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Advanced Economics For Engineers ININ 6030 Leemary Berrios Irving Rivera Wilfredo Robles

Cost indexes

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Correlation and Regression Analysis

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Page 1: Cost indexes

Advanced Economics For Engineers ININ 6030

Leemary Berrios Irving Rivera

Wilfredo Robles

Page 2: Cost indexes

Agenda

Correlation and Regression Analysis

Time Series

Cost Index

Page 3: Cost indexes

Correlation and Regression Analysis Definition & Background

In statistics, regression analysis refers to techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables.

Regression analysis helps us understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed.

In statistics, correlation indicates the strength and direction of a linear relationship between two random variables.

In general statistical usage, correlation or co-relation refers to the departure of two random variables from independence.

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Correlation and Regression Analysis

• The earliest form of regression was the method of least squares published by Legendre in 1805, and by Gauss in 1809. Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun.

• Sir Francis Galton was the first who used the term regression analysis. Galton fit a least squares line and used it to predict the son’s height from the father’s height.

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Importance and Applications

Regression can be useful when we have multiple independent variable affecting the dependent variable (e.g. Demand of a product) as a function of other parameters (e.g. interest rates, growth in GNP, housing starts.)

Regression methods continue to be an area of active research. In recent decades, new methods have been developed for

Robust Regression

Time Series and Growth Curves

Bayesian Methods for regression

Regression is widely used and frequently misused

e.g. Relate the shear strength of spot welds with the number of parking spaces.

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Importance and Applications • Design of experiments

It helps to determine the level of each factor in the model

• Forecasting in time series

Linear regression finds a target

• Epidemiology

Early evidence relating tobacco smoking to mortality and morbidity came from studies employing regression

• Finance

The capital asset pricing model uses linear regression as well as the concept of Beta for analyzing and quantifying the systematic risk of an investment.

• Environmental science

Linear regression finds application in a wide range of environmental science applications.

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Glossary

Independent variable ,predictor or regressor

Dependent variable , response.

MSR Mean square regression

MSE Mean square error

ρ Correlation coefficient

Intercept

Slope

^

y

x

1

^

0

^

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Linear Regression Assumptions

Errors are uncorrelated random variables with mean zero and constant variance .

Errors behave normally distributed.

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Simple Linear Regression Equations

2

1

)(

xxSn

i

ixx

))((1

xxyyS i

n

i

ixy

n

i

n

i

i

i

n

i

n

i

iin

i

ii

n

x

x

n

xy

xy

xy

1

2

12

1 1

11

^

1

^

0

)(

*

xy o 1

^

Regression Equation Intercept and Slope

Errors

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Simple Linear Regression Equations Analysis of variance for testing significance of a regression

Source of Variation Sum of Squares

Degrees of Freedom Mean Square Fo

Regression SSR= xyS1

^

1 MSR=SSR/1 E

R

MS

MS

Error SSE= SS T - xyS1

^

n-2 MSE=SSE/(n-2)

Total SST n-1

0:

0:

01

00

BH

H If p-value < .05 reject Ho

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Simple Linear Regression Equations

10 2

2

R

SS

SSR

T

R The coefficient is often used to judge the adequacy of a regression Model. The square correlation between X and Y.

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Correlation

-1<ρ<1

-1 inverse dependency

0 independence

+1 direct relation

General Rules 1. A coefficient of correlation r >.87 or <-.87 will mean a strong relation

between x and Y

2. The effectiveness o the study will depend on the sample size

Hypothesis test Ho: The data is independent (there’s not relation)

Ha: The data is dependent

If p-value < .05 reject Ho

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Correlation

0

100

200

300

400

500

600

0 10 20 30

r=+1

0

100

200

300

400

500

600

0 10 20 30

r=-1

0

5

10

15

20

25

30

0 5 10 15 20 25

r=0

Page 14: Cost indexes

Simple Linear Regression Equations

2/1

00

20

)3)(arctan(arctan

25

1

2

nhhRZ

n

R

nRT

0:

0:

1

0

H

H

2/1)*( Txx

xy

SSS

Sr

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Multiple Linear Regression

With many independent variables we will apply ordinary least squares which is a method to estimate unknown parameters

1

1

1

432

432

432

432

1^

110

^

.

1

1

1

*)*(

....

n

i

i

iiiin

iii

iii

iii

TT

nn

y

y

y

Y

bbbb

bbb

bbb

bbb

X

YXXX

xxy

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Example 1 Make a regression

model for the following

data.

Week Price Sales

1 199 25

2 199 27

3 199 24

4 179 35

5 199 21

6 199 26

7 199 29

8 199 28

9 199 32

10 169 48

11 169 45

12 199 30

13 199 38

14 199 37

15 199 38

16 199 39

17 179 45

18 199 40

19 199 39

20 199 42

Page 17: Cost indexes

Time Series Models

Definition

• Predict a future parameter as a function of past values of that parameter.

• What TSM do is to try to capture past trends and extrapolate them into the future.

• E.G. Demand of a product is a parameter that can be described based on the historical demand reported. So past demand is often a good predictor of future demand.

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Applications/Importance

• Whenever we want to follow the development of some random quantity over time, we are dealing with a Time Series.

• Time series are very common, and are familiar from the general media: charts of stock prices, popularity ratings of politicians, and temperature curves are all examples.

• Whenever somebody uses the word “trend”, you know we are dealing with a time series (Janert, P.K. 2006).

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Equations and Calculations

• Although there are many different time series models, the basic procedure is the same for all.

• We treat in time periods (e.g., months), labeled i=1,2,…,t, where period t is the most recent data observation.

• The actual observation are denoted as A(i) and the forecast for periods t+τ , τ=1,2,…, be represented by f(t+ τ).

• A time series model takes as input the past observations A(i) and generate predictions for future values f(t+ τ).

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Moving Average

The best well-known and most commonly applied smoothing technique is the Moving Average.

The idea is very simple:

only average the last m observations and use this average for all future forecast.

m

iAtF

t

mti 1)(

)(

F(t + τ) = F(t) τ = 1,2…

Page 21: Cost indexes

Example of Moving Average Model: Month

t

Demand

A(t)

Forecast f(t)

m = 3 m = 5

1 10

2 12

3 12

4 11 11.33

5 15 11.67

6 14 12.67 12.0

7 18 13.33 12.8

8 22 15.67 14.0

9 18 18.00 16.0

10 28 19.33 17.4

Observation:

The moving average approach gives equal weight to each of the m most recent observations and no weight to observations older than these.

33.113

121210)3(

F

67.113

111212)4(

F

Using m = 3;

Page 22: Cost indexes

Example: Moving Average with m=3

and m=5

0

5

10

15

20

25

30

0 5 10 15

Month (t)

Demand Demand

Moving Average m=3

Moving Average m=5

Page 23: Cost indexes

Exponential Smoothing

Computes a smoothed estimate as a weighted average of the most recent observation and the previous smoothed estimate, and it works as follows. We compute the smoothed estimate and forecast at time t as

where α is a smoothing constant between 0 and 1 chosen by the user. The best value will depend on the particular data.

)1()1()()( tFtAtF

F(t + τ) = F(t) τ = 1,2…

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Example of Exponential Smoothing with α =

0.2 and α = 0.6 Month

t

Demand

A(t)

Forecast f(t)

α = 0.2 α = 0.6

1 10 ---- ----

2 12 10.00 10.00

3 12 10.40 11.20

4 11 10.72 11.68

5 15 10.78 11.27

6 14 11.62 13.51

7 18 12.10 13.80

8 22 13.28 16.32

9 18 15.02 19.73

10 28 15.62 18.69

The simplest possible initialization method is to set F(1)=A(1)=10 and start the process.

Observation: Lower values of α make the

model more stable, but less responsive. The model will tend to underestimate parameters with an increasing trend and the opposite also.

40.10)2(

)10)(2.01()12)(2.0()2(

)1()1()2()2(

F

F

FAF

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Example: Exponential Smoothing

with α=0.2 and α=0.6

0

5

10

15

20

25

30

0 5 10 15

Month (t)

Demand

Demand

Exponential Smoothingwith α=0.2

Exponential Smoothingwith α=0.6

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Exponential Smoothing with a Linear Trend (Double) • Estimates the smoothed

estimate in a manner similar to exponential smoothing, but also computes a smoothed trend, or slope in the data.

• Specifically designed to track data with upward or downward trends (model assumes it is linear).

• The basic method updates a smoothed estimate F(t) and a smoothed trend T(t) each time a new observation becomes available.

• Where α and β are smoothing constants between 0 and 1 to be chosen by the user.

)()()(

)1()1()]1()([)(

)]1()1()[1()()(

tTtFtf

tTtFtFtT

tTtFtAtF

Page 27: Cost indexes

Example of Exponential Smoothing with Linear Trend, α = 0.2 and β = 0.2

Month

t

Demand

A(t)

Smoothed

Estimate

F(t)

Smoothed

Trend

T(t)

Forecast

f(t)

1 10 10.00 0.00 ----

2 12 10.40 0.08 10.00

3 12 10.78 0.14 10.48

4 11 10.94 0.14 10.92

5 15 11.87 0.30 11.08

6 14 12.53 0.37 12.17

7 18 13.93 0.58 12.91

8 22 16.00 0.88 14.50

9 18 17.10 0.92 16.88

10 28 20.02 1.32 18.03

The simplest initialization method is to set F(1)=A(1) and T(1)=0.

08.0)2(

)0)(2.01()104.10(2.0)2(

)1()1()]1()2([)2(

4.10)2(

)010)(2.01()12(2.0)2(

)]1()1()[1()2()2(

T

T

TFFT

F

F

TFAF

Page 28: Cost indexes

Example: Double Exponential

Smoothing α=0.2 and β=0.2

0

5

10

15

20

25

30

0 5 10 15

Month (t)

DemandDemand

Double ExponentialSmoothing

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Quantitative Measures for evaluating models

The three most common quantitative measures are the mean absolute deviation (MAD), mean square deviation (MSD), and bias (BIAS).

Each of these takes the differences between the forecast and the actual values, f(t)-A(t), and computes a numerical score.

Objective: Find model coefficients

that make MAD and/or MSD small as possible and make BIAS close to zero. Zero BIAS does not mean that the forecast is accurate, only that the errors tend to be balanced high and low.

n

tAtfBIAS

n

tAtfMSD

n

tAtfMAD

n

t

n

t

n

t

11

2

1)()()]()([|)()(|

Page 30: Cost indexes

When to use?

Moving Average • Commonly used with time series data to smooth out short-

term fluctuations and highlight longer-term trends or cycles.

• For example, it is often used in technical analysis of financial data, like stock prices, returns or trading volumes. It is also used in economics to examine gross domestic product, employment or other macroeconomic time series. Many accounting processes and chemical processes fit into this categorization.

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When to use?

Exponential Smoothing

• Stationary data with no trend or seasonality. It is a technique that can be applied either to produce smoothed data for presentation, or to make forecasts.

• Commonly applied to financial market and economic data, but it can be used with any discrete set of repeated measurements. Very common for small samples of data.

Double Exponential Smoothing

• Data with a trend but no seasonality.

• Examples: Tourist arrivals, drugs demand.

Page 32: Cost indexes

Interactive Example Suppose the monthly sales

for a particular product for the past 20 months have been as follows:

Using Minitab run a five-period (m=5) moving average model, an exponential smoothing model with smoothing constant α=0.2, and a double exponential smoothing model with smoothing constants α=0.4 and β=0.2. Determine which model fits better for this data.

Month Sales

1 22

2 21

3 24

4 30

5 25

6 25

7 33

8 40

9 36

10 39

11 50

12 55

13 44

14 48

15 55

16 47

17 61

18 58

19 55

20 60

Page 33: Cost indexes

Results

Page 34: Cost indexes

Cost Indexes Definition: Cost indexes are numerical values that reflect historical

change in engineering cost. They compare cost or price changes between two points in time for a fixed quantity of goods or services. On conclusion cost index are just dimensionless numbers for a given year showing the cost at that time relative to a certain base year.

History: Italian G. R. Carli, devised the index numbers on the 1750; to investigated the effects of the discovery of America on the purchasing power of money in Europe.

Relevance: because prices vary across time due to economic conditions indexes are useful to engineer as a base of reference to evaluate different alternative on a given project since it convert applicable costs on the past to equivalents costs now or in the future. They are mostly use to calculate materials and labor costs. Their more popular use are in construction industries to compare the cost of building now using previous designs and in government agencies to forecast the state of the economy. Indexes Costs are publish on the Engineering News Record.

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Cost Indexes (cont.) Equation: Cc = Cr(Ic/Ir) Where: Cc = present or future or past cost, dollars Cr = original reference cost, dollars Ic = index number at the present or future or past time Ir = index number at the time reference cost was obtained Example: Construction of a 70,000 square foot warehouse is planned for a future

period. Several years ago a similar warehouse was constructed for a unit estimate of $162.50 when the index was 118. The index for the construction period is forecast as 143; at construction time what will be the cost per square foot?

Cc = ? Cr = $162.50 Ic = 143 Ir = 118 Cc = 162.50(143/118) = $196.93/ft²

Page 36: Cost indexes

References Hopp, W.J., Spearman, M.L. (2008). Factory Physics, 3rd Edition,p.415-

430, NY: McGraw-Hill.

Janert, P.K. “Exponential Smoothing.” toyproblems.org. Feb. 2006<http://www. toyproblems.org>.

Marshall, G. "Time-Series Data." A Dictionary of Sociology. 1998. Encyclopedia.com. 9 Sep. 2009 <http://www.encyclopedia.com>.

Montgomery, D.C., Runger, G (2003). Applied Statistics and Probability for Engineers, 3rd Edition, p.391-426.

Newnan, D. G., Lavelle J. P. & Eschenbach, T. G. (2000). Engineering Cost and Cost Estimating. Engineering Economic Analysis (8th Ed.) (pp. 50-51). Texas: Engineering Press.

Ostwald, P. F. (1992). Forecasting. Engineering Cost Estimating (pp. 170-176) (3rd Ed.),. New Jersey: prentice Hall.