Lithium-Ion Battery Analysis for Reliability and Accelerated Testing Using Logistic Regression

Transcription

1 for Reliability and Accelerated Testing Using Logistic Regression Travis A. Moebes, PhD Dyn-Corp International, LLC Houston, Texas

2 Biography Dr. Travis Moebes has a B.S. in Mathematics with Engineering Applications - University of Texas-Austin, a M.S. in Applied Mathematics from Sam Houston State University - Huntsville, and a PhD in Mathematics - Analytic Topology from the University of Houston. He was nominated in 1984 for the Fields Medal (Mathematics, International Union of Mathematicians, WARSAW, POLAND) for his work in applying combinatory to Topology and Geophysics. His over 100 publications includes papers in Mathematics, Geophysics, Computer Science, Education and Reliability Analysis in the American Mathematical Society, IEEE, NASA Journals, and RAMS. He worked for SAIC at NASA for 25 years as a Reliability Engineer. He joined Dyn-Corp International, LLC in 2012 where he works in reliability analysis development for aviation at NASA/JSC/AOD in Houston. ASTR 2013 Oct 9-11, San Diego, CA

3 Purpose Report on BASICS and Methods To Perform Logistic Regression For Obtaining Probabilities of Failures On Lithium-Ion Battery Cells That Have Gone Through Cycle-Down/Up Accelerated Testing Give A Road Map To Follow Good Analysis Practices Through The Steps Of Performing Logistic Regression Using SAS Enterprise Miner (EM) and TIBCO Miner(TM) Demonstrate How Statistical Based Algorithms Improved Accelerated Testing

4 Topics Testing for the significance of the logistic regression model Insightful Miner (IM) and SAS EM Lithium-Ion Battery Cell Test Data Vendor 1, Vendor 2, Vendor 3 Logistic Regression Assessment results for Selection of Vendor

5 The Logistic Regression Model In logistic regression, you model the probability of a binary event occurring as a linear function of a set of independent variables. Logistic regression models are a special type of linear model in which the dependent variable is categorical and has exactly two levels.

6 The Logistic Regression Model Linear Regression Analysis A linear model provides a way of estimating a dependent variable Y, conditional on a linear function of a set of independent variables, X1, X2... Xp. Mathematically, this is written as:

7 The Logistic Regression Model continued In this equation, the terms are the coefficients βi of the linear model; the intercept of the model is β0 and e is the residual. Estimates of the coefficients,, are computed from the training data from which an estimate of the dependent variable, is computed by substituting the estimated coefficients into Equation for Y above. An estimate of the residual, is then the difference between the observed dependent variable and its estimate.

8 The Logistic Regression Model continued Logistic regression is a predictive analysis, like linear regression, but logistic regression involves prediction of a dichotomous dependent variable. The predictors can be continuous or dichotomous, just as in regression analysis, but ordinary least squares regression (OLS) is not appropriate if the outcome is dichotomous.

9 The Logistic Regression Model continued Whereas the OLS regression uses normal probability theory, logistic regression uses binomial probability theory. This makes things a bit more complicated mathematically, so we will only cover this topic fairly superficially (believe me; I'm mixing it with ease!).

10 The Logistic Regression Model continued Chi-square and Logistic Regression Because the binomial distribution is used, we might expect that there will be a relationship between logistic regression and chi-square analysis. It turns out that the 2 X 2 contingency analysis with chi-square is really just a special case of logistic regression, and this is analogous to the relationship between ANOVA and regression

11 The Logistic Regression Model continued Chi-square and Logistic Regression(continued) With chi-square contingency analysis, the independent variable is dichotomous and the dependent variable is dichotomous. We can also conduct an equivalent logistic regression analysis with a dichotomous independent variable predicting a dichotomous dependent variable.

12 The Logistic Regression Model continued Chi-square and Logistic Regression(continued) Logistic regression is a more general analysis, however, because the independent variable (i.e., the predictor) is not restricted to a dichotomous variable. Nor is logistic regression limited to a single predictor. Let's take an example. Coronary heart disease (CHD) is an increasing risk as one's age increases. We can think of CHD as a dichotomous variable (although one can also imagine some continuous measures of this). For this example, either a patient has CHD or not. If we were to plot the relationship between age and CHD in a scatter plot, we would get something that looks like this: (next slide)

13 The Logistic Regression Model continued Chi-square and Logistic Regression(continued) Figure 1: Plot of CHD by AGE

14 The Logistic Regression Model continued Chi-square and Logistic Regression(continued) We can see from the graph that there is somewhat of a greater likelihood that CHD will occur at older ages. But this figure is not very suitable for examining that. If we tried to draw a straight (best fitting) line through the points, it would not do a very good job of explaining the data. One solution would be to convert or transform these numbers into probabilities.

15 The Logistic Regression Model continued Chi-square and Logistic Regression(continued) The y values can only be 0 or 1, so an average of them will be between 0 and 1 (.2,.9,.6 etc.). This average is the same as the probability of having a value of 1 on the y variable, given a certain value of x (notated as P(y xi). So, we could then plot the probabilities of y at each value of x and it would look something like this: (next slide)

16 The Logistic Regression Model continued Chi-square and Logistic Regression(continued) Figure 2 Cumulative Probability Curve For The Logistic Distribution

17 Testing for the significance of the logistic regression model IM and SAS EM TIBCO Miner- continued The p-value for each t-statistic indicates if the corresponding coefficient is significant in the model. In general, if the p-value is less than 0.05 the t- statistic is greater than This suggests that the coefficients are significant. In Figure 3 below, the small p-values for Start implies the term is very significant and the variables Age and Number contribute to the model but less so. Generally, a test for the intercept is uninformative since we rarely expect the regression surface to intersect with the origin.

18 Testing for the significance of the logistic regression model IM and SAS EM TIBCO Miner- continued Figure 3 TIBCO Miner Work flow Diagram

19 Testing for the significance of the logistic regression model IM and SAS EM TIBCO Miner- continued Figure 4 Model Assessment output From TM

20 Testing for the significance of the logistic regression model IM and SAS EM TIBCO Miner- continued In Figure 4 above, the small p-values for Start implies the term is very significant and the variables Age and Number contribute to the model but less so. Generally, a test for the intercept is uninformative since we rarely expect the regression surface to intersect with the origin.

21 Testing for the significance of the logistic regression model IM and SAS EM SAS Enterprise Miner SAS Enterprise Miner uses the Likelihood Ratio Test for Global Null Hypothesis to evaluate the entire model. See Figure 6 below. The Null hypothesis is that are no significant variables in the model whose variation explains the variation in the dependent variable. One may reject the Null Hypothesis if Pr is less than (95% confidence) and Beta = 0. Models with higher Likelihood Ratio Chi-Square statistics are the better models.

22 Testing for the significance of the logistic regression model IM and SAS EM SAS Enterprise Miner Figure 5 : SAS EM Work Flow Diagram

23 Testing for the significance of the logistic regression model IM and SAS EM SAS Enterprise Miner - continued Figure 6: The SAS EM display for the Logistic Regression node SAS EM also displays t-statistics for term importance with p- values

24 Lithium-Ion Battery and Data Fourteen-hundred rows by 52 columns of Electrovaya Cell Acceptance Data were obtained from JSC/EP. A portion of the data appears below in Figure 7 Figure 7 Snapshot of the Vendor 1 Cell Acceptance Data

25 Lithium-Ion Battery and Data Figure 7 Snapshot of the Vendor Cell Acceptance Data ASTR 2013, Oct. 9-11, San Diego, CA

26 Lithium-Ion Battery and Data - continued The LIB No. column refers to the battery, the Module No. refers to the Modules containing the cells, and the Serial column refers to the cells. The P/F column indicates if a particular cell (serial number) passed or failed the vendor test. We assume the possibility of building a Logistic Regression classification model with this data using P/F as the dependent parameter and all other parameters as possible independent parameters.

27 Lithium-Ion Battery and Data - continued Our scope was limited to the Logistic Regression classification for model determination. Other models such as Classification Trees and Classification Neural Networks were not used at this time. Our study was also limited to the above data set. Different data sets and models are planned for the future.

28 The Lithium Battery Lithium-Ion Battery and Data - continued Figure 8 Cells (cell 5) and (cell 4) Damage Due To Debris

29 Lithium-Ion Battery and Data - continued JSC/EP tests showed there are 2 separate internal shorts, both on bottom corners of cell #11577occurred. JSC/EP The short tests at GZ showed affected several there cell fold are layers 2 more separate deeply than internal the more superficial shorts, damage both caused on by bottom short at the opposite corners of on cell # #11577occurred. The short at GZ affected several cell fold layers more deeply than the more superficial damage caused by short at the opposite corner on cell #11577.

30 Analysis and Assessment results Data from five cells selected by JSC/EP were processed through three model options to determine the best model and to indicate a known cell that failed. The output of the best model showed good acceptability statistics and an indication of the failed cell as less acceptable than the other cells. All results were similar in both TM and EM processing and model building.

31 Analysis and Assessment results-continued Option 1 All Rows and Columns As Input A total of 1400 rows of data and all usable columns (53) were used as input to initiate the process of finding a best model. There were 52 independent variables. Option 2. This option was like Option 1, except six independent variables were dropped because of less significant statistical indications based on p-values and t-statistics from Option 1. Option 3. This option was also like Option 1, except deleting the following: Columns with too many blanks, namely the soft short test parameters with the exception of column ACR8C. Non-relevant columns according to JSC Engineering (EP). Relevant columns with good t- Statistics were kept. Categorical Columns. These had less than desirable t-statistics. ASTR 2013, Oct. 9-11, San Diego, CA

32 Analysis and Assessment results-continued R-Square LR Ratio Pr/Beta /0.0 Option 1 Option /0.0 Option /0.0 TM SAS EM SAS EM Figure 8 Model Evaluation Analysis results from both TM and SAS EM.

33 Analysis and Assessment results-continued Model Analysis Results - TM Figure 9 Pass/Fail Output From TM and Option 3. Predict.prob is the model prediction that the cell should pass vendor tests. Predict.class is the model indication of Pass or Fail.

34 Analysis and Assessment results-continued Model Analysis Results SAS EM Figure 10 Pass/Fail Output From SAS EM and Option 3. Predict.prob is the model prediction that the cell should pass vendor tests. Predict.class is the model indication of Pass or Fail.

35 Analysis and Assessment results-continued Our best statistical model showed that cells11577 and in module FGM-91 and battery 1010 had a probability of 0.79 to pass. This was the lowest probability of passing than any other cells in the module. Cell showed very similar results. All other cells showed a 0.90 probability or higher to pass. Acceptance Statistics were good.

36 Analysis and Assessment results-continued Further Logistic Regression Models were applied to more Electrovaya Battery Test Cell Data. More low probability indicators were found and confirmed by hardware testing Electrovaya was dropped as the vendor. Sony (MoliJ) and Motorola (LV) were chosen as the two possible vendors to furnish the Lithium-Ion battery cells. The use of the LR Models were used to cherry pick out the potential cells that may fail without returning to more hardware, non-destructive testing.