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
S Regression Statistics
Multiple R
0.778
R Square
0.605
A Adjusted R Square
0.530
Standard Error
0.471
Observations
20.000
M ANOVA P Regression
Residual
L Total
df 3.000 16.000 19.000
E Log(Y) Intercept
(X2) Discharge rate
R (X3) Depth of discharge E (X4) Temperature
Coefficients 1.352 -0.134 -0.003 0.050
Figure 6: Initial Regression Analysis for Log(Y) and X2, X3, X4
SS 5.429 3.551 8.980
Standard Error 0.510 0.077 0.007 0.011
MS 1.810 0.222
t Stat 2.651 -1.730 -0.399 4.584
F-value is moderately high
P-value is moderately low
P-value
F
Significance F
8.154
0.002
VIF
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
from Minitab
0.017
0.271
2.434
0.271
2.434
0.103
-0.298
0.030
-0.298
0.030
1.039
0.695
-0.018
0.012
-0.018
0.012
1.043
0.000
0.027
0.073
0.027
0.073
1.005
The P-value for depth of discharge is high, which indicates that we may want to discard.
The P-value for discharge rate is also high, which indicates that we may want to discard.
PO The regression equation is: RT Log Cycles to Failure = 1.35 - 0.134 Discharge Rate - 0.00285 Depth of Discharge + 0.0497 Temperature
6
The observations resulting from the initial regression results above are as follows:
- The variance inflation factors (VIF) values obtained from Minitab for each independent
variable are all in the range of 1, so there is little to no colinearity among independent
variables and the estimates for coefficients are considered stable.
- The R-Squared = 60.5%, which is moderately high. Ultimately, we would like a higher R-
Squared value, indicating increased "goodness of fit" for the model. - The Durbin-Watson statistic = 2.02425, indicating very little to no presence of auto-
correlation among observations. - The critical F-value is moderately high, but not significantly high. Ultimately, we would
S prefer a higher F-value. - The statistical significance, or P-value is low, but not extremely low. Ultimately, we would prefer a lower P-value that is closer to zero. A- When looking at the individual P-values for the independent variables, that X2 and X3 have high P-values. In particular, the P-value for X3 is very high. This indicates that we M may want to consider discarding X3 from the model. - The residual analysis appears to support the assumption of normality for residuals. - The normal probability plot of the residuals shows some deviation from normality. However, deviations do not invalidate the assumption of normality for the residuals. P - We do see a high P-value for the residuals probability plot (0.243), which indicates goodness of fit for normality test. - There is 1 influential observation (outlier) identified within the probability plot for the L residuals. This observation may require review or possible removal. - There is no apparent heteroscedasticity in the plot of the residual versus fitted values E for Log(Y). So, there is evidence to support constant variance in residuals. The following figures support the observations listed above:
Figure 7: Residual Plots for Log (Y)
R Residual Plots for Log Cycles to Failure
E Normal Probability Plot
99
Versus Fits
0.6
P 90
0.3
50
Residual
O 10
1
-1.0
-0.5
0.0
0.5
1.0
R Residual
0.0 -0.3 -0.6
1.0
1.5
2.0
2.5
3.0
Fitted Value
Percent
T Histogram
Versus Order
0.6 4.8
Residual
Frequency
0.3 3.6
0.0
2.4 -0.3
1.2 -0.6
0.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Residual
2 4 6 8 10 12 14 16 18 20 Observation Order
7
Figure 8: Probability Plot of Residuals
Probability Plot of RESI5
Normal - 95% CI
99
Percent
95 90
S 80
70
A60 50 40 M30 20
10
P 5
L 1
-1.5
-1.0
-0.5
0.0
0.5
1.0
ERESI5
Mean StDev N AD P-Value
-4.99600E-16 0.471 20 0.654 0.243
Figure 9: Plot of Residuals versus Fitted Values for Log(Y)
R Residuals Versus Fitted Values E 0.50 P 0.25 O 0.00 RT -0.25
RESI5
-0.50
-0.75
1.0
1.5
2.0
2.5
3.0
FITS5
8
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