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.

V Bandi and R Lahdelma

1

Forecasting


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

Hours Days Weeks Months Year 5 Years 20-50 Years


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


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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


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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


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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


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Let us try some simple forecast models


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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.

Month Number of units

Jan 46

Feb 56

Mar 43

Apr 43

May 60

Jun 72


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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


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Forecasting applications in Engineering

• Planning and optimization- example: coordination of cogeneration

• Simulation - Planning new systems

- Improving existing systems

- To understand the behavior of systems


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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


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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


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What regression does

• 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 yt = a0+a1xt

• a0= 261 MW

• a1= -11.3 MW/Co

- The curve fits badly at high temperatures• therefore it is misaligned also for cold temperatures


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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


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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- yt = a0+a1xt, 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 et


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Linear Regression model, Determining parameters

• We seek for parameters a values that minimize the square sum of the error variables

Min s.t.

• If we introduce the vector/matrix notations,

And


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Linear Regression model, Determining parameters

• The problem in vector/matrix format

• Substituting e into the objective function yields an unconstrained optimization problem

Min (Xa – y)T(Xa – y) = aTXTXa – 2aTXTy + yTy

• Derivative w.r.t to a gives the solution

2XTXa – 2XTy = 0

a = (XTX)-1 XTy


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Generalizations of linear regressionMultiple independent (explaining)

variables• Linear regression model with multiple parameters xi

yt = a0+a1x1,t+a2x2,t+….+anxn,t , where t= 1,...,T

• Now there are more unknown parameters ai and the X-matrix becomes wider

And

• The matrix formulas and solution remain the same

a = (XTX)-1 XTy


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Heat Demand Forecast

• Heat demand depends- Weather

• Outside temperature, wind, solar radiation, seasons

- Building properties

- Residents behavior

• Forecasting requires identification of independent variables


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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


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Heat demand forecast using Regression based on outside temperature

• Dependent variable- Heat consumption (historical data)

• Independent variable- Outside temperature (historical data)

• Forecasting model yt = a0 + a1xt

• The curve fits badly at high temperatures, therefore it is misaligned also for cold temperatures


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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(eTe/T)


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Forecast based on outdoor temperatureForecast 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%


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Forecast based on outdoor temperatureForecast 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


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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.


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Correlation coefficient


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Correlation between outside temperature and heat consumption for a single building

• Correlation coefficient for a building r = -0.956- Strong negative relation ship

- Model could have been more accurate if r = -1


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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


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Modified Forecast Model

• Original forecast model

yt = a0 + a1xt

• y intercept a0 also has a negative on accuracy, as it influences the forecast being a constant

• Modified forecast modelyt = ah(t) + a1xt

Where is a social component based on weekly rhythm


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Forecast based using weekly rhythm

RMSE (Root-Mean-Squared-Error) or standard deviation for annual forecast is 13%


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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

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.

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