Computing Fundamentals 1 Equations and Reduction in CafeOBJ Lecturer: Patrick Browne.
Volume 8 Number 1 COMPUTING PRESSURE REDUCTION …
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Volume 8 Number 1 COMPUTING PRESSURE REDUCTION CONSTANTS USING INDICATOR VARIABLES Bernard Augustine (1) Suitland House 4500 Silver Hill Road Suitland, MD 20746 ABSTRACT (b) Cody, Wyoming (COD) (c) Wendover, Utah (ENV) 2. METHODOLOGY 1. INTRODUCTION There are two reasons for writing this paper: (a) To discuss the indicator variable technique for developing models; y TABLE 1 x 1 1 if the station is BPI a otherwise 1 if the station is COD o otherwise r 3) yl is the dependent var iable (reduc- tion constant) The indicator variables Xl and x2 are defined as: 2) Xl and x2 are the indicator var i- abIes 1) T is the independent var iable (12 hour mean temperature) To illustrate the indicator variable pro- cess, consider the data in Table I where: To use this technique to develop a single regression model that can com- pute a reduction constant for more than one weather station. (b) Since computers are an integral part oE a weather station, more standard procedures can be automa- ted. A procedure is now available Eor calculating sea-level pressure Eor manned and unmanned !Wea ther stations. Each weather station has a unique equa- tion that is used to compute reduction cons tan ts. Sea-level pressure is calculated by multiplying station pressure by the reduction constant (2). This paper discusses an alternative method Eor computing reduction constants. The independent var iables used in regres- sion models are quantitative; i.e., they are actual numerical values. However, many var iables cannot be defined numer ic- ally (e.g., marital status, sex, race, color, shape, religion, etc.). These are qualitative variables and serve to identi- fy class membership. It is necessary to examine a method to quantify the levels of a qualitative variable for regression ana- lysis. The qualitative variables are com- monly referred to as indicator variables and take the values 0 and 1. If a quali- tative variable has n levels, it is repre- sented by n-l indicator variables. For example, if three weather stations are grouped geographically, there will be two indicator variables each being assigned the value a or 1. If this rule is viola- ted, a computational difficulty - multi- collinearity - occurs. It is beyond the scope of this paper to discuss multicol- linearity, but its importance cannot be over emphasized. Serious errors will pro- duce unwanted results. 3. EXAMPLE The follOWing three weather stations are grouped geographically: (a) Big piney, wyoming (BPI) - 50 -40 -)0 -20 -10 o 10 20 )0 40 50 60 70 80 90 -50 -40 -)0 -20 -10 o 10 20 )0 40 50 60 70 80 90 100 -)0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 o 0.)294 0.)228 0.)168 O. )11) 0.)064 0.)020 0.2977 0.29)6 0.2895 0.2848 0.2810 0.2779 0.2751 0.2727 0.2707 0.2)65 0.2)24 0.2284 0.2250 0.2217 0.2186 0.2157 0.2128 0.2100 0.2070 0.2041 0.2016 0.1995 0.1977 0.1959 0.1950 0.1860 7
Transcript of Volume 8 Number 1 COMPUTING PRESSURE REDUCTION …
Volume 8 Number 1
COMPUTING PRESSURE REDUCTION CONSTANTSUSING INDICATOR VARIABLES
Bernard Augustine (1)Suitland House
4500 Silver Hill RoadSuitland, MD 20746
ABSTRACT (b) Cody, Wyoming (COD)
(c) Wendover, Utah (ENV)
2. METHODOLOGY
1. INTRODUCTION
There are two reasons for writing thispaper:
(a) To discuss the indicator variabletechnique for developing models;
y
TABLE 1
x 1
1 if the station is BPIa otherwise
1 if the station is CODo otherwise
r
3) yl is the dependent var iable (reduction constant)
The indicator variables Xl and x2 aredefined as:
2) Xl and x2 are the indicator var iabIes
1) T is the independent var iable (12 hourmean temperature)
To illustrate the indicator variable process, consider the data in Table I where:
To use this technique to develop asingle regression model that can compute a reduction constant for morethan one weather station.
(b)
Since computers are an integral part oE a weatherstation, more standard procedures can be automated. A procedure is now available Eor calculatingsea-level pressure Eor manned and unmanned !Wea therstations. Each weather station has a unique equation that is used to compute reduction cons tan ts.Sea-level pressure is calculated by multiplyingstation pressure by the reduction constant (2).This paper discusses an alternative method Eorcomputing reduction constants.
The independent var iables used in regression models are quantitative; i.e., theyare actual numerical values. However,many var iables cannot be defined numer ically (e.g., marital status, sex, race,color, shape, religion, etc.). These arequalitative variables and serve to identify class membership. It is necessary toexamine a method to quantify the levels ofa qualitative variable for regression analysis. The qualitative variables are commonly referred to as indicator variablesand take the values 0 and 1. I f a qualitative variable has n levels, it is represented by n-l indicator variables. Forexample, if three weather stations aregrouped geographically, there will be twoindicator variables each being assignedthe value a or 1. If this rule is violated, a computational difficulty - multicollinearity - occurs. It is beyond thescope of this paper to discuss multicollinearity, but its importance cannot beover emphasized. Serious errors will produce unwanted results.
3. EXAMPLE
The follOWing three weather stations aregrouped geographically:
(a) Big piney, wyoming (BPI)
- 50-40-)0-20-10
o1020)0405060708090
-50-40-)0-20-10
o1020)0405060708090
100-)0
111111111111111ooooooooooooooooo
ooooooooooooooo1111111111111111o
0.)2940.)2280.)168O. )11)0.)0640.)0200.29770.29)60.28950.28480.28100.27790.27510.27270.27070.2)650.2)240.22840.22500.22170.21860.21570.21280.21000.20700.20410.20160.19950.19770.19590.19500.1860
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National Weather Digest
-20 0 0 0.1826 x4-10 0 0 0.1797
0 0 0 0.1773 x510 0 0 0.175120 0 0 0.1727 X630 0 0 0.170440 0 0 0.1685 x750 0 0 0.167060 0 0 0.1656 x870 0 0 0.1642SO 0 0 0.1623 x990 0 0 0.1620
100 0 0 0.1('13
Winslow, Ar izona INW
Junction, Texas JCT
Mar fa, Texas MRF
Prescott, Arizona PRC
Truth or Consequences TCS
Caliente, Ar izona P38
page, Arizona PGA
0.18606296
-0.00044666
1.8289148E-06
0.02453614
-0.00635929
-0.01649633
0.02090837
-0.11830744
0.01594893
0.02810267
0.01837490
-0.00278536
0.00011796
0.00017853
0.00020567
Regression Coefficients
0.00013467
0.00035055
0.00012151
0.00014068
0.00013679
0.00016779
-9.4330045";-07
-1.1593344,,-06
-1. 3199238E-06
-1.0491946E-06
-1. 5973964E-06
-7. 8462579E-07
Parameter
Intercept
(1.0) X 5
x6
x 7x8
x9TX 1
rX 2rX
3
(1. J)
CMN
DUG
CAO
CALLLETTERS
TABLE 2
X3 Douglas, Arizona
X2 Deming, New Mexico
Xl Clayton, New Mexico
bO • 0.1772102) b1 • 0.000271 78 b2 • 1.1887078" 10-6
;"0 solve reduction constants ~or station Cl./'J, use,
where'
Xo solve reduction constants (or station • i. use,
b) • 0.12456728 b4 - 0.04114))9 b5
• 0.00019448
b6 • 4.7676655- 10- 5 b7
• 4.06)198911 10-7
It is desired to find the best set of independent variables to predict the reduction constants. Therefore, an interactioneffect must be included in the model. Theinteraction effect involves a cross-product term between the independent variablesand the indicator variables. In thiscase, the cross-product terms are TXI'TX2' T2 xl , and T2 x2 . If thecross-product terms are not significant,they are not included in the model. Theregression analysis is done by using thestatistical computer package SAS (Statistical Analysis System).
ro solve reduction constants for station c"v, use,
y. bo - b11 • b2 r1 because both xl and x2 • 0
INDICATORVARIABLES STATION NAME
Table 2 shows the results of five othergroups of weath~r stations. Included arethe station name, call letters, significant variables and their regression coefficients.
The results of fitting the model are:
rX5
rX6
rX7
h SThe constant bO is the intercept.---------------------1 TX 9
2T xl
2r x 22
T x3
2r x42T x
52
T x6
B
Volume 8 Number 1
r 2x7
-1.2479958E-06 '~x3 0.000083382 ~2
r X8
-1.0407030E-06 ' Xl -0.000000302 ,,2
r x9
-9. 6294037E-07 ..: ..~~ m ~~:.??~~??~~'''''''",,''''.''.I ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
ENN
BIG
ANI
COE
FYU
CALLLETTERS
Xs Fort Yukon, Alaska
X3 Cape Decision, Alaska
X4 Nenana, Alaska
Xl Aniak, Alaska
X2 Big Delta, Alaska
INDICATORVARIABLES STATION NAME==~= ~!::f..':c~
INDICATOR CALLVARIABLES STATION NAME LETTERS
Xl Greenville, Maine 6B2
x2 Johnbury, Vermont 9B2
x3 Wor cester , Massachuset ts ORH
Rome, Geor gia RMG
Middleton Island, Alaska MOO
Port Heiden, Alaska PTH
Parameter Regression Coefficient
Intercept 0.00375817
T -5.0274260,';-06
?2 1.6202833;';-08
xl -0.00033310
x 2 0.04926333
x3
-0.00178684
x4 0.01102643
x5
0.01406149
x6 0.06186686
x 70.00262816
x8 -0.00116935
TX2 -0.00010611
TX3
1. 4689853E-06
TX4 -1. 5963286E-05
TX5
-2.0043523E-05
TX6 -0.00012322
TX7 -3. 6784392E-06
'rx8 1. 0552560E-062T x2 2.8712619,,:-072T x4 4.8038654,,-08
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Parameter Regression Coefficient
Intercept 0.02505938
r -3.8256313E-05
r 2 1.2106106E-07
xl 0.01604374
x2 0.01604374
x3
0.01548032
TX1 -1.7049487E-05
rX2 -2. 2802562E-06
TX3
-1.7665513E-05•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• o••• -••••••••
INDICATOR CALLVARIABLES STATION NAME LETTERS
xl Blue Canyon, California BLU
x2 Meacham, Oregon MEH
x3 Mount Shasta, California MHS
x4 Sandberg, california SDB
Sexton Summit, Oregon SXT
Parameter i{egression Soe:ficient
Intercept 0.16275247
or -0.00030873
T2 0.00000073
xl 0.06578299
Xo 0.00936228L
x3
-0.01417618
x4 0.02709380
TX l -0.00009081
X6 Gulkana, Alaska
X7 Illiama, Alaska
GKN
ILl
National Weather Digest
T2X
56.2099541~-08
.:~ ~O ~:.~.~~~:.~~ ~~.~.~ .
INDICATOR CALLVARIABLES STATION NAME LETTERS
Xl Chamberlain, South Dakota CHB
Xz Warroad, Minnesota D45
x3 Marseilles, Il1inois MMO
x4 poplar Bluff, Missouri POZ
x5 Devils Lake, Nor th Dakota Pll
x6 Roseglen, North Dakota PZ4
x7 Medicine Lodge, Kansas P28
x8 pequot Lake, Minnesota P39
x9 Redig, South Dakota REJ
xlD Sidney, Nebraska SNY
xlI valentine, Nebraska VTN
Elkhart, Kansas IRS
FOOTNOTES AND REFERENCES
1. Bernard Augustine is a gradua te of MarshallUniversity, Huntington, West virginia. He hasbeen with the Communications Division since 1967.Mr. Augustine has a wide variety of experiencewith the National Weather Service. Before comingto the Washington, D.C., area, he worked in weather stations at Barter Island and Nome, Alaska;Beckley and Huntington, West Virginia; and Norfolk, Virginia.
2. Augustine, Bernard G, 1981: "Computer Generated Sea-Level Pressure," National Weather Digest,6:2. p.34.
Parameter
Intercept
X3
x4
x5
x6
x 7x8
x 9x 10x 11
'rX3TX4
Tx 5
TX6
rX7
TX8
Tx 10rX 11
Regression Coefficient
0.12261314
-0.00016899
3. 1476108E-07
-0.05258632
-0.07984645
-0.09382217
-0.10797337
-0.06307131
-0.03947410
-0.06161888
-0.07129986
0.00256523
0.05959343
-0.01600669
6.7954851';;-05
9.9562546~-05
0.00011157
0.00012494
8.0444l90E-05
5.2752462E-05
7. 3588753E-05
8.9498260E-05
-8. 5144597il-05
1. 9748260E-05
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