Fedorova, Valentina

Gammerman, Alexander

Ivin, Evgeny

Nouretdinov, Ilia

The paper considers an application of martingales for testing the hypothesis that the data are generated by the Gauss linear model. In addition, the martingales are also used for suggesting some possible clustering of the data. A Russian household dataset is taken to show results of this approach. In particular, with the help of martingales the households were split into groups with a similar structure of expenditure. It was assumed that within each group dependence of the household total consumption on its income can be described by the Gauss linear model. The structure of expenditure was tested for the households within the groups.<br/

Royal Holloway - Pure

Testing and Clustering with Gauss linear assumptionfor a household dataValentina Fedorova,Ilia Nouretdinov,Alex GammermanComputer Learning Research CentreRoyal Holloway, University of London{name}@cs.rhul.ac.ukEvgeny IvinMoscow School of EconomicsLomonosov Moscow State UniversityAbstractThe paper considers an application of martingales for testing the hypothesis thatthe data are generated by the Gauss linear model. In addition, the martingales arealso used for suggesting some possible clustering of the data. A Russian householddataset is taken to show results of this approach. In particular, with the helpof martingales the households were split into groups with a similar structure ofexpenditure. It was assumed that within each group dependence of the householdtotal consumption on its income can be described by the Gauss linear model. Thestructure of expenditure was tested for the households within the groups.Keywords: testing assumption, martingales, conformal prediction, clustering.1 IntroductionIn many applications an assumption is made that the data are generated by the Gausslinear model - that our response variables y are linearly dependent on the vectors ofexplanatory variables x with the normally distributed random noise ξ:y = β1 + β2 · x+ ξwhere β1 is unknown coefficient and β2 is a vector of coefficients of the same length asx. Our aim is to test this assumption for a special class of predictors, called conformalpredictors (Vovk et al., 2005). Conformal predictors allow us to estimate confidence of thepredictions and their main property is validity: for a given significance level  the numberof errors would not exceed . It has been proved that this property of validity holds formany different models including i.i.d. model1, Gauss linear model and some other models.The underlying models (assumptions) can be tested using martingales: they allow us toaccumulate evidence against the assumptions. That is if the martingale value is largethen we reject the assumption. In this paper we use the martingale testing for the Gausslinear assumption and present results of the testing for a Russian household dataset.1independent and identically distributed data1The martingales can also help to split the data into certain clusters when a mar-tingale changes its behavior. In the work Ho (2005) a similar problem of splitting wasconsidered for the i.i.d. assumption: how the data sequence may be split by a martingaleinto several i.i.d. clusters. Within each of them data examples followed a random order.However, the Gauss linear assumption does not require anything of the order of examples(x1, y1), . . . , (xn, yn), so we can sort them in a way convenient for testing our assumption.In this work we use ranking by xi (that is one-dimensional). As we will see, this allowsus to find deviations from the assumption effectively and to clusterize the data by thedeviations so that a cluster is a group of households with a similar level of income anda structure of expenditure. This direction of study was mostly motivated by the workAivazian and Kruglyak (2010) where authors describe the data as a mixture of severalclusters with the normal distributions.The rest of the paper is organized as following: in section 2 we briefly describe thedata and used model, in section 3 we outline the concepts of martingale testing, in section4 we explain the ideas of our clustering, in section 5 we present results of analysis of thedata and finally we summarize the work in section 6.2 DataWe will study the part of the Russian households dataset (http://www.micro-data.ru)collected during the first quarter of 2003 year. It consists of 53149 households, eachhousehold is described by several attributes. The attributes could be numerical (i.e.income, number of members of family etc.) or categorized (i.e. householder’s profession).To split the dataset into clusters we consider only links between the income budget ofhousehold and its total consumption. We use the following Gauss linear model:log10(consumption) = β1 + β2 · log10(income) + ξ, (1)where β = (β1, β2) is a real number vector of coefficients and ξ ∼ N(0, σ) is independentand normally distributed noise.Figure 1 plots the data on the logarithmic scale of income and consumption. Thedark dashed line represents function log10(consumption) = log10(income).The paper will study households with the income budget more than 500 rubles duringthe first quarter of 2003 year. We suppose that for the rest of households studied timeperiod does not give enough information on their income. The dark solid vertical line infigure 1 shows the income level of 500 = 102.7 rubles. We can see that the points thatlie to the left of the line look like outliers. From this considerations for our analysis wewill use the dataset of 53082 households with income greater than 500 rubles during thequarter.3 TestingIn this section we describe the testing of the hypothesis that data agree with the Gausslinear model. It proceeds in two steps. The first one is a calculation of p-values as anoutput of conformal predictors. And the second one is a calculation of martingales thatreflect the “correctness” of the assumption.2Figure 1: The Russian household data is given for the first quarter of 2003 year: income and total consumption are on thelogarithmic scale. The dark dashed line is for log10(consumption) = log10(income) and the dark solid vertical line is forincome level of 500 rubles (log10(500) = 2.7 in the scale).3.1 Conformal predictorsWe start testing by obtaining a sequence of p-values using conformal predictor for theGauss linear assumption. Recalling the model (1) we can denote i-th example of thedata as (xi, yi), where xi =(1, log10(income))is a vector of p = 2 attributes and yi =log10(consumption) is a label. To describe the process of p-value calculation we willuse the following notation: yˆli is the Least Square prediction for xi, based on examples{(x1, y1), . . . , (xl, yl)}; yˆn is shorthand for yˆn−1n ; s2n−1 =1n−p−1∑n−1i=1 (yi − yˆn−1i )2 is thestandard estimation of variance of ξ from the first l examples; Xl is a matrix consists ofrows x1, . . . , xl; Yl is a column-vector of labels y1, . . . yn; symbol′ denotes operation oftransposition.To calculate p-values by a conformal predictor we specify a “non-conformity” measureof an example (xi, yi) with respect to the given examples{(x1, y1), . . . , (xl, yl)}asαi = |yi − yˆli|. (2)The “non-conformity” measure is a way of scoring how i-th example is different from therest of given examples. It is known that the random variable:ti =yi − yˆisi−1√1 + x′i(X′i−1Xi−1)−1xihas a t-distribution with i − p − 1 degrees of freedom (see, eg. Ryan, 1997, p. 17).Consider the value ti provided by the given data and denote t|ti|i−p−1 for the |ti|-th quintileof t-distribution with i − p − 1 degrees of freedom. Then p-value is estimated by theequationpi = 2 · (1− t|ti|i−p−1).3This equation for p-value agrees with the equation (2) for “non-conformity” measure (fordetails see Vovk et al., 2005, pp. 202 - 203). A p-value is a number from [0, 1]. It is closeto 1 if the pair (xi, yi) fits well to the model (1) with the parameters estimated by thegiven examples{(x1, y1), . . . , (xl, yl)}.To obtain a sequence of p-values we use conformal predictor in on-line mode (exam-ples arrive one after another) and the algorithm 1 summarizes this process.Algorithm 1 Generating p-valuesInput:{(x1, y1), . . . , (xn, yn)}– data, sorted by xiOutput:(p1, . . . , pn)– sequence of p-valuesfor i = p+ 1 to n doXi−1 = x1...xi−1Yi−1 = (y1 . . . yi−1)′(xi, yi) is a new exampleβˆ = (X′i−1Xi−1)−1X′i−1Yi−1yˆi = xi · βˆs2i−1 =1i−p−1∑i−1j=1(yj − yˆi−1j )2ti =yi−yˆisi−1√1+x′i(X′i−1Xi−1)−1xifind quintile of t-distribution t|ti|i−p−1pi = 2 · (1− t|ti|i−p−1)end forIn the paper we assume only the Gauss linear model (1) for dependence between xiand yi. It allows us to put data in the order relevant for the problem (sort the householdsby the value of income). Output of the algorithm 1 is a sequence of p-values: that is thep-value pi that corresponds to the data point (xi, yi). It was proved that the p-valuesdistribute independently and uniformly in [0,1] if the assumption of conformal predictoris satisfied (see Vovk et al., 2005, Theorem 8.1).3.2 MartingalesThe main tool for testing our assumption is martingales. Denote zi = (xi, yi), i = 1, . . . nand assume that each zi is generated by the probability distribution P. Then a sequenceof non-negative random variables M0,M1, . . . is a martingale if each Mn is a measurablefunction of z1, . . . , zn andMn = E(Mn+1|M1, . . . ,Mn),where E refers to the expected value in the probability space P. We set M0 = 1. Afterobserving a new example, the martingale value reflects the strength of evidence againstthe assumption that probability distribution P generates the data.4According to Ville’s inequality (see Ville, 1939, p. 100)P{∃n : Mn ≥ C}≤ 1/C, ∀C > 0.it is unlikely that any Mn would have a large value.3.2.1 Martingale based on p-valuesIn this paper we use the so called sleepy jumper martingale introduced in Vovk et al.(2003) as a generalization of power martingales.A family of power martingales is defined asM εn =n∏i=1(εpε−1i ), (3)where ε ∈ [0, 1] and pi-s are the p-values output by algorithm 1. If the p-values aredistributed independently and uniformly then the power martingales keep value close to0.The sleepy jumper martingale approximates the performance of the best power mar-tingale (martingale from the family of power martingales with the largest final value).For a sequence ε˜ = (ε1, . . . , εn), the equation (3) is generalized asM ε˜n =n∏i=1(εipεi−1i ),where εi ∈ [0, 1], i = 1, . . . n. Then sleepy jumper martingale is defined asMn =∫[0,1]∞M ε˜nµdε˜, (4)where µ is a probability distribution in [0, 1]∞. The sleepy jumper distribution is specifiedby two parameters: R for probability of awake/asleep and S for probability of changecurrent εi to another value from [0, 1]. To obtain the distribution µ sleepy jumper createssequences of states. When the state is (s, j) the new state is generated from the Cartesianproduct {awake, asleep} × [0, 1] with respect to the following conditions:• if s is asleep then the new state is (asleep, j) with the probability R or it is (awake, j)with the probability 1−R;• if s is awake then the new state is (s, j): s is awake with the probability 1−R or it isasleep with the probability R, j takes random value from [0, 1] with the probabilityJ or it stays the same j with the probability 1− J .Sleepy jumper starts from the state (s1, j1) = (asleep, 1) and further generates se-quences (s1, j1), (s2, j2), . . .. From a sequence of states corresponding sequence ε˜ for equa-tion (4) is defined asεi ={ji, if si = awake;1, otherwise.This process gives the distribution µ and we can calculate the sleepy jumper martingale(4).54 ClusteringBy studying a martingale performance we can split the data into clusters. Since the dataare sorted by values xi the martingale tests whether yi can be reliably predicted from xiand previous data according to the Gauss linear assumption. We assume that the pointswhere the martingale performance becomes different (for example by visual inspection)can reflect changes in the structure of expenditure. We split the data by those points andthen each cluster is analyzed separately.We will present results and additional comments about clustering for the data in thesection 5.5 ResultsWe study the data on the logarithmic scale and sorted it by value of income.5.1 Testing and clusteringFigure 2 shows sleepy jumper martingale performance for the household data. The x-axisis for the number of a household (as sorted by income) and y-axis is for the value of themartingale.Figure 2: Sleepy jumper martingale is performing on the household data. The final value of the martingale is higher than10300. The dark dashed lines roughly split the household data according to the different growth rate of the martingale.The large final value of the martingale means that the Gauss linear assumption isfalsified for the data.According to the martingale performance in figure 2 we split the data into threeclusters:• in the first cluster the martingale grows insignificantly except of several points wherep-values occasionally appear to be very close to zero;6• in the second cluster the martingale tends to grow faster;• in the third cluster the growth of the martingale becomes very fast.To illustrate the splitting on the data the dashed vertical lines in figure 3 show the bordersthat we see in the figure 2 for the martingale performance. But unlike figure 2 this pictureis plotted in terms of income value itself (not the number of the household). There we cansee that, for example, in the first cluster (the income of households less than 14000 rubles)the whole income is used for the consumption (up to the noise). For the households withhigher income the consumption is usually smaller than the income. It is clear from thefigure where the diagonal line corresponds to the equality between the income and theconsumption.Figure 3: The household data are split according to the clusters found in the figure 2.5.2 Expenditure structure for clustersThe martingale performance for the whole household data showed that the Gauss linearassumption is falsified. But the martingale values were changing differently. Apparentlyfor each cluster the assumption is violated in a different way. The clusters can be usefulif they correspond to a certain expenditure structure. We will check within each clusterwhether the spending on food, non-food goods, services and alcohol is proportional to thetotal consumption.For the testing, each example is represented by the total consumption as an attributeand a category of spending as a label. We use the conformal predictor for the Gauss linearassumption to generate the p-values and finally calculate performance of the martingalefor each cluster.Figure 4 shows the analysis of the expenditure structure inside the three clusters. Alarge martingale value shows that the linear dependence between the category of spendingand the total consumption is falsified or normality of the noise is broken.7The results in figure 4 show that most of martingales take large values. Only in thesecond cluster most of the martingale values do not increase. There hypotheses about theexpenditure structure can work. An exception is the spending on alcohol, it seems to beless predictable than other spendings.5.3 Conformal prediction and validity for clustersLet us now compare the clusters by applicability of the Gauss linear model for predictions.We use the on-line conformal predictor for the Gauss linear assumption and we checkwhether its predictions are valid.If the assumption were satisfied for data then predictions would be automaticallyvalid, i.e. the error rate does not exceed beforehand chosen significance level (for detailssee Vovk et al., 2005, section 8.5). The question is whether the predictions are valid withinthe clusters.The error rate for the significance level of 5% and median accuracy of obtainedpredictions are plotted in figure 5. The dashed grey line on the graphs of errors (leftcolumn in figure 5) shows the allowed level of 5% for errors.So the Gauss linear model is practically applicable inside the first, second and mostof the third clusters.6 Discussion and conclusionsIn the paper we use the martingale approach for testing the Gauss linear assumption andclustering data according to the martingale performance. From what we presented herethe following results can be concluded:• The Gauss linear assumption for the whole dataset is falsified.• According to the martingale behavior we split the data into the clusters. Because thedataset was sorted by income the clusters correspond to some interval of householdincome values. We have seen that different clusters show different consumptionbehavior. Within each of the clusters conformal predictor for the Gauss linearassumption produce practically valid predictions.For further research we can try to split clusters into smaller ones taking into accountmore attributes.Acknowledgements:This work has been supported by EraSysBio+ grant funds from the European Union, BB-SRC and BMBF “Living with uninvited guests: comparing plant and animal responses toendocytic invasions” to the Salmonella Host Interactions Project European Consortium,SHIPREC; Animal Health and Veterinary Laboratories Agency of Department for En-vironment, Food and Rural Affairs on Machine learning algorithms for analysis of largeveterinary datasets.We are grateful to S.A.Aivazian and M.V.Kruglyak for providing the data and usefulinformation for this work. We also would like to thank Royal Holloway University ofLondon for support and funding.8ReferencesS. A. Aivazian and M. V. Kruglyak. Typology of the Russian Household’s ConsumptionBehaviour (Metodology of Investigation, Informational Maintenance and ExperimentalApprobation) (Rus). CEMI Russian Academy of Science, Moscow, 2010.S.-S. Ho. A martingale framework for concept change detection in time-varying datastreams. In Proc. Int. Conf. on Machine Learning (ICML 2005), 2005.T. P. Ryan. Modern Regression Methods. John Willey & Sons, New York, 1997.http://www.micro-data.ru.J. Ville. Etude critique de la notion de collectif. Gauthier-Villars, Paris, 1939.V. Vovk, I. Nouretdinov, and A. Gammerman. Testing exchangeability on-line. In Pro-ceedings of the 20th International Conference on Machine Learning (ICML 2003), pages768–775, Washington, DC, 2003.V. Vovk, A. Gammerman, and G. Shafer. Algoritmic learning in a random world. Springer,New York, 2005.9Figure 4: Martingale is plotted for testing structure of expenditure. The columns from left to right are for the first, secondand third clusters respectively. The rows from top to bottom are for different categories of spending. If a spending takesfixed share of the total consumption (up to normally distributed noise) martingale should not grow.10Figure 5: Error rate and median accuracy for on-line conformal prediction are plotted for the significance level of 5% for thefirst, second and third clusters from top row to bottom. Obtained predictions are practically valid with the only exceptionof the end of the third cluster.11

Testing and Clustering with Gauss Linear Assumption for a Household Data

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Testing and Clustering with Gauss Linear Assumption for a Household Data

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