Last Updated 27 Mar 2020

# Time Series Models

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TIME SERIES MODELS Time series analysis provides tools for selecting a model that can be used to forecast of future events. Time series models are based on the assumption that all information needed to generate a forecast is contained in the time series of data. The forecaster looks for patterns in the data and tries to obtain a forecast by projecting that pattern into the future. A forecasting method is a (numerical) procedure for generating a forecast. When such methods are not based upon an underlying statistical model, they are termed heuristic.

A statistical (forecasting) model is a statistical description of the data generating process from which a forecasting method may be derived. Forecasts are made by using a forecast function that is derived from the model. WHAT IS A TIME SERIES? A time series is a sequence of observations over time. A time series is a sequence of data points, measured typically at successive time instants spaced at uniform time intervals. 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 annual freshman enrollment in a department of a university, and the daily volume of flows in a river. Forecasting time series data is important component of operations research because these data often provide the foundation for decision models. An inventory model requires estimates of future demands, a course scheduling and staffing model for a university requires estimates of future student inflow, and a model for providing warnings to the population in a river basin requires estimates of river flows for the immediate future. * TWO MAIN GOALS:

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There are two main goals of time series analysis: (a) identifying the nature of the phenomenon represented by the sequence of observations, and (b) forecasting (predicting future values of the time series variable). Both of these goals require that the pattern of observed time series data is identified and more or less formally described. Once the pattern is established, we can interpret and integrate it with other data (e. g. , seasonal commodity prices). Regardless of the depth of our understanding and the validity of our interpretation (theory) of the phenomenon, we can extrapolate the identified pattern to predict future events.

Several methods are described in this chapter, along with their strengths and weaknesses. Although most are simple in concept, the computations required to estimate parameters and perform the analysis are tedious enough that computer implementation is essential. The easiest way to identify patterns is to plot the data and examine the resulting graphs. If we did that, what could we observe? There are four basic patters, which are shown in Figure 1. Any of these patterns, or a combination of them, can be present in a time series of data: 1. Level or horizontal

This pattern exists when data values fluctuate around a constant mean. This is the simplest pattern and easiest to predict. A horizontal pattern is observed when the values of the time series fluctuate around a constant mean. Such time series is also called stationery. In Retail data, stationery time series can be found easily since there are products which sales roughly the same amount of items every period. In the stock market however, it's difficult (if not impossible) to find horizontal patterns. Most of the time series there are non-stationery.

Time series with horizontal patterns are very easy to forecast. 2. Trend When data exhibit an increasing or decreasing pattern over time, we say that they exhibit a trend. The trend can be upward or upward. The trend pattern is straightforward. It consists of a long-term increase or decrease of the values of the time series. Trend patterns are easy to forecast and are very profitable when found by stock traders. 3. Seasonality Any pattern that regularly repeats itself and is of a constant length is a seasonal pattern is.

Such seasonality exists when the variable ewe are trying to forecast is influenced by seasonal factors such as the quarter or month of the year or day of the week. A time series with seasonal patterns are more difficult to forecast but not too difficult. The values of these time series are influenced by seasonal factors, such as the turkey in Christmas period. Also, ice cream sales are affected by seasonality. People buy more ice creams during the summer. Forecasting algorithms which can deal with the seasonality can be used for forecasting such time series. Holt-Winters' method is one such algorithm. 4.

Cycles Cyclical patterns are usually confused with the seasonal patterns. While seasonal patterns are influenced by seasonal factors, cyclical patterns do not necessarily have a fixed period. A seasonal pattern can be cyclical, but a cyclical is not necessarily seasonal. Cyclical patterns are the most difficult to forecast. Most forecasting tools can deal with seasonality, trend and horizontal time series but very few can offer acceptable forecasts to cyclical patterns unless there is some sort of indication as to how the cycle evolves. Random Variation is unexplained variation that cannot be predicted.

The more random variation a data set has, the harder it is to forecast accurately. In practice, forecasts derived by these methods are likely to be modified by the analyst upon considering information not available from the historical data. We should understand that to obtain a good forecast the forecasting model should be matched to the patterns in the available data. TIME SERIES METHODS The Naive Method Among the time-series models, the simplest is the naive forecast. A naive forecast simply uses the actual demand for the past period as the forecasted demand for the next period.

This, of course, makes the assumption that the past will repeat. An example of naive forecasting is presented in Table 1. Table 1 Naive Forecasting Period| Actual Demand (000's)| Forecast (000's)| January| 45| | February| 60| 45| March| 72| 60| April| 58| 72| May| 40| 58| June| | 40| This model is only good for a level data pattern. One of the advantages of this model is that only two historical pieces of information need to be carried: the mean itself and the number of observations on which the mean was based. Averaging Method Another simple technique is the use of averaging.

To make a forecast using averaging, one simply takes the average of some number of periods of past data by summing each period and dividing the result by the number of periods. This technique has been found to be very effective for short-range forecasting. Variations of averaging include the moving average, the weighted average, and the weighted moving average. A moving average takes a predetermined number of periods, sums their actual demand, and divides by the number of periods to reach a forecast. For each subsequent period, the oldest period of data drops off and the latest period is added.

Assuming a three-month moving average and using the data from Table 1, one would simply add 45 (January), 60 (February), and 72 (March) and divide by three to arrive at a forecast for April: 45 + 60 + 72 = 177 ? 3 = 59 To arrive at a forecast for May, one would drop January's demand from the equation and add the demand from April. Table 2 presents an example of a three-month moving average forecast. Table 2 Three Month Moving Average Forecast Period| Actual Demand (000's)| Forecast (000's)| January| 45| | February| 60| | March| 72| | April| 58| 59| May| 40| 63|

June| | 57| A weighted average applies a predetermined weight to each month of past data, sums the past data from each period, and divides by the total of the weights. If the forecaster adjusts the weights so that their sum is equal to 1, then the weights are multiplied by the actual demand of each applicable period. The results are then summed to achieve a weighted forecast. Generally, the more recent the data the higher the weight, and the older the data the smaller the weight. Using the demand example, a weighted average using weights of . 4, . 3, . , and . 1 would yield the forecast for June as:  60(. 1) + 72(. 2) + 58(. 3) + 40(. 4) = 53. 8 Forecasters may also use a combination of the weighted average and moving average forecasts. A weighted moving average forecast assigns weights to a predetermined number of periods of actual data and computes the forecast the same way as described above. As with all moving forecasts, as each new period is added, the data from the oldest period is discarded. Table 3 shows a three-month weighted moving average forecast utilizing the weights . 5, . 3, and . 2. Table 3

Three–Month Weighted Moving Average Forecast Period| Actual Demand (000's)| Forecast (000's)| January| 45| | February| 60| | March| 72| | April| 58| 55| May| 40| 63| June| | 61| | | | Exponential Smoothing Exponential smoothing takes the previous period's forecast and adjusts it by a predetermined smoothing constant, ? (called alpha; the value for alpha is less than one) multiplied by the difference in the previous forecast and the demand that actually occurred during the previously forecasted period (called forecast error). To make a forecast for the next time period, you eed three pieces of information: 1. The current period’s forecast 2. The current period’s actual value 3. The value of a smoothing coefficient, alpha, which varies between 0 and 1. Exponential smoothing is expressed formulaically as such: New forecast = previous forecast + alpha (actual demand ? previous forecast) A forecast for February is computed as such: New forecast (February) = 50 + . 7(45 ? 50) = 41. 5 Next, the forecast for March: New forecast (March) = 41. 5 + . 7(60 ? 41. 5) = 54. 45 This process continues until the forecaster reaches the desired period.

In Table 4 this would be for the month of June, since the actual demand for June is not known. Table 4 Period| Actual Demand (000's)| Forecast (000's)| January| 45| 50| February| 60| 41. 5| March| 72| 54. 45| April| 58| 66. 74| May| 40| 60. 62| June| | 46. 19| Forecasting Trend There are many ways to forecast trend patterns in data. Most of the models used for forecasting trend are the same models used to forecast the level patterns, with an additional feature added to compensate for the lagging that would otherwise occur. Trend-Adjusted Exponential Smoothing

When a trend exists, the forecasting technique must consider the trend as well as the series average ignoring the trend will cause the forecast to always be below (with an increasing trend) or above (with a decreasing trend) actual demand Double exponential smoothing smooths (averages) both the series average and the trend forecast for period t+1: Ft+1 = At + Tt average: At = aDt + (1 - a) (At-1 + Tt-1) = aDt + (1 - a) Ft average trend: Tt = B CTt + (1 - B) Tt-1 current trend: CTt = At - At-1 forecast for p periods into the future: Ft+p = At + p Tt here: At = exponentially smoothed average of the series in period t Tt = exponentially smoothed average of the trend in period t CTt = current estimate of the trend in period t a = smoothing parameter between 0 and 1 for smoothing the averages B = smoothing parameter between 0 and 1 for smoothing the trend Linear Trend Line Linear trend line is a time series technique that computes a forecast with trend by drawing a straight line through a set of data. The forecasting equation for the linear trend model is: Y= a + bX where t is the time index.

The parameters alpha and beta (the "intercept" and "slope" of the trend line) are usually estimated via a simple regression in which Y is the dependent variable and the time index t is the independent variable. Forecasting Seasonality Recall that any regularly repeating pattern is a seasonal pattern. We are all familiar with quarterly and monthly seasonal patterns. For example, seasonality includes sales of Christmas tree before Christmas, sales of jackets, hotel registrations and sales of greeting cards. The procedure for computing seasonality consists of the following steps: 1. Calculate the average demand per season . Calculate a seasonal index for each season of each year: 3. Average the indexes by season 4. Forecast demand for the next year & divide by the number of seasons 5. Multiply next year’s average seasonal demand by each average seasonal index. Selecting a Forecasting Method The selection of a forecasting method is a difficult task that must be base in part on knowledge concerning the quantity being forecast. With forecasting procedures, we are generally trying to recognize a change in the underlying process of a time series while remaining insensitive to variations caused by purely random effects.

The goal of planning is to respond to fundamental changes, not to spurious effect. Bibliography: Box, G. E. P and G. M. Jenkins and G. D. Reinsel, Time Series Analysis, Forecasting, and Control, Third Edition, Prentice Hall, Englewood Cliffs, NJ, 1993. Brockwell, Peter J. and Davis, Richard A. (2002). Introduction to Time Series and Forecasting, 2nd. ed. , Springer-Verlang. Chatfield, C. , The Analysis of Time Series: An Introduction, Fifth Edition, Chapman &Hall, Boca Raton, FL, 1996. Fuller, W. A. , Introduction to Statistical Time Series, Second Edition, John Wiley & Sons, New York, 1996. (Electronic Version): StatSoft, Inc. 2012). Electronic Statistics Textbook. Tulsa, OK: StatSoft. WEB: http://www. statsoft. com/textbook/. (Printed Version): Hill, T. & Lewicki, P. (2007). STATISTICS: Methods and Applications. StatSoft, Tulsa, OK. (Electronic Version): A First Course on Time Series Analysis - an open source book on time series analysis with SAS WEB: http://www. statistik-mathematik. uni-wuerzburg. de/wissenschaftforschung/time_series/ (Electronic Version): Forecasting - levels, examples, manager, definition, model, type, company  WEB:http://www. referenceforbusiness. com/management/Ex-Gov/Forecasting. html#b#ixzz28ty2DePJ

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