Regression is A statistical procedure used to find relationships among a set of variables.

Linear regression is graphically depicted using a straight line of best fit with the slope defining how the change in one variable impacts a change in the other. The y-intercept of a linear regression relationship represents the value of the dependent variable when the value of the independent variable is zero.

1. Statistical Modelling for Business Analytics

2. Approaches to demand forecasting Understanding the objective of forecasting Integrate demand planning and forecasting throughout the supply chain. Understanding and identifying customer segment. Identifying the major factors influence the demand forecast Determine the appropriate forecast technique Establish performance and error measures for forecasting. 2

3. COMPONENTS OF A FORECAST • Past demand • Lead time of product replenishment • Planned advertising or marketing efforts • Planned price discounts • State of the economy • Actions that competitors have taken 3

4. Types of Forecasts 4 Moving Average Exponential Smoothing Holt’s Model Time-Series Methods: include historical data over a time interval Forecasting Techniques No single method is superior Delphi Methods Jury of Executive Opinion Sales Force Composite Consumer Market Survey Qualitative Models : attempt to include subjective factors Causal Methods: include a variety of factors Regression Analysis Multiple Regression Winter’s Model Trend Projections

5. Qualitative Methods Delphi Method interactive group process consisting of obtaining information from a group of respondents through questionnaires and surveys Jury of Executive Opinion obtains opinions of a small group of high-level managers in combination with statistical models Sales Force Composite allows each sales person to estimate the sales for his/her region and then compiles the data at a district or national level Consumer Market Survey solicits input from customers or potential customers regarding their future purchasing plans 5

6. Decomposition of a Time-Series Time series can be decomposed into: Trend (T): gradual up or down movement over time Seasonality (S) : pattern of fluctuations above or below trend line that occurs every year Cycles(C): patterns in data that occur every several years Random variations (R): “ blips”in the data caused by chance and unusual situations OBSERVED DEMAND = Systematic Component + Random Component ( Forecast error) 6

7. Decomposition of Time-Series The goal of any forecasting method is to predict the systematic component of demand and estimate the random component. In its most general form, the systematic component of demand data contains a level, a trend, and a seasonal factor Multiplicative model assumes demand is the product of the four components Demand = T * S * C * R Additive model assumes demand is the summation of the four components Demand = T + S + C + R 7

8. Moving Averages 8 Simple moving average = Moving average methods consist of computing an average of the most recent n data values for the time series and using this average for the forecast of the next period.

9. Three-Month Moving Average 9 Month Actual Shed Sales Three-Month Moving Average January 10 February 12 March 13 April 16 May 19 June 23 July 26 (10+12+13)/3 = 11 2 / 3 (12+13+16)/3 = 13 2 / 3 (13+16+19)/3 = 16 (16+19+23)/3 = 19 1 / 3

10. Weighted Moving Averages 10 Weighted moving averages use weights to put more emphasis on recent periods. Weighted moving average =    

11. Weighted Moving Averages 11 Period 3 Last month 2 Two months ago 1 Three months ago 3*last month demand+2* two months ago demand+1*three months ago demand 6 Sum of weights Weights Applied

12. Weighted Three-Month Moving Average 12 Month Actual Sales Three-Month Weighted Moving Average 10 12 13 16 19 23 January February March April May June July 26 [3*13+2*12+1*10]/6 = 12 1 / 6 [3*16+2*13+1*12]/6 =14 1 / 3 [3*19+2*16+1*13]/6 = 17 [3*23+2*19+1*16]/6 = 20 1 / 2

13. Exponential Smoothing Exponential smoothing is a type of moving average technique that involves little record keeping of past data. New forecast = previous forecast + (previous actual –previous forecast) Mathematically this is expressed as: F t = F t-1 + (D t-1 - F t-1 ) F t-1 = previous forecast  = smoothing constant (0<  <1) F t = new forecast D t-1 = previous period actual

14. Exponential Smoothing Qtr Actual Rounded Forecast using  =0.10 1 180 175 2 168 175.00+0.10(180-175)= 175.5 3 159 175.50+0.10(168-175.50)= 174.75 4 175 174.75+0.10(159-174.75)= 173.18 5 190 173.18+0.10(175-173.18)= 173.36 6 205 173.36+0.10(190-173.36)= 175.02 7 180 175.02+0.10(205-175.02)= 178.02 8 182 178.02+0.10(180-178.02)=

15. Exponential Smoothing Qtr Actual Tonnage Unloaded Rounded Forecast using  =0.50 1 180 175 2 168 175.00+0.50(180-175)= 177.50 3 159 4 175 5 190 6 205 7 180 8 182 9 ?

16. Exponential Smoothing with Trend Adjustment( Holt’s model) Simple exponential smoothing - first-order smoothing Trend adjusted smoothing - second-order smoothing Low  gives less weight to more recent trends, while high  gives higher weight to more recent trends. Simple exponential smoothing fails to respond to trends, so a more complex model is necessary with trend adjustment.

17. Example: Compute the adjusted exponential forecast for the first week of march for a firm with the following data. Assume the forecast for the first week of January (F ) as 600 and the corresponding initial trend (T ) as 0. let = 0.1 and =0.2. 17 Month Jan. Feb. Week 1 2 3 4 1 2 3 4 Demand 650 600 550 650 625 675 700 710

18. Solution: first week of jan . F t =  D t-1 +(1-  )(F t-1 + T t-1 ) = 0.1 (650) + 0.9 (600 +0) = 605 T t = (F t – F t-1 )+ (1 - )T t-1 = 0.2(605 - 600)+0.8(0)=1.00 F t+1 = F t + T t = 605+1=606, 18

19. 19 So forecast for first week of march is 644.04, i.e 644 units.

20. Trend- and Seasonality-Corrected Exponential Smoothing (Winter’s Model) Appropriate when the systematic component of demand is assumed to have a level, trend, and seasonal factor Systematic component = ( level+trend )(seasonal factor) Assume periodicity of demand to be p. Obtain initial estimates of level (L ), trend (T ), seasonal factors (S 1 ,…,S p ) using procedure for static forecasting In period t, the forecast for future periods is given by: F t+1 = ( L t +T t )(S t+1 ) and F t+n = (L t + nT t ) S t+n

21. Trend- and Seasonality-Corrected Exponential Smoothing (continued) After observing demand for period t+1, revise estimates for level, trend, and seasonal factors as follows: L t+1 = a (D t+1 /S t+1 ) + (1- a )( L t +T t ) T t+1 = b (L t+1 - L t ) + (1- b ) T t S t+p+1 = g (D t+1 /L t+1 ) + (1- g )S t+1 a = smoothing constant for level b = smoothing constant for trend g = smoothing constant for seasonal factor

22. Regression Analysis: A statistical procedure used to find relationships among a set of variables. Linear regression is graphically depicted using a straight line of best fit with the slope defining how the change in one variable impacts a change in the other. The y-intercept of a linear regression relationship represents the value of the dependent variable when the value of the independent variable is zero.

23. Linear regression In a simple regression analysis the relationship between the dependent variable y and some independent variable x can be represented by a straight line y= a+bx Where, b is the slope of the line a is the y-intercept a = ∑y/ N b = ∑ xy / ∑x 2 23

24. Example: the following data gives the sales of the company for various years. Fit the straight line. Forecast the sales for the year 2016. 24 year 2007 2008 2009 2010 2011 2012 2013 2014 2015 Sales (000) 13 20 20 28 30 32 33 38 43

25. Year Sale (y) Deviation (x) x 2 xy 1 13 -4 16 -52 2 20 -3 9 -60 3 20 -2 4 -40 4 28 -1 1 -28 5 30 6 32 1 1 32 7 33 2 4 66 8 38 3 3 114 9 43 4 16 172 N=9 ∑y= 257 ∑x=0 ∑x 2 =60 ∑ xy = 204 25 a = 28.56, b= 3.4 The equation of the straight line of best fit is y= 28.56 + 3.4 x So, sale for the year 2016 = 28.56 + 3.4 X 5 = 45.56= 45560

26. Logistic regression Logistic regression is a data analysis technique that uses mathematics to find the relationships between two data factors. A logistic regression model predicts a dependent data variable by analyzing the relationship between one or more existing independent variables. For example, logistic regression could be used to predict whether a political candidate will win or lose an election or whether a high school student will be admitted to a particular college. These binary outcomes enable straightforward decisions between two alternatives.

27. Logistic regression formula and model An example of a logistic function formula can be the following. P = 1 ÷ (1 + e^ − (a + bx)) P is the probability of the dependent variable being 1. e is the base of the natural logarithm. a is the intercept or the bias term. b is the coefficient for the independent variable. x is the value of the independent variable.

28. Applications of logistic regression

29. Linear Regression Logistic Regression Linear regression is used to predict the continuous dependent variable using a given set of independent variables. Logistic Regression is used to predict the categorical dependent variable using a given set of independent variables. Linear Regression is used for solving Regression problem. Logistic regression is used for solving Classification problems. In Linear regression, we predict the value of continuous variables. In logistic Regression, we predict the values of categorical variables. In linear regression, we find the best fit line, by which we can easily predict the output. In Logistic Regression, we find the S-curve by which we can classify the samples. Least square estimation method is used for estimation of accuracy. Maximum likelihood estimation method is used for estimation of accuracy. The output for Linear Regression must be a continuous value, such as price, age, etc. The output of Logistic Regression must be a Categorical value such as 0 or 1, Yes or No, etc. In Linear regression, it is required that relationship between dependent variable and independent variable must be linear. In Logistic regression, it is not required to have the linear relationship between the dependent and independent variable. In linear regression, there may be collinearity between the independent variables. In logistic regression, there should not be collinearity between the independent variable.

30. Example of logistic regression A company wants to predict whether customers will purchase a product or not based on their age and income level. The following dataset is provided, where: Age is the customer's age in years. Income is the customer's annual income in thousands of dollars. Purchased is whether the customer purchased the product (1) or not (0). Predict whether a 42-year-old customer with an income of $70,000 will purchase the product. Age Income Purchased 25 30 30 50 35 60 1 40 80 1 45 90 1 50 100 1 55 120 1 60 150 1

31. Example of logistic regression For a 42-year-old customer with an income of $70,000:Plugging in the values into the logistic regression equation:𝑦^=1/1+𝑒−(−10+0.15⋅42+0.08⋅70) y^​ = 1/1+e −(−10+0.15⋅42+0.08⋅70) 𝑦^=1/1+𝑒−(−10+6.3+5.6)=11+𝑒−(1.9) y^​ = 1/1+e −(−10+6.3+5.6) ​ Calculating the value of 𝑦^​ ≈0.87 Therefore, the probability that the customer will purchase the product is approximately 87%.Since 0.87 is closer to 1, the model predicts that the customer will purchase the product.