Lukas Felsberger

Dieter Kranzlmüller

Benjamin Todd

Field-Reliability Predictions based on Statistical System Lifecycle Models

Keywords: Reliability Prediction, System Lifecycle, Bayesian Learning

Reliability measures the ability of a system to provide its intended level of service. It is influenced by many factors throughout a system lifecycle. A detailed understanding of their impact often remains elusive since these factors cannot be studied independently. Formulating reliability studies as a Bayesian regression problem allows to simultaneously assess their impact and to identify a predictive model of reliability metrics. The proposed method is applied to currently operational particle accelerator equipment at CERN. Relevant metrics were gathered by combining data from various organizational databases. To obtain predictive models, different supervised machine learning algorithms were applied and compared in terms of their prediction error and reliability. Results show that the identified models accurately predict the mean-time-between-failure of devices -an important reliability metric for repairable systems -and reveal factors which lead to increased dependability. These results provide valuable inputs for early development stages of highly dependable equipment for future particle accelerators.

Introduction

Reliability measures the ability of a system to perform as expected during its intended lifetime. The field-reliability of complex repairable systems is a result of all actions during all stages of its system lifecycle. These stages are (1) conceptual design, (2) detailed design and testing, (3) manufacturing, (4) installation, [START_REF] Denson | The history of reliability prediction[END_REF] operation and maintenance, and (6) phase-out and disposal. At each stage an interplay of complex technical, organizational, and human processes leads to a more or less desirable outcome in terms of system reliability. An assessment of all stages and processes is not feasible, since models capturing the interactions between all relevant processes in system development do not exist. Therefore, most common reliability methods focus on certain stages and This work has been sponsored by the Wolfgang Gentner Programme of the German Federal Ministry of Education and Research (grant no. 05E12CHA) aspects during a system lifecycle, which can be modeled and understood -we provide an overview in section 2. However, such methods struggle to quantify the overall uncertainty of reliability predictions in a systematic way since relevant contributions during a system lifecycle might have been disregarded and are not straight-forward to include. Instead of focusing on models for certain stages and aspects of a system we propose to learn a statistical model of the whole product lifecycle to predict the observed field-reliability with machine learning techniques as depicted in figure 1. For a set of existing comparable systems with known field-reliability so-called quantitative reliability indicators are gathered. Using the reliability indicators as input variables and the field-reliability metric as target variables, a statistical reliability model is learned by a supervised machine learning algorithm. The learned model will always be an approximation of the true underlying system lifecycle processes. The lost accuracy due to the statistical model and the limited granularity of the reliability indicators can be quantified by Bayesian methods. Thereby, the overall predictive certainty can be quantified in an efficient way based on the available data. Fig. 1. Illustration of the proposed approach. The achieved field-reliability (c) can be seen as the result of relevant processes during the whole product lifecycle [START_REF] Barnard | What is wrong with reliability engineering?[END_REF][START_REF] Bishop | Pattern Recognition and Machine Learning (Information Science and Statistics)[END_REF][START_REF] Blondel | Scikit-learn user guide[END_REF][START_REF] Cawley | On over-fitting in model selection and subsequent selection bias in performance evaluation[END_REF][START_REF] Denson | The history of reliability prediction[END_REF]. It is not feasible to capture and model all of the relevant processes. Instead, it is proposed to learn a reduced-order statistical lifecycle model (b) with machine-learning algorithms based on quantitative reliability indicators (a).

We demonstrate that the learned models accurately predict reliability metrics even with a limited set of reliability indicators (as is the case at early stages of a system's lifecycle). Compared to traditional reliability assessment methods, this leads to a reduced workload for reliability predictions and to a systematic quantification of uncertainties. Furthermore by an appropriate choice of reliability indicators and machine learning algorithms one can study the influence of each individual reliability indicator. This information assists engineers in design decisions for highly reliable systems. The rest of the paper is structured as follows: In section 2, we present related methods to reliability predictions. In section 3, we explain the methodology of our approach and in section 4 we apply it to a use-case.

Literature Review

A general review of the challenges in reliability studies is given in [START_REF] Zio | Reliability engineering: Old problems and new challenges[END_REF]. The author of [START_REF] Zio | Reliability engineering: Old problems and new challenges[END_REF] concludes that the two major challenges in reliability studies are complexity and uncertainty. Reliability studies must consider technical, organizational and human factors each of which influences the field-reliability of systems. In the following paragraph a selection of reliability prediction methods to tackle these problems is given.

Reliability Engineering Methods Scientific literature on reliability engineering prediction methods of electronic systems is numerous. An attempt to classify and evaluate the existing methods is given in the IEEE standard 1413 [START_REF] Pecht | The IEEE standards on reliability program and reliability prediction methods for electronic equipment[END_REF][START_REF] Elerath | IEEE 1413: A standard for reliability predictions[END_REF] and its successors. In this standard they have been classified as based on handbooks, stress and damage models (often referred to as physics-of-failure based), and field-data.

Most methods are based on early designs of the considered system and the selected components. A common criticism for handbook based models is that they do not consider interactions of components but only single-component faults. However, faults due to single-component failures are not dominant [18,[START_REF] Denson | The history of reliability prediction[END_REF][START_REF] Foucher | A review of reliability prediction methods for electronic devices[END_REF][START_REF] Barnard | What is wrong with reliability engineering?[END_REF][START_REF] Leonard | How failure prediction methodology affects electronic equipment design[END_REF]. As a result the actual field-reliabilities can deviate from the predicted ones by orders of magnitude [START_REF] Jones | A comparison of electronic-reliability prediction models[END_REF]. The author of [START_REF] Denson | The history of reliability prediction[END_REF] argues that some methods should not be used to predict the field-reliabilities but rather as part of a review process at a stage when limited information on the final design is available. Stress-and damage models are in general more accurate than handbook-based methods. However, the development of such methods requires more effort [18]. Instead of assessing the system on the component level, some approaches use a top-down approach in which the field-reliability of new systems is estimated from field-data of similar systems in operation [START_REF] Gullo | In-service reliability assessment and top-down approach provides alternative reliability prediction method[END_REF][START_REF] Johnson | Improvements in reliability assessment and prediction methodology[END_REF].

Reliability Program Assessment A different approach to evaluate the field-reliability of systems is taken in [START_REF] Miller | Assess your program for probability of success using the reliability scorecard tool[END_REF]. The likelihood of achieving the required field-reliability is estimated by a review of the design processes. Each system is assigned a score depending on its design processes and it is shown that this score correlates with the probability of fulfilling field-reliability requirements. Thereby organizational aspects of reliability are taken into account.

Organizational and Human Reliability Analysis

In the review article [START_REF] Zio | Reliability engineering: Old problems and new challenges[END_REF] section 3.1.3 is dedicated to non-technical factors in reliability studies since its contribution to the field-reliability can be significant.

In our work we propose to infer the most relevant processes or factors in a system lifecycle from the field-reliability data of a set of systems. This includes organizational and human reliability factors. The method can be applied at any stage of a system lifecycle to guide engineering decisions.

Methodology

In this section we define the relevant terms, explain the methods used and describe the general methodology.

Definitions

System Reliability It is generally defined as the ability of a system to provide its intended level of services for a specified time t. For a constant failure rate and repairable systems, it is usually measured as availability A, which is defined by

A = M T BF M T BF + M T T R (1)
with M T BF being the mean-time-between-failure and M T T R being the meantime-to-repair. The M T BF is being calculated as

M T BF = t operation n f aults (2)
with t operation being the cumulative operational time of the considered devices and n f aults being the total number of faults within the operational time. The M T T R can be evaluated by

M T T R = t inrepair n f aults (3)
with t inrepair being the total time a system is in repair and n f aults the total number of faults during the operational time. The un-availability U A is given by

U A = 1 -A.
System Lifecycle It is the overall process describing the lifetime of a system. It is a concept from systems engineering to address all stages of a product from its beginning to end. Here these stages shall be divided into (1) conceptual design, (2) detailed design and testing, (3) manufacturing, (4) installation, (5) operation and maintenance, (6) and phase-out and disposal. 3System Definition This discussion is focused on repairable electronic systems. A more precise definition will be given for the use-case in section 4. 4

Method

The central assumption is that the observed field-reliability is the outcome of all technical, organizational and human processes during all stages of a system's lifecycle. It is unfeasible to model all these interactions due to their complexity and non-linearity. Therefore, we restrict ourselves to learning statistical models of the observed field-reliability of comparable systems based on reliability indicators collected throughout the system lifecycle. Modern machine learning algorithms are capable of learning accurate predictive models of field-reliability based on the relevant reliability indicators. The loss of information due to the limited availability of data and the intrinsic uncertainty of the problem can be assessed by using Bayesian machine learning methods.

Lifecycle Analysis by Machine Learning

To arrive at a firm mathematical description of the proposed method, let us hypothesize the existence of a deterministic model F : Z → Y to determine any field-reliability metric Y ∈ Y from all relevant input variables Z ∈ Z in the form of

Y = F(Z). (4)
This would be a model to quantify the contribution of all relevant processes towards the field-reliability during the whole system lifecycle. Since it is not possible to derive such a formula or to gather all relevant inputs, we try to approximate the true field-reliability metrics Y by a reduced model

Y ≈ y = f (x), (5)
with x ∈ X , dim(X ) dim(Z) being the set of collected reliability indicators and f : x → y, y ∈ Y being an approximate model. When supplied with pairs of input and output data D = {(x 1 , Y 1 ), ..., (x N , Y N )}, a statistical learning algorithm can learn such a model by minimizing a certain loss function l : Y ×Y → R. This is essentially a regression problem which can be studied with a vast range of learning algorithms.

There are three additional requirements, which render algorithms fitter for the intended purpose. Firstly, to quantify the uncertainty of the predictions of the reliability metrics, probabilistic models shall be learned

p(Y|x). ( 6
)
Our method is based on an arbitrary non-linear mapping from reliability indicators to features Φ : X → R n . Since it is of interest which features are relevant, secondly, algorithms of parametric form will yield that additional information,

p(Y|w • Φ(x)), (7)
with w ∈ R n being a weight vector indicating the relevance of each feature. Thirdly, methods learning sparse models based on fewer features are preferred from a practical point of view, since they require a reduced data collection effort for predicting field-reliability. A general justification of such methods on philosophical grounds is given by Occam's razor [START_REF] Gauch | Scientific method in practice[END_REF].

We present concrete algorithms fulfilling these criteria in section 4. Even though the outlined requirements are not mandatory, they facilitate the data collection and model assessment process by providing additional feedback.

Data Collection, Model Selection and Reliability Prediction

The collection of data and the training and selection of a model should be seen as an integrated process. The problem domain and a-priori available expert knowledge allows to draw guidelines for the data collection. We present these guidelines in the paragraphs below. After that, we show how to learn a predictive model with the collected data and how further refinements of the data collection are assisted by properly selected learning algorithms.

Collection of Training Systems

Since the method is based on the field-reliability of existing comparable systems, the choice of the collected systems will have an influence on the accuracy of the predictions for future systems. Two general recommendations can be given for this selection:

-Only systems which have been in use for a significant exploitation period with accurately monitored reliability metrics shall be used. -The choice of systems for which a field-reliability model is learned shall include systems which are comparable to the system for which a field-reliability shall be predicted. In reliability studies, comparable systems are similar in terms of technical, organizational, and human factors throughout their lifecycle.

Collection of Reliability Indicators

The choice of these indicators largely influences the quality of the models in terms of their accuracy and interpretability.

The following statements can be made:

-Based on expert knowledge, recommendations can be given for indicators which carry important reliability information; e.g. operational conditions such as load, temperature or humidity can contribute significantly to the failure rate. Systems, which are mass-produced, will achieve different fieldreliabilities than prototypes. Different organizational structures or project management strategies influence the reliability of a final system. [START_REF] O'connor | Practical reliability engineering[END_REF][START_REF] Womack | Machine that changed the world[END_REF].

-In engineering practice the collection of data is facing practical limitations due to time or other restrictions. Therefore, a natural choice is to begin to collect the indicators based on a trade-off between collection effort and expected information content. For the use-case in section 4, we show that accurate predictions can be obtained from a very limited set of meta-variables as reliability indicators. Furthermore, one always needs to consider the availability of the indicators for the systems in the data-set. 5Collection of System Reliability Metrics The choice of reliability metrics is usually given by the system under study. For our choice of system and assuming a constant failure rate,6 these are given by M T BF and M T T R. Based on these other metrics can be derived.

Model Selection and Validation

Using the collected data, one is able to compile a data-set D = {(x 1 , Y 1 ), ..., (x N , Y N )} for which x i and Y i are the collected reliability indicators and the field-reliability metrics for system i, respectively. A reliability model shall be learned with this data-set. We use a general model selection and assessment approach as is e.g. discussed in chapter 7 of [START_REF] Hastie | The Elements of Statistical Learning[END_REF] with minor modifications due to the particularities of the problem setting. The first step is to split the data-set into a training data-set D train and a test data-set D test . This splitting is not performed arbitrarily. Instead the training data-set shall contain systems with an age higher than a certain threshold age a s and the test data-set shall exclusively contain systems younger than the threshold age. Thereby, we test the approach for its applicability to future systems. For the model selection and assessment the training data-set will be used exclusively. 7 With a five-fold cross-validation method we compare different learning algorithms in terms of their applicability to the problem setting and their prediction errors. For algorithms which additionally require the tuning of hyperparameters, we used so-called nested cross-validation in which the hyperparameters are optimized in a five-fold inner cross-validated grid-search nested within each of the five outer cross-validation folds [START_REF] Cawley | On over-fitting in model selection and subsequent selection bias in performance evaluation[END_REF]. The expected mean and variance of the cross-validation error Err CV is reported for each of the evaluated models. It serves as an estimate for the expected generalization error [START_REF] Hastie | The Elements of Statistical Learning[END_REF].

The confidence of the predictions and the relevance of the selected reliability indicators can be studied with a learning algorithm which satisfies equations 6 and 7. Investigating the identified model parameters and predictions obtained by such an algorithm for one or several cross-validation folds gives this additional information. The confidence or uncertainty of the predictions provides feedback on the amount and quality of the collected data. The weight vector w indicates the relevance of the features and the reliability indicators. Depending on the complexity of the mapping Φ : X → R n from the reliability indicators to the features we can identify the most important reliability indicators. Using this information and expert knowledge, we can refine our data-set (choice of systems and reliability indicators) and feature mapping Φ to obtain more precise models. This idea is illustrated in figure 3.2 Obtaining Reliability Predictions Once satisfying models in terms of their predictive errors and interpretability are found with the procedure described above, they are tested with the full data-set. Since the data-set is split by the age of the systems, this testing simulates a prediction scenario -we identify a model based on data of systems in the past and evaluate its applicability to future systems. The predictive models are now trained with the whole training data-set. 8 Based on the input values of the test data-set x test the models can predict the expected field-reliability y test . As the prediction is simulated, we know the observed fieldreliabilities Y test and can compare these to the predicted ones to obtain the test error Err test . 9 When the test error is of the order of the expected generalization error Err CV obtained during model selection and validation one can conclude that the model is capable of predicting the field-reliability for new systems. The overall data collection, model selection and reliability prediction process is summarized in the pseudo-algorithm below. The use-case in section 4 follows the presented procedure closely.

Pseudoalgorithm illustrating the overall model selection and reliability prediction process:

1. D = {(x 1 , Y 1 ), ..., (x N , Y N )} ← Initial

Use-Case

This section describes how the proposed method was used to learn a model for the expected field-reliability of accelerator power converters. The system of interest, the collected data and features, the used learning algorithms and the results are discussed.

System Definition

The considered systems are magnet power converters at the CERN particle accelerator facilities. A power converter is a device to transform electrical energy. The conversion is in terms of voltage, current and waveform. Magnet power converters control the flow of current through particle accelerator magnets. In order to achieve precise magnetic fields these converters generally need to control the output current very precisely.

Dataset, Reliability Metrics and Reliability Indicators

Set of Systems At CERN there are currently more than 6000 power converters of approximately 600 different types in use. Their field-reliability is continuously tracked by a centralized computerized maintenance management system (CMMS). After removal of converter types with a cumulative operational time t operation of less than ten years and cleaning the data, approximately 300 power converter types remained for reliability analysis. Table 1 gives an overview of minimal and maximal characteristic attributes of power converters in the dataset. Considering the vast range of converter types one would not expect a global model to accurately predict the field-reliability. Therefore, both local-and global-models were trained. Reliability Indicators for the Systems The initial choice of reliability indicators depends on

the system development stage at which the prediction shall be carried out, recommendations from system experts, the time or effort which can be attributed to the data collection.

The following list shows the collected reliability indicators. The selection is based on recommendations from CERN engineers in charge of the complete lifecycle of the studied systems. Naturally, the selection is also limited by the availability of data:

-I: Rated current of the converter. Depending on the rated current different converter technologies have to be used. One major stress effect of high currents in terms of reliability is an increased heat load which requires a proper heat management [START_REF] O'connor | Practical reliability engineering[END_REF][START_REF] Kapur | Reliability engineering[END_REF]. -U: Rated voltage of the converter. Higher voltages require the appropriate electrical insulation and can cause failure mechanisms such as arcing or corona discharge [START_REF] O'connor | Practical reliability engineering[END_REF][START_REF] Kapur | Reliability engineering[END_REF]. -P: Rated power of the converter. Similarly to currents increased power leads to increased heat loads.

-Quantity: This refers to the quantity of each type of power converter that is used at CERN. The quantity of a power converter is not related to a physical wear-out mechanism. However, throughout the lifecycle converter types produced and operated in large quantities are treated differently than power converters of small quantities both in terms of technical and organizational matters. -Avg. Age: The average age of converters for each converter type. Depending on the maintenance strategy a decreasing or constant availability as a function of the age is expected, respectively. -Cum. Age: The cumulative age of converters for each converter type. A dependency of the availability on the cumulative age could indicate both a organizational learning curve in terms of a more efficient maintenance and a degradation with age of the converters.

-Pol 0-9: The polarity of the converter. This indicates the operating modes, technology and complexity of the converter.10 -Acc. 1-9: The accelerator in which the converter type is used. Depending on the accelerator the converter type is exposed to different operating conditions 11 and operation modes. in Acc.: The number of different particle accelerators in which each power converter is used.

We probed different indicators for their information content by appropriate Bayesian learning methods. The required learning algorithms are introduced later in this section.

Reliability Metrics for the Systems The studied field-reliability metrics are M T BF and M T T R as defined in section 3. 12 These are directly computed in the CMMS with the necessary variables for power converter type i which are defined as follows:

t operation,i : Cumulative time in operation of all converters of converter type i. Note that commissioning and testing times are not counted towards operation time. n f aults,i : Cumulative number of faults of all converters of converter type i during the operational time t operation,i . Note that only internal faults of the system which require an external action to alleviate the problem are included. Internal faults which are automatically resolved or are very short and faults due to external reasons are not included. This ensures that a model for the reliability of the considered systems itself is learned and not of its surroundings. t inrepair,i : Cumulative time in repair of all converters of converter type i during the operational time t operation,i . The repair time starts by a request from the system operators to the system experts and ends when the problem was resolved and the system can continue to operate.

Algorithms By formulating the reliability prediction problem as a supervised machine learning problem we can choose from a range of existing learning algorithms to generate the desired statistical model for predictive purposes. Since the uncertainty in the field-reliability predictions shall be quantified (i.e. finding a model as presented in equation 6), the choice of algorithms is narrowed down. Furthermore, sparse parametric models (as in equation 7) are preferred since they potentially require fewer reliability indicators to be collected and -more importantly -since they allow an estimation of the relevance of the choice of reliability indicators and the generated features.

A summary of the chosen algorithms is given in table 2. Note that the scikitlearn python implementations of the algorithms were used [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF]. A detailed description of each algorithm can be found on their website and in their user-guide [START_REF] Blondel | Scikit-learn user guide[END_REF]. Since the algorithms are standard implementations, only references to detailed documentation are given: T . • The categorical indicators x cat were split into binary features, whereas the number of binary variables corresponds to the number of categories per categorical variable. A feature vector of 34 dimensions was obtained by combining all features. This was the first choice for the feature mapping and we refer to it as firstorder feature mapping. The second choice of feature mapping accounts for second-order interactions of the numeric variables and we refer to it as second-order feature mapping:

Φ(x num ) = x num , log(x num ), [x num , log(x num )] • [x num , log(x num )] T T . (8)
By this more complex mapping we obtain 629 features. One could expect that a more accurate model can be learned when including second-order interactions which is balanced by a lack of interpretability of the individual feature weights.

In the following we report the results of our model selection procedure. For each algorithm the cross-validation error Err CV 14 is reported and the feature weights w of the learned models and the obtained predictions are plotted for the last cross-validation fold. 15 All results are provided in terms of the two chosen reliability metrics M T BF and M T T R.

Reference Configuration

The first configuration we studied is based on the complete set of power converters, the complete set of reliability indicators and the first-order feature mapping. The cross-validation errors Err CV are given in table 3a for the M T BF and in table 4a for the M T T R. As the values for the target variables were not scaled to unit-variance but simply by a logarithmic function, the values from the M T BF table cannot be compared with values of the M T T R table. Only values within a table are comparable. We noted that all algorithms yielded models with comparable cross-validation errors. The obtained parameter weights w for the last cross-validation fold are shown in figure 3b for the M T BF and in figure 3d for the M T T R. All algorithms identified similar models. For the M T BF the dominant parameter was the logarithm of the quantity of converters per type log(Qty) for all models and the rated power P was dominant for the M T T R. From the predictions obtained with the BAR algorithm for the last fold, figure 3a for the M T BF and figure 3c for the M T T R, we noted that the model for the M T T R did not identify a significant variation whereas the M T BF was predicted properly. We concluded that a precise model for the M T BF had been learned with the collected data, the selected feature mappings, and algorithms. For the M T T R no such model could be identified and a further refinement would be necessary. In the following we present variations of the reference configuration in terms of selected systems, reliability indicators and feature mappings.

Reduced Set of Training Systems

The second configuration is similar to the reference configuration except for using a random sub-selection of only 42 converter types in the training data-set. This illustrates the dependence of the confidence levels of the identified feature weights w and the predictions for the Bayesian algorithms (ARD and BAR) on the amount of training data. The cross-validation errors Err CV in table 3b for the M T BF and table 4b for the M T T R were larger than those of the reference configuration. The obtained parameters weights w for the last cross-validation fold in figure 3f (M T BF ) and in figure 3h (M T T R) deviated slightly in absolute terms and largely in terms of their confidence levels from the reference configuration for the ARD and BAR algorithm. The predictive uncertainties of the BAR algorithm in figure 3e (M T BF ) and figure 3g (M T T R) increased only slightly in comparison with the reference configuration. Again, no predictive model of the M T T R could be identified. We concluded that a reduced set of training data manifests itself in increased uncertainties in parameters or predictions. Reduced Set of Reliability Indicators The third configuration resembles the reference configuration except for the removal of the variable indicating the quantity of systems per converter type. This variable had been identified as the single most important reliability indicator for M T BF predictions. The cross-validation errors Err CV in table 3c for the M T BF were much larger than those of the reference configuration and slightly larger for the M T T R models (table 4c). The obtained parameter weights w for the last cross-validation fold in figure 4b for the M T BF models were totally different than the reference configuration. The weights for the M T T R models (figure 4d) were similar to the reference configuration. The predictive uncertainties of the BAR algorithm in figure 3e increased drastically for the M T BF and only slightly for the M T T R (figure 3g) in comparison with the reference configuration. This is consistent with our expectation, since we removed the most important reliability indicator for the M T BF models. This time no proper predictive model of either the M T T R or the M T BF could be identified. We concluded that the choice of reliability indicators has a strong influence on the quality of the models.

Second-Order Feature Mapping In the fourth configuration the second-order feature mapping replaces the first-order mapping of the reference configuration. The cross-validation errors Err CV in table 3d for the M T BF and table 4d for the M T T R were of the same order as those of the reference configuration except for the model learned with the ARD algorithm. The 629 obtained parameter weights w were not illustrated. The predictions of the BAR algorithm in figure 4e (M T BF ) and in figure 4f (M T T R) were comparable with the reference configuration. No model could be identified for the M T T R. We concluded that the extended feature mapping does not improve the predictive errors and complicates the interpretation of the models.

Prediction The reference configuration was used for the prediction scenario as it had shown to be interpretable and predicted the M T BF in the model selection procedure properly. The prediction of the M T T R was not carried out since no suitable model had been identified. The predictive models were learned with the full training data-set and their predictions were evaluated with the test data-set. 16 Due to the splitting of the training and test data by the age of the systems this simulated a prediction scenario. The test errors Err test in table 3e were of the same order as the cross-validation errors Err CV of the reference configuration (3a). We concluded that the learned models generalize to newer power converters. The feature weights (5b) and the predictions (5a) were consistent with our expectations and demonstrate that we could predict the M T BF accurately.

Discussion One of the major insights created by applying the methods to the use-case is that the field-reliabilities are strongly dependent on the quantity of converters per converter type. This fact can lead to an increased reliability for future systems. However, explanations for this dependence are plentiful and a more detailed analysis will have to be carried out.

The method is capable of learning more detailed statistical models for the whole lifecycle of systems. This requires to collect more reliability indicators than were available in this work. However, the purpose of this work was to illustrate that even with very coarse high-level data a good predictive model can be trained. The selected Bayesian algorithms which learn sparse parametric models were especially fit for this purpose. It has to be pointed out that the approach is empirical and that causal relationships have to be identified or confirmed by further studies or expert judgment.

Conclusion and Outlook

An approach was presented to predict the field-reliability of complex electronic systems at an early development stage based on a statistical lifecycle model learned from data collected for similar operational systems. It was demonstrated that the field-reliability can be predicted accurately based on very few reliability indicators. Compared to existing methods this implies a reduced data collection effort and an integrated quantification of predictive uncertainty based on the granularity of the available information and the implicit randomness of the investigated processes. The results of such a study uncover reliability relevant factors which lead to improved system designs at very early stages of design. Sparse Bayesian Regression methods are the key to efficiently learn accurate models. The confidence in field-reliability predictions is automatically quantified with respect to the available data and the randomness inherent in the problem. Future research can focus on more detailed and potentially incomplete data-sets.

Based on that, further relevant processes for the field-reliability of systems may be uncovered.

Fig. 2 .

 2 Fig. 2. Illustration of the iterative data collection and reliability prediction process. The choice of (1) systems, (2) reliability indicators and (3) feature mappings influences the quality of the predictive model (4). The learning algorithm provides feed-back in the form of an expected prediction error (a), relevance weights for the reliability indicators (b) and uncertainty bounds for the field-reliability predictions (c).

Fig. 3 .

 3 Fig. 3. (a),(c),(e),(g):Prediction of the log(M T BF )/log(M T T R) for the last fold of the cross-validation procedure. The orange line depicts the mean of the predictive distribution and the orange shaded area the 95% confidence intervals. The blue dots mark the actual observed field-reliabilites. Note that the different converter types were ordered by the mean of the predictive distribution for illustration purposes.(b),(d),(f),(h):Estimated feature weights for the parametric models. Figures (a),(b),(c),(d) are for the reference configuration and figures (e),(f),(g),(h) for a reduced set of data items in the learning data.

Fig. 4 .

 4 Fig. 4. (a),(c),(e),(f):Prediction of the log(M T BF )/log(M T T R) for the last fold of the cross-validation procedure. The orange line depicts the mean and the orange shaded area the 95% confidence intervals. The blue dots mark the actual observed field-reliabilites. Note that the different converter types were ordered by the mean of the predictive distribution.(b),(d):Estimated feature weights for the parametric models.Figures (a),(b),(c),(d) are for the configuration with a reduced set of reliability indicators and figures (e),(f) for the second-order feature mapping. Note that the illustrations of the 629 second-order feature weights were omitted.

Fig. 5 .Table 3 .Table 4 .

 534 Fig. 5. (a):Predictions of the log(M T BF ) with the final models for the test data-set. The orange line depicts the mean and the orange shaded area the 95% confidence intervals. The blue dots mark the actual observed field-reliabilites. Note that the different converter types were ordered by the mean of the predictive distribution.(b):Estimated feature weights for the predictive models.

  data collection. 2. Sort D by system age. 3. Split D in D train and D test with a test < a s ≤ a train . 4. While satisfying predictive model has not been found do: (a) Shuffle D train randomly. (b) Evaluate Err CV by (nested) CV. (c) Evaluate parameter weights w and predictive uncertainty for one fold. (d) If Model has large Err CV or predictive uncertainty then -Change set of systems, reliability indicators, or feature mapping. (e) Else jump to 5. 5. Train predictive model with D train . 6. Evaluate Err test and compare with Err CV . 7. Evaluate parameter weights w and predictive distributions.

Table 1 .

 1 Illustration of characteristic power converter attributes of the studied dataset.

	Power [W] Current [A] Voltage [U] Age [yrs] MTBF [hrs]
	Minimum 10 -6	10 -4	10 -3	2.2	10 3
	Maximum 10 8	4 • 10 4	10 5	49.7	6 • 10 5

Table 2 .

 2 Summary of learning algorithms.Model Selection and ValidationThis section closely follows the procedure presented in section 3. The data-set D was compiled from the data collection described above including 281 collected systems, nine reliability indicators and two field-reliability metrics. To simulate a prediction scenario the whole dataset of 281 different converter types was split into a training set D train with 210 converter types which are at least fifteen years old and a test set with 71 converter types which are less than fifteen years old.13 For the model selection and validation we restricted ourselves to the training data which we shuffled randomly. A scaling operator re-scaled the features or inputs x train to zero mean and unit variance. The same scaling operator was later applied to the features in the test data-set x test . Furthermore, the logarithms of the reliability metrics log(Y) were taken instead of their nominal value for the full data-set. Based on the introduced (nested) cross-validation we compared the follfowing different choices of the set of systems, reliability indicators and feature mappings for all the introduced algorithms:-Choice of systems: We trained models with the complete set of power converter types and with a random sub-selection of only 42 converter types. -Choice of reliability indicators: We trained models with the complete set of reliability indicators and a set in which the quantity of converters per type was removed.

		UQ (6) Feature Weights (7) Sparsity Global/Local
	ARD yes	yes	yes	Global
	BAR yes	yes	balanced Global
	GP	yes	no	no	Local
	ENCV no	yes	yes	Global
	SVR no	only for linear kernel no	Local
	-ARD -Automatic Relevance Determination Regression: Sparse Bayesian re-
	gression technique as described in [2] -Chapter 7.2.1. The implementation
	is taken from [3] -Chapter 1.1.10.2.
	-BAR -Bayesian Ridge Regression: A Bayesian regression method as intro-
	duced in [15]. It is similar to the ARD Regression but fewer parameters have
	to be determined from the data. The implementation is taken from [3] -
	Chapter 1.1.10.1.
	-GP -Gaussian Process Regression. A kernel-trick based Bayesian Regression
	technique. The implementation is described in [21] -Algorithm 2.1 and was
	taken from [3] -Chapter 1.7.1. The kernel is based on a combination of a
	radial-basis-function kernel and a white-kernel. The kernel parameters were
	optimized in the learning process.
	-EN: Elastic Net Regression. The implementation is taken from [3] -Chapter
	1.1.5 -which includes a description of the algorithm. Hyperparameters were
	optimized in a cross-validated grid-search.
	-SVR -Support Vector Machine Regression: A kernel-trick based regression
	method. A description is given in [3] -Chapter 1.4.2. Linear basis functions
	were used and the hyperparameters were optimized by a cross-validated grid-
	search.

-Choice of feature mapping: Based on the reliability indicators, following features were generated:

• Based on the numeric indicators x num linear features and logarithmic features were chosen -Φ(x num ) = [x num , log(x num )]

  )

	Feature Weight	0.0 0.5 1.0 1.5	EN SVR ARD BAR
		0.5	Quantity Cum. Age Avg. Age I U P Pol 0 Pol 1 Pol 2 Pol 3 Pol 4 Pol 5 Pol 6 Pol 7 Pol 8 Pol 9 constant Acc. 1 Acc. 2 Acc 3. Acc. 4 Acc. 5 Acc. 6 Acc. 7 Acc. 8 Acc. 9 Acc. 10 in Acc. log(Qty.) log(Cu. A.) log(Av. A.) log(I) log(U) log(P)
			(b)

Depending on the system under study the definitions of the stages may change. The proposed methodology is not restricted to this specific choice of stages.

There is no implicit restriction for the proposed method to electronic repairable systems. It can also be used for non-repairable systems and for mechanic, electric, electronic, or software systems. However, the definitions of the fault metrics must be adapted.

If availability indicators are unavailable for some of the selected systems, supervised learning techniques for incomplete data-sets can be employed.

This assumption can be relaxed by e.g. predicting a parameterized failure rate distribution over time. Then, instead of M T BF and M T T R the reliability metrics are the parameters of the distribution. This requires a different data collection and can be considered for future work.

[START_REF] Foucher | A review of reliability prediction methods for electronic devices[END_REF] Using the test data-set would lead to an over-fitting of the models and an underestimation of the generalization error.

Again, hyperparameters are optimized by a cross-validated grid search over a hyperparameter grid.

[START_REF] Gullo | In-service reliability assessment and top-down approach provides alternative reliability prediction method[END_REF] We note that in a realistic application scenario the true observed field-reliabilities are not available. However, the available data can always be split by system age to test the generalizability of the identified models to newer systems.

The discrete set of polarities is given by: (1) Unipolar, (2) Bipolar Switch Mechanic, (3) Bipolar I -Unipolar U -2 Quadrants, (4) Unipolar I Bipolar U 2 Quadrants, (5) Bipolar Pulse-Width-Modulation, (6) Bipolar Relay, (7) Bipolar Electronic I/U, (8) Bipolar Anti-Parallel 4 Quadrants, (9) Bipolar I-circulation 4 Quadrants and, (0) un-specified or other Polarity.

E.g. the radiation levels differ on the kind of accelerator. However, there is also different operation conditions within each of the accelerators.

Note that the Availability A and Un-Availability UA can be directly obtained from the M T BF and the M T T R.

In other words we pretended to be in 2003 and tried to predict the field-reliability of power converters between 2003 and 2018.

Note that the mean-squared-error was used throughout.

Note that only predictions obtained by the BAR algorithm are illustrated due to space limitations. It assigns relevance weights to the feature functions and and it quantifies uncertainties of both the field-reliability predictions and the feature function weights. Therefore, it is suited to study the earlier mentioned additional information provided by algorithms which satisfy equations 6 and 7.


Guido Bologna
email: guido.bologna@hesge.chguido.bologna@unige.ch

A Rule Extraction Study Based on a Convolutional Neural Network

Keywords: Convolutional Neural Networks, Rule Extraction, Sentiment Analysis

Convolutional Neural Networks (CNNs) lack an explanation capability in the form of propositional rules. In this work we define a simple CNN architecture having a unique convolutional layer, then a Max-Pool layer followed by a full connected layer. Rule extraction is performed after the Max-Pool layer with the use of the Discretized Interpretable Multi Layer Perceptron (DIMLP). The antecedents of the extracted rules represent responses of convolutional filters, which are difficult to understand. However, we show in a sentiment analysis problem that from these "meaningless" values it is possible to obtain rules that represent relevant words in the antecedents. The experiments illustrate several examples of rules that represent n-grams.

Introduction

A natural way to explain neural network responses is by means of propositional rules [START_REF] Holzinger | What do we need to build explainable ai systems for the medical domain?[END_REF]. Andrews et al. introduced a taxonomy to characterize all rule extraction techniques [START_REF] Andrews | Survey and critique of techniques for extracting rules from trained artificial neural networks[END_REF]. Diederich and Dillon presented a rule extraction study on the classification of four emotions from SVMs [START_REF] Diederich | Sentiment recognition by rule extraction from support vector machines[END_REF]. Extracting rules in Sentiment Analysis (SA) is very challenging with thousands of words represented in the inputs [START_REF] Diederich | Sentiment recognition by rule extraction from support vector machines[END_REF]. As a consequence Diederich and Dillon restricted their study to only 200 inputs and 914 samples. Bologna and Hayashi used the Quantized Support Vector Machine to generate rules explaining Tweets' sentiment polarities [START_REF] Bologna | A rule extraction study from svm on sentiment analysis[END_REF]. In this work we propose a model to generate rules from a Convolutional Neural Network architecture (CNN) that is trained with a dataset related to SA. To the best of our knowledge this problem has not been tackled.

Since Convolutional Neural Networks (CNNs) started to be broadly used less than ten years ago, very few works have proposed to determine the acquired knowledge in these models. Several authors proposed to interpret CNNs in the neighborhood of the instances. As an example, Ribeiro et al. presented LIME whose purpose is to learn an interpretable model in the local region close to an input instance [START_REF] Ribeiro | Why should i trust you?: Explaining the predictions of any classifier[END_REF]. Koh et al. determined the training instances that are the most important for the prediction [START_REF] Koh | Understanding black-box predictions via influence functions[END_REF]. Finally, Zhou et al. presented CAM whose purpose is to determine discriminative image regions using the average pooling of CNNs [START_REF] Zhou | Interpreting deep visual representations via network dissection[END_REF].

This work illustrates rule extraction from a CNN architecture shaped by an input layer representing sentences of word embeddings [START_REF] Mikolov | Efficient estimation of word representations in vector space[END_REF], a convolutional layer with filters, and max-pooling layer followed by a fully connected layer. After the training phase, the fully connected layer is replaced by a Discretized Interpretable Multi Layer Perceptron (DIMLP) [START_REF] Bologna | Rule extraction from a multilayer perceptron with staircase activation functions[END_REF] that allows us to extract rules. This subnetwork approximates the fully connected part of the original CNN to any desired precision. Nevertheless, the antecedents of the extracted rules represent maximal responses of convolutional filters, which can not be understood directly. Thankfully, it is possible to go back to the words that are relevant in the decision-making process. Specifically, these words are structured into n-grams, that depend on the size of the convolutional filters. In the following sections we first describe the used model, then we present the experiments, followed by the conclusion.

The Interpretable CNN Architecture

The CNN architecture used here is similar to those typically used in SA problems [START_REF] Kim | Convolutional neural networks for sentence classification[END_REF]. Before rule extraction, we train a CNN having only a fully connected layer between the last two layers of neurons. Then, we perform rule extraction by replacing these last two layers of neurons by a special subnetwork that can be translated into symbolic rules [START_REF] Bologna | Rule extraction from a multilayer perceptron with staircase activation functions[END_REF]. Hence, after training the modified CNN for rule extraction has two components:

a CNN without the fully connected layer; a DIMLP with only a hidden layer that approximates the CNN fully connected layer.

Rules generated from the DIMLP component are related to filter thresholds in the convolutional layer of the CNN. From these values relevant words represented in the input layer can be determined. More details are given in the following paragraphs.

The CNN Subnetwork

Our CNN subnetwork is quite similar to that proposed in [START_REF] Kim | Convolutional neural networks for sentence classification[END_REF] and [START_REF] Cliche | Bb twtr at semeval-2017 task 4: Twitter sentiment analysis with cnns and lstms[END_REF]. A smaller representation of the adopted CNN architecture is described in Figure 1. The inputs are words represented by word embeddings of dimensionality d = 300, calculated with word2vec [START_REF] Mikolov | Efficient estimation of word representations in vector space[END_REF]. Here, the maximal number of words in a sentence is equal to s = 59, thus the size of an instance in the left part of Figure 1 is equal to 17700(= 59 * 300). Then, an instance is convolved with filters of different size. Specifically, for each filter we define a filtering matrix of size f • d, with f = 1, 2, 3, corresponding to the number of words on which it operates.

After convolution we apply a Relu function (Relu(x) = Max(0, x)), which is non-linear. For each filter, the result of the convolution provides a vector of size s -f + 1. Afterward, the max-pooling operator is applied and the maximal value is retained, independently of where it is located in a sentence. All the maxpooling units are concatenated (right most layer in Figure 1) and then follows a fully connected layer. In the output layer, a Softmax activation function is used. The CNN subnetwork has an input layer with data provided as word embeddings; these word vectors are represented horizontally, while sentences are represented vertically. Then follows a convolutional layer with filters of different size. The convolution of the input with filters provides one-dimensional vectors (represented vertically). Finally, the "max" function reduces the size of these vectors that are finally concatanated (last layer on the right). The fully connected layer is not represented.

Specifically, for N a i scalars it calculates an N-dimensional vector with values between 0 and 1. It is given as:

S i = exp(a i ) ∑ N k=1 exp(a k ) ; ∀i ∈ 1 . . . N. ( 1
)
We use the Lasagne Library Package [START_REF] Dieleman | Lasagne: First release[END_REF] to train our CNNs. The training phase is achieved with the Cross-Entropy loss function. After learning the fully connected layer, which is not represented in Figure 1 is replaced by a DIMLP subnetwork having two layers of weights, with the second being equal to the fully connected layer of the CNN.

The DIMLP Subnetwork

DIMLP differs from standard Multi Layer Perceptrons in the connectivity between the input layer and the first hidden layer. Specifically, any hidden neuron receives only a connection from an input neuron and the bias neuron, as shown in Figure 2. After the first hidden layer, neurons are fully connected. Note that very often DIMLPs are defined with two hidden layers; the number of neurons in the first hidden layer being equal to the number of input neurons. The key idea behind rule extraction from DIMLPs is the precise localization of axis-parallel discriminative hyperplanes. In other words, the input space is split into hyperrectangles representing propositional rules. Specifically, the first hidden layer creates for each input variable a number of axis-parallel hyperplanes that are effective or not, depending on the weight values of the neurons above the first hidden layer. More details on the rule extraction algorithm can be found in [START_REF] Bologna | A model for single and multiple knowledge based networks[END_REF]. The activation function in the output layer of a standard DIMLP [START_REF] Bologna | Rule extraction from a multilayer perceptron with staircase activation functions[END_REF] is a sigmoid function given as

σ(x) = 1 1 + exp(-x) . ( 2
)
However, here since the CNN is trained with a Softmax function in the output layer, we replace the sigmoid by it. In the first hidden layer the activation function is a staircase function S(x) with Q stairs that approximate the Identity function I(x) = x; ∀x ∈ (R min ..R max ), with two constants R min and R max .

S(x) = R min , if x ≤ R min ; (3)
R min represents the abscissa of the first stair. By default R min = 0.

S(x) = R max , if x ≥ R max ; (4)
R max represents the abscissa of the last stair. By default R max = 1. Between R min and R max S(x) is given as

S(x) = I(R min + [ q • x -R min R max -R min ] ( R max -R min q )). ( 5
)
Square brackets indicate the integer part function and q = 1, . . . Q. The step function t(x) is a particular case of the staircase function with only one step:

t(x) = { 1 if x > 0; 0 otherwise. ( 6
)
The approximation of the Identity function by a staircase function depends on the number of stairs. The larger the number of stairs the better the approximation. Note that the step/staircase activation function makes it possible to precisely locate possible discriminative hyperplanes.

As an example, in Figure 2 assuming two different classes, the first is being selected when y 1 > σ(0) = 0.5 (black circle) and the second with y 1 ≤ σ(0) = 0.5 (white squares). Hence, two possible hyperplane splits are located in -w 10 /w 1 and -w 20 /w 2 , respectively. As a result, the extracted unordered rules are:

-(x 1 < -w 10 /w 1 ) → square -(x 2 < -w 20 /w 2 ) → square -(x 1 ≥ -w 10 /w 1 ) and (x 2 ≥ -w 20 /w 2 ) → circle

From meaningless rules to meaningful rules

Each antecedent extracted from the max-pooling-layer is given as a < t, or a ≥ t; t being a filter threshold involving in the input layer a number of activated words. Specifically, these words correspond to bigrams and trigrams when the filter size in the convolutional layer is equal to two or three, respectively. In practice, filters of size equal to one are convolved with all possible single words. Then with the obtained values, we retain all the single words that make true a rule antecedent related to the max-pool-layer. This is repeated for filters of size two with respect to bigrams and so on with other filter sizes.

Generally, the condition expressed in a rule antecedent is true with more than one n-gram; thus a rule antecedent which is true for the DIMLP subnetwork implies a disjunction of one or more n-grams represented in the input layer (one or more n-grams connected by a logical or). Nevertheless, with the use of the "Max" function a unique n-gram becomes dominant (the one with the highest activation) and cancels the others.

Disjunctions of n-grams related to a rule antecedent extracted from the maxpooling-layer involving thousands of words could be considered too numerous. However, in practice these words are not necessarily encountered, especially for rules activated by a small number of examples. These words will be useful for determining possible contradictions, such as the simultaneous presence of words/ngrams that are clearly of positive polarity and others of negative polarity.

Experiments

In the experiments we use a well-known binary classification problem describing movie reviews with Tweets [START_REF] Pang | A sentimental education: Sentiment analysis using subjectivity summarization based on minimum cuts[END_REF]. 3 The number of examples is equal to 10662, with half of them belonging to the positive class. Note that the total number of single words in the dataset is equal to 21426, while the total number of bigrams is equal to 111590. Words are coded into vectors of word embeddings of size 300 [START_REF] Mikolov | Efficient estimation of word representations in vector space[END_REF]. Two CNN architectures were defined; one with 50 filters of size one and two (f = 1, 2) and another with 40 filters of size one, two and three. Hence, the last layer in Fig. 1 has 100 and 120 neurons, respectively. For deep learning training we use Lasagne libraries, version 0.2 [START_REF] Dieleman | Lasagne: First release[END_REF]. The loss function is the categorical cross-entropy and the training parameters are:

learning rate: 0.02; momentum: 0.9; dropout = 0.2;

From the whole dataset we selected the first 10% of the samples of each class as a testing set and the rest as training examples. Moreover, a subset of the training set representing 10% of it was used as a tuning set for early-stopping. Table 1 shows the results for the first CNN architecture. The first row of this Table is related to the original CNN, while the other rows provide results of the approximations obtained with the CNN-DIMLP combination by varying the number of stairs in the staircase activation function. Columns from left to right designate:

train accuracy; predictive accuracy on the testing set; fidelity, which is the degree of matching between rules and the model; predictive accuracy of the rules; predictive accuracy of the rules when rules and model agree; number of extracted rules and total number of rule antecedents.

Note that these rules involve filter responses in the antecedents from which ngrams are determined (cf. Sect. 2.3). Table 2 presents the results obtained with the second CNN architecture. The best predictive accuracy of rules was obtained with the second architecture, which takes into account trigrams. Note however that accuracy performance in this work is not a priority, since our purpose is to demonstrate how to generate meaningful propositional rules from CNNs. Overall, fidelity on the testing set is above 90%, meaning that rules explain CNN responses in a large majority cases.

The antecedents of rules extracted from the DIMLP subnetwork involve long lists of n-grams. Here we illustrate several rules extracted from the first architecture with q = 200 (the one with the highest fidelity). Rules are ranked according to their support with respect to the training set, which is the proportion of covered samples. Note that rules are not disjointed, which means that a sample can activate more than a rule. The first rule has a support of 765 training samples and 76 testing samples; it is given as:

-(f 51 < 0.285) (f 65 < 0.09) (f 70 < 0.275) (f 74 ≥ 0.185) (f 87 < 0.305) (f 99 < 0.06) Class = POSITIVE
Here, f designates maximal values of filters with respect to the max-pooling layer. Indexes between one and 50 are related to single words, while those between 51 and 100 correspond to bigrams. The accuracy of this rule is 93.9% on the training set and 88.1% on the testing set. Antecedent f 74 ≥ 0.185 involves the presence of at least one bigram in a list of 1218 possible bigrams. Generally, the possible bigrams are not necessarily encountered in the tweets activating a rule. By negating all other antecedents of this rule we obtain a list of 12730 bigrams that are required to be absent. We illustrate a number of sentences with possible bigrams including the dominant bigram, represented in bold and related to the rule shown above. Note that three consecutive in bold represent two consecutive bigrams:

1. offers that rare combination of entertainment and education . 2. a thoughtful , provocative , insistently humanizing film . 3. a masterful film from a master filmmaker , unique in its deceptive grimness , compelling in its fatalist worldview . 4. cantet perfectly captures the hotel lobbies , two-lane highways , and roadside cafes that permeate vincent's days . 5. the film makes a strong case for the importance of the musicians in creating the motown sound . 6. a compelling coming-of-age drama about the arduous journey of a sensitive young girl through a series of foster homes and a fierce struggle to pull free from her dangerous and domineering mother's hold over her . 7. this delicately observed story , deeply felt and masterfully stylized , is a triumph for its maverick director . 8. a portrait of alienation so perfect , it will certainly succeed in alienating most viewers . 9. it's sincere to a fault , but , unfortunately , not very compelling or much fun .

The first eight tweets are correctly classified and we can clearly recognize words of positive polarity. The two tweets at the end of the list are classified as positive, whereas their class is negative. The last tweet is wrongly classified, because "not" before "very compelling" has been ignored. Regarding the ninth tweet, "so perfect" contributes without any doubt to a positive polarity; then, at the end of the sentence "alienating" contributes to its true negative classification, but it is not considered at all by the rule.

Rule number 17 is given as:

-(f 7 < 0.315) (f 39 < 0.295) (f 54 ≥ 0.05) (f 65 < 0.12) (f 75 < 0.25) (f 83 < 0.13) (f 95 ≥ 0.135) (f 96 ≥ 0.16) Class = POSITIVE
It has a support of 404 samples in the training set and 47 samples in the testing set with an accuracy of 91.6% and 83.0%, respectively. The presence of one or more bigrams is imposed by f 54 , f 95 , and f 96 ; it contains 8266 elements. Ninetysix single words must be absent; they are related to f 7 , and f 39 . Moreover, 6307 mandatory absent bigrams depends on f 65 , f 75 , and f 83 . Ten tweets with their possible bigrams including the dominant bigram are shown here; the last two tweets are wrongly classified:

1. an utterly compelling 'who wrote it' in which the reputation of the most famous author who ever lived comes into question .

2. between the drama of cube ? s personal revelations regarding what the shop means in the big picture , iconic characters gambol fluidly through the story , with charming results . 3. it's an old story , but a lively script , sharp acting and partially animated interludes make just a kiss seem minty fresh . 4. this is simply the most fun you'll ever have with a documentary ! 5. one the best , most understated performances of [jack nicholson's] career . 6. tadpole is a sophisticated , funny and good-natured treat , slight but a pleasure . 7. a smart , sweet and playful romantic comedy . 8. mr . parker has brilliantly updated his source and grasped its essence , composing a sorrowful and hilarious tone poem about alienated labor , or an absurdist workplace sitcom . 9. beautifully filmed and well acted . . . but admittedly problematic in its narrative specifics . 10. one of the oddest and most inexplicable sequels in movie history .

The following list illustrate negative tweets correctly classified by a rule reaching an accuracy of 100% on the testing set:

1. it's an 88-minute highlight reel that's 86 minutes too long . 2. such an incomprehensible mess that it feels less like bad cinema than like being stuck in a dark pit having a nightmare about bad cinema . 3. during the tuxedo's 90 minutes of screen time , there isn't one true 'chan moment' . 4. the script becomes lifeless and falls apart like a cheap lawn chair . 5. the script falls back on too many tried-and-true shenanigans that hardly distinguish it from the next teen comedy . 6. a close-to-solid espionage thriller with the misfortune of being released a few decades too late .

Finally, we show another example of negative tweets correctly classified by another rule yielding predictive accuracy equal to 100% :

1. maybe leblanc thought , " hey , the movie about the baseball-playing monkey was worse . " 2. a muddled limp biscuit of a movie , a vampire soap opera that doesn't make much sense even on its own terms . 3. the script becomes lifeless and falls apart like a cheap lawn chair . 4. a baffling subplot involving smuggling drugs inside danish cows falls flat , and if you're going to alter the bard's ending, you'd better have a good alternative . 5. given the fact that virtually no one is bound to show up at theatres for it , the project should have been made for the tube . 6. jonathan parker's bartleby should have been the be-all-end-all of the modernoffice anomie films .

Conclusion

We presented a model that allowed us to extract rules of high fidelity from a typical trained CNN architecture for Sentiment Analysis. Rule extraction was first applied to the layer before the output layer and then relevant words were determined from convolutional filters thresholds. Generated rules are described by disjunctions of n-grams that must be present or conjunctions of n-grams that must be absent. Moreover, extracted n-grams do not depend on particular positions in sentences. In the experiments, several examples of tweets with discriminatory bigrams that explained CNN responses were illustrated. These discriminatory words are important, as they can be used to understand how correct/wrong classifications are obtained by the classifier.

Fig. 1 .

 1 Fig. 1.The CNN subnetwork has an input layer with data provided as word embeddings; these word vectors are represented horizontally, while sentences are represented vertically. Then follows a convolutional layer with filters of different size. The convolution of the input with filters provides one-dimensional vectors (represented vertically). Finally, the "max" function reduces the size of these vectors that are finally concatanated (last layer on the right). The fully connected layer is not represented.

Fig. 2 .

 2 Fig. 2.A DIMLP network creating two discriminative hyperplanes. The activation function of neurons h1 and h2 is a step function, while for output neuron y1 it is a sigmoid.

Table 1 .

 1 Results obtained with the fist CNN architecture including filters of size one and two, involving unigrams and bigrams. In the the DIMLP subnetwork the number of stairs of the staircase activation function is varied from 20 to 200.

		Train Acc. Test Acc. Fidelity Test Acc. (r1) Test Acc. (r2) #Rules/#Ant.
	CNN	81.5	74.6	-	-	-	-
	CNN (q=20)	81.2	74.0	93.6	72.9	75.0	747/6159
	CNN (q=50)	81.5	74.0	94.8	74.3	75.5	636/5730
	CNN (q=100)	81.4	74.1	95.3	73.0	74.7	669/5619
	CNN (q=200)	81.4	74.5	95.8	73.4	75.0	562/5131

Table 2 .

 2 Results obtained with the second CNN architecture including filters of size one, two and three, involving unigrams, bigrams and trigrams.

		Train Acc. Test Acc. Fidelity Test Acc. (r1) Test Acc. (r2) #Rules/#Ant.
	CNN	83.0	75.4	-	-	-	-
	CNN (q=20)	83.0	75.0	94.6	73.4	75.6	637/5633
	CNN (q=50)	82.9	75.3	95.2	72.8	75.3	559/5065
	CNN (q=100)	82.9	75.4	95.5	75.2	76.5	568/4885
	CNN (q=200)	83.0	75.3	94.0	73.5	75.9	555/4798

Available at: http://www.cs.cornell.edu/people/pabo/movie-review-data/.