Ahmad Mazyad

Fabien Teytaud

Cyril Fonlupt

A Comparative Study on Term Weighting Schemes for Text Classification

Text Classification (or Text Categorization) is a popular machine learning task. It consists in assigning categories to documents. In this paper, we are interested in comparing state of the art classifiers and state of the art feature weights. Feature weight methods are classic tools that are used in text categorization. We extend previous studies by evaluating numerous term weighting schemes for state of the art classification methods. We aim at providing a complete survey on text classification for fair benchmark comparisons.

Introduction

Nowadays, at the time of the rapid growth of the internet, the volume of text documents becomes more and more important. Consequently, effective document retrieval may be a really hard task, especially without any organization. Text classification has become a state of the art solution to this problem. Over time, several classification methods appear [START_REF] Deng | A comparative study on feature weight in text categorization[END_REF], such as k-nearest neighbor [START_REF] Yang | Expert network: Effective and efficient learning from human decisions in text categorization and retrieval[END_REF], Naïve Bayes [START_REF] Mccallum | A comparison of event models for naive bayes text classification[END_REF], decision trees [START_REF] Apte | Text mining with decision rules and decision trees[END_REF], neural networks [START_REF] Ng | Feature selection, perceptron learning, and a usability case study for text categorization[END_REF], boosting methods [START_REF] Schapire | Boostexter: A boosting-based system for text categorization[END_REF] and Support Vector Machines [START_REF] Cortes | Support-vector networks[END_REF]. In this paper we are interested in finding a good term weighting method for state of the art classification algorithms. The paper is organized as follows: in Sect. 2 we present text classification and in particular state of the art term weighting method definitions. In Sect. [START_REF] Crammer | Online passiveaggressive algorithms[END_REF] we present the state of the art classifiers used in our study. In Sect. 4, we compare the different term weighting methods applied to 3 famous text categorization benchmarks. Finally, we discuss and present future works in Sect. 5.

Text Classification

Text Classification (TC) aims at automatically assigning a set of predefined categories to a text document. Depending on the text corpus being classified, each document can be in one or multiple categories. This task is achieved by 1 using a classifier learned on a training set of labeled documents. A fundamental step in learning a classifier is to represent text documents in a suitable format recognizable by this classifier. In Vector Space Model (VSM), each text document is represented as a vector of index terms in which each term is associated with a weight (score) that measures how informative/discriminative the correspondent term is. The method which assigns a weight to a term is called Term Weighting Scheme (TWS).

To the best of our knowledge, no complete survey exists on how effective TWS performs with different state of the art classifiers. In this paper, we focus on this comparison in order to have a fair and complete study.

One of the most famous TWS is tf.idf proposed by Jones in [START_REF] Jones | A statistical interpretation of term specificity and its application in retrieval[END_REF] and stands for term frequency-inverse document frequency. tf.idf is an unsupervised term weighting method. It is the product of the Term Frequency component (TF) by the Collection Frequency component (CF): Term Frequency (tf) and Inverse Document Frequency (idf) respectively. We use the logarithmically scaled tf defined as:

tf t,d = 1 + log(f t,d ) .
where f t∈d stands for the occurrence of term t in the document d. And idf defined as:

idf (t) = log |D| |{d ∈ D|t ∈ d }| .
where |D| is the total number of documents and |{d ∈ D|t ∈ d }| is the number of documents that contains the term t.

TC is a supervised learning task, such that document membership (class information) is known in advance. We call Supervised Term Weighting (STW), the term weighting that incorporates the class information. In that context, researchers proposed various supervised term weighting methods that replace the unsupervised collection frequency component idf by a supervised component. For instance, Chi-square (χ 2 ) is a test of independence between two variables and it was first used as a TWS in text categorization in [START_REF] Deng | A comparative study on feature weight in text categorization[END_REF][START_REF] Debole | Supervised term weighting for automated text categorization[END_REF]. Gain ratio (gr) in [START_REF] Debole | Supervised term weighting for automated text categorization[END_REF], odds ratio (or) in [START_REF] Deng | A comparative study on feature weight in text categorization[END_REF], relevance frequency (rf ) was proposed by Lan et al. in [START_REF] Lan | Supervised and traditional term weighting methods for automatic text categorization[END_REF], inverse category frequency (icf ) proposed by Wang et al.in [START_REF] Wang | Inverse category frequency based supervised term weighting scheme for text categorization[END_REF], and term relevance ratio (trr) by Youngjoong in [START_REF] Youngjoong | A study of term weighting schemes using class information for text classification[END_REF].

Thus, the general formula for the different TWS in this paper, could be defined as:

w t,d = tf t,d × CF (t) .
Table 1 shows all the CF included in this study that are used in almost all TC works. All these TWS are used in classic machine learning tools for TC. We present some of these tools in the next section.

Classifiers

To study the effect of each Supervised Term Weighting (STW) on classification tasks, we use five known learning algorithms: Support Vector Machine (SVM), Passive-Aggressive (PA), Stochastic Gradient Descent (SGD), Nearest Centroid (NC), C4.5 (C4.5). However it's important to note that we are studying the effectiveness of STW rather than the performance of the learning algorithms. SVM is a supervised machine learning algorithm used for both classification and regression. SVM has been proposed by Cortes and Vapnik in [START_REF] Cortes | Support-vector networks[END_REF]. Joachims [START_REF] Joachims | Text categorization with support vector machines: Learning with many relevant features[END_REF] was the first to use SVM for text categorization in which he shows the superiority of SVM over other traditional learning methods. PA introduced in [START_REF] Crammer | Online passiveaggressive algorithms[END_REF] is an online learning algorithm for large scale dataset. The algorithm watches a stream of instances. Once a new instance is received, the algorithm outputs a prediction. Later, the instance true label is uncovered and the algorithm updates its prediction function. SGD [START_REF] Zhang | Solving large scale linear prediction problems using stochastic gradient descent algorithms[END_REF] is another learning algorithm for large scale classification task. It is used to learn linear models such as linear SVM, by mimimizing its objective function.

NC [START_REF] Tibshirani | Diagnosis of multiple cancer types by shrunken centroids of gene expression[END_REF] is a simple neighborhood-based classification algorithm. The algorithm computes a centroid for each class. It then outputs the label of the nearest centroid to the test instance as the predicted label. C4.5 [START_REF] Quinlan | C4.5: Programs for Machine Learning[END_REF] is a supervised tree-based learning algorithm. In 2008, C4.

Evaluation

To assess the performance of STW, we use the standard F1 measure. The F1 score considers both precision (true positive over true positive plus false positive) p and recall (true positive over true positive plus false negative) r and can be formally defined as: F 1(p, r) = 2rp r+p . We also report the precision and recall. The precision and recall results of the multiple binary tasks are averaged using the micro-(µ) and macro-(m) averaged measures. Tables 3, 4, 5, 6, 7, 8, 9, 10, 11 show the µ/m-averaged precision, recall and f-score, for reuters, oshumed and 20newsgroups datasets, respectively. In these tables, the highest µ/m score over a column is underlined, and the best pair of

Results

In tables 3, 4 and 5, we present the µ/m-averaged precision, recall, and f-score, respectively, for Reuters-21578 dataset. In tab.5, NC shows the lowest performance, considering both µ and m scores. PA have the highest µ-score (87.22%) and the second highest m-score (48.51%) preceded only by C4.5 with a m-score of 54.24%. Regarding TWS, even though, tf.idf shows higher scores, the results are very close.

Tables 6, 7 and 8 shows the µ/m-averaged precision, recall, and f-score, respectively, for Oshumed dataset. Considering both precision and recall scores in 8, PA shows the best performance, followed by SGD, SVM. Strangely C4.5 shows the lowest performance. Regarding TWS, tf.or outperforms clearly all other methods except when used in conjunction NC. tf.rf , tf.gr and tf.ig have close results, and come second, followed by tf and tf.idf . tf.icf performs poorly. For these two datasets, we can note that, in comparison with the other algo- Overall, in our study, we find that tf.or is the best TWS. tf.idf , tf.gr and tf.ig are also good choices for weighting features. tf.χ 2 , tf.icf and tf.rf are the worst methods.

Conclusion

The aim of this paper is to give an insight into the different TWS available for TC. These schemes are used in conjunction with five classifers tested on Reutrers-21578, Oshumed and 20newsgroups datasets. Our work aims at extending previous surveys and establishing a clean and fair basis for TC benchmarks.

To sum up, we find that the superiority of supervised term weighting methods over unsupervised methods is still not clear. Even though, in our experiment, tf.or gives better results than tf.idf , we find no consistent superiority. In addition, tf.idf is shown to be superior that the three supervised methods tf.rf , tf.χ 2 and tf.icf . We find also that, alongside with tf.or which gave the best results, tf.idf , tf.gr and tf.ig are good choices for weighting features.

2 N

 2 × ((w × z -x × y) 2 )/((w + y)(x + z)(w + x)(y + z)) ig ((w/N ) × log(w × N )/((w + x)(w + y))) + ((y/N ) × log(y × N )/((y + z)(w + y))) +((x/N ) × log(x × N )/((w + x)(b + z))) + ((z/N ) × log(z × N )/((y + z)(b + z))) gr ig/((-(w + y)/N )(log(w + y)/N ) -((x + z)/N )(log(x + z)/N )) or log (2 + (w * z)/(x * y)) rf log (2 + (w/ max(1, x))) icf log 2 (C/Ci)

Table 1 :

 1 Collection Frequency Components. Given a term t and a category c, N stands for the total number of documents, |C| is the total number of categories and |C

t | is the number of categories where the term t occurs, w is the number of documents that contain t and belong to category c, x is the number of documents that contain t and do not belong to c, y is the number of documents that do not contain t and belong to cj, z is the number of documents that do not contain t and do not belong to c.

  is one of the most used test collection for TC research. We use the "ModApte" split which contains 90 categories. The second dataset is extracted from the Oshumed 2 collection compiled by William Hersh. The last test collection used in our experiment is the 20 Newsgroups. The dataset "20news-bydate" 3 is sorted by date and splitted into training set (about 60%) and test set (about 40%). Duplicates are removed. Newsgroup-identifying headers (Xref, Newsgroups, Path, Followup-To, Date) are also removed. In all three test collections, we applied lower case transformation, word stemming and stop word removal. No additional preprocessing steps or feature selection is performed. Reuters-21578 and Oshumed are multi-labelled datasets. 20Newsgroups is a multi-class dataset. In all cases, we transform the task into multiple binary single label tasks using the one-vs.-all transformation strategy aka one-vs.-rest. Table2shows some statistics on the three collections.

	4 Results and Discussion
	4.1 Experiments
	Three widely-used datasets are used to evaluate the classifiers: Reuters, Os-
	humed and 20 Newsgroups.
	Reuters-21578 1 Reuters	Oshumed	Newsgroups
	# documents	7769/3019 6286/7643 11314/7532
	# terms	26000	30198	101322
	# categories	90	23	20
	size of the smallest category	1/1	65/70	377/251
	size of the largest category	2877/1087 1799/2153	600/399
				5 has re-
	ceived a considerable amount of attention after being ranked first in the Top 10
	Algorithms in Data Mining [16].

Table 2 :

 2 Statistics on the three test collections (train data/test data).

Table 3 :

 3 µ/m-averaged precision results (%) on Reuters-21578 dataset using different weighting methods.

		PA	SVM	SGD	NC	C4.5
	tf	91.50/62.69 94.37/56.75 94.40/56.64 39.22/30.28 82.17/57.17
	tfchi2	91.35/63.23 94.37/56.75 94.46/55.54 39.22/30.28 82.18/56.23
	tfgr	91.26/61.37 94.37/56.75 94.48/57.21 39.22/30.28 82.44/55.26
	tficf	93.21/64.03 94.95/57.31 94.69/61.25 48.87/50.00 81.64/55.34
	tfidf	93.12/64.14 95.17/56.95 94.45/58.85 63.40/47.57 81.82/56.65
	tfig	91.56/62.63 94.37/56.75 94.48/56.68 39.22/30.28 82.45/58.53
	tfor	91.73/63.42 94.37/56.75 94.47/56.62 39.22/30.28 82.07/56.24
	tfrf	91.51/60.75 94.37/56.75 94.45/55.52 39.22/30.28 81.93/55.63

Table 4 :

 4 µ/m-averaged recall results (%) on Reuters-21578 dataset using different weighting methods.

		PA	SVM	SGD	NC	C4.5
	tf	82.27/42.74 78.85/33.51 79.73/35.08 89.93/61.76 81.62/53.79
	tfchi2 81.76/42.64 78.85/33.51 79.62/34.82 89.93/61.76 81.41/52.91
	tfgr	82.27/41.55 78.85/33.51 79.54/34.81 89.93/61.76 81.62/51.81
	tficf	79.59/39.44 75.27/30.64 77.19/33.20	86.75/52.96 80.26/51.91
	tfidf	82.02/41.81 78.37/33.60 80.02/36.29	87.55/55.60 80.80/53.65
	tfig	82.61/41.93 78.85/33.51 79.51/34.81 89.93/61.76 81.70/53.51
	tfor	82.10/42.59 78.85/33.51 79.46/34.66 89.93/61.76 81.68/53.32
	tfrf	82.00/41.15 78.85/33.51 79.57/34.75 89.93/61.76 81.97/52.97

µ/m scores considering all classifiers and all TWS are bolded. The pair that have the highest average is choosen as the best.

Table 5 :

 5 µ/m-averaged f-score results (%) on Reuters-21578 dataset using different weighting methods.

		PA	SVM	SGD	NC	C4.5
	tf	86.64/48.48 85.91/39.74 86.45/41.15 54.61/34.75	81.90/53.63
	tfchi2	86.29/48.51 85.91/39.74 86.41/40.80 54.61/34.75	81.79/53.24
	tfgr	86.53/47.14 85.91/39.74 86.37/41.14 54.61/34.75	82.03/51.79
	tficf	85.87/46.42 83.97/37.77 85.05/40.28 62.52/46.43	80.94/52.05
	tfidf	87.22/48.20 85.95/40.32 86.64/42.73 73.55/47.05	81.31/53.36
	tfig	86.86/47.76 85.91/39.74 86.35/40.97 54.61/34.75 82.08/54.24
	tfor	86.65/48.48 85.91/39.74 86.32/40.85 54.61/34.75	81.87/52.82
	tfrf	86.49/46.74 85.91/39.74 86.37/40.76 54.61/34.75	81.95/52.82

Table 6 :

 6 µ/m-averaged precision results (%) on Oshumed dataset using different weighting methods.

		PA	SVM	SGD	NC	C4.5
	tf	71.13/73.55	78.77/81.13 79.85/82.42 39.22/35.64 57.09/53.40
	tfchi2 64.72/61.40	72.81/71.56 71.57/69.70 47.34/45.23 58.22/56.02
	tfgr	76.17/78.21 81.04/80.14 80.67/79.39 58.65/58.85 56.72/52.89
	tficf	74.27/75.04	80.80/81.07 77.92/77.81 69.32/68.58 55.63/53.09
	tfidf	75.76/78.26	80.83/80.36 80.48/79.11 54.40/53.06 57.54/53.61
	tfig	76.14/77.84 81.04/80.14 80.81/79.49 58.65/58.85 56.84/53.65
	tfor	74.25/76.45	79.74/81.91 79.44/81.19 53.58/53.61 57.34/54.62
	tfrf	74.08/76.38	80.29/83.20 80.39/82.11 52.24/52.12 57.64/54.63

Table 7 :

 7 µ/m-averaged recall results (%) on Oshumed dataset using different weighting methods.

		PA	SVM	SGD	NC	C4.5
	tf	52.91/44.32 46.21/35.96 47.27/37.76 68.04/66.55 56.08/51.73
	tfchi2 56.50/51.22 54.83/48.33 50.77/45.94	64.82/65.07 56.70/52.42
	tfgr	54.89/47.80 48.61/40.01 52.08/44.79	66.60/64.62 56.89/52.70
	tficf	45.57/40.16 35.50/29.46 42.02/36.32	51.58/47.07 57.43/52.71
	tfidf	53.55/45.80 46.82/37.43 50.19/42.00	67.71/65.54 56.35/52.50
	tfig	54.76/47.84 48.61/40.01 51.96/44.73	66.60/64.62 57.36/53.67
	tfor	58.15/53.84 54.68/48.36 56.53/51.26	66.14/66.29 55.89/51.46
	tfrf	56.38/50.68 52.32/44.55 53.69/46.72	66.32/65.98 55.53/51.79

Table 8 :

 8 µ/m-averaged f-score results (%) on Oshumed dataset using different weighting methods.

		PA	SVM	SGD	NC	C4.5
	tf	60.68/53.95 58.25/47.02 59.39/48.78 49.76/44.48 56.58/52.42
	tfchi2	60.33/55.51 62.55/55.27 59.40/52.01 54.72/51.83 57.45/53.88
	tfgr	63.80/58.11 60.77/51.71 63.29/56.05 62.37/60.16 56.80/52.65
	tficf	56.48/51.32 49.33/41.93 54.60/48.32 59.15/55.25 56.51/52.67
	tfidf	62.75/56.42 59.30/49.08 61.83/53.41 60.33/57.43 56.94/52.88
	tfig	63.71/58.12 60.77/51.71 63.25/56.02 62.37/60.16 57.10/53.47
	tfor	65.22/62.37 64.87/58.78 66.05/60.57 59.20/57.43 56.60/52.76
	tfrf	64.03/60.08 63.36/55.52 64.38/57.19 58.44/56.05 56.56/53.00

Table 9 :

 9 µ/m-averaged precision results (%) on 20newsgroups dataset using different weighting methods. Scores for Newsgroups dataset are presented in tables 9, 10 and 11. tf.ig and tf.gr record the best scores (70%/69%) in conjunction with both SVM and SGD. Overall, in this dataset, SVM performs the best, followed by SGD and PA. C4.5 records very low scores. As for TWS, tf.ig and tf.gr give the best results, followed closely by tf.or, tf.idf and tf.icf . tf.rf shows the lowest scores. In constrast to the high recall scores and low precision scores registered by NC algorithm on Reuters-21578 and Oshumed datasets, NC registered approximately equal results on both precision and recall. Concerning C4.5, we can note that precision and recall results are approximately equal on the three datasets.

		PA	SVM	SGD	NC	C4.5
	tf	63.91/63.77 66.94/66.58	61.05/62.78 55.91/62.22 44.07/44.12
	tfchi2 58.55/60.54 60.26/60.35	59.33/59.51 47.73/60.20 38.20/38.16
	tfgr	68.43/68.37 69.69/69.41 70.19/70.06 62.85/71.44 43.07/43.37
	tficf	68.14/68.23 69.15/69.23	69.24/68.85 59.43/71.87 49.19/51.77
	tfidf	68.31/68.10 69.69/69.29	61.26/66.59 64.27/69.19 43.65/43.67
	tfig	68.97/68.86 70.14/69.79	70.26/69.85 63.64/71.56 44.16/44.40
	tfor	68.57/68.19 69.80/69.26	69.54/69.24 56.44/69.03 45.13/44.77
	tfrf	56.00/55.42 57.73/56.95	56.57/55.36 36.56/46.22 42.22/42.58
	Table 10: µ/m-averaged recall results (%) on 20newsgroups dataset using dif-
	ferent weighting methods.
		PA	SVM	SGD	NC	C4.5
	tf	63.91/62.87 66.94/65.81	61.05/59.69 55.91/55.17 44.07/43.01
	tfchi2 58.55/57.05 60.26/58.74	59.33/57.82 47.73/47.05 38.20/37.17
	tfgr	68.43/67.33 69.69/68.52	70.19/68.92 62.85/62.10 43.07/42.07
	tficf	68.14/66.96 69.15/67.90	69.24/67.95 59.43/58.62 49.19/48.15
	tfidf	68.31/67.20 69.69/68.48	61.26/59.95 64.27/63.32 43.65/42.73
	tfig	68.97/67.86 70.14/68.93 70.26/68.94 63.64/62.78 44.16/43.15
	tfor	68.57/67.42 69.80/68.52	69.54/68.26 56.44/55.81 45.13/44.05
	tfrf	56.00/54.86 57.73/56.38	56.57/55.10 36.56/36.06 42.22/41.33

http://disi.unitn.it/moschitti/corpora.htm

http://disi.unitn.it/moschitti/corpora.htm

http://qwone.com/ ~jason/20Newsgroups/


Ahmad Mazyad

Fabien Teytaud

Cyril Fonlupt

Learning new Term Weighting Schemes with Genetic Programming

Text Classification (or Text Categorization) is a popular machine learning task which consists in assigning categories to documents. Feature weight methods are classic tools that are used in text categorization in order to assign a score to each term of a document based on a mathematical formula. In this paper, we are interested in automatically generating these formulas based on genetic programming. We experiment the generated formulas on three well-known benchmarks and state of the art classifiers.

Introduction

Text classification is an important task, with numerous applications, such as web search, document classification, information retrieval [START_REF] Deerwester | Indexing by latent semantic analysis[END_REF][START_REF] Pang | Opinion mining and sentiment analysis[END_REF][START_REF] Nigam | Text classification from labeled and unlabeled documents using em[END_REF]. Over time, numerous text classification methods have appeared [START_REF] Deng | A comparative study on feature weight in text categorization[END_REF], such as k-nearest neighbor [START_REF] Yang | Expert network: Effective and efficient learning from human decisions in text categorization and retrieval[END_REF], Naïve Bayes [START_REF] Mccallum | A comparison of event models for naive bayes text classification[END_REF], decision trees [START_REF] Apte | Text mining with decision rules and decision trees[END_REF], neural networks [START_REF] Ng | Feature selection, perceptron learning, and a usability case study for text categorization[END_REF], boosting methods [START_REF] Schapire | Boostexter: A boosting-based system for text categorization[END_REF] and Support Vector Machines [START_REF] Cortes | Support-vector networks[END_REF].

For text classification a crucial point is to represent text documents in a suitable format recognizable by a classifier. In the Vector Space Model (VSM) representation, a document is represented by a vector of terms and each term is associated with a weight. This weight represents how informative/discriminative the correspondent term is. The method which assigns a weight to a term is called Term Weighting Scheme (TWS). Numerous TWS exist and the most used are presented in Section 2. They are generated according to human a priori and mathematical rules. TWS are usually simple mathematical expressions. Unfortunately, depending on the application, it is not easy to know a priori which TWS will be effective.

As expression discovery may naturally be addressed by genetic programming, we are interested in this paper to study whether it is possible to generate TWS automatically using this programming approach. We are interested to know if a stochastic evolutionary process with no information about the complexity, the shape and the size of the expression can find at least competitive discriminative TWS.

The paper is organized as follows : Section 2 presents TWS and the main state-of-the-art methods. In section 3 we present Genetic Programming. Section 3.2 presents how we apply genetic programming to TWS. Section 4 presents the experiments and the results, and then we conclude in section 5.

Term weighting scheme

First, we present Term Frequency-Inverse Document Frequency (tf.idf ) which is the most popular TWS method, proposed by Jones in [START_REF] Jones | A statistical interpretation of term specificity and its application in retrieval[END_REF]. This method is unsupervised (does not take into account the class of a document) and can be formally defined as :

w t,d = tf t,d × log N N t , (1)
where, w The intuition behind this formula is to have a large weight for terms which are frequent in only a few number of documents (having a term present in all documents is not informative).

In [START_REF] Salton | Term-weighting approaches in automatic text retrieval[END_REF], Salton et al. pointed out three main considerations for a text retrieval system that are believed to improve both recall and precision:

• Term Frequency (TF) factor : The TF factor is used to capture the relative importance of terms in a document. This is the first term in the previous formula.

• Collection Frequency (CF) factor : Also called term discrimination. The importance of words in a document (TF factor) does not provide enough discrimination ability. A common word like 'The' is frequent in almost all documents, and then it could not separate a group of documents from the remainder of the collection. Hence a discrimination factor is needed to favor those terms that are concentrated in a few documents of the collection. This corresponds to the second part of the previous formula.

Main known CF factors are presented in Table 1.

• Normalization factor : In text retrieval, all documents are considered equally important, however TF factor will favor large documents over shorter ones, as large documents contains more unique terms and/or greater occurrence values. The normalization factor can eliminate this length effect.

Text classification is a supervised learning task, consequently a document class is known in advance. To compute weights some methods used this information. Such methods are called Supervised Term Weighting (STW). In that context and based on the considerations discussed above, researchers proposed various supervised term weighting to use in feature selection methods namely, χ 2 [START_REF] Deng | A comparative study on feature weight in text categorization[END_REF][START_REF] Debole | Supervised term weighting for automated text categorization[END_REF], information gain [START_REF] Deng | A comparative study on feature weight in text categorization[END_REF][START_REF] Debole | Supervised term weighting for automated text categorization[END_REF], gain ratio [START_REF] Debole | Supervised term weighting for automated text categorization[END_REF], odds ratio [START_REF] Deng | A comparative study on feature weight in text categorization[END_REF], relevance frequency [START_REF] Lan | Supervised and traditional term weighting methods for automatic text categorization[END_REF], and recently inverse category frequency [START_REF] Wang | Inverse category frequency based supervised term weighting scheme for text categorization[END_REF].

Table 1: Four traditional CF factors. Given a term t and a category cat, N stands for the total number of documents, a is the number of documents that contain t and belong to cat, c is the number of documents that contain t and do not belong to cat, b is the number of documents that do not contain t and belong to cat and d is the number of documents that do not contain t and do not belong to cat.

CF

Defined by

idf log( N Nt ) χ 2 N * (a * d-b * c) * (a * d-b * c) (a+c) * (b+d) * (a+b) * (c+d) or log(2 + a * d b * c ) rf log(2 + a max(1,c) )
3 Genetic programming

In this section, we first introduce the Genetic Programming model, and then we present how we adapt the GP model to the TWS.

Presentation

Evolutionary computing is based on the Darwin's theory of "survival of the fittest". The main scheme of evolutionary algorithms is to evolve a population of individuals that are randomly generated. Each individual represents a candidate solution that undergoes a set of genetic operators that allow to mix and alter partial solutions. One of the key features of evolutionary algorithms is that they are stochastic approaches.

Genetic Programming (GP) belongs to the family of evolutionary algorithms. It was first proposed by Cramer in [START_REF] Cramer | A representation for the adaptive generation of simple sequential programs[END_REF] and then popularized by Koza [START_REF] Koza | Genetic programming: on the programming of computers by means of natural selection[END_REF]. Unlike genetic algorithms where the aim is to find out a solution, the goal of GP is to find out a computer program that is able to solve a problem.

In GP, a set of random expressions that usually represent computer programs is generated. As in all evolutionary computation algorithms, this set of programs will evolve and change dynamically during the course of evolution. What makes GP suitable for a number of different applications is that these computer programs can represent many different structures, such as mathematical expressions for symbolic regression, decision trees, programs that control a robot to fulfill a certain task or programs that are able to predict defibrillation success in patients ans so on.

The quality of a candidate solution (i.e. a program) is usually assessed by confronting it with a set of fitness cases. This step is usually the most time-consuming step as the programs may get huge and several thousands of candidate programs are usually evaluated at each generation. These computer programs will undergo one or several evolutionary operators that will alter in a hopefully beneficial way. The most classical evolutionary operators are usually the crossover operator that allows the exchange of genetic material (in our case subtrees) and the mutation operator that allows a small alteration to the program.

In the most conventional GP approach, programs are usually depicted by trees. In GP terminology, the set of nodes are split into two sets, inner nodes of the tree are drawn from a set of functions while the terminal nodes (leaves) are drawn from a so-called terminal set. Depending on the problem, the set of functions can be mathematical functions, Boolean functions, control flow functions (if,...), or any functions that may be suitable to solve the given problem. The terminal set is usually the set of inputs of the problem, e.g., parameters and constants for symbolic regression problems, sensors for robot planning... If the stopping criteria is reached, then the best individual is returned, otherwise, the loop continues and the best individuals are selected (according to their fitness). There exist numerous ways for selecting the population, the mutation and the crossover operators. This is beyond the scope of this paper and the reader can refer to [START_REF] Koza | Genetic programming: on the programming of computers by means of natural selection[END_REF][START_REF] Karakus | Function identification for the intrinsic strength and elastic properties of granitic rocks via genetic programming (gp)[END_REF] for more information.

Term Weighting Scheme using genetic programming

A CF factor is a combination of statistical information. It is intended to measure the discriminative power of a term, that is, it tells how much a term is related to a certain category. These statistical information are combined by means of mathematical operators and functions.

We are interested in automatically evolving a CF factor (an individual) using GP (the learned CF factor combined to the TF factor form a term weighting method)

In our context of automatically evolving term weighting methods, an individual is a combination of the function set that is built with simple arithmetical operators (+,-,*,/,log,...) and the terminal set (constant values and inputs to our problem). A presentation of the implementation and the parameters used in this study will be presented in Section 4.1.

Tables 2 and3 show the statistical information used as terminal set, and the operators for generating formulas (the function set) which represent CF factors. As it can be seen, the function set is made of very simple arithmetical functions while the terminal set includes to the best of our knowledge all the statistical information used to build a TWS.

Experiments and results

In order to validate our approach, we compare the four most well-known traditional weighting methods with GP evolved formulas on the ten most frequent categories of the popular dataset Reuters-21578 using linear Support Vector Machine (SVM) [START_REF] Fan | Liblinear: A library for large linear classification[END_REF]. In all our experiments, we perform a cross-validation process.

Experimental setup

Reuters-21578 is a multi-labeled dataset and one of the most frequently used test collections for text classification research. We use the traditional "ModApte" split which contains 90 categories. The dataset is transformed into multiple single-label binary classification tasks using the binary relevance transformation strategy.

The point is that the ten binary tasks corresponding to the ten most frequent categories are used. No preprocessing steps or feature selection are performed on the resulting dataset.

For our the Liblinear classification library [START_REF] Fan | Liblinear: A library for large linear classification[END_REF] is used with a L2-regularized L2-loss Support Vector classification (dual) solver. The other parameters are set to their default values.

Concerning the GP, we use a population of 100 individuals (candidate solutions, i.e. formulas), and 100 generations (corresponding to the stopping criterion).

The crossover probability is set to 0.85 and the mutation probability is defined as one over the number of terminals and operators.

A validation phase is needed as one cannot learn directly over the test set. Thus, the population of formulas are evaluated by means of stratified 3-fold cross validation over the training dataset.

To assess the performance of STW, we report in the result tables the precision p, the recall r and the standard F 1 measure.

The precision is the true positive tp over the true positive plus the false positive f p. The recall is defined as the true positive over the true positive plus the false negative f n.

The F 1 measure is defined as

F 1(tp, f n, f p) = 2 * tp 2 * tp + f n + f p .
In order to obtain one general score for the multiple tasks, the F1 measure is usually averaged using micro-/macro-averaging methods. Macro-averaging gives equal weight to each category, thus, it gives an insight on the efficiency/effectiveness of a classifier on small categories. In contrast, micro-averaging gives weight depending on the category size (the larger the category, the larger the weight), thus results are dominated by large categories.

The micro-F 1 (µ -F 1) is defined as

µ -F 1 = F 1( C l=1 tp l , C l=1 f p l , C l=1 f n l ) .
The macro-F 1 (m -F 1) is defined as

m -F 1 = 1 C C l=1 F 1(tp l , f p l , f n l ) .
It is important to understand that the macro measure m gives equal weight to each category. The micro measure µ gives equal weight to each per-document classification decision. The main problem of the F1 measure is that it ignores true negatives and its magnitude is mostly determined by the number of true positives, then large classes dominate small classes in terms of micro measure [START_REF] Manning | Introduction to information retrieval[END_REF][START_REF] Van Asch | Macro-and micro-averaged evaluation measures[END_REF].

Results

Table 4 shows the results of the four traditional TWS on the ten categories. The CF column is the best of the four traditional TWS used in this paper (the best between idf, rf, or, χ 2 ). We report the precision, the recall and the F 1-score for the ten categories experimented.

In comparison, in Table 6 we report the same measures but for formulas generated with GP. For each category, the corresponding formulas generated by GP are presented in Table 5.

First, we can notice that for each category the generated formulas are different. The generated formulas (f 1 to f 10 ) perform better on seven out of ten tasks, and the traditional TWS performed better on three tasks (interest, wheat and corn).

In Table 7 we compare µ -F 1 and m -F 1 for the best validated TWS between idf, rf, or, χ 2 for each category against the generated formulas f 1 to f 10 based on GP. In terms of micro score (i.e. µ) the performance is slightly better for the generated formula, and for the macro score (i.e. m) we are able to improve the performance by 1%. Even if the results seem close, it is impressive that we are able to have such performances with automatically generated TWS.

In Table 8 we compare the performances of each TWS with f 11 in terms of micro-/macro-averaged F 1 scores over all categories. f 11 is a formula which has been generated by GP by considering all ten categories. The goal is to experiment whether GP is able to generate a formula which is good on all ten datasets. In term of micro-averaged score, the generated f 11 formula is better than all the other formulas. For the macro-averaged score, results are similar.

Finally, in Table 9 it is interesting to see that five from ten generated formulas (f 4 , f 5 , f 6 , f 8 and f 10 ), which have been generated on one specific category, generalize well on the nine other categories, and are slightly better than the four traditional TWS, (see Table 8).

b * (N t -N c ) acq f 2 C t -(b/(log 1p((N t * d * a * log 2p(log 2p(N t ))) + b)/d)) money-fx f 3 (N + c) -(log 1p(N t ) * d) grain f 4 N t /(c -C t ) crude f 5 d * (N t + N + C + C t -(d * d)) trade f 6 N * (N t -N t + d) interest f 7 d -(((N c + C) + ((c * ((N c * N c ) + a))/N t ))/N t ) ship f 8 d * N t * (N t + N c + C t + b)/c wheat f 9 log 2p(N c + ((N c * N t )/(log 2p((b + C)/C t ) * b)) corn f 10 (c * (N c + N c ) + N t * a * a)/c All f 11 log 1p(d/c) + (N t /(log 1p(log 2p(N c )) * c))

Conclusion

The aim of this paper is to test the ability of GP in automatically generating efficient TWS. Experiments are conducted and the generated TWS are compared with four traditional TWS. In the experiments, the generated formulas are able to compete with the traditional schemes and are even slightly better in some cases.

We also find that the generated formulas have a good generalization ability.

Our future work will focus on generating a TWS that can generalize even better over different datasets. That can be done by learning directly over the test data of different datasets using the GP.

Table 2 :

 2 Statistical information (Terminals) used to evolve a TWS.

	Label Description
	N	Total number of documents
	C	Number of categories
	C t	Number of categories that contain the term t
	N cat	Number of documents in the positive category cat
	N cat	Number of documents that do not belong to cat
	N t	Number of documents that contain t
	N t	Number of documents that do not contain t
	a	Number of documents that contain t and belong to the positive category cat
	b	Number of documents that do not contain t and belong to cat
	c	Number of documents that contain t and do not belong to cat
	d	Number of documents that do not contain t and do not belong to cat

Table 3 :

 3 Arithmetic operators and functions used to evolve a TWS.

	Operator Description
	+	Arithmetic addition operator
	-	Arithmetic subtraction Operator
	*	Arithmetic multiplication operator
	/	Arithmetic division operator

log 1p(x) Natural (base e) logarithm of 1 plus ( ln(1 + x)) log 2p(x) Natural (base e) logarithm of 2 plus ( ln(2 + x))

Table 4 :

 4 Best traditional scheme for each category in top 10 Reuters-21578 categories evaluated by 3-fold cross validation over the training set.

		CF	Precision	Recall	F1
	earn	idf	98.98% 97.79% 98.38%
	acq	idf	98.71% 95.55% 97.10%
	money-fx idf	80.23% 77.09% 78.63%
	grain	rf	96.43% 90.60% 93.43%
	crude	rf	89.19% 87.30% 88.24%
	trade	rf	82.08% 74.36% 78.03%
	interest	idf	88.00% 67.18% 76.19%
	ship	or	95.24% 67.42% 78.95%
	wheat	chi	84.15% 97.18% 90.20%
	corn	chi	91.53% 96.43% 93.91%

Table 5 :

 5 Formulas generated by GP. f 11 is a formula generated on all the ten categories.

		Denoted by Defined by
	earn	f 1

Table 6 :

 6 Formulas generated by GP and evaluated by 3-fold cross validation for each category of the ten biggest categories in Reuters-21578.

		CF Precision	Recall	F1
	earn	f 1	98.70%	98.16% 98.43%
	acq	f 2	98.85%	95.83% 97.32%
	money-fx f 3	81.82%	80.45% 81.13%
	grain	f 4	94.04%	95.30% 94.67%
	crude	f 5	90.22%	87.83% 89.01%
	trade	f 6	83.49%	77.78% 80.53%
	interest	f 7	87.63%	64.89% 74.56%
	ship	f 8	92.00%	77.53% 84.15%
	wheat	f 9	86.84%	92.96% 89.80%
	corn	f 10	86.15% 100.00% 92.56%

Table 7 :

 7 µ -F 1 and m -F 1 for the best validated TWS between idf, rf, or, χ 2 for each category against the generated formulas f 1 to f 10 based on GP. Results are really close, especially for the µ score. It is interesting to see that GP is able to compete with a priori usual known TWS.

		Best rules	GP
	µ-F1	93.19% 93.53%
	m-F1	87.31% 88.42%

Table 8 :

 8 µ-F 1 and m-F 1 for all TWS over all categories against the generated formula f 11 validated over all categories.

		µ -F 1 m -F 1
	idf 92.52%	84.90%
	rf	92.56%	85.68%
	or	92.54%	85.93%
	χ 2	91.59%	87.18%
	f 11 93.13%	86.55%

Table 9 :

 9 µ -F 1 and m -F 1 over all categories for the generated formula f 1 to f 10 . We can note that the generated formula reach good performance even on the other categories, which is important in term of finding a global rule.

		µ -F 1 m -F 1
	f 1	91.51%	80.77%
	f 2	91.06%	79.40%
	f 3	91.77%	83.32%
	f 4	92.13%	86.30%
	f 5	92.82%	85.59%
	f 6	92.68%	85.24%
	f 7	92.44%	84.65%
	f 8	92.39%	86.96%
	f 9	89.47%	75.78%
	f 10 91.10%	86.82%


P J Masset
email: masset@dechema.de

D Texier

M Schu

Coefficients of thermal expansion of (Ni 0?5 Al 0?5 ) (12x) Hf x alloys (x50…0?2)

Keywords: Coefficients of thermal expansion, CTE, NiAl alloys, Al-Ni-Hf, high temperature

The coefficients of thermal expansion (CTEs) of Ni-Al-Hf alloys have been investigated due to their potential as corrosion resistant coatings in aggressive environments at high temperature. The CTEs of (Ni 0?5 Al 0?5 ) (12x) Hf x alloys (x50…0?2) were measured between 473 and 1073 K (200-800uC) and range between 8 and 15610 26 K 21 depending on temperature and alloy composition. The temperature dependence of the CTE was found to be properly described by a simple expression: (19?982-0?24016X Hf )[0?15182z1?8287610 23 (T/K)22?12378610 26 (T/ K) 2 z8?87011610 210 (T/K) 3 ]. It was shown that the CTE of (Ni 0?5 Al 0?5 ) (12x) Hf x alloys varies linearly with the amount of Hf in the alloy which allows the control of the CTE. Within the (Ni 0?5 Al 0?5 ) (12x) Hf x alloy composition range (x50…0?2) it covers the CTE range of most of the austenitic and ferritic steels and the CTE of the coatings can be adjusted to those of the substrate to be protected without inducing additional strains at the coatings/substrate interface.

Introduction

In high temperature processes, materials can face aggressive chemical environments sometimes even under mechanical load. Usually, the construction materials are designed to offer the best mechanical properties to fit the industrial requirements. However, they do not always sustain aggressive environments, e.g. chlorine containing atmospheres in waste combustion processes. As an alternative, coatings may be applied onto the material surface and act as a physical barrier against the environment to reduce the degradation rate. The selection of the coating material should fit the following two requirements:

(i) it should be more resistant against the environment than the base material (ii) the coefficient of thermal expansion (CTE) should match that of the material to be coated. For processes where the environment contains chlorine, it has been shown that NiAl based thermal sprayed coatings were resistant against high temperature corrosion attack. 1 However the CTE of NiAl does not fit all of the CTEs range of the ferritic and austenitic stainless steels. 1,2 The properties of NiAlzX alloys (X5Hf, Cr …) have been extensively investigated for their use as high temperature structural materials. Hafnium was found to be an effective solid solution strengthener for NiAl alloys at high temperature. [3][4][START_REF] Darolia | Field:Scr[END_REF] The phase diagram of the Al-Ni-Hf system 6-9 exhibits three ternary intermetallic phases which can be in equilibrium with the b-NiA1 phase, namely, NiAlHf (Laves), Ni 2 AlHf (Heusler) and Ni 7 (Hf, Al) 2 . The solubility of Hf in the b-NiAl phase was found to be ,5 at.-%. [START_REF]Ordered intermetallics -physical metallurgy and mechanical behaviour[END_REF] Very recently, the ternary system has been optimised and thermodynamic data were made available, especially in the Ni rich corner. [START_REF] Zhang | [END_REF] Although these alloys have been widely investigated for their mechanical properties, reported values of the CTEs of Al-Ni-Hf alloys are scarce [START_REF] Haynes | [END_REF][12][START_REF] Touloukian | Thermophysical properties of matter thermal expansion[END_REF] and need further investigation. The evolution of CTEs of NiAl, NiAl 3 and NiAlHf alloys with temperature is plotted in Fig 1 with data taken from the literature. [START_REF] Haynes | [END_REF][12][START_REF] Touloukian | Thermophysical properties of matter thermal expansion[END_REF] The available CTE data for the ternary alloys are only for very low Hf content and does not show any influence of the hafnium content. In addition predictive models developed to calculate CTEs of multi-element alloys do not reproduce the experimental data with a high level of confidence.

It is apparent that new experimental measurements may bring new data to be used as a basis for further modelling. Therefore, the aim of the present work is to investigate the composition and temperature dependence of the CTE of (Ni 0?5 Al 0?5 ) (12x) Hf x alloys (x varies from 0 to 0?2) through the substitution of NiAl by Hf.

Experimental

Alloys were prepared from 99?99%Ni, 99?99%Al and Hf (containing ,1 at.-%Zr). Five compositions of (Ni 0?5 Al 0?5 ) (12x) Hf x alloys were prepared by centrifugal casting in an induction furnace and solidified in an Ar cooled copper mould. Cylindrical ingots (10 cm in length and 15 mm in diameter) of ,40 g were obtained. Their nominal compositions are reported in Table 1. For each a small portion was used for microstructural analyses. For the microstructural characterisation the specimens were mechanically polished with 4000 grit SiC abrasive paper and then ultrasonically washed in acetone and ethanol for 10 min. The samples were post-experimentally investigated by metallography and scanning electron microscopy (SEM). Before observation, the samples were gold sputtered, electrochemically coated with Ni and embedded in an epoxy resin. Scanning electron microscopy images were taken by an XL40 electronic microscope from Philips coupled with energy dispersive analysis of X-rays (EDAX). In addition, X-ray diffraction (XRD) patterns were recorded on a Siemens D500 using a Cu K a cathode (H/2 H mode) every 0?02u.

The as cast specimens for thermal expansion measurements were machined via electrodischarge machining to parallelograms of dimension 2610615 mm. Expansion measurements were made between room temperature and 800uC and with a heating and cooling rate of 3 K min 21 using a Linseis1500 dual push rod differential dilatometer. Data were recorded at 30 s intervals. The dilatometer was of horizontal configuration with push rods and sample holder constructed of high density alumina. A platinum/Pt-10%Rh type S thermocouple was used to monitor the sample temperature. Oxidation of the samples during the dilatometer runs was minimised by maintaining under a reductive Ar-5%H 2 atmosphere (flow: 5 mL min 21 ) a slight overpressure. The error due to the growth of an oxide layer was found to be ,3% and was neglected. Three measurements from 20 to 800uC were made on all the specimens to ensure repeatability.

Results and discussion

Thermodynamic calculations

Thermodynamic calculations were carried out using the software FactSage. The stability diagrams of the pure elements (Ni, Al and Hf) were calculated as a function of the chlorine and oxygen partial pressures at 1073 K (800uC). This temperature represents the highest temperature for the application envisaged and from the point of view of corrosion. The computed stability diagrams were superimposed on each other in Fig. 2 to evidence the area of stability of the alloys studied. In these calculations, the activity of the elements was not taken into account. The stability area of HfO 2 is even slightly larger that that of alumina. It is therefore thought that hafnium should have a good corrosion resistance in chlorine containing atmospheres, even for low oxygen partial pressures. Considering the amount of Al in the prepared alloys the growth of a pure alumina layer is expected.

Alloy characterisation

The alloys of general composition (Ni 0?5 Al 0?5 ) (12x) Hf x were prepared by centrifugal casting under vacuum or argon. The nominal and measured compositions of the prepared alloys were measured by means of EDAX and are summarised in Table 2. The differences in composition may be ascribed to the vaporisation of the pure metals during alloy processing at high temperature.

The microstructures of the as cast alloys are shown in Figs. 3 and4. The local composition of each phase has been determined by EDAX measurements (Table 2). The microstructures of the alloys are of the b-NiAl and of the Ni 2 AlHf Heusler type phases evidenced by XRD analysis (Fig. 5).

Coefficient of thermal expansion measurements

The CTEs of the (Ni 0?5 Al 0?5 ) (12x) Hf x alloys were measured using a L75/1550 dilatometer from Linseis. The measurements were carried out between room temperature and 1073 K (800uC) under reducing Ar-5%H 2 atmosphere. This was conducted in order to minimise the formation of oxides which would hamper the determination of the true CTE of the materials. Reproducible results were obtained with heating rates (3 K min 21 . A blank measurement was carried out as reference by using the same thermal cycle and the measurements were repeated three times. The provided values are the mean values of three measurements and are reported in Table 3. Between room temperature and 1073 K (800uC) the CTEs of the alloys vary from 8 to 16610 26 K 21 depending on the temperature and alloy composition. In the temperature range of the CTE measurements, no microstructural changes were expected, since the kinetics of the diffusion processes should be slow for this temperature range. This was confirmed by the good reproducibility of the CTE versus temperature curves. From the data analysis it was found that the temperature and composition dependence could be considered separately. It means that the analytical expression of the CTE of the alloys investigated could be expressed as the product of a temperature and composition contribution (equation. ( 1)). The parameters were optimised using all the experimental data and the curves were fitted using a multilinear regression procedure.

a T NiAlHf ~(a 0 za 1 X Hf )(b 0 zb 1 Tzb 2 T 2 zb 3 T 3 ) ( 1
)
where T is the temperature (in K); a 0 and a 1 are the parameters which describe the concentration depen- For each alloy composition the analytical expression of the CTE was derived from equation ( 2) and is reported in Table 4. The experimental points as well as the fitted curves are plotted in Fig. 6. The same trend was observed in all five curves. Between room temperature and 1073 K (800uC) the CTE values increase monotonously with the temperature. The curvature of the curve is given by equation (3).

The curvature changes and becomes positive for higher temperatures when the second derivative annuls, i.e.

L 2 a Tc NiAlHf LT 2 X Hf ~CTE ~0
This occurs at temperature T c independent of hafnium content, i.e. X Hf , when T c is equal to 22b 2 /6b 3 . Using the optimised set of parameters, the calculation of T c leads to the value of 797 K (524uC). One of the reasons for these 96 (1) changes with temperature may be ascribed to a possible disordering/ordering transition due to the magnetic contribution in the NiAl based alloy. However at that time no evidence is available to sustain this hypothesis. The dependence of the CTE of the (Ni 0?5 Al 0?5 ) (12x) Hf x alloys on the hafnium content in the alloys was investigated as well. Figure 7 shows the evolution of the CTE versus Hf content for different temperatures. A linear dependence was observed between the CTE values and the Hf content in the alloy. The slopes for the variations of the CTE with Hf content appear almost constant with temperature and equal to 20?2402¡0?007 K 21 /at.-%Hf. According to these measurements the CTE of the (Ni 0?5 Al 0?5 ) (12x) Hf x alloys can be tuned by adjusting the hafnium content in the alloys. In addition, the variations of the CTE with the hafnium content appeared to be independent of the microstructure of the alloys and the phase arrangement.

The improvement of the corrosion behaviour is primarily linked to the alloy composition and its chemistry. The alloys produced can be considered as alumina formers as the aluminium content in the alloys is high. For the corrosion behaviour itself no change is expected from the modification of the CTEs. However, the CTEs of the coating can be adjusted to those of the substrate by adding hafnium. Since the stresses at the coating/substrate interface are proportional to the difference of the CTEs between coating and substrate, it can be expected that tendency for coating delamination or spalling can be reduced, especially under thermal cycling. In this respect, the whole system should offer a better corrosion protection of the substrate in chlorine containing atmospheres by combining optimised chemical and physical properties of the coating.

Conclusions

The main conclusions of this work are as follows.

1. (Ni 0?5 Al 0?5 ) (12x) Hf x alloys were cast using a centrifugal inductive furnace and their microstructures were investigated. For a substitution level of x50?2 the alloy exhibits an eutectic type microstructure.

2. The coefficients of thermal expansion of (Ni 0?5 Al 0?5 ) (12x) Hf x alloys were measured between room temperature and 800uC and can be described by (19?982-0?24016X Hf )[0?15182z1?8287610 23 (T/K)2 2?12378610 26 (T/K) 2 z8?87011610 210 (T/K) 3 ] 3. The CTEs of the alloys vary linearly with the hafnium content in the alloys.

4. The CTEs of the alloys may be tuned by adjusting the hafnium content.

5. The CTEs of the alloys seem to be independent of the alloy microstructure.

  dence; b 0 , b 1 , b 2 and b 3 are the parameters which represent the temperature dependence and X Hf the hafnium molar fraction is expressed in at.-%. The expression of the CTE for the alloys (Ni 0?5 Al 0?5 ) (12x) Hf x (x50…0?2) is given by equation (2).a T NiAlHf ~(19:982{0:24016X Hf ) 0:15182z0:8287|10 {3 (T=K){2:12378 Â |10 {6 (T=K) 2 z8:87011|10 {10 (T=K) 3 Ã (2)

2 !

 2 a x50?02; b x50?05; c x50?1; d x50?2 3 Images (SEM) of alloy microstructures of composition (Ni 0?5 Al 0?5 )(12x)Hf x L 2 a T NiAlHf LT X Hf ~CTE ~(a 0 za 1 X Hf )(2b 2 z6b 3 T)

5 X

 5 -ray diffraction spectra of alloys (Ni 0?5 Al 0?5 ) (12x) Hf x 4 Detailed SEM images of alloy microstructure of alloy (Ni 0?5 Al 0?5 ) 0?8 Hf 0?2

Table 1

 1 Nominal and measured (in brackets) compositions of produced alloys by centrifugal casting

		Nominal composition
		(measured composition), at.-%
	Alloy	Ni	Al	Hf
	Ni 0?5 Al 0?5 Ni 0?49 Al 0?49 Hf 0?02 Ni 0?475 Al 0?475 Hf 0?05 47 . 5 (44 . 6) 50 . 0 (50 . 5) 49 . 0 (50 . 4) Ni 0?45 Al 0?45 Hf 0?1 45 . 0 (45 . 1) Ni 0?4 Al 0?4 Hf 0?2 40 . 0 (39 . 8) 40 (42 . 0) 50 . 0 (49 . 5) 49 . 0 (47 . 5) 47 . 5 (50 . 6) 45 (44 . 4)	-10 (10 . 6) 2 (2 . 1) 5 (4 . 8) 20 (18 . 2)
	Table 2 Measured compositions of phases in alloys (Ni 0?5 Al 0?5 ) (12x) Hf x by EDAX
		Phase 1			Phase 2			Phase 3
	Alloy	Ni	Al	Hf	Ni	Al	Hf	Ni	Al	Hf
	Ni 0?5 Al 0?5 Ni 0?49 Al 0?49 Hf 0?02 Ni 0?475 Al 0?475 Hf 0?05 Ni 0?45 Al 0?45 Hf 0?1 Ni 0?4 Al 0?4 Hf 0?2	50 . 5 50 . 3 55 . 4 45 . 2 39 . 8	4 9 . 5 4 9 . 7 4 4 . 6 5 3 . 3 4 2	---1 . 5 1 8 . 2	-5 8 . 7 3 4 . 3 3 5 . 1 -	-3 0 . 2 3 8 . 3 3 5 . 4 -	-1 1 . 1 2 7 . 4 2 9 . 5 -	-5 6 . 7 4 6 . 2 4 6 . 8 -	-3 9 . 4 3 1 3 0 . 9 -	-2 2 . 8 3 . 9 2 2 . 3 -

Table 3

 3 Experimental values of CTE of (Ni

	0?5 ) (12x) Hf x alloys
	0?5 Al
	CTE, 10 26
	Nominal composition,	at.-%

Coefficients of thermal expansion values of NiAl, Ni 3 Al and NiAl-Hf alloys taken from literature

Stability diagrams of pure metals Al (green dotted line), Hf (red dashed line), Ni (black full line) function of chlorine and oxygen partial pressures at 1073 K (800uC) (For the full colour version of this paper go to www.ingentaconnect.com)

Acknowledgements

The authors wish to sincerely thank Mrs Ellen Berghof-Hasselba ¨cher and Mrs Daniela Hasenflug (metallographic examinations) and Mr Michael Jusek (XRD), Mr Peter Gawenda (SEM analysis). One of the authors (DT) acknowledges the financial support for a mobility grant from the region Languedoc-Roussillon (France).