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Statistics & Forecasting

 

1. Introduction

1.1 Forecasting Basics

  • Forecasts are usually wrong (Forecasts should contain error measure)

  • Aggregate Forecasts are more accurate

  • The longer the horizon, the lower the accuracy

  • Common Sense Compatibility (Include other known Information)

  • A forecast for one function in a company might not be useful to another function

1.2 Key questions which must be answered

  • what is the purpose of the forecast?

  • what specifically do we wish to forecast?

  • how important is the past in predicting the future?

  • what system will be used to make the forecast

1.3 Forecasting Horizons

  • Long-term: more than 2 years

  • Medium-term: 3 months to 2 years

  • Short-term: 0 to 3 months

2. Definitions

  • Demand Estimation

  • Finding current values of demand for various values of price & the other determinant variables.

  • Is used to evaluate the optimality of current pricing & promotional policies

  • Demand Forecasting

  • Finding values for demand in the future time periods.

  • Is used to plan production, inventories, new product & investment

  • Time Series

    A set of numbers which are observed on a regular, recurring basis. Historical observations are known; future values must be forecasted.

  • Stationary

    Values of a series hover or cluster around a fixed mean, or level.

  • Trend

    Values of a series show persistent movement in one direction, either up or down. Trend may be linear or non-linear.

  • Seasonality

    Values of a series move up or down on a periodic basis which can be related to the calendar.

  • Cycle

    Values of a series move through long term upward and downward swings which are not related to the calendar.

  • Pattern + Noise

    A Time Series can be thought of as two components: Pattern, which can be used to forecast, and Noise, which is purely random and cannot be forecasted.

  • Generating Process

    The "equation" which actually creates the time series observations. In most real situations, the generating process is unknown and must be inferred.

  • Accuracy and Bias

    Accuracy is a measure of how closely forecasts align with observations of the series. Bias is a persistent tendency of a forecast to over-predict or under-predict. Bias is therefore a kind of pattern which suggests that the procedure being used is inappropriate.

  • Focus Forecasting

    develops forecasts by various techniques, then picks the "best" by some measure of forecast error.

  • Fit versus Forecast

    A forecast model which has been "tuned" to fit a historical data set very well will not necessarily forecast future observations more accurately. A more complicated model can always be devised which will fit the old data well --but which will probably work poorly with new observations.

  • Forecast Optimality

    A forecast is optimal if all the actual pattern in the process has been discovered. As a result, all remaining forecast error is attributable to "unforecastable" noise. In more technical terms, the forecast is optimal if the mean squared forecast error equals the variance of the noise term in the long run. Note that an optimal forecast is not necessarily a "perfect" forecast. Some forecast error is expected to occur.

3. Management of Forecasting

3.1 Purpose of the Forecast

Planning

  • Resource Allocation, decision in Marketing, Capital Budgeting, Cash Management, Manpower Planning, Material Procurement and Production Planning

  • Level of detail and precision of forecast should be responsive to the intended purpose of forecast

  • Avoid using a forecast created for one purpose for a significantly different purpose

Control

  • Predicting changes in basic Patterns is an Early Alarming Tool that help us to be more more responsive

3.2 Premises of the Forecast

  • A forecast should be accompanied by an Explicit list of Internal and External premises (assumptions) at the Time of Forecast and Afterward

  • Managers should also recognize Implicit premises (Underlying Factors that could affect forecast)

  • Premises themselves are Quasi Forecasts

  • What is your Internal Premises: e.g. increase sales force with 10% next year)

  • What is your External Premises: e.g. 5% economic growth during the next year)

3.3 Priorities for Forecasting

  • Forecasting all variables is not economic

  • Cost effectiveness decide Priorities of forecasting

  • Effectiveness is a function of usefulness:

    • Accuracy

    • Appropriateness

    • Timelines

    • Importance of the forested Item

  • Cost of forecast

    • Cost of development

    • Data acquisition & Storage

    • Operating & Maintenance Cost

3.4 Performance Measurement

  • Forecast Error

    • Mean Absolute Deviation

    • Mean Percentage of Error

    • Mean absolute percentage of Error

    • Mean Squared Error

  • Forecasting Assessment with respect to data not used in developing the forecast

3.5 Accuracy and Bias Measures

  • Bias

    A forecast is biased if it errs more in one direction than in the other

  • Accuracy

    Forecast accuracy refers to the distance of the forecasts from actual demand ignore the direction of that error.

  • MD

    cancels out the over and under – good measure of bias not accuracy

  • MAD

    fixes the canceling out, but statistical properties are not suited to probability based dss

  • MSE

    fixes canceling out, equivalent to variance of forecast errors, HEAVILY USED statistically appropriate measure of forecast errors

  • RMSE

    easier to interpret. Relative metrics are weighted by the actual demand

  • MPE

    shows relative bias of forecasts

  • MAPE

    shows relative accuracy

3.6 Periodic Review

  • Cost Effectiveness

  • Priorities

  • Premises

  • Competent Personnel

  • Valid reliable Timely Data

  • Correct Forecasting Method

  • Forecasting is communicated effectively

  • Adjusting forecasting process if it is unsatisfactory

  • Forecast fulfills requirements of Stakeholders

  • Corrective Actions are Tekken if any step is not fulfilled

4. Forecasting Methods

Subjective or Qualitative

  • Judgmental

    • Sales force surveys

    • Delphi techniques

    • Jury of experts

  • Experimental

    • Customer surveys

    • Focus group sessions

    • Test Marketing

Objective or Quantitative

  • Causal / Relational

    • Econometric Models

    • Leading Indicators

    • Input-Output models

  • Time Series

    • “Black Box” approach

    • Uses past to predict the future

4.1 Qualitative Forecasting Methods

  • Delphi Method

    forecast is developed by a panel of experts who anonymously answer a series of questions; responses are fed back to panel members who then may change their original responses

  • very time consuming and expensive

  • new groupware makes this process much more feasible

  • Market Research

    panels, questionnaires, test markets, surveys, etc.

  • Product Life-cycle Analogy

    forecasts based on life-cycles of similar products, services, or processes

  • Expert Judgment

    by management, sales force, or other knowledgeable persons

4.2 Causal Models

  • Regression

    mathematical equation relates a dependent variable to one or more independent variables that are believed to influence the dependent variable

  • Econometric Models

    system of interdependent regression equations that describe some sector of economic activity

  • Input-output Models

    describes the flows from one sector of the economy to another, and so predicts the inputs required to produce outputs in another sector

  • Simulation Modeling

Regression Analysis

It describes the response of the forecast (dependant) variable to changes in one or more explanatory (independent) variables

 

Y=a0+a1x1+a2x2+..anxn
  • Estimates aim to minimize “Sum of Least Squares”

 

Demand (Sales) is a Function in:

  • Controllable Variables:

    4 P's: Price, Promotional efforts (advertising), Product Design & Place of Sales (Distribution)

  • Uncontrollable Variables:

    Consumer Incomes, Consumer Tastes & Preferences, the actions of competitors, population, political environment .....

4.3 Time Series Models

  • Moving Averages

    (simple moving average, weighted moving average): forecast is based on arithmetic average of a given number of past data points

  • Exponential Smoothing

    (single exponential smoothing, double exponential smoothing): a type of weighted moving average that allows inclusion of trends, etc.

  • Mathematical Models

    (trend lines, log-linear models, Fourier series, etc.): linear or non-linear models fitted to time-series data, usually by regression methods

  • Box-Jenkins methods:

    autocorrelation methods used to identify underlying time series and to fit the "best" model

Time Series Patterns

  • Random, Horizontal or Stationary (No Pattern)

  • Trend,  Linear (default) or Nonlinear (Product Life-Cycle)

  • Seasonality (Recurring Seasonal Pattern)

    Repetition at Fixed Intervals (days of the week, days of the month, months of the year, quarters of the year)

  • Cyclic

    • Long Term (Longer than One year)

    • Cycles vary in length

  • Combination

  • of trend, seasonal and cyclic

Random Patterns

 

Increasing Linear Trend

 

Curve Linear Trend

 

Seasonal & Linear Trend

Stationary Series

Moving Average

  • Forecast demand by calculating an average of actual demands from a specified number of prior periods

  • Each new forecast drops the demand in the oldest period and replaces it with the demand in the most recent period; thus, the data in the calculation "moves" over time

  • Higher value of N - greater smoothing, lower responsiveness

  • Lower value of N - less smoothing, more responsiveness

  • A large value of N is appropriate if the underlying pattern of demand is stable

  • A smaller value of N is appropriate if the underlying pattern is changing or if it is important to identify short-term fluctuations

Weighted Moving Average

 

At = W1 Dt + W2 Dt-1 +... + Wn Dt-N+1
  • Each historical demand may be weighted differently

  • N = total number of periods in the average

  • Wt = weight applied to period t's demand

  • Sum of all the weights = 1

  • Forecast: Ft+1 = At = forecast for period t+1

Exponential Smoothing

  • New Forecast = Weighted Average of Last Demand and Last Forecast

  • Small a => Stability up

  • Large a => Responsiveness up

Moving Average vs. Exponential Smoothing

Trends

Exponential Smoothing with Linear Trend

 

Regression Analysis

Regression in Excel

  • Formal Regression Analysis”

  1. Tools, Data Analysis, Regression

  2. Input Y Range: Dependent Variable

  3. Input X Range: Independent Variable

  4. Specify where you want Output

  5. Output is Table with Regression Statistics

Trend Tool

  • =Trend (y- range,x- range,x- value)

  • Output: Trend Estimate for “x-value”

  • Y-range: Observed Data (Demand)

  • X-range: Independent Variable (Time)

  • X-value: Value (Date) for which to estimate Y (Demand)

Non Linear Trend

  • Regression

  • Same Procedure as Linear Regression

  • Difference: Change in (Time)             

                              

  • Step 1: calculate the average demand y per period for each year (y) of past data by dividing total demand for the year by the number of periods in the year

  • Step 2: divide the actual demand Dy,t for each period (t) by the average demand y per period (calculated in Step 1) to get a seasonal factor fy,t for each period; repeat for each year of data

  • Step 3: calculate the average seasonal factor t for each period by summing all the seasonal factors fy,t for that period and dividing by the number of seasonal factors

  • Step 4: determine the forecast for a given period in a future year by multiplying the average seasonal factor t by the forecasted demand in that future year

Actual Demand 

Year 

 Q1  

Q2  

Q3    

Q4 

Total 

Avg.   

1

100

 70

60

90 

320

80

2

120

80

 70

 110

380

95

3

134

80

70

100

381

96

 

Seasonal Factor 

Year 

 Q1  

Q2  

Q3    

Q4 

1

1.25

 0.875

0.75

1.125

2

1.26

0.84

0.74

1.16

3

1.4

0.83

0.73

1.04

Avg. Seasonal Factor

 

1.3

0.85

0.74

1.083

  • Seasonal Factor - the percentage of average quarterly demand that occurs in each quarter.

  • Annual Forecast for year 4 is predicted to be 400 units.

  • Average forecast per quarter is 400/4 = 100 units.

  • Quarterly Forecast = avg. forecast seasonal factor.

  • Q1: 1.303(100) = 130

  • Q2: .85(100) = 85

  • Q3: .74(100) = 74

  • Q4: 1.083(100) = 108

Error Measures

  • Mean Absolute Deviation MAD

  • Mean Square Deviation MSD

  • Bias

 

Criteria for selecting a forecasting method

Objectives: 1. Maximize Accuracy and 2. Minimize Bias
Select the method that:

  1. gives the smallest bias, as measured by cumulative forecast error; or

  2. gives the smallest mean absolute deviation (MAD); or

  3. gives the smallest tracking signal; or

  4. supports management's beliefs about the underlying pattern of demand

or both accuracy and bias should be used together.

What about the number of periods to be sampled?

  • if demand is inherently stable, low values of a and higher values of N are suggested

  • if demand is inherently unstable, high values of a and and lower values of N are suggested

  • Demand Predictions are dependent on Life Cycle

Decisions Made During PLC

  • Product Development

Analysis

  • Development Effort

Delphi / expert

  • Market Entry

Comparisons

  • Product Specs

QFD

  • Product Introduction

Analysis

  • facility size

market tests

  • supply chain design

consumer survey life cycle analysis

  • Growth

Analysis

  • capacity expansion

Causal Models

  • statistical tech

 

  • production planning

Simulation

  • promotions

 
  • Steady State

Analysis

  • production planning

time series

  • inventory models

causal models