Cost indexes
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The document provides an overview of correlation and regression analysis, time series models, and cost indexes. It defines correlation, regression analysis, and their importance and applications. It discusses simple linear regression equations, assumptions, and hypothesis testing. It also covers multiple linear regression, moving averages, exponential smoothing, and quantitative measures for evaluating time series models. The document is serving as the agenda for the Advanced Economics for Engineers course taught by Leemary Berrios, Irving Rivera, and Wilfredo Robles.
![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.
F (t ) A(t ) (1 )[ F (t 1) T (t 1)]
T (t ) [ F (t ) F (t 1)] (1 )T (t 1)
f (t ) F (t ) T (t )
• Where
α and β are
smoothing
constants
between 0 and 1 to be
chosen by the user.](https://image.slidesharecdn.com/costindexes-131230191939-phpapp02/85/Cost-indexes-26-320.jpg)
![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.
F (2) A(2) (1 )[ F (1) T (1)]
F (2) 0.2(12) (1 0.2)(10 0)
F (2) 10.4
T (2) [ F (2) F (1)] (1 )T (1)
T (2) 0.2(10.4 10) (1 0.2)(0)
T (2) 0.08](https://image.slidesharecdn.com/costindexes-131230191939-phpapp02/85/Cost-indexes-27-320.jpg)
![Quantitative Measures for evaluating
models
The three most common
Objective:
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.
t 1| f (t ) A(t ) |
n
MAD
n
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.
t 1[ f (t ) A(t )]2
n
MSD
n
n
BIAS
t 1
f (t ) A(t )
n](https://image.slidesharecdn.com/costindexes-131230191939-phpapp02/85/Cost-indexes-29-320.jpg)
Cost indexes
- 1. Advanced Economics For Engineers ININ 6030 Leemary Berrios Irving Rivera Wilfredo Robles
- 2. Agenda Correlation and Regression Analysis Time Series Cost Index
- 3. 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.
- 4. 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.
- 5. 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.
- 6. 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.
- 7. Glossary x ^ y MSR MSE ρ Independent variable ,predictor or regressor Dependent variable , response. Mean square regression Mean square error Correlation coefficient ^ 0 Intercept ^ 1 Slope
- 8. Linear Regression Assumptions Errors are uncorrelated random variables with mean zero and constant variance . Errors behave normally distributed.
- 9. Simple Linear Regression Equations Regression Equation Intercept and Slope ^ y o 1 x ^ 0 y 1 x n n Errors n S xx ( xi x) ^ 1 2 S xy ( yi y )( xi x) i 1 i 1 y * x i i 1 i i i 1 n n n i 1 n yx n ( xi ) 2 i 1 n xi2 i 1 i
- 10. 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 1 MSR=SSR/1 MS R MS E n-2 n-1 MSE=SSE/(n-2) ^ Regression SSR= 1 S xy ^ Error Total H 0 : 0 0 H1 : B0 0 SSE= SS T - 1 S xy SST If p-value < .05 reject Ho
- 11. Simple Linear Regression Equations R2 SS R SST 0 R2 1 The coefficient is often used to judge the adequacy of a regression Model. The square correlation between X and Y.
- 12. 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
- 14. Simple Linear Regression Equations H0 : 0 H1 : 0 T0 R n2 1 R2 n 25 Z 0 (arctan hR arctan h 0 )(n 3)1/ 2 r S xy 1/ 2 ( S xx * SST )
- 15. Multiple Linear Regression With many independent variables we will apply ordinary least squares which is a method to estimate unknown parameters y x .... x ^ 0 1 1 n n ^ ( X T * X ) 1 X T * Y 1 1 X 1 bin yi1 y Y i1 . yn1 bi 2 bi 3 bi 2 bi 3 bi 2 bi 3 bi 2 bi 3 bi 4 bi 4 bi 4 bi 4
- 16. Example 1 Make a regression model for the following data. Week Price 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sales 199 199 199 179 199 199 199 199 199 169 169 199 199 199 199 199 179 199 199 199 25 27 24 35 21 26 29 28 32 48 45 30 38 37 38 39 45 40 39 42
- 17. 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.
- 18. 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).
- 19. 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).
- 20. Moving Average The best well-known and most commonly applied smoothing technique is the Moving Average. F (t ) t i t m 1 The idea is very simple: only average the last m observations and use this average for all future forecast. F(t + τ) = F(t) A(i ) m τ = 1,2…
- 21. Example of Moving Average Model: Month t Demand A(t) Using m = 3; Forecast f(t) m=3 m=5 1 10 F (3) 2 12 F (4) 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 10 12 12 11.33 3 12 12 11 11.67 3 Observation: The moving average approach gives equal weight to each of the m most recent observations and no weight to observations older than these.
- 22. Example: Moving Average with m=3 and m=5 30 25 Demand 20 15 Demand 10 Moving Average m=3 5 Moving Average m=5 0 0 5 10 Month (t) 15
- 23. 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 F (t ) A(t ) (1 ) F (t 1) F(t + τ) = F(t) τ = 1,2… where α is a smoothing constant between 0 and 1 chosen by the user. The best value will depend on the particular data.
- 24. 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 The simplest possible initialization method is to set F(1)=A(1)=10 and start the process. 1 10 ---- ---- 2 12 10.00 10.00 F (2) (0.2)(12) (1 0.2)(10) 3 12 10.40 11.20 F (2) 10.40 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 F (2) A(2) (1 ) F (1) 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.
- 25. Example: Exponential Smoothing with α=0.2 and α=0.6 30 25 Demand Demand 20 15 Exponential Smoothing with α=0.2 10 5 Exponential Smoothing with α=0.6 0 0 5 10 Month (t) 15
- 26. 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. F (t ) A(t ) (1 )[ F (t 1) T (t 1)] T (t ) [ F (t ) F (t 1)] (1 )T (t 1) f (t ) F (t ) T (t ) • Where α and β are smoothing constants between 0 and 1 to be chosen by the user.
- 27. 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. F (2) A(2) (1 )[ F (1) T (1)] F (2) 0.2(12) (1 0.2)(10 0) F (2) 10.4 T (2) [ F (2) F (1)] (1 )T (1) T (2) 0.2(10.4 10) (1 0.2)(0) T (2) 0.08
- 28. Example: Double Exponential Smoothing α=0.2 and β=0.2 30 25 Demand 20 Demand 15 10 Double Exponential Smoothing 5 0 0 5 10 Month (t) 15
- 29. Quantitative Measures for evaluating models The three most common Objective: 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. t 1| f (t ) A(t ) | n MAD n 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. t 1[ f (t ) A(t )]2 n MSD n n BIAS t 1 f (t ) A(t ) n
- 30. When to use? Moving Average • Commonly used with time series data to smooth out shortterm 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.
- 31. 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.
- 32. Interactive Example Suppose the monthly sales for a particular product for the past 20 months have been as follows: Using Minitab run a fiveperiod (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
- 33. Results
- 34. 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.
- 35. 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
- 36. 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. 5051). Texas: Engineering Press. Ostwald, P. F. (1992). Forecasting. Engineering Cost Estimating (pp. 170176) (3rd Ed.),. New Jersey: prentice Hall.