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# Quantitative Techniques In Forecasting Finance Essay

A forecast is an estimate of future performance based on past experiences made by analyst. Past data are systematically combined in predetermined way to obtain this estimate. Forecasting is not guessing or prediction. It helps manager to Plan the system and plan the use of the system. There are many departments in an organization that are affected by the forecasting decision and activities.

Accounting

Finance

Human Resources

Marketing

Operations

Product/Service Design

Forecasting situations vary widely in their time horizons, factors determining actual outcomes, types of data patterns, and many other aspects. Forecasting methods can be very simple such as using the most recent observation as a forecast (which is called the "naive method"), or highly complex such as neural nets and econometric systems of simultaneous equations. The choice of method depends on what data are available and the predictability of the quantity to be forecast.

## Forecasting Method

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This can be classified as qualitative and quantitative. Qualitative methods generally involve the expert

judgment to develop forecast. Such methods are appropriate when historical data on the variable being forecast are either not applicable or unavailable. Quantitative forecasting methods can be used when

(1) past information about the variable being forecast is available,

(2) the information can be quantified, and

(3) it is reasonable to assume that the pattern of the past will continue into the future. In such cases a forecast can be developed using a time series method or a causal method.

When a historical data is restricted to past values of the variable to be forecast , the forecast value is called time series method and he data is called as time series data. Here ,a pattern is discovered and then pattern is extrapolated into future; the forecast is solely based on past values of the variable and past forecast errors.

Causal forecasting methods are based on the assumption that the variable forecasted has a cause-effect relationship with one or more other variables. Consider the relationship between advertising expenditures and sales, a marketing manager might attempt to predict sales for a given level of advertising expenditures. Some may rely on intuition to judge how two variables are related. However, if data can be obtained, a statistical procedure called regression analysis can be used to develop an equation showing how the variables are related.

In regression terminology, the variable being predicted is called the dependent variable. The variable or variables being used to predict the value of the dependent variable are called the independent variables. For example, in analyzing the effect of advertising expenditures on sales, a marketing manager's desire to predict sales would suggest making sales the dependent variable. Advertising expenditure would be the independent variable used to help predict sales. In statistical notation, y denotes the dependent variable and x denotes the independent variable. In this report , we have mainly covered Quantitative Methods of Forecasting.

There are five basic steps in Forecasting Which Organization need to follow :

Problem definition: Often this is most difficult part of forecasting. Defining the problem carefully requires an understanding of how the forecasts will be used, who requires the forecasts, and how the forecasting function fits within the organization requiring the forecasts. A forecaster needs to spend time talking to everyone who will be involved in collecting data, maintaining databases, and using the forecasts for future planning.

Gathering information: Always, two kinds of information are must : (a) statistical data, and (b) the people experience who collected the data and use for forecasts. Many a times, Problem is obtaining enough historical data to be to fit to a good statistical model. However, much older data will not be so useful due to changes in the system being forecast.

Preliminary (exploratory) analysis :This start by graphing the data and looking to for consistent patterns, significant trend, seasonal importance, evidence of the presence of business cycles and how strong are the relationships among the variables available for analysis.

Choosing and fitting models : Selecting the model to use depends on the availability of historical data, the strength of relationships between the forecast variable and any explanatory variables, and the way the forecasts are to be used. It is common to compare two or three potential models.

Using and evaluating a forecasting model : Once you have selected the model and its parameters estimated, the model is to be used to make forecasts. The performance of the model can only be evaluated after the data for the forecast period have become available. A number of methods have been developed to help in assessing the accuracy of forecasts, as discussed in the next section.

Next ,I present you three methods of Quantitative Forecasting :

## Learning Objective

This chapter discusses about time series analysis and forecasting. Here ,you would learn ,

1) About different time-series forecasting models-moving averages, exponential smoothing, the linear trend, the quadratic trend, the exponential trend-and the autoregressive models and least-squares models for seasonal data

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2) To choose the most appropriate time-series forecasting model.

Since we know that Quantitative forecasting methods can be used when :

(1) past information about the variable being forecast is available,

(2) the information can be quantified, and

(3) it is reasonable to assume that the pattern of the past will continue into the future.

In such cases, a forecast can be developed using a time series method or a causal method. If the historical data is only restricted to past values of the variable to be forecast, the forecasting procedure is called a time series method and the data are referred to as a time series data. The objective of time series analysis is to discover a pattern in the historical data or time series and then extrapolate the pattern into the future; the forecast is based solely on past values of the variable and/or on past forecast errors.

## Time Series Patterns

A time series is a sequence of observations on a variable measured at successive points in time or over successive periods of time. The measurements may be taken every hour, day, week, month, or year, or at any other regular interval.1 The pattern of the data is an important factor in understanding how the time series has behaved in the past. If such behavior can be expected to continue in the future, we can use the past pattern to guide us in selecting an

appropriate forecasting method.

To identify the underlying pattern in the data, a useful first step is to construct a time series plot. A time series plot is a graphical presentation of the relationship between time and the time series variable; time is on the horizontal axis and the time series values are shown on the vertical axis. Let us see it through a example.

## Horizontal Pattern

A horizontal pattern exists when the data fluctuate around a constant mean. To illustrate a time series with a horizontal pattern, consider the 12 weeks of data in Table 1 as shown. These data show the number of gallons of gasoline sold by a gasoline distributor in Bennington, Vermont, over the past 12 weeks. The average value or mean for this time series is 19.25 or 19,250 gallons per week. Figure1 shows a time series plot for these data. Note how the data fluctuate around the sample mean of 19,250 gallons. Although random variability is present, we would say that these data follow a horizontal pattern.

## Table 1 Figure 1

Gasoline Sales Time Series Gasoline Sales Time Series Plot

1

17

2

21

3

19

4

23

5

18

6

16

7

20

8

18

9

22

10

20

11

15

12

22

## Trend Pattern

Although time series data generally exhibit random fluctuations, a time series may also show gradual shifts or movements to relatively higher or lower values over a longer period of time. If a time series plot exhibits this type of behavior, we say that a trend pattern exists. A trend is usually the result of long-term factors such as population increases or decreases, changing demographic characteristics of the population, technology, and/or consumer preferences.

For example, consider the time series of bicycle sales for a particular manufacturer over the past 10 years, as shown in Table 2 and Figure 2. Note that 21,600 bicycles were sold in year one, 22,900 were sold in year two, and so on. In year 10, the most recent year, 31,400 bicycles were sold. Visual inspection of the time series plot shows some up and down movement over the past 10 years, but the time series also seems to have a systematically increasing or upward trend.

The trend for the bicycle sales time series appears to be linear and increasing over time, but sometimes a trend can be described better by other types of patterns. For instance, the data in Table 18.4 and the corresponding time series plot in Figure 18.4 show the sales for a cholesterol drug since the company won FDA approval for it 10 years ago. The time series increases in a nonlinear fashion; that is, the rate of change of revenue does not increase by a constant amount from one year to the next. In fact, the revenue appears to be growing in an exponential fashion. Exponential relationships such as this are appropriate when the percentage change from one period to the next is relatively constant.

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1

21.6

2

22.9

3

25.5

4

21.9

5

23.9

6

27.5

7

31.5

8

29.7

9

28.6

10

31.4

## Source : Statistics for Business and Economics, ch 18 Time Series Analysis and Forecasting

The trend for the bicycle sales time series appears to be linear and increasing over time, but sometimes a trend can be described better by other types of patterns.

## Seasonal Pattern :

The trend of a time series can be identified by analyzing multiyear movements in historical data. Seasonal patterns are recognized by seeing the same repeating patterns over successive periods of time. For example, a manufacturer of swimming pools expects low sales activity in the fall and winter months, with peak sales in the spring and summer months. Manufacturers of snow removal equipment and heavy clothing, however, expect just the opposite yearly pattern. Not surprisingly, the pattern for a time series plot that exhibits a repeating pattern over a one-year period due to seasonal influences is called a seasonal pattern. While we generally think of seasonal movement in a time series as occurring within one year, time series data can also exhibit seasonal patterns of less than one year in duration. For example, daily traffic volume shows within-the-day "seasonal" behavior, with peak levels occurring during rush hours, moderate flow during the rest of the day and early evening, and light flow from midnight to early morning.

As an example of a seasonal pattern, consider the number of umbrellas sold at a clothing store over the past five years. Table 3 shows the time series and Figure 3 shows the corresponding time series plot. The time series plot does not indicate any long-term trend in sales. In fact, unless you look carefully at the data, you might conclude that the data follow a horizontal pattern. But closer inspection of the time series plot reveals a regular pattern in the data. That is, the first and third quarters have moderate sales, the second quarter has the highest sales, and the fourth quarter tends to have the lowest sales volume. Thus, we would conclude that a quarterly seasonal pattern is present.

## Selecting a Forecasting Method

The underlying pattern in the time series is an important factor in selecting a forecasting method. Thus, a time series plot should be one of the first things developed when trying to determine what forecasting method to use. If we see a horizontal pattern, then we need to select a method appropriate for this type of pattern. Similarly, if we observe a trend in the data, then we need to use a forecasting method that has the capability to handle trend effectively.

The next two sections illustrate methods that can be used in situations where the underlying pattern is horizontal; in other words, no trend or seasonal effects are present. We then consider methods appropriate when trend and/or seasonality are present in the data.

## Forecast Accuracy

Here, we begin by developing forecasts for the gasoline time series shown in Table 1 using the simplest of all the forecasting methods: an approach that uses the most recent week's sales volume as the forecast for the next week. For instance, the distributor sold 17 thousand gallons of gasoline in week 1; this value is used as the forecast for week 2. Next, we use 21, the actual value of sales in week 2, as the forecast for week 3, and so on. The forecasts

obtained for the historical data using this method are shown in Table 4 in the column labeled Forecast. Because of its simplicity, this method is often referred to as a naive forecasting method.

Forecast methods are used to determine how well a particular forecasting method is able to reproduce the time series data that are already available. By selecting the method that has the best accuracy for the data already known, we hope to increase the likelihood that we will obtain better forecasts for future time periods.

The key concept associated with measuring forecast accuracy is forecast error, defined as

Forecast Error = ActualValue - Forecast

Table 4 COMPUTING FORECASTS AND MEASURES OF FORECAST ACCURACY USING THE

## Source : Statistics for Business and Economics, ch 18 Time Series Analysis and Forecasting

For instance, because the distributor actually sold 21 thousand gallons of gasoline in week 2 and the forecast, using the sales volume in week 1, was 17 thousand gallons, the forecast error in week 2 is

## Forecast Error in week 2 = 21 -17= 4

The fact that the forecast error is positive indicates that in week 2 the forecasting method underestimated the actual value of sales. Next, we use 21, the actual value of sales in week 2, as the forecast for week 3. Since the actual value of sales in week 3 is 19, the forecast error for week 3 is 19 -21=-2. In this case, the negative forecast error indicates that in week 3 the forecast overestimated the actual value. Thus, the forecast error may be positive or negative, depending on whether the forecast is too low or too high. A complete summary of the forecast errors for this naive forecasting method is shown in Table 4 in the column labeled Forecast Error.

A simple measure of forecast accuracy is the mean or average of the forecast errors. Table 4 shows that the sum of the forecast errors for the gasoline sales time series is 5; thus, the mean or average forecast error is 5/11 _ .45. Note that although the gasoline time series consists of 12 values, to compute the mean error we divided the sum of the forecast errors by 11 because there are only 11 forecast errors. Because the mean forecast error is positive, the method is under forecasting; in other words, the observed values tend to be greater than the forecasted values. Because positive and negative forecast errors tend to offset one another, the mean error is likely to be small; thus, the mean error is not a very useful measure of forecast accuracy.

The mean absolute error, denoted MAE, is a measure of forecast accuracy that avoids the problem of positive and negative forecast errors offsetting one another. As you might expect given its name, MAE is the average of the absolute values of the forecast errors. Table 4 shows that the sum of the absolute values of the forecast errors is 41; thus,

## MAE= average of the absolute value of forecast errors = 41/11 = 3.73

Another measure that avoids the problem of positive and negative forecast errors offsetting each other is obtained by computing the average of the squared forecast errors. This measure of forecast accuracy, referred to as the mean squared error, is denoted MSE. From Table 4, the sum of the squared errors is 179; hence,

## MSE =average of the sum of squared forecast errors = 179/11 = 16.27

The size of MAE and MSE depends upon the scale of the data. As a result, it is difficult to make comparisons for different time intervals, such as comparing a method of forecasting monthly gasoline sales to a method of forecasting weekly sales, or to make comparisons across different time series. To make comparisons like these we need to work with relative or percentage error measures. The mean absolute percentage error, denoted MAPE, is such a measure. To compute MAPE we must first compute the percentage error for each forecast. For example, the percentage error corresponding to the forecast of 17 in week 2 is computed by dividing the forecast error in week 2 by the actual value in week 2 and multiplying the result by 100. For week 2 the percentage error is computed as follows:

## Percentage error for week 2 = (4/21)*100 = 19.05

Thus, the forecast error for week 2 is 19.05% of the observed value in week 2. A complete summary of the percentage errors is shown in Table 4 in the column labeled Percentage Error. In the next column, we show the absolute value of the percentage error. Table 4 shows that the sum of the absolute values of the percentage errors is 211.69; thus,

## MAPE = average of the absolute value of percentage forecast errors = 211.69/11=19.24

Summarizing, using the naive (most recent observation) forecasting method, we obtained the following measures of forecast accuracy:

## MAPE =19.24%

These measures of forecast accuracy simply measure how well the forecasting method is able to forecast historical values of the time series.

Measures of forecast accuracy are important factors in comparing different forecasting methods, but we have to be careful not to rely upon them too heavily. Good judgment and knowledge about business conditions that might affect the forecast also have to be carefully considered when selecting a method. And historical forecast accuracy is not the only consideration, especially if the time series is likely to change in the future.

## Moving Averages and Exponential Smoothing

In this section we discuss three forecasting methods that are appropriate for a time series with a horizontal pattern: moving averages, weighted moving averages, and exponential smoothing. These methods also adapt well to changes in the level of a horizontal pattern However, without modification they are not appropriate when significant trend, cyclical, or seasonal effects are present. Because the objective of each of these methods is to "smooth out" the random fluctuations in the time series, they are referred to as smoothing methods. These methods are easy to use and generally provide a high level of accuracy for short range forecasts, such as a forecast for the next time period.

## Moving Averages

Moving averages for a chosen period of length L consist of a series of means, each computed over time for a sequence of L observed values. Moving averages, represented by the symbol MA(L) can be greatly affected by the value chosen for L, which should be an integer value that corresponds to, or is a multiple of, the estimated average length of a cycle in the time series.

To illustrate, suppose you want to compute five-year moving averages from a series that has n=11 years. Because L=5, the five-year moving averages consist of a series of means computed by averaging consecutive sequences of five values. You compute the first five-year moving average by summing the values for the first five years in the series and dividing by 5.

## MA(5) =( Y1 + Y2 + Y3 + Y4 + Y5/)5

You compute the second five-year moving average by summing the values of years 2 through 6 in the series and then dividing by 5.

## MA(5) =( Y2 + Y3 + Y4 + Y5+ Y6)/5

You continue this process until you have computed the last of these five-year moving averages by summing the values of the last 5 years in the series and then dividing by 5. When you have annual time-series data, L should be an odd number of years. By following this rule, you are unable to compute any moving averages for the first (L-1)/2 years or the last (L-1)/2 years of the series. Thus, for a five-year moving average, you cannot make computations for the first two years or the last two years of the series.

When plotting moving averages, you plot each of the computed values against the middle year of the sequence of years used to compute it. If n=5 and L=11, the first moving average is centered on the third year, the second moving average is centered on the fourth year, and the last moving average is centered on the ninth year.

To show how moving averages can be used to forecast gasoline sales, we use a three-week moving average (k =3) and begin by computing the forecast of sales in week 4 using the average of the time series values in weeks 1-3.

F4 = average of weeks 1-3 = (17+21+19)/3 = 19

Thus, the moving average forecast of sales in week 4 is 19 or 19,000 gallons of gasoline. Because the actual value observed in week 4 is 23, the forecast error in week 4 is 23 -19 =4.

A complete summary of the three-week moving average forecasts for the gasoline sales time series is provided in Table 5. Figure 5 shows the original time series plot and the three-week moving average forecasts. Note how the graph of the moving average forecasts has tended to smooth out the random fluctuations in the time series.

## Forecast Accuracy :

Using the three-week moving average calculations in Table 5, the values for these three measures of forecast accuracy are:

## Weighted moving Average

In the moving averages method, each observation in the moving average calculation receives the same weight. One variation, known as weighted moving averages, involves selecting a different weight for each data value and then computing a weighted average of the most recent k values as the forecast. In most cases, the most recent observation receives the most weight, and the weight decreases for older data values. Let us use the gasoline sales time series to illustrate the computation of a weighted three-week moving average. We assign a weight of 3/6 to the most recent observation, a weight of 2/6 to the second most recent observation, and a weight of 1/6 to the third most recent observation. Using this weighted average, our forecast for week 4 is computed as follows:

## Forecast for week 4 = 1/6(17) + 2/6(21) + 3/6(19) = 19.33

Note that for the weighted moving average method the sum of the weights is equal to 1. Forecast accuracy To use the weighted moving averages method, we must first select the number of data values to be included in the weighted moving average and then choose weights for each of the data values. In general, if we believe that the recent past is a better predictor of the future than the distant past, larger weights should be given to the more recent observations. However, when the time series is highly variable, selecting approximately equal weights for the data values may be best. The only requirement in selecting the weights is that their sum must equal 1. To determine whether one particular combination of number of data values and weights provides a more accurate forecast than another combination, we recommend using MSE as the measure of forecast accuracy. That is, if we assume that the combination that is best for the past will also be best for the future, we would use the combination of number of data values and weights that minimizes MSE for the historical time series to forecast the next value in the time series.

## Exponential Smoothing

Exponential smoothing also uses a weighted average of past time series values as a forecast; it is a special case of the weighted moving averages method in which we select only one weight-the weight for the most recent observation. The weights for the other data values are computed automatically and become smaller as the observations move farther into the past. The exponential smoothing equation follows.

Ft+1 = Î±Yt+(1 +Î±)Ft

Ft+1 = forecast of the time series for period t +1

Yt =actual value of the time series in period t

Ft= forecast of the time series for period t

Î± =smoothing constant (0â‰¤ Î±â‰¤ 1)

Above Equation shows that the forecast for period t + 1 is a weighted average of the actual value in period t and the forecast for period t. The weight given to the actual value in period t is the smoothing constant Î± and the weight given to the forecast in period t is 1 - Î±. It turns out that the exponential smoothing forecast for any period is actually a weighted average of all the previous actual values of the time series. Let us illustrate by working with a time series involving only three periods of data: Y1, Y2, and Y3.

To initiate the calculations, we let F1 equal the actual value of the time series in period 1; that is, F1 =Y1. Hence, the forecast for period 2 is :

F2 = Î±Y1 + (1 - Î±)F1

= Î±Y1 + (1 - Î±)Y1 =Y1

We see that the exponential smoothing forecast for period 2 is equal to the actual value of the time series in period 1.

The forecast for period 3 is

F3 = Î±Y2 +(1 -Î±)F2 =Î±Y2 +(1 -Î±)Y1

Finally, substituting this expression for F3 in the expression for F4, we obtain ,

F4 = Î±Y3 +Î±(1-Î±)Y2 +(1 -Î±)2Y1

We now see that F4 is a weighted average of the first three time series values. The sum of the coefficients, or weights, for Y1, Y2, and Y3 equals 1. A similar argument can be made to show that, in general, any forecast Ft_1 is a weighted average of all the previous time series values.

Despite the fact that exponential smoothing provides a forecast that is a weighted average of all past observations, all past data do not need to be saved to compute the forecast for the next period. In fact, above equation shows that once the value for the smoothing constant Î± is selected, only two pieces of information are needed to compute the forecast: Yt, the actual value of the time series in period t, and Ft, the forecast for period t.

## Simple Linear Regression Analysis

Regression analysis is a statistical technique that model the relationship between two or more variable. Here , the variable you wish to predict is called the dependent variable. The variables used to make the prediction are called independent variables. In addition to predicting values of the dependent variable, regression analysis also allows you to identify the kind of mathematical relationship between a dependent variable and an independent variable, to quantify the effect that changes in the independent variable have on the dependent variable, and to identify unusual observations. In this chapter ,we discuss simple Linear Progression in which a single numerical independent variable, X, is used to predict the numerical dependent variable Y.

Simple Regression Analysis Equation

## Linear Regression

Yi = a + bXi

where:

Y = dependent variable

X = independent variable

a = Y-intercept of the line

b = slope of the line

Let us understand it with the help of a company Sunflowers Apparel.

Scenario :

The sales for Sunflowers Apparel, a chain of upscale clothing stores for women, have increased during the past 12 years as the chain has expanded the number of stores. Until now, Sunflowers managers selected sites based on subjective factors, such as the availability of a good lease or the perception that a location seemed ideal for an apparel store. As the new director of planning, you need to develop a systematic approach that will lead to making better decisions during the site-selection process. As a starting point, you believe that the size of the store significantly contributes to store sales, and you want to use this relationship in the decision-making process. How can you use statistics so that you can forecast the annual sales of a proposed store based on the size of that store?

The business objective of the director of planning is to forecast annual sales for all new stores, based on store size. To examine the relationship between the store size in square feet and its annual sales, data were collected from a sample of 14 stores as shown in table 6

## Source : Statistics for Business and Economics, ch 18 Time Series Analysis and Forecasting

Figure 6 displays the scatter plot for the data in Table 6. Observe the increasing relationship between square feet (x) and annual sales (y). As the size of the store increases, annual sales increase approximately as a straight line.

## Scatter Diagram for Sunflower Apparel Data

Excel and Minitab simple linear regression models for the Sunflowers Apparel data

## Source : Statistics for Business and Economics, ch 18 Time Series Analysis and Forecasting

In above figure ,that b0 =.9645 and b1=1.6999. Using equation of Linear Regression ,the prediction for these data is Yi = .9645 + 1.6699Xi.

The slope b1 is +1.699. This means that for each increase of 1unit in X ,the predicted value of Y is estimated to increase by 1.699 units. In other words, for each increase of 1.0 thousand square feet in the size of the store, the predicted annual sales are estimated to increase by 1.6699 millions of dollars. Thus, the slope represents the portion of the annual sales that are estimated to vary according to the size of the store. The Y intercept, b0 is .9645. The Y intercept represents the predicted value of Y when X equals 0. Because the square footage of the store cannot be 0, this Y intercept has little or no practical interpretation. interpretation. Moreover, the Y intercept for this example is outside the range of the observed values of the X variable, and therefore interpretations of the value of should be made cautiously.

## Conclusion

We see that forecasting has become a important tool for any organization to excel. Predicting the future (forecasting) is must for success today's competitive economy. From manufacturing and inventory to pricing and finance, all organization prepared for the future if it has a better understanding today of what may happen tomorrow. Forecasting allows companies to reduce costs. For Instance, a company may hold fewer inventories, hire fewer people, or build fewer plants if management knows what the future holds. Companies may increase revenues by optimizing manufacturing capacity, making better decisions that are closer to the customer base, or by improving the efficiency and productivity of their marketing budget to drive volume and profits. Further, companies may use forecasting to understand the effects of today's activities on future results. For example, a company may predict the effects of promotions, capital investments, or economic shifts. Organizations normally adhere to a variety of forecasting philosophies such as top-down, bottom-up, straight-line, or accelerated (to name a few). While there is no crystal ball, it is paramount to understand the options that are available. In fact, some companies conduct forecasting by utilizing different methods concurrently to verify the results. If forecasts created by different methods cluster around a certain number (the target), confidence normally builds around that figure. Some organizations launch forecasts from the lowest level. This bottom-up based approach takes individual outlets, and accumulates sales or production through every channel and division, all the way up through the corporate hierarchy. Other companies utilize a top-down based approach. Starting with the highest echelons of the organization, forecasts are decomposed across the various divisions and channels. Each individual forecast project can be tackled by using one or more statistical methods.

## Statistical Method Employed

By utilizing a statistical analysis a Organization can develop forecasting models. An organization can choose the actual methods based on either the type of data that is available, or by the type of information the business requires. Some of the more popular methods available are:

Regression analysis can predict the outcome of a given key business indicator (the dependent variable) based on the interactions of other related business drivers (the explanatory variables).

Trend analysis relies on determining trends in the time-series to predict future results.

Exponential smoothing uses a weighted average of past and current values, adjusting the weight on current values to account for the effects of fluctuation in the data (such as seasonality). Using an alpha term (in-between 0 and 1) the method allows adjusting the sensitivity of the smoothing effects. This model is often used for large-scale forecasting projects, as it is both robust and easy to apply.

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