Kolmogorov-Smirnov test

Abstract. Leakage detection seeking the evidence of sensitive data de- pendencies in the side-channel traces instead of trying to recover the sensitive data directly under the enormous efforts with numerous leak- age models and state-of-the-art distinguishers can provide a fast pre- liminary security assessment on the cryptographic devices for designers and evaluators. Therefore, it is a popular topic in recent side-channel research of which the Welch’s t-test-based Test Vector Leakage Assess- ment (TVLA) methodology is the most widely used one. However, the TVLA is not always the best option under all kinds of conditions (as we can see in the latter section of this paper). Kolmogorov-Smirnov test is a well-known nonparametric method for statistical analysis to determine whether the samples are from the same distribution by analyzing the cumulative distribution. It has been proposed into side-channel analysis as a successful distinguisher. This paper proposes—to our knowledge, for the first time—Kolmogorov-Smirnov test as a new method for leak- age detection. Besides, we propose two implementations to speed up the KS leakage detection procedure. Experimental results on simulated leak- age with various parameters and the practical traces verify that KS is an effective and robust leakage detection tool and the comprehensive comparison with TVLA shows that KS-based leakage detection can be a right-hand supplement to TVLA when performing the side-channel assessment.

ABSTRACT: Current emergency communications services basically rely on public networks. However, these networks may fail in extreme situations, so emergency services become unreliable in emergency cases. The cognitive radio architecture can make the communication more efficient and reliable with dynamic network configuration. Spectrum sensing is considered as one of the most important and challenging issues for the establishment of dynamic spectrum access with Cognitive Radio (CR) architecture. This paper proposes a spectrum sensing algorithm based on Kolmogorov-Smirnov test on OFDM based wireless emergency networks. The paper also covers the implementation of the algorithm and performance analysis on real data.

In this paper we propose an improvement of the Kolmogorov-Smirnov test for normality. In the current implementation of the Kolmogorov-Smirnov test, a sample is compared with a normal distribution where the sample mean and the sample variance are used as parameters of the distribution. We propose to select the mean and variance of the normal distribution that provide the closest fit to the data. This is like shifting and stretching the reference normal distribution so that it fits the data in the best possible way. If this shifting and stretching does not lead to an acceptable fit, the data is probably not normal. We also introduce a fast easily implementable algorithm for the proposed test. A study of the power of the proposed test indicates that the test is able to discriminate between the normal distribution and distributions such as uniform, bi-modal, beta, exponential and log-normal that are different in shape, but has a relatively lower power against the student t-distribution that is similar in shape to the normal distribution. In model settings, the former distinction is typically more important to make than the latter distinction. We demonstrate the practical significance of the proposed test with several simulated examples.

able to operate on-line a and it is based on the statisti- cal characterization of non-epidemic influenza incidence rates. A major advantage of this approach is its statistical meaningfulness, and thus detection is not only a binary result but the confidence in its outcome can be assessed in terms of probabilities resorting to hypothesis testing theory. Non-epidemic data is modeled with an exponen- tial distribution, in the vein of [14]. The unique parameter of the distribution is the decaying factor, that is initially estimated by a training data set and sequentially updated as new observations are recorded. Such statistical char- acterization is used to design a detector based on the one-sample Kolmogorov-Smirnov test. The method was applied to data in the Diagnosticat database to successfully detect influenza activity.

The KS test diagnostic focuses on distributional differences between the observed and imputed data. In the MI literature, there are other proposed diagnostics that target differences between the observed and imputed data in more specific characteristics, such as the location and spread. For example, Stuart et al. [22] propose flagging imputation models if i) the absolute difference in means between the observed and imputed data is greater than 2 standard deviations, or ii) the ratio of the variances of the observed and imputed values is less than 0.5 or greater than 2. Similarly, classical tests of differences in variances or means (e.g. F-test, t-test and non-parametric counterparts) could be used. Although we did not assess these tests in our study, we believe that our general conclusions will still apply. Under MAR, we do not expect the imputed data to resemble the observed data. In fact, we may be relying on MI to recover these differences based on information in the observed data. It may be useful to explore how the observed and imputed data differ (e.g. through plotting or tabulating summary statistics). However, we do not recommend auto- matic flagging of differences using numerical tests, because the discrepancies between observed and imputed data do not necessarily signal a problem.

Let T be the value of the test statistics at the specified case and t i be the value of the test statistics for i th method (i=1 for bootstrap and i=2 for half-sample). The pointwise MSE was the measure computed to compare the simulation results. It is computed as

It stems from Table 3 that significantly higher percentages of rejected null hypothe- ses of the Kolmogorov-Smirnov test are only for variables of Category I (from the mul- tidimensional analysis) and for all the variables jointly (one-dimensional analysis). These results are consistent with those of Hack ethal (2001), who identified differences in prof- itability between groups of European banks, but only using of market based variables (cf. Chapter 3). However, the percentages are higher only for the years 2000-2004, and in par- ticular for the income from banking activity. No major differences have been observed in the distributions of profitability ratios: pre-tax earnings / total assets and net income from banking activity / total assets for other dimensions, i.e. for variables of Categories II and III. This means that the membership of a bank in a given group may be important for the explana tion of differences in profitability after the year 2000, but only for groups identi- fied of the basis of all variables jointly or on the basis of variables of Category I. In terms of the net income from banking activity, the percentage of rejected hypotheses is also higher between 1997 and 1999; however, the existence of strategic groups that would be meaningful for the explanation of differences in profitability in those years, cannot be confirmed when using pre-tax earnings / total assets instead.

Before we applied the Permutation Test combined with Kolmogorov-Smirnov Test and Wilcoxon Rank Sum Test to the real data, we carry a simulation study to investigate the power of the test near the estimated parameter values. Furthermore, we check the powers at the null hypothesis if they coincide with the significance levels. Hence, the simulation results reflect the goodness of the method applied to our data. We first create several animals’ data for each distribution group assuming they have bi-variate normal distribution. After we standardize them into a Cartesian system, we applied the three steps introduced in Chapter two of this thesis. We find the power of the test under the null hypothesis situation when the two groups have same distribution. We also create the situations of increasing numbers of observation to see how good our method will be when the sample size increases. From Table 11, we can see the powers at null hypothesis match the significance level we pre-assigned. Just a median deviation of means or/and variance-covariance matrices, we obtain reasonably powers, which are better than we originally expected. Therefore, we are comfortable to use this method to test the equality of two- dimensional distributions. (Table 11)

Data analysis was performed using Stata version 12.0 (Stata Corp. LP, College Station, United States of Amer- ica). We described the socio-economic status, EQ-5D- 5 L profiles, utility score and VAS scores according to age groups and gender. Due to the non-normal distribu- tion of utility and VAS scores (Kolmogorov-Smirnov test, p < 0.05), the differences of utility and VAS scores between different groups were tested by employing Mann-Whitney (for gender, living location, having acute symptoms in the past 4 weeks, having chronic diseases in the past 3 months and using health service in the past 12 months), and Kruska-wallis tests (for age groups, marital status, education, occupation, income quintiles, and number of health issues). Mann-whitney test was also used to test the differences of EQ-5D-5 L index and EQ-VAS score according to different dimensions of the EQ-5D-5 L instrument. Spearman’s correlation coeffi- cient was also conducted to identify the relationship be- tween utility score and VAS score. Correlations were classified in three categories: weak (rh0 < 0.3); moderate (0.3 < rh0 < 0.5); and strong (>0.5) [37]. P -value <0.05 was considered statistical significance.

To test for the null hypotheses of the study, we run Kolmogorov-Smirnov test for mean difference among two groups. The test is nonparametric in nature, unlike parametric tests such as independent sample t-test which has assumption relation to sample, population, and normality, Kolmogorov-Smirnov test does not require such assumptions. Therefore, the aim here is to achieve two objectives. On one hand, the study will examine whether there is no significant statistical difference in tax compliance under pre and post implementation of m-filing in Malaysia, on the other hand, the study will examine whether there is no significant statistical difference in compliance complexity under pre and post implementation of m-filing in Malaysia. It is worth noting that the introduction of m-filing enables the taxpayers to file their tax returns using their mobile smartphones which are simpler and convenience to the taxpayers and which hitherto not possible before 2012. Therefore, it is possible to assume that tax compliance might have significantly improved after the introduction of the new system due to enhancement in simplicity and convenience of payment. Similarly, it is also logical to assume that as individuals become much more familiar with the usage of smartphones, filing tax return using such smartphones will reduce the compliance complexity. Thus, there is a possibility that both tax compliance and compliance complexity will be different before and after the implementation of m-filing in Malaysia. The result of the analysis is depicted in Table 1, 2 and 3 below.

Based on the measurement of water content obtained, it can be seen that before the use of Blemish Balm Cream the face of volunteers experiencing dry skin (dehydration), after the use of Blemish Balm Cream Anti-aging there is a gradual increase in skin moisture percentage which makes the skin of volunteers normal. This is supported by the results of statistical analysis calculated by the Two Way Anova and Kolmogorov-Smirnov Test methods. The statistical analysis results of the Kolmogorov-Smirnov Test from the measurement of water content show a significant difference (p ≤ 0.05) in F0, F1, F2 and F3 after using Blemish Balm Cream for 4 weeks which shows that there is a significant difference in the increase in water content and the influence of skin conditions between Blemish Balm Cream without vitamin E (blank) with Formula containing vitamin E (F1, F2 and F3) Vitamin E activity easily penetrates through the skin layers and increases the water content in the skin to be more effective. Blemish Balm Cream preparations without containing Vitamin E (blank) has a percentage increase in water content that is longer than F1, F2 and F3, where the average percentage of water content obtained before use until after 4 weeks of use is F0: 27.3- 49.6%., F1: 22.6- 50.16%., F2: 28-56.1% and F3: 26-56.1%. The percentage limit of normal water levels on facial skin is 30-44% and the percentage of moisture> 45% can prolong the process of skin aging 16 .

The data consist of the number of million revolutions before failure for each of 23 ball bearings used in a life test. The background details about the data can be found in Lieblein and Zelen (1956). Chhikara and Folks (1989) showed by means of the Kolmogorov-Smirnov test that the inverse Gaussian distribution in (3) provides a good fit to the data in Table 1. If we assume that the parameters ( λ , φ ) have the prior given by (5) with n = 1/2, corresponding to unit standard de- viation, then the posterior for φ , the shape parameter, is

...275 Table 40 (6.0): Distribution of scores for the MAAQ factors based on the Kolmogorov-Smirnov test in the Italian sample ...287 Table 41 (6.0): Distribution of scores for the MAAQ [r]

The present research identified the factors that influence the adoption of performance-based budgeting (PBB). A 38-item questionnaire was used for data collection, which was distributed among 30 managers, assistants, and experts active in budgeting processes. The results of Kolmogorov-Smirnov test suggested that the data distribution is non-normal; thus, a set of non-parametric tests were used for data analysis.First, the results of chi-squared test indicated a significant positive relationship between PBB adoption and ability, acceptance, and authority. Then, binomial test was applied and the results suggested that all the variables were at a desirable level. Finally, Friedman test was applied for ranking the variables. The results showed that acceptance, ability, and authority were respectively considered as the most important variables. Moreover, political acceptance, technical ability, and procedural authority were ranked as the most important subscales. Given the positive relationship between PBB adoption and the independent variables, the following conclusions can be made: 1. Special technical abilities are requiredfor collecting performance information and providing a commonly available database where performance information is readily accessible to a variety of users.

Two independent samples (pooled) t-test and one-way analysis of variance (ANOVA) were used for comparison of means. Since the Kolmogorov-Smirnov test showed that the main dependent variable (i.e. first 2.5 years med school GPA) data was not normally distributed (p>0.05), the linear correlation between variables was assessed by Spearman coefficients of correlation, To determine predic- tors of academic success, Multiple Linear Regression (GPA score as dependent factor HSGD-GPA, PUC-GPA, Biology, Chemistry and Math scores in NUEE, raw and T scores of NEO-FFI domains as explanatory variables) and Logistic Regression Model (being “A grade” student as dependent factor, and the rest of demographic, cognitive and non-cognitive characteristics as predictors) were used. All analyses were performed by SPSS statistical software (version 16.0, SPSS Inc., Chicago, Ill., USA).

Using the SPSS v.20 statistical package, a series of analyses were run: 1) sample size adequacy (Kaiser- Meyer-Olkin [KMO] index) and data normal distribu- tion (Kolmogorov-Smirnov test, skewness and kurtosis) for each scale; 2) correlations between each item with its corresponding mean factor score and mean scale score; 3) internal consistency of each scale and factor with Cronbach’s α, and with scale-factor correlations; and 4) construct validity, estimating the underlying factors with an exploratory principal component analysis with orthogonal varimax rotation, retaining factors agreeing with Kaiser’s criterion (a.k.a. root latent criterion) (eigenvalue > 1) and items with at least a 0.40 commu- nality value.

Abstract When dealing with classical spike train analysis, the practitioner often per- forms goodness-of-fit tests to test whether the observed process is a Poisson process, for instance, or if it obeys another type of probabilistic model (Yana et al. in Bio- phys. J. 46(3):323–330, 1984; Brown et al. in Neural Comput. 14(2):325–346, 2002; Pouzat and Chaffiol in Technical report, arXiv:0909.2785, 2009). In doing so, there is a fundamental plug-in step, where the parameters of the supposed underlying model are estimated. The aim of this article is to show that plug-in has sometimes very un- desirable effects. We propose a new method based on subsampling to deal with those plug-in issues in the case of the Kolmogorov–Smirnov test of uniformity. The method relies on the plug-in of good estimates of the underlying model that have to be consis- tent with a controlled rate of convergence. Some nonparametric estimates satisfying those constraints in the Poisson or in the Hawkes framework are highlighted. More- over, they share adaptive properties that are useful from a practical point of view. We show the performance of those methods on simulated data. We also provide a com- plete analysis with these tools on single unit activity recorded on a monkey during a sensory-motor task.

The mean and standard deviation of MTR values were calculated for descriptive analysis and the Kolmogorov– Smirnov test was applied to test for normality. Using a repeated measurement analysis of variance, the mean values of all ROI of all off-resonance frequencies and flip angles of the transplanted and not transplanted lungs were compared to evaluate the following factors: local- isation of ROI placement, off-resonance frequency; flip angle; and transplantation/no transplantation. After checking for normality using Kolmogorov–Smirnov test, Student’s t-test was performed for all flip angles and off- resonance frequencies for the following different groups: comparison A: the transplanted well-ventilated vs trans- planted infiltrated lung tissue; comparison B: the trans- planted lung with CAF (including well-ventilated and infiltrated lung tissue) vs the transplanted lung without CAF (including well-ventilated and infiltrated lung Table 2 Acquisition parameters of the MT-prepulse and the ZTE sequence

alternatives; Yu (1971) computed upper and lower bounds of the Pitman asymptotic efficiency of the Kolmogorov-Smirnov test with respect to the Neyman test and lo- cally most powerful rank test; Rothe (1983) obtained a lower bound for the Pitman ef- ficiency of Friedman type tests; for the group of integral tests of homogeneity, general- izing the omega-square tests, Nikitin (1984) found a lower bound of the Pitman asymp- totic relative efficiency relative to the Student test for the shift alternative; Weissfeld and Wieand (1984) presented bounds on the Pitman efficiency for two-sample scale and location statistics along with densities for which these bounds are sharp; Jansen and Ramirez (1993) derived bounds for the Pitman efficiency for efficiency comparisons in linear inference; Tsai (2009) established a lower bound on the Pitman efficiency of the spherical Wilcoxon rank test relative to the spherical T 2 -test; Ermakov (2011) de-

The Kolmogorov-Smirnov test, on the other hand, is used to test whether two independent distri- butions of continuous, unbinned numerical data are different (Conover, 1999). For example, the Kolmogorov-Smirnov test can indicate whether the distribution of county-level loss ratios for the ALV category is different from the distribution of the ACT category. This means that hypotheses for the Kolmogorov-Smirnov test are not rooted in a mean or median (as in the Wilcoxon-Mann-Whitney test), which implies that a statistical difference between the distributions being compared may be due to a variety of reasons (i.e. difference in means, standard deviations, skewness, kurtosis, etc.). The Kolmogorov-Smirnov test does not provide any in- sight as to what caused the difference in the distri- butions. The advantage of the Kolmogorov-Smirnov test, on the other hand, is the fact that it does not impose normality and equality of variance assump- tions for the test to be valid.