclic effect
A seasonal effect
2. Plot the 3-year moving balance. Does
this change any of your answers from #
1?
56
57. 20-47 - Answers
Seems to be a long term trend – going
upward.
Does not seem to be a cyclic effect.
Because we know the data is in quarter
years, this seems to be a strong seasonal
effect.
The three year moving average plot
strengthens these observations.
57
58. 20-47
Why is exponential smoothing not
recommended as a forecasting tool in this
problem?
Look at the exponential smoothing plot (w
= 0.3) Does this confirm your answer
above?
58
59. 20-47 - Answers
Exponential smoothing is not
recommended as a forecasting tool for
this problem because of the strong
seasonal effect.
The 0.3 gives a large smoothing effect,
and the seasonality is no larger as
evident.
59
60. 20-47
Look at the trend analysis plots. Does the
Linear or the Quadratic trend model seem
to be better? Why? What is the equation
of the trend line?
60
61. 20-47 - Answers
The linear model seems to fit well. The
quadratic model doesn’t show much of a
curve. Checking the MAD for both models
shows 6.8716 for the linear model and
6.8784 for the quadratic model. Since the
linear model has a slightly smaller MAD
we will use it.
The equation of the trend line, then, is
Y = 20.2105 + 0.732331t
61
62. 20.47
Predict the sales for the next 4 quarters
using the trend analysis line.
62
64. 20-47
Using the seasonal indexes and the trend
line, forecast revenues for the next four
quarters.
64
65. 20-47 - Answer
The Fitted trend equation for the
decomposition plot is Y = 21.7511 +
0.567732 t
The next time period is number 21.
Y21 = [21.75 + 0.568 (21)] * 0.63581 =
21.41
Y22 = [21.75 + 0.568 (22)] * 1.05732 =
36.21
Y23 = [21.75 + 0.568 (23)] * 1.36851 = 65
66. 20-47 (continued)
If the actual sales numbers for the next
four quarters were 24, 40, 48, 36.
Calculate the MAD for the predictions
from the trend line and seasonal indexes.
66
68. 20-47
Go back to the linear trend equation
forecasts. If the actual sales numbers for
the next four quarters were 24, 40, 48, 36.
Calculate the MAD.
Which method of forecasting did a better
job in this case?
68
69. 20-47 Answers
MAD = |35.58 - 24| + |36.31 – 40| + |37.05 – 48|
+ |37.78 – 36| / 4
= (11.58 + 3.69 + 10.95 + 1.78 ) / 4
= 28/4 = 7 This is the MAD for the trend
analysis.
The MAD for the seasonal trend analysis was
2.43.
The seasonal trend analysis did the better job of
forecasting.
69