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Module 4. Forecasting MGS3100

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Forecasting

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**Quantitative Forecasting**

--Forecasting based on data and models Casual Models: Price Population Advertising …… Causal Model Year 2000 Sales Time Series Models: Sales1999 Sales1998 Sales1997 …… Time Series Model Year 2000 Sales

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**Causal forecasting Regression**

Find a straight line that fits the data best. y = Intercept + slope * x (= b0 + b1x) Slope = change in y / change in x Best line! Intercept

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**Causal Forecasting Models**

Curve Fitting: Simple Linear Regression One Independent Variable (X) is used to predict one Dependent Variable (Y): Y = a + b X Given n observations (Xi, Yi), we can fit a line to the overall pattern of these data points. The Least Squares Method in statistics can give us the best a and b in the sense of minimizing (Yi - a - bXi)2: Regression formula is an optional learning objective

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**Curve Fitting: Simple Linear Regression**

Find the regression line with Excel Use Function: a = INTERCEPT(Y range; X range) b = SLOPE(Y range; X range) Use Solver Use Excel’s Tools | Data Analysis | Regression Curve Fitting: Multiple Regression Two or more independent variables are used to predict the dependent variable: Y = b0 + b1X1 + b2X2 + … + bpXp

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**Time Series Forecasting Process**

Look at the data (Scatter Plot) Forecast using one or more techniques Evaluate the technique and pick the best one. Observations from the scatter Plot Techniques to try Ways to evaluate Data is reasonably stationary (no trend or seasonality) Heuristics - Averaging methods Naive Moving Averages Simple Exponential Smoothing MAD MAPE Standard Error BIAS Data shows a consistent trend Regression Linear Non-linear Regressions (not covered in this course) R-Squared Data shows both a trend and a seasonal pattern Classical decomposition Find Seasonal Index Use regression analyses to find the trend component

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**Evaluation of Forecasting Model**

BIAS - The arithmetic mean of the errors n is the number of forecast errors Excel: =AVERAGE(error range) Mean Absolute Deviation - MAD No direct Excel function to calculate MAD

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**Evaluation of Forecasting Model**

Mean Square Error - MSE Excel: =SUMSQ(error range)/COUNT(error range) Standard error is square root of MSE Mean Absolute Percentage Error - MAPE R2 - only for curve fitting model such as regression In general, the lower the error measure (BIAS, MAD, MSE) or the higher the R2, the better the forecasting model

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**Stationary data forecasting**

Naïve I sold 10 units yesterday, so I think I will sell 10 units today. n-period moving average For the past n days, I sold 12 units on average. Therefore, I think I will sell 12 units today. Exponential smoothing I predicted to sell 10 units at the beginning of yesterday; At the end of yesterday, I found out I sold in fact 8 units. So, I will adjust the forecast of 10 (yesterday’s forecast) by adding adjusted error (α * error). This will compensate over (under) forecast of yesterday.

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**Naïve Model The simplest time series forecasting model**

Idea: “what happened last time (last year, last month, yesterday) will happen again this time” Naïve Model: Algebraic: Ft = Yt-1 Yt-1 : actual value in period t-1 Ft : forecast for period t Spreadsheet: B3: = A2; Copy down

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**Moving Average Model Simple n-Period Moving Average Issues of MA Model**

Naïve model is a special case of MA with n = 1 Idea is to reduce random variation or smooth data All previous n observations are treated equally (equal weights) Suitable for relatively stable time series with no trend or seasonal pattern

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**Smoothing Effect of MA Model**

Longer-period moving averages (larger n) react to actual changes more slowly

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**Moving Average Model Weighted n-Period Moving Average**

Typically weights are decreasing: w1>w2>…>wn Sum of the weights = wi = 1 Flexible weights reflect relative importance of each previous observation in forecasting Optimal weights can be found via Solver

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**Weighted MA: An Illustration**

Month Weight Data August 17% 130 September 33% 110 October 50% 90 November forecast: FNov = (0.50)(90)+(0.33)(110)+(0.17)(130) = 103.4

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**Exponential Smoothing**

Concept is simple! Make a forecast, any forecast Compare it to the actual Next forecast is Previous forecast plus an adjustment Adjustment is fraction of previous forecast error Essentially Not really forecast as a function of time Instead, forecast as a function of previous actual and forecasted value

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**Simple Exponential Smoothing**

A special type of weighted moving average Include all past observations Use a unique set of weights that weight recent observations much more heavily than very old observations: weight Decreasing weights given to older observations Today

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Simple ES: The Model New forecast = weighted sum of last period actual value and last period forecast : Smoothing constant Ft : Forecast for period t Ft-1: Last period forecast Yt-1: Last period actual value

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**Simple Exponential Smoothing**

Properties of Simple Exponential Smoothing Widely used and successful model Requires very little data Larger , more responsive forecast; Smaller , smoother forecast (See Table 13.2) “best” can be found by Solver Suitable for relatively stable time series

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**Time Series Components**

Trend persistent upward or downward pattern in a time series Seasonal Variation dependent on the time of year Each year shows same pattern Cyclical up & down movement repeating over long time frame Each year does not show same pattern Noise or random fluctuations follow no specific pattern short duration and non-repeating

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**Time Series Components**

Cycle Trend Random movement Time Time Seasonal pattern Trend with seasonal pattern Demand Time Time

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Trend Model Curve fitting method used for time series data (also called time series regression model) Useful when the time series has a clear trend Can not capture seasonal patterns Linear Trend Model: Yt = a + bt t is time index for each period, t = 1, 2, 3,…

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**Pattern-based forecasting - Trend**

Regression – Recall Independent Variable X, which is now time variable – e.g., days, months, quarters, years etc. Find a straight line that fits the data best. y = Intercept + slope * x (= b0 + b1x) Slope = change in y / change in x Best line! Intercept

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**Pattern-based forecasting – Seasonal**

Once data turn out to be seasonal, deseasonalize the data. The methods we have learned (Heuristic methods and Regression) is not suitable for data that has pronounced fluctuations. Make forecast based on the deseasonalized data Reseasonalize the forecast Good forecast should mimic reality. Therefore, it is needed to give seasonality back.

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**Pattern-based forecasting – Seasonal**

Example (SI + Regression) Actual data Deseasonalized data Deseasonalize Forecast Reseasonalize

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**Pattern-based forecasting – Seasonal**

Deseasonalization Deseasonalized data = Actual / SI Reseasonalization Reseasonalized forecast = deseasonalized forecast * SI

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**Seasonal Index What’s an index? Suppose Ratio**

SI = ratio between actual and average demand Suppose SI for quarter demand is 1.20 What’s that mean? Use it to forecast demand for next fall So, where did the 1.20 come from?!

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**Calculating Seasonal Indices**

Quick and dirty method of calculating SI For each year, calculate average demand Divide each demand by its yearly average This creates a ratio and hence a raw index For each quarter, there will be as many raw indices as there are years Average the raw indices for each of the quarters The result will be four values, one SI per quarter

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**Classical decomposition**

Start by calculating seasonal indices Then, deseasonalize the demand Divide actual demand values by their SI values y ’ = y / SI Results in transformed data (new time series) Seasonal effect removed Forecast Regression if deseasonalized data is trendy Heuristics methods if deseasonalized data is stationary Reseasonalize with SI

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Causal or Time series? What are the difference? Which one to use?

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**Can you… describe general forecasting process?**

compare and contrast trend, seasonality and cyclicality? describe the forecasting method when data is stationary? describe the forecasting method when data shows trend? describe the forecasting method when data shows seasonality?

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