V Bandi and R Lahdelma 1 Forecasting

V Bandi and R Lahdelma 2 Forecasting? Decision-making deals with future problems -Thus data describing future must be needed Representation of what occurs in future Time Operative decisions Tactical decisions Strategic decisions HoursDaysWeeksMonthsYear5 Years20-50 Years

V Bandi and R Lahdelma 3 Time horizons of Forecast Depending on the purpose, the time horizon may differ -Operational planning Day – week level -Tactical planning Week – month– year -Strategic planning Year – 10 year – 50 years

V Bandi and R Lahdelma 4 Requirments of forecasting model Sufficient accuracy -Depends on purpose of the forecast Operative decisions requires high degree of accracy Necessary Input data availability -Having access to real data is always a challenge Model must be easy to update and maintain the model -when the system changes -Not overly complex and specialized

V Bandi and R Lahdelma 5 Different approaches to forecasting Theory-oriented -The laws of physics determine how the system behaves; therefore the model is formed based on theoretical laws Example: Heat is transferred through radiation, conduction and convection... Data-oriented -History data is analyzed in order to find out dependencies Requires applied mathematical techniques

V Bandi and R Lahdelma 6 Different approaches to forecasting In practice it is wise to use both (theortical and data- oriented) approaches together -forecast model structure is planned based on theory but the parameters are estimated from history data -Sometimes observing the data can reveal dependences that are otherwise missed in theoretical analyses -Understanding the laws of physics allows making the model more generic and accurate

V Bandi and R Lahdelma 7 Let us try some simple forecast models

V Bandi and R Lahdelma 8 Forecasting demand for Cars The demand for Toyota cars over first six months in helsinki region is summarized in following table. Forcast the demand for car in next 6 months.

V Bandi and R Lahdelma 9 Forecast demand for cars Simple modeling techniques -Based on a averages, weighing averages In the example, dependency between month and net unit of sales is hard to identify -It is very difficult to forecast accurately

V Bandi and R Lahdelma 10 Forecasting applications in Engineering Planning and optimization -example: coordination of cogeneration Simulation -Planning new systems -Improving existing systems -To understand the behavior of systems

V Bandi and R Lahdelma 11 Forecasting methods Based on averages -Moving averages Smoothing techniques Regression -Linear regression In simplest form: Y = aX+b Y dependent variable, X independent variable -Non-linear regression -Dynamic regression Neural networks and many more

V Bandi and R Lahdelma 12 Regression analysis A regression analysis is for forecasting one variable from another -we must decide which variable will be independent variable and which is dependent variable Y -This choice is usually motivated by a theory or hypothesis of causality The alleged “cause” is X and the alleged “effect” is Y

V Bandi and R Lahdelma 13 What regression does -A regression analysis produces a straight line that estimates the average value of Y at any specific value of X -Example: Heat demand forecast in a year based on out door temperature y t = a 0 +a 1 x t a 0 = 261 MW a 1 = MW/C o -The curve fits badly at high temperatures therefore it is misaligned also for cold temperatures

V Bandi and R Lahdelma 14 Forecast regression model The model aims to explain the behavior of the unknown quantity y in terms of known quantities x, parameters a and random noise e -y = f(x, a) + e The structure of the model (shape of function f()) can be determined based on theory, based on intuition or by exploring history data - The parameters a are estimated from history data so that the noise e is minimized When the model has a good structure, e is white noise Forecasting models can be classified according to the shape of function f

V Bandi and R Lahdelma 15 Linear Regression model based on one dependent and one independent variable A model where a single dependent variable y is explained by a single independent variable x is fitted to history data -y t = a 0 +a 1 x t, where t= 1,...,T This is a linear equation system with two unknowns -The equation can be solved in the least squares sense (2- norm) -To solve it we augment it with a error variable e t

V Bandi and R Lahdelma 16 Linear Regression model, Determining parameters

V Bandi and R Lahdelma 17 Linear Regression model, Determining parameters

V Bandi and R Lahdelma 18 Generalizations of linear regression Multiple independent (explaining) variables

V Bandi and R Lahdelma 19 Heat Demand Forecast Heat demand depends -Weather Outside temperature, wind, solar radiation, seasons -Building properties -Residents behavior Forecasting requires identification of independent variables

V Bandi and R Lahdelma 20 Heat Demand Forecast Accurate heat demand forecast -Weather, resident behavior, building properties can be considered as independent variables -Forecast modelling with all independent variables requires data Obtaining data is challenging According to previous studies, outside temperature has most influence on heat demand

V Bandi and R Lahdelma 21 Heat demand forecast using Regression based on outside temperature Dependent variable -Heat consumption (historical data) Independent variable -Outside temperature (historical data) Forecasting model y t = a 0 + a 1 x t The curve fits badly at high temperatures, therefore it is misaligned also for cold temperatures

V Bandi and R Lahdelma 22 Standard Deviation (SD) or RMSE (Root- Mean-Squared-Error) The square root of the mean/average of the square of all of the error -The use of SD or RMSE is very common and it makes an excellent general purpose error criteria for forecasts stdev(e) = sqrt(e T e/T)

V Bandi and R Lahdelma 23 Forecast based on outdoor temperature Forecast vs actual for sample week The forecast is on good on average, but does not quite satisfactory, RMSE (Root-Mean-Squared-Error) or standard deviation for annual forecast is 20%

V Bandi and R Lahdelma 24 Forecast based on outdoor temperature Forecast vs actual for sample week RMSE = 20% (out of average demand) -not a good forecast Reason for low accuracy -Outside temperature alone cannot explain heat consumption completely -Outside temperature alone cannot explain heat consumption completely. This can be explained by correlation coefficient between outside temperature and heat consumption

V Bandi and R Lahdelma 25 Correlation coefficient The correlation coefficient is a number between -1 and 1 that indicates the strength of the linear relationship between two variables -Very strong positive linear relationship between X and Y r ≈ 1: -No linear relationship between X and Y. Y does not tend to increase or decrease as X increases. r ≈ 0: -Very strong negative linear relationship between X and Y. Y decreases as X increases r ≈ -1 The sign of r (+ or -) indicates the direction of the relationship between X and Y. The magnitude of r (how far away from zero it is) indicates the strength of the relationship.

V Bandi and R Lahdelma 26 Correlation coefficient

V Bandi and R Lahdelma 27 Correlation between outside temperature and heat consumption for a single building Correlation coefficient for a building r = Strong negative relation ship -Model could have been more accurate if r = -1

V Bandi and R Lahdelma 28 Residents behavior in a building People behavior usually have a rhythm (a strong, regular repeated pattern) Lets hypothesis residents behavior has similar rhythm or on weekdays (Monday to Friday) and weekends (Saturday and Sunday) Let us modify the forecast model using these week rhythms

V Bandi and R Lahdelma 29 Modified Forecast Model

V Bandi and R Lahdelma 30 Forecast based using weekly rhythm RMSE (Root-Mean-Squared-Error) or standard deviation for annual forecast is 13%

V Bandi and R Lahdelma 31 Improving accuracy of the model The weekly rhythm model does not consider that some weeks and days are different -E.g. during holiday seasons, religious holidays etc the demand is different from the normal weekday The days can be classified e.g. working day, Saturday, holiday Possible to include more independent variables -solar radiation, wind speed and direction, cloudiness,... In general these affect the precision only a little History data from multiple years -Weighted regression – recent history can obtain more weight