SAMPLE Using Linear Regression REPORT
SSatellite Applications Motivated by the Development of a Silver-Zinc Battery A Battery Performance Analysis MPLEUsing Linear Regression REP By: Leslie Gillespie-Marthaler O EMSE 271 RT December 18, 2009
1
Introduction: Satellite manufacturers recently proposed replacing battery technology with a silver-zinc technology. Since satellite applications require reliable and long-lasting battery technology, the manufacturing association requested an analysis of the following:
1. Develop a model for linear regression based on battery performance data, using the Log
of (Cycles to Failure); the model should be based on the best predictors available to
characteristic the behavior of the battery throughout its lifecycle; 2. Perform diagnostic analysis of the fitted model; and 3. Forecast the Cycles to Failure with a 95% confidence interval, using the model for the
following independent variables: X1 = 1.5, X2 = 4.5, X3 = 50, X4 = 25, X5 = 2.
SThe table below provides the original battery performance data provided by the manufacturing
association.
AThe Dependent Variable is:
- Cycles to Failure is the dependent variable (Y)
M - The Log of (Cycles to Failure) is represented as Log(Y)
The Independent Variables are: - Charge Rate (X1)
P - Discharge Rate (X2)
- Depth of Discharge (X3) - Temperature (X4)
L - End of Charge (X5) ETable 1: Original Performance Data
Data 1 2 3 4 5 6 7 8 9 10 11
Cycles to
Failure
Y 101.000 141.000 96.000 125.000 43.000 16.000 188.000 10.000
3.000 386.000 45.000
Log Cycles
to Failure
Log(Y) 2.004 2.149 1.982 2.097 1.633 1.204 2.274 1.000 0.477 2.587 1.653
Charge Rate
(Amps)
X1 0.375 1.000 1.000 1.000 1.625 1.625 1.625 0.375 1.000 1.000 1.000
Discharge
RRate
(Amps) X2
E 3.130
3.130
P 3.130
3.130 3.130
O 3.130
3.130 5.000
R 5.000
5.000
T 5.000
Depth of Discharge
(% of rated amperehours)
X3
60.000 76.800 60.000 60.000 43.200 60.000 60.000 76.800 43.200 43.200 100.000
Temperature (Celsius)
X4 40.000 30.000 20.000 20.000 10.000 20.000 20.000 10.000 10.000 30.000 20.000
End of charge (Volts)
X5 2.000 1.990 2.000 1.980 2.010 2.000 2.020 2.010 1.990 2.010 2.000
12
2.000 0.301
1.625
5.000
76.800
10.000
1.990
13
76.000 1.881
0.375
1.250
76.800
10.000
2.010
14
78.000 1.892
1.000
1.250
43.200
10.000
1.990
15 160.000 2.204
1.000
1.250
76.800
30.000
2.000
16
3.000 0.477
1.000
1.250
60.000
0.000
2.000
17 216.000 2.334
1.625
1.250
43.200
30.000
1.990
18
73.000 1.863
1.625
1.250
60.000
20.000
2.000
19 314.000 2.497
0.375
3.130
76.800
30.000
1.990
20 170.000 2.230
0.375
3.130
60.000
20.000
2.000
2
When initially analyzing the performance data, the following observations were made concerning the Dependent Variable (Y) and its relationship with the Independent Variables (X15):
- There is large variability in the original cycles to failure (Y) data. In the histogram of the
dependent variable (Y), we can see that it is skewed toward the left. This could be
problematic in conducting the regression analysis. - When we conduct a probability plot for this data, the standard deviation is also very
large.
SThese observations are displayed in the histogram and probability plot generated by Minitab
below:
A Figure 1: Histogram of Cycles to Failure (Y)
Histogram of Cycles to Failure
M Normal
5
Mean 112.3
StDev 104.7
N
20
P4
LE 3
Frequency
2 1 0
99 95 90 80
RE -100
0
100
200
300
400
Cycles to Failure
P Figure 2: Probability Plot of Cycles to Failure (Y)
O Probability Plot of Cycles to Failure RT Normal - 95% CI
Mean StDev N AD P-Value
112.3 104.7
20 0.668 0.069
70
Percent
60
50
40
30
20
10 5
1
-300 -200 -100
0
100 200 300 400 500
Cycles to Failure
3
We would prefer a more normalized distribution for the dependent variable. When comparing the original dependent variable (Y) to the Log (Y), we do see some improvement in the distribution, indicating increased normality. The following observations were made when analyzing Log (Y):
- The standard deviation for Log cycles to failure is much smaller, but the P-value has
decreased. - In general, we would prefer to have a larger p-value in order to indicate greater
normality of the distribution. - At this point, it is difficult to discern the greater normality expressed by the Log (Y).
S- For the purposes of this project (and to meet the client's request), we will choose (Log cycles to failure) as the dependent variable for the regression model. Choosing the Log(Y) allows for clear interpretation in that constant changes to Log(Y) translate to Aconstant percentage changes in Y.
These observations are displayed in the histogram and probability plot generated by Minitab
M below:
Figure 3: Histogram of Log Cycles to Failure (Log(Y))
P Histogram of Log Cycles to Failure
Normal
L 7
Mean 1.737
E StDev 0.6875
Frequency
6 5 4 3 2 1 0
0.4
0.8
N
20
REPO 1.2 1.6 2.0 2.4 2.8 3.2 RT Log Cycles to Failure
4
Figure 4: Probability Plot of Log Cycles to Failure (Log(Y))
Probability Plot of Log Cycles to Failure
Normal - 95% CI
99
Mean
1.737
Percent
95
90
80
S 70 60 50 A 40 30 20 M10 5 P1 0
1
2
3
Log Cycles to Failure
StDev N AD P-Value
0.6875 20
1.046 0.007
4
LE Correlation Analysis: In order to determine the best predictors for the regression model, we
completed a correlation analysis of the dependent variable Log(Y) and the independent variables (X1-5). The figure below displays the correlation strengths between the dependent and independent variables.
R Figure 5: Correlation between Log(Y) and X1-5
EP Log(Y)
X1
O X2
X3
R X4 T X5
Log Cycles to Failure
Log(Y) 1 -
0.175377126 -
0.291453599 -
0.068901748 0.718930287
0.101140168
Charge Rate
X1
1 -0.08686 -0.31402 -0.13537 0.007163
Discharge Rate
X2
1 0.191942 -0.00283 0.064439
Depth Discharge
X3
1 0.066934 0.019973
Temp X4
1 0.11434
End of Charge
X5
1
The threshold chosen to indicate significant correlation is (0.19). The highlighted values represent significant correlation. Based on these findings, we should keep the following independent variables as best predictors for the regression model: (X2) Discharge Rate, (X3) Depth of Discharge, and (X4) Temperature.
Initial Regression Analysis: Based on this decision, we then move forward with regression analysis using the informed outcome from the correlation analysis. The results of the initial regression analysis are displayed below.
5
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- multiple linear regression excel template
- multiple linear regression analysis using microsoft
- multiple linear regression excel 2010 tutorial for use
- developing a competitive salary structure
- regression step by step using microsoft excel
- calculating and displaying regression statistics in excel
- simple linear regression open university
- sample using linear regression report
- chapter 14 simple linear regression
- simple linear regression university of sheffield
Related searches
- simple linear regression test statistic
- linear regression coefficients significance
- linear regression test statistic calculator
- linear regression without a calculator
- linear regression significance
- linear regression coefficient formula
- linear regression significance test
- linear regression slope significance testing
- linear regression statistical significance
- linear regression hypothesis example
- simple linear regression hypothesis testing
- simple linear regression null hypothesis