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• 1.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish Univ Agr Sci, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Univ Rwanda, Rwanda. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Mean-Squared errors of small area estimators under a multivariate linear model for repeated measures data2019In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 48, no 8, p. 2060-2073Article in journal (Refereed)

In this paper, we discuss the derivation of the first and second moments for the proposed small area estimators under a multivariate linear model for repeated measures data. The aim is to use these moments to estimate the mean-squared errors (MSE) for the predicted small area means as a measure of precision. At the first stage, we derive the MSE when the covariance matrices are known. At the second stage, a method based on parametric bootstrap is proposed for bias correction and for prediction error that reflects the uncertainty when the unknown covariance is replaced by its suitable estimator.

• 2.
Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania.
Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania.
Asset liability management for Tanzania pension funds by stochastic programming2018In: Afrika Statistika, ISSN 2316-090XArticle in journal (Refereed)
• 3.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. University of Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Mean-Squared errors of small area estimators under a multivariate linear model for repeated measures data2018In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, p. 1-23Article in journal (Refereed)
• 4.
Linnaeus University, Växjö, Sweden.
Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
On n/p-Asymptotic Distribution of Vector of Weighted Traces of Powers of Wishart Matrices2018In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 33, p. 24-40Article in journal (Refereed)

The joint distribution of standardized traces of $\frac{1}{n}XX'$ and of $\Big(\frac{1}{n}XX'\Big)^2$, where the matrix $X:p\times n$ follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition $\frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$. Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of $n$ and $p$.

• 5.
Department of Mathematics, University of Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Small area estimation with missing data using a multivariate linear random effects model2018In: Japanese Journal of Statistics and Data Science, ISSN 2520-8756Article in journal (Refereed)

In this article small area estimation with multivariate data that follow a monotonic missing sample pattern is addressed. Random effects growth curve models with covariates are formulated. A likelihood based approach is proposed for estimation of the unknown parameters. Moreover, the prediction of random effects and predicted small area means are also discussed.

• 6.
Linköping University, Department of Mathematics, Mathematical Statistics .
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Weather Derivatives Pricing Using Regime Switching Model2018In: Monte Carlo Methods and Applications, ISSN 0929-9629, Vol. 24, no 1, p. 13-27Article in journal (Refereed)

In this study we discuss the pricing of weather derivatives whose underlying weather variable is temperature. The dynamics of temperature in this study follows a two state regime switching model with a heteroskedastic mean reverting process as the base regime and a shifted regime defined by Brownian motion with nonzero drift. We develop mathematical formulas for pricing futures and option contracts on heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. The local volatility nature of the model in the base regime captures very well the dynamics of the underlying process, thus leading to a better pricing processes for temperature derivatives contracts written on various index variables. We use the Monte Carlo simulation method for pricing weather derivatives call option contracts.

• 7.
Integrated Polytechnic Regional Centre, Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. University of Rwanda, Rwanda.
Estimation of Parameters in the Growth Curve Model with a Linearly Structured Covariance Matrix: A Simulation Study2017In: International Journal of Scientific Engineering and Technology, ISSN 2277-1581, Vol. 6, no 1, p. 45-49Article in journal (Refereed)

In this paper, the implementation of algorithm proposed in (Nzabanita, J., et al. 2012) for some known linear structures on the covariance matrix Σ is performed and simulations for different sample sizes are repeated many times. For these simulations, the percentages of non positive definite estimates are produced, and the linear structures are identified and classified.

• 8.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, College of Science and Technology, University of Rwanda, Kigali, Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden.
Mean-squared errors of small area estimators under a multivariate linear model for repeated measures data2017Report (Other academic)

In this paper, we discuss the derivation of the first and second moments for the proposed small area estimators under a multivariate linear model for repeated measures data. The aim is to use these moments to estimate the mean-squared errors (MSE) for the predicted small area means as a measure of precision. A two stage estimator of MSE is obtained. At the first stage, we derive the MSE when the covariance matrices are known. To obtain an unbiased estimator of the MSE, at the second stage, a method based on parametric bootstrap is  proposed for bias correction and for prediction error that reects the uncertainty when the unknown covariance is replaced by its suitable estimator.

• 9.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
More on Estimation of Banded and Banded Toeplitz Covariance Matrices2017Report (Other academic)

In this paper we consider two different linear covariance structures, e.g., banded and bended Toeplitz, and how to estimate them using different methods, e.g., by minimizing different norms.

One way to estimate the parameters in a linear covariance structure is to use tapering, which has been shown to be the solution to a universal least squares problem. We know that tapering not always guarantee the positive definite constraints on the estimated covariance matrix and may not be a suitable method. We propose some new methods which preserves the positive definiteness and still give the correct structure.

More specific we consider the problem of estimating parameters of a multivariate normal p–dimensional random vector for (i) a banded covariance structure reflecting m–dependence, and (ii) a banded Toeplitz covariance structure.

• 10.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
On E [Pi(k)(i=0) Tr{W-mi}], where W similar to Wp (l, n)2017In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 46, no 6, p. 2990-3005Article in journal (Refereed)

In this paper, we give a general recursive formula for $\small E[\prod_{i=0}^k Tr\{W^{m_i}\}]$, where $\small W \sim \mathscr{W}_p(I,n)$ denotes a real Wishart matrix. Formulas for fixed n, p  are presented as well as asymptotic versions when $\small \frac{n}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c$i.e. when the so called Kolmogorov condition holds. Finally, we show  application of the asymptotic moment relation when deriving moments for the Marchenko-Pastur distribution (free Poisson law). A numerical  illustration using implementation of the main result is also performed.

• 11.
Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania.
Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania.
Projecting Tanzania Pension Fund System2017In: African Journal of Applied Statistics, ISSN 2316-0861, Vol. 4, no 1, p. 193-218Article in journal (Refereed)

A mandatory Tanzania pension fund with a final salary defined benefit is analyzed. This fund is a contributory pay-as-you-go defined benefit pension system which is much affected by the change in demography. Two kinds of pension benefit, a commuted (at retirement) and a monthly (old age) pension are considered. A decisive factor in the analysis is the increased life expectancy of members of the fund. The projection of the fund’s future members and retirees is done using expected mortality rates of working population and expected longevity. The future contributions, benefits, asset values and liabilities are analyzed. The projection shows that the fund will not be fully sustainable on a long term due to the increase in life expectancy of its members. The contributions will not cover the benefit payouts and the asset value will not fully cover liabilities. Evaluation of some possible reforms of the fund shows that they cannot guarantee a long-term sustainability. Higher returns on asset value will improve the funding ratio, but contributions are still insufficient to cover benefit payouts.

• 12.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department od Mathematics, University of Dar el Salaam, Tanzania.
Regime Switching models on Temperature Dynamics2017In: International Journal of Applied Mathematics and Statistics, ISSN 0973-1377, E-ISSN 0973-7545, Vol. 56, no 2Article in journal (Refereed)

Two regime switching models for predicting temperature dynamics are presented in this study for the purpose to be used for weather derivatives pricing. One is an existing model in the literature (Elias model) and the other is presented in this paper. The new model we propose in this study has a mean reverting heteroskedastic process in the base regime and a Brownian motion in the shifted regime. The parameter estimation of the two models is done by the use expectation-maximization (EM) method using historical temperature data. The performance of the two models on prediction of temperature dynamics is compared using historical daily average temperature data from five weather stations across Sweden. The comparison is based on the heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. The expected HDDs, CDDs and CAT of the models are compared to the true indices from the real data. Results from the expected HDDs, CDDs and CAT together with their corresponding daily average plots demonstrate that, our model captures temperature dynamics relatively better than Elias model.

• 13.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, College of Science and Technology, University of Rwanda, Kigali, Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Small area estimation under a multivariate linear model for incomplete repeated measures data2017Report (Other academic)

In this paper, the issue of analysis of multivariate repeated measures data that follow a monotonic sample pattern for small area estimation is addressed. Random effects growth curve models with covariates for both complete and incomplete data are formulated. A conditional likelihood based approach is proposed for estimation of the mean parameters and covariances. Further, the prediction of random effects and predicted small area means are also discussed. The proposed techniques may be useful for small area estimation under longitudinal surveys with grouped response units and drop outs.

• 14.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Department of Mathematics, University of Rwanda. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Small Area Estimation under a Multivariate Linear Model for Repeated measures Data2017In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 46, no 21, p. 10835-10850Article in journal (Refereed)

In this article, Small Area Estimation under a Multivariate Linear model for repeated measures data is considered. The proposed model aims to get a model which borrows strength both across small areas and over time. The model accounts for repeated surveys, grouped response units and random effects variations. Estimation of model parameters is discussed within a likelihood based approach. Prediction of random effects, small area means across time points and per group units are derived. A parametric bootstrap method is proposed for estimating the mean squared error of the predicted small area means. Results are supported by a simulation study.

• 15.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, College of Science and Technology, University of Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Swedish University of Agricultural Sciences. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Small area estimation with missing data using a multivariate linear random effects model2017Report (Other academic)

In this article small area estimation with multivariate data that follow a monotonic missing sample pattern is addressed. Random effects growth curve models with covariates are formulated. A likelihood based approach is proposed for estimation of the unknown  parameters. Moreover, the prediction of random effects and predicted small area means are also discussed.

• 16.
University of Toronto, Department of Statistics, Canada.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Test for the mean matrix in a Growth Curve model for high dimensions2017In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 46, no 13, p. 6668-6683Article in journal (Refereed)

In this paper we consider the problem of testing (a) sphericity and (b) intraclass covariance structure under a Growth Curve model. The maximum likelihood estimator (MLE) for the mean in a Growth Curve model is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. The MLE for the covariance matrix is based on the MLE for the mean, which can be very poor for p close to N. For both structures (a) and (b), we modify the MLE for the mean to an unweighted estimator and based on this estimator we propose a new estimator for the covariance matrix. This new estimator leads to new tests for (a) and (b). We also propose two other tests for each structure, which are just based on the sample covariance matrix.

To compare the performance of all four tests we compute for each structure (a) and (b) the attained significance level and the empirical power. We show that one of the tests based on the sample covariance matrix is better than the likelihood ratio test based on the MLE.

• 17.
Linnaeus University, Växjö, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Testing Independence via Spectral Moments2017In: Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009, Vol. 192, p. 263-274Article in journal (Refereed)

Assume that a matrix X : p × n is matrix normally distributed and that the Kolmogorov condition, i.e., limn,p→∞ n = c > 0, holds. We propose a test for identity of the covariance matrix using a goodness-of-fit approach. Calculations are based on a recursive formula derived by Pielaszkiewicz et al. The test performs well regarding the power compared to presented alternatives, for both c < 1 or c ≥ 1.

• 18.
Department of Statistics, University of Toronto, Toronto, Canada.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Testing sphericity and intraclass covariance structures under a Growth Curve model in high dimension2017In: Communications in statistics. Simulation and computation, ISSN 0361-0918, E-ISSN 1532-4141, Vol. 46, no 7, p. 5740-5751Article in journal (Refereed)

In this article, we consider the problem of testing (a) sphericity and (b) intraclass covariance structure under a growth curve model. The maximum likelihood estimator (MLE) for the mean in a growth curve model is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. The MLE for the covariance matrix is based on the MLE for the mean, which can be very poor for p close to N. For both structures (a) and (b), we modify the MLE for the mean to an unweighted estimator and based on this estimator we propose a new estimator for the covariance matrix. This new estimator leads to new tests for (a) and (b). We also propose two other tests for each structure, which are just based on the sample covariance matrix.

To compare the performance of all four tests we compute for each structure (a) and (b) the attained significance level and the empirical power. We show that one of the tests based on the sample covariance matrix is better than the likelihood ratio test based on the MLE.

• 19.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Weather derivatives pricing using regim switching models2017Report (Other academic)

In this study we discuss the pricing of weather derivatives whose underlying weather variable is temperature. The dynamics of temperature in this study follows a two state regime switching model with a heteroskedastic mean reverting process as the base regime and a shifted regime defined by Brownian motion with mean different from zero. We develop the mathematical formulas for pricing futures contract on heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. We also present the mathematical expressions for pricing the corresponding options on futures contracts for the same temperature indices. The local volatility nature of the model in the base regime captures very well the dynamics of the underlying process, thus leading to a better pricing processes for temperature derivatives contracts written on various index variables. We provide the description of Montecarlo simulation method for pricing weather derivatives under this model and use it to price a few weather derivatives call option contracts.

• 20.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Crop yield estimation at district level for agricultural seasons 2014 in Rwanda2016In: African Journal of Applied Statistics, ISSN 2316-0861, Vol. 3, no 1, p. 69-90Article in journal (Refereed)

In this paper, we discuss an application of Small Area Estimation (SAE) tech- niques under a multivariate linear regression model for repeated measures data to produce district level estimates of crop yield for beans which comprise two varieties, bush beans and climbing beans in Rwanda during agricultural seasons 2014. By using the micro data of National Institute of Statistics of Rwanda (NISR) obtained from the Seasonal Agricul- tural Survey (SAS) 2014 we derive efficient estimates which show considerable gain. The considered model and its estimates may be useful for policy-makers or for further analyses.

• 21.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, University of Dar es Salaam, Tanzania.
Regime Switching models on Temperature Dynamics2016Report (Other academic)

Two regime switching models for predicting temperature dynamics are presented in this study for the purpose to be used for weather derivatives pricing. One is an existing model in the literature (Elias model) and the other is presented in this paper. The new model we propose in this study has a mean reverting heteroskedastic process in the base regime and a Brownian motion in the shifted regime. The parameter estimation of the two models is done by the use expectation-maximization (EM) method using historical temperature data. The performance of the two models on prediction of temperature dynamics is compared using historical daily average temperature data from five weather stations across Sweden. The comparison is based on the heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. The expected HDDs, CDDs and CAT of the models are compared to the true indices from the real data. Results from the expected HDDs, CDDs and CAT together with their corresponding daily average plots demonstrate that, our model captures temperature dynamics relatively better than Elias model.

• 22.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, University of Rwanda, Rwanda. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Bilinear regression model with Kronecker and linear structures for the covariance matrix2015In: Afrika Statistika, ISSN 2316-090X, Vol. 10, no 2, p. 827-837Article in journal (Refereed)

In this paper, the bilinear regression model based on normally distributed random matrix is studied. For these models, the dispersion matrix has the so called Kronecker product structure and they can be used for example to model data with spatio-temporal relationships. The aim is to estimate the parameters of the model when, in addition, one covariance matrix is assumed to be linearly structured. On the basis of n independent observations from a matrix normal distribution, estimating equations in a flip-flop relation are established and the consistency of estimators is studied.

• 23.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Closed Form of the Asymptotic Spectral Distribution of Random Matrices Using Free Independence2015Report (Other academic)

The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. Random matrix theory is the main eld placing its research interest in the diverse properties of matrices, with a particular emphasis placed on eigenvalue distribution. The aim of this article is to point out some classes of matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider matrices, later denoted by $\mathcal{Q}$, which can be decomposed into the sum of asymptotically free independent summands.

Let $(\Omega,\mathcal{F},P)$ be a probability space. We consider the particular example of a non-commutative space$(RM_p(\mathbb{C}),\tau)$, where $RM_p(\mathbb{C})$ denotes the set of all  $p \times p$ random matrices, with entries which are com-plex random variables with finite moments of any order and $\tau$ is tracial functional. In particular, explicit calculations are performed in order to generalize the theorem given in [15] and illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a particular form of matrix$Q\in\mathcal{Q}$.

Finally, the main result is a new theorem pointing out classes of the matrix $Q$ which leads to a closed formula for the asymptotic spectral distribution. Formulation of results for matrices with inverse Stieltjes transforms, with respect to the composition, given by a ratio of 1st and 2nd degree polynomials, is provided.

• 24.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. University of Rwanda, PO.Box 3900 Kigali, Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE–750 07 Uppsala, Sweden.. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Extended GMANOVA Model with a Linearly Structured Covariance Matrix2015Report (Other academic)

In this paper we consider the extended generalized multivariate analysis of variance (GMANOVA) with a linearly structured covariance matrix. The main theme is to find explicit estimators for the mean and for the linearly structured covariance matrix. We show how to decompose the residual space, the orthogonal complement to the mean space, into m + 1 orthogonal subspaces and how to derive explicit estimators of the covariance matrix from the sum of squared residuals obtained by projecting observations on those subspaces. Also an explicit estimator of the mean is derived and some properties of the proposed estimators are studied.

• 25.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. University of Rwanda, Department of Mathematics.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology Swedish University of Agricultural Sciences Uppsala, Sweden.. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Extended GMANOVA Model with a Linearly Structured Covariance Matrix2015In: Mathematical Methods of Statistics, ISSN 1066-5307, E-ISSN 1934-8045, Vol. 24, no 4, p. 280-291Article in journal (Refereed)

In this paper we consider the extended generalized multivariate analysis of variance (GMANOVA) with a linearly structured covariance matrix. The main theme is to find explicit estimators for the mean and for the linearly structured covariance matrix. We show how to decompose the residual space, the orthogonal complement to the mean space, into m + 1 orthogonal subspaces and how to derive explicit estimators of the covariance matrix from the sum of squared residuals obtained by projecting observations on those subspaces. Also an explicit estimator of the mean is derived and some properties of the proposed estimators are studied.

• 26.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. University of Rwanda, PO.Box 3900 Kigali, Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. bDepartment of Energy and Technology, Swedish University of Agricultural Sciences, SE–750 07 Uppsala, Sweden.. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Maximum Likelihood Estimation in the Tensor Normal Model with a Structured Mean2015Report (Other academic)

There is a growing interest in the analysis of multi-way data. In some studies the inference about the dependencies in three-way data is done using the third order tensor normal model, where the focus is on the estimation of the variance-covariance matrix which has a Kronecker product structure. Little attention is paid to the structure of the mean, though, there is a potential to improve the analysis by assuming a structured mean. In this paper, we introduce a 2-fold growth curve model by assuming a trilinear structure for the mean in the tensor normal model and propose an algorithm for estimating parameters. Also, some direct generalizations are presented.

• 27.
Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
More on explicit estimators for a banded covariance matrix2015In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 19, no 1, p. 49-62Article in journal (Refereed)

The problem of estimating mean and covariances of a multivariate normally distributed random vector has been studied in many forms. This paper focuses on the estimators proposed by Ohlson et al. (2011) for a banded covariance structure with m-dependence. We rewrite the estimator when m = 1, which makes it easier to analyze. This leads to an adjustment, and an unbiased estimator can be proposed. A new and easier proof of consistency is then presented.

This theory is also generalized to a general linear model where the corresponding theorems and propositions are stated to establish unbiasedness and consistency.

• 28.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE-750 07 Uppsala, Sweden.. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Recursive formula for E(∏i Tr{(WΣ-1)mi}), where W~Wp(∑; n) in finite and asymptotic regime2015Report (Other academic)

In this paper, we give a general recursive formula for E(∏i Tr{(WΣ-1)mi}), where W~Wp(∑; n) denotes a real Wishart matrix. Formulas for xed n; p are presented as well as asymptotic versions when n/p→c, when n,p→∞ i.e., when the so called Kolmogorov condition holds. Finally, we show application of the asymptotic moment relation when deriving moments for the Marchenko-Pastur distribution (free Poisson law). A numerical illustration using implementation of the main result is also performed.

• 29.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Mathematics, College of Science and Technology, University of Rwanda, P.O. Box 3900 Kigali, Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Mathematics, College of Science and Technology, University of Rwanda, P.O. Box 3900 Kigali, Rwanda. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE- 750 07 Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Small Area Estimation under a Multivariate Linear Model for Repeated Measures Data2015Report (Other academic)

In this paper, we consider small area estimation under a multivariate linear regression model for repeated measures data. The aim of the proposed model is to get a model which borrows strength across small areas and over time, by incorporating simultaneously the area effects and time correlation. The model accounts for repeated surveys, group individuals and random effects variations. Estimation of model parameters is discussed within a restricted maximum likelihood based approach. Prediction of random e ects and the prediction of small area means across time points and per group units for all time points are derived. The results are supported by a simulation study.

• 30.
Department of Statistics, University of Toronto, 100 St. George Street, Toronto, Canada, M5S 3G3.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Testing Some Covariance Structures under a Growth Curve Model in High Dimension2015Report (Other academic)

fIn this paper we consider the problem of testing (a) sphericity and (b) intraclass covariance structure under a Growth Curve model. The maximum likelihood estimator (MLE) for the mean in a Growth Curve model is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. The MLE for the covariance matrix is based on the MLE for the mean, which can be very poor for p close to N. or both structures (a) and (b), we modify the MLE for the mean to n unweighted estimator and based on this estimator we propose a new estimator for the covariance matrix. This new estimator leads to new tests for (a) and (b). We also propose two other tests for each structure, which are just based on the sample covariance matrix.    To compare the performance of all four tests we compute or each structure (a) and (b) the attained signicance level and the empirical power. We show that one of the tests based on the sample covariance matrix is better than the likelihood ratio test based on the MLE.

• 31.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Cumulant-moment relation in free probability theory2014In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 18, no 2, p. 265-278Article in journal (Refereed)

The goal of this paper is to present and prove a cumulant-moment recurrent relation formula in free probability theory. It is convenient tool to determine underlying compactly supported distribution function. The existing recurrent relations between these objects require the combinatorial understanding of the idea of non-crossing partitions, which has been considered by Speicher and Nica. Furthermore, some formulations are given with additional use of the Möbius function. The recursive result derived in this paper does not require introducing any of those concepts. Similarly like the non-recursive formulation of Mottelson our formula demands only summing over partitions of the set. The proof of non-recurrent result is given with use of Lagrange inversion formula, while in our proof the calculations of the Stieltjes transform of the underlying measure are essential.

• 32.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Energy and Technology, Swedish University of Agricultural Sciences, SE-750 07 Uppsala, Sweden.. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
On Free Moments and Free Cumulants2014Report (Other academic)

The concepts of free cumulants and free moments are indispensably related to the idea of freeness introduced by Voiculescu [Voiculescu, D., Proc. Conf., Buşteni/Rom., Lect. Notes Math. 1132(1985), pp. 556-588] and studied further within Free probability theory. Free probability theory is of great importance for both the developing mathematical theories as well as for problem solving methods in engineering.

The goal of this paper is to present theoretical framework for free cumulants and moments, and then prove a new free cumulant-moment relation formula. The existing relations between these objects will be given. We consider as drawback that they require the combinatorial understanding of the idea of non--crossing partitions, which has been considered by Speicher [Speicher, R., Math. Ann., 298(1994), pp. 611-628] and then widely studied and developed by Speicher and Nica [Nica, A. and Speicher, R.:  Lectures on the Combinatorics of Free Probability, Cambridge University Press, Cambridge, United Kingdom, 2006]. Furthermore, some formulations are given with additional use of the Möbius function. The recursive result derived in this paper does not require introducing any of those concepts, instead the calculations of the Stieltjes transform of the underlying measure are essential.

The presented free cumulant--moment relation formula is used to calculate cumulants of degree 1 to 5 as a function of the moments of lower degrees. The simplicity of the calculations can be observed by a comparison with the calculations performed in the classical way using non-crossing partitions. Then, the particular example of non-commutative space i.e., space of p×p matrices X=(Xij)ij, where Xij has finite moments, equipped with functional E(TrX)∕p is investigated.

• 33.
Department of Statistics, University of Toronto, 100 St. George Street, Toronto, Canada, M5S 3G3.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Test for the mean matrix in a Growth Curve model for high dimensions2014Report (Other academic)
• 34.
Swedish University of Agricultural Sciences, Uppsala, Sweden and Department of Statistics, Uppsala University, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
A note on mean testing for high dimensional multivariate data under non-normality2013In: Statistica neerlandica (Print), ISSN 0039-0402, E-ISSN 1467-9574, Vol. 67, no 1, p. 81-99Article in journal (Refereed)

A test statistic is considered for testing a hypothesis for the mean vector for multivariate data, when the dimension of the vector, p, may exceed the number of vectors, n, and the underlying distribution need not necessarily be normal. With n,p→∞, and under mild assumptions, but without assuming any relationship between n and p, the statistic is shown to asymptotically follow a chi-square distribution. A by product of the paper is the approximate distribution of a quadratic form, based on the reformulation of the well-known Box's approximation, under high-dimensional set up. Using a classical limit theorem, the approximation is further extended to an asymptotic normal limit under the same high dimensional set up. The simulation results, generated under different parameter settings, are used to show the accuracy of the approximation for moderate n and large p.

• 35.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
The Multilinear Normal Distribution: Introduction and Some Basic Properties2013In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 113, no S1, p. 37-47Article in journal (Refereed)

In this paper, the multilinear normal distribution is introduced as an extension of the matrix-variate normal distribution. Basic properties such as marginal and conditional distributions, moments, and the characteristic function, are also presented.

The estimation of parameters using a flip-flop algorithm is also briefly discussed.

• 36.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Swedish University of Agricultural Sciences.
Estimation of parameters in the extended growth curve model with a linearly structured covariance matrix2012In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 16, no 1, p. 13-32Article in journal (Refereed)

In this paper the extended growth curve model with two terms and a linearly structured covariance matrix is considered. We propose an estimation procedure that handles linear structured covariance matrices. The idea is first to estimate the covariance matrix when it should be used to define an inner product in a regression space and thereafter reestimate it when it should be interpreted as a dispersion matrix. This idea is exploited by decomposing the residual space, the orthogonal complement to the design space, into three orthogonal subspaces. Studying residuals obtained from projections of observations on these subspaces yields explicit consistent estimators of the covariance matrix. An explicit consistent estimator of the mean is also proposed and numerical examples are given.

• 37.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
More on the Kronecker Structured Covariance Matrix2012In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 41, no 13-14, p. 2512-2523Article in journal (Refereed)

In this paper, the multivariate normal distribution with a Kronecker product structured covariance matrix is studied. Particularly focused is the estimation of a Kronecker structured covariance matrix of order three, the so called double separable covariance matrix. The suggested estimation generalizes the procedure proposed by Srivastava et al. (2008) for a separable covariance matrix. The restrictions imposed by separability and double separability are also discussed.

• 38.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Royal Institute of Technology, Sweden.
On the Distribution of Matrix Quadratic Forms2012In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 41, no 18, p. 3403-315Article in journal (Refereed)

A characterization of the distribution of the multivariate quadratic form given by XAX′, where X is a p×n normally distributed matrix and A is an n×n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of noncentralWishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.

• 39.
University of Toronto, Department of Statistics.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Profile Analysis with Random-Effects Covariance Structure2012In: Journal of the Japan Statistical Society, ISSN 1882-2754, E-ISSN 1348-6365, Vol. 42, no 2, p. 145-164Article in journal (Refereed)

In this paper, we consider a parallel profile model for several groups. Given the parallel profile model we construct tests based on the likelihood ratio, without any restrictions on the parameter space, testing the covariance matrix for random-effects structure or sphericity. Furthermore, given both the parallel profile and random- effects covariance structure the level hypothesis is tested. The attained significance levels and the empirical powers for the given tests in this paper are compared with the tests given by Yokoyama and Fujikoshi (1993) and Yokoyama (1995).

• 40.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
A U-statistics Based Approach to Mean Testing for High Dimensional Multivariate Data Under Non-normality2011Report (Other academic)

A test statistic is considered for testing a hypothesis for the mean vector for multivariate data, when the dimension of the vector, p, may exceed the number of vectors, n, and the underlying distribution need not necessarily be normal. With n, p large, and under mild assumptions, the statistic is shown to asymptotically follow a normal distribution. A by product of the paper is the approximate distribution of a quadratic form, based on the reformulation of well-known Box's approximation, under high-dimensional set up.

• 41.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Department of Energy and Technology, Swedish University of Agricultural Sciences, Box 7032, SE–750 07 Uppsala, Sweden. Department of Energy and Technology, Swedish University of Agricultural Sciences, Box 7032, SE–750 07 Uppsala, Sweden.
Explicit Estimators under m-Dependence for a Multivariate Normal Distribution2011In: Annals of the Institute of Statistical Mathematics, ISSN 0020-3157, E-ISSN 1572-9052, Vol. 63, no 1, p. 29-42Article in journal (Refereed)

The problemof estimating parameters of amultivariate normal p-dimensional random vector is considered for a banded covariance structure reflecting mdependence. A simple non-iterative estimation procedure is suggested which gives an explicit, unbiased and consistent estimator of the mean and an explicit and consistent estimator of the covariance matrix for arbitrary p and m.

• 42.
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics .
More on the Kronecker Structured Covariance Matrix2011Report (Other academic)

In this paper the multivariate normal distribution with a Kronecker product structured covariance matrix is studied. Particularly, estimation of a Kronecker structured covariance matrix of order three, the so called double separable covariance matrix. The estimation procedure, suggested in this paper, is a generalization of the procedure derived by Srivastava et al. (2008), for a separable covariance matrix.

Furthermore, the restrictions imposed by separability and double separability are discussed.

• 43.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
University of Toronto, Department of Statistics.
Profile Analysis for a Growth Curve Model2011Conference paper (Other academic)

In this talk, we consider profile analysis of several groups where subvectors of the mean vectors are equal. This leads to a profile analysis in a growth curve model. The likelihood ratio statistics are given for the three hypotheses known in literature as parallelism, level hypothesis and flatness. Furthermore, exact and asymptotic distributions are given in the relevant cases.

• 44.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Energy and Technology, Swedish Univerity of Agricultural Sciences, SE-750 07 Uppsala, Sweden.
Some Tests of Covariance Matrices for High Dimensional Multivariate Data2011Report (Other academic)

Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p >> n. The statistics are derived under very general conditions, particularly avoiding any strict assumptions on the traces of the unknown covariance matrix. Neither any relationship between n and p is assumed. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. The validity of the commonly used assumptions for high-dimensional set up is also briefly discussed.

• 45.
Linköping University, Department of Mathematics. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
The Multilinear Normal Distribution:Introduction and Some Basic Properties2011Report (Other academic)

In this paper, the multilinear normal distribution is introduced as an extension of the matrix-variate normal distribution. Basic properties such as marginal and conditional distributions, moments, and the characteristic function, are also presented. The estimation of parameters using a flip-flop algorithm is also briefy discussed.

• 46.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Department of Energy and Technology, Swedish University of Agricultural Sciences, Box 7032, SE–750 07 Uppsala, Sweden.
Explicit Estimators of Parameters in the Growth Curve Model with Linearly Structured CovarianceMatrices2010In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 101, no 5, p. 1284-1295Article in journal (Refereed)

Estimation of parameters in the classical Growth Curve model when the covariance matrix has some specific linear structure is considered. In our examples maximum likelihood estimators can not be obtained explicitly and must rely on optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linear structured covariance matrix.

• 47.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
More on the Kronecker structured covariance matrix2010Conference paper (Other academic)

The Kronecker structured covariance matrix in multivariate normal distribution will be studied. Particularly, the mapping and parametrization which are induced by the Kronecker product are considered.

Furthermore, estiamtion and the uniqueness of the estimators will be discussed in the case of a covariance matrix which is a Kronecker product of several matrices.

• 48.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
University of Toronto, Department of Statistics.
Profile Analysis for a Growth Curve Model2010Conference paper (Other academic)

In this talk, we consider profile analysis of several groups where the groups have partly equal means. This leads to a profile analysis for a growth curve model. The likelihood ratio statistics are given for the three hypotheses known in literature as parallelism, level hypothesis and flatness. Furthermore, exact and asymptotic distributions are given in the relavant cases.

• 49.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Univerity of Toronto, Department of Statistics.
Profile Analysis for a Growth Curve Model2010In: Journal of the Japan Statistical Society, ISSN 1882-2754, E-ISSN 1348-6365, Vol. 40, no 1, p. 1-21Article in journal (Refereed)

In this paper, we consider profile analysis of several groups where the groups have partly equal means. This leads to a profile analysis for a growth curve model. The likelihood ratio statistics are given for the three hypotheses known in literature as parallelism, level hypothesis and flatness. Furthermore, exact and asymptotic distributions are given in the relevant cases.

• 50.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Explicit Estimators of Parameters in the Growth Curve Model with Linearly Structured Covariance Matrices2009Report (Other academic)
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# Single source of error seasonal local level model: is this correct?

I am trying to gradually build up some single source of error state space models for revenue forecasting. I’m a relative newbie to STAN and want to ensure that my code is correct. I think it is, but I need feedback.

I’m building upon the multiple source of error (MSOE) state space code here:

data {
int<lower=1> n;
vector[n] y;
}
parameters {
vector[n] mu;
vector[n] seasonal;
real<lower=0> sigma_level;
real<lower=0> sigma_seas;
real<lower=0> sigma_irreg;
}
transformed parameters {
vector[n] yhat;
yhat = mu + seasonal;
}
model {
for(t in 4:n)
seasonal[t] ~ normal(- sum(seasonal[t-3:t-1]), sigma_seas);

for(t in 2:n)
mu[t] ~ normal(mu[t-1], sigma_level);

y ~ normal(yhat, sigma_irreg);
}

However, that does not use a single source of error, as I want. As a first pass, I want to use Hyndman’s exponential smoothing models as templates (e.g. here). The key difference between the SSOE and MSOE models is that in the code above, sigma_level and sigma_seas are multiples of the residual sigma_irreg. To use just a level example (omitting the seasons for space):

y_t = l_{t-1} + e_t
l_t = l_{t-1} + \alpha e_t

So if we know what l_{t-1} is, the expectation and variance of l_t are l_{t-1} and \alpha^2 \sigma^2, if e_t \sim N(0, \sigma^2). I think. I hope.

Because it lets me update the above code to:

data {
int<lower=1> n;
vector[n] y;
int<lower=1> S;              // Seasons
}
parameters {
vector[n] mu;
vector[n] seasonal;
real<lower=0, upper=1> alpha;                  // Level smoother
real<lower=0, upper=1> gamma;                  // Seasonal smoother
real<lower=0> sigma_irreg;
}
transformed parameters {
real<lower=0> sigma_level;
sigma_level = alpha * sigma_irreg;
real<lower=0> sigma_seas;
sigma_seas  = gamma * sigma_irreg;
vector[n] yhat;
yhat = mu + seasonal;
}
model {
// Priors
alpha ~ beta(2, 1);
gamma ~ beta(2, 2);
// Model
for(t in S:n)
seasonal[t] ~ normal(- sum(seasonal[t-S+1:t-1]), sigma_seas);

for(t in 2:n)
mu[t] ~ normal(mu[t-1], sigma_level);

y ~ normal(yhat, sigma_irreg);
}

Anyway, I would greatly appreciate feedback. I deeply hope this is correct, because there is no way in hell I’m ever going to program a Kalman filter and I would love to actually use Stan and Bayes in my work.

1 Like

My first question would be does the local level and seasonal base model run with simulated data that looks like your real data and recovers the known parameters. Second would be does it run with your real data?

Good question. I will create fake data with parameters and try to recover it today.

The model does run with real data and it does seem to fit well. I feel optimistic but want to check it to be sure. Worst case scenario, I might try to rewrite it in error-correction form.

I’ll update soon.

1 Like

format: plain


text:
##### Volume 01 (2012)
Research Paper Approximations and expansions
##### 1. Smooth biproximity spaces and P-smooth quasi-proximity spaces

O. A. Tantawy; S. A. El-Sheikh; R. A. Majeed

Volume 06, Issue 02 , Spring 2017, Pages 91-107

##### Abstract
The notion of smooth biproximity space  where $\delta_1,\delta_2$ are gradation proximities defined by Ghanim et al. [10]. In this paper, we show every smooth biproximity space $(X,\delta_1,\delta_2)$ induces a supra smooth proximity space $\delta_{12}$ finer than $\delta_1$ and $\delta_2$. We ...  Read More

Research Paper Fuzzy
##### 2. Generalized inverse of fuzzy neutrosophic soft matrix

R. Uma; P. Murugadas; S. Sriram

Volume 06, Issue 02 , Spring 2017, Pages 109-123

##### Abstract
Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Aim of this article is to find the maximum and minimum solution of the fuzzy neutrosophic soft relational equations xA=b and Ax=b, where ...  Read More

Research Paper General topology
##### 3. Fuzzy $\bigwedge_{e}$ Sets and Continuity in Fuzzy Topological spaces

Volume 06, Issue 02 , Spring 2017, Pages 125-134

##### Abstract
We introduce a new class of fuzzy open sets called fuzzy $\bigwedge_e$ sets which includes the class of fuzzy $e$-open sets. We also define a weaker form of fuzzy $\bigwedge_e$ sets termed as fuzzy locally $\bigwedge_e$ sets. By means of these new sets, we present the notions of fuzzy $\bigwedge_e$ ...  Read More

Research Paper Combinatorics
##### 4. On the energy of non-commuting graphs

M. Ghorbani; Z. Gharavi-Alkhansari

Volume 06, Issue 02 , Spring 2017, Pages 135-146

##### Abstract
For given non-abelian group G, the non-commuting (NC)-graph $\Gamma(G)$ is a graph with the vertex set $G$\ $Z(G)$ and two distinct vertices $x, y\in V(\Gamma)$ are adjacent whenever $xy \neq yx$. The aim of this paper is to compute the spectra of some well-known NC-graphs.  Read More

Research Paper Partial differential equations
##### 5. An implicit finite difference scheme for analyzing the effect of body acceleration on pulsatile blood flow through a stenosed artery

A. Haghighi; N. Aliashrafi; N. Asghary

Volume 06, Issue 02 , Spring 2017, Pages 147-161

##### Abstract
With an aim to investigate the effect of externally imposed body acceleration on two dimensional,pulsatile blood flow through a stenosed artery is under consideration in this article. The blood flow has been assumed to be non-linear, incompressible and fully developed. The artery is assumed to be an ...  Read More

Research Paper Difference and functional equations
##### 6. Initial value problems for second order hybrid fuzzy differential equations

Volume 06, Issue 02 , Spring 2017, Pages 163-170

##### Abstract
Usage of fuzzy differential equations (FDEs) is a natural way to model dynamical systems under possibilistic uncertainty. We consider second order hybrid fuzzy differentia  Read More

Research Paper Linear and multilinear algebra; matrix theory
##### 7. Minimal solution of fuzzy neutrosophic soft matrix

M. Kavitha; P. Murugadas; S. Sriram

Volume 06, Issue 02 , Spring 2017, Pages 171-190

##### Abstract
The aim of this article is to study the concept of unique solvability of max-min fuzzy neutrosophic soft matrix equation and strong regularity of fuzzy neutrosophic soft matrices over Fuzzy Neutrosophic Soft Algebra (FNSA). A Fuzzy Neutrosophic Soft Matrix (FNSM) is said to have Strong, Linear Independent ...  Read More


text:
This is post #3 of my Vector Angle series. Make sure you check out the first post for a quick introduction!

# Method #3: Projection⌗

Projection is a cool topic. Explained visually, we’re finding the best approximation for the purple vector, if it had to be on the same line as the orange vector. It’s the closest approximation for a vector constrainted to a given subspace.

If you don’t quite get it yet, don’t worry, we’ll build up some intuition before we dive into the math.

Projection is useful because it allows us to use one of its cool properties. Any vector, when projected onto a line, plane, or n-1 dimensional space is guaranteed1 to form a right angle with the line/plane/whatever.

Once we compute the projection, we will have two pieces of information. Two sides of a triangle with an identical $\theta$, but now with angle $A$ guaranteed to be 90 deg.

That sounds good, but before we go any further, let’s build an intuition for what projection does.

# Building Intuition⌗

When we’re projecting one vector onto another vector, visualize that we’re projecting the vector’s tip (the point) onto another vector’s subspace (the line). A subspace is basically any point that we can reach by scaling a vector by a constant.

The shortest line from a point to a subspace (line) forms a perpendicular angle with the subspace.

This is helpful since a projection is exactly that - the projected point has the shortest possible distance from the subspace to the original point.

We can also reason about the process by looking at the unit vectors of the space.

When we project one unit vector onto this subspace, we calculate the length of the projected vector. Remember, the projection is the closest we can get from the tip of the green vector (the point) to the subspace defined by the pink vector (the line).

Next, we calculate the length for the projection of the other unit vector.

Now, when projecting a vector onto that subspace, we can just split it into unit vectors, and multiply it by the projection multiples (1.2x and 0.98x).

A simple way of thinking about this is that we want the projected vector to reside on the pink vector’s line. To do that, we want to multiply the pink vector by a constant so that it is the closest it can get to the purple vector without leaving the pink vector’s subspace.

With the process above, we’re getting the closest we can for each component of the vector and adding those components together to calculate the answer.

(P.S. If you understand what we’re doing there, try combining that with your understanding of matricies as linear transformations. You can really understand what projection matricies are doing!2)

Now let’s use this right angle in our calculations!

# Practical Application⌗

Let’s do it

Given two vectors $a$ and $b$, let’s project vector b onto the subspace defined by vector $a$.

Here’s the formula for the projection of vector $b$ onto vector $a$:

$$b_{proj} = \frac{a \cdot b}{|a|^2} a = \frac{a \cdot b}{a \cdot a} a$$

Now we have a way to get the three sides that we need, and we can choose which sides to use in our calculation.

Side A length formula:

$|v|$

Side B length formula:

$|\frac{u \cdot v}{u \cdot u}u|$

Side C length formula:

$|v - \frac{u \cdot v}{u \cdot u}u|$

Now we could technically choose any of these 2 lengths, so let’s try all 3 possibilities and see which one is simplest.

Sin $\frac{Opp}{Hyp}$

$$\frac{|v - \frac{u \cdot v}{u \cdot u}u|}{|v|}$$

$$\frac{\sqrt{{(v - \frac{u \cdot v}{u \cdot u}u)}^2}}{\sqrt{(v)^2}}$$

$$\sqrt{\frac{{(v - \frac{u \cdot v}{u \cdot u}u)}^2}{v^2}}$$

$$\sqrt{\frac{{v^2 - 2(v\frac{u \cdot v}{u \cdot u}u)} + (\frac{u \cdot v}{u \cdot u}u)^2}{v^2}}$$

Cos $\frac{Adj}{Hyp}$

$$\frac{|\frac{u \cdot v}{u \cdot u}u|}{|v|}$$

$$\frac{\sqrt{(\frac{u \cdot v}{u \cdot u}u)^2}}{\sqrt{v^2}}$$

$$\sqrt{\frac{(\frac{u \cdot v}{u \cdot u}u)^2}{v^2}}$$

Tan $\frac{Opp}{Adj}$

$$\frac{|v - \frac{u \cdot v}{u \cdot u}u|}{|\frac{u \cdot v}{u \cdot u}u|}$$

$$\frac{\sqrt{(v - \frac{u \cdot v}{u \cdot u}u)^2}}{\sqrt{(\frac{u \cdot v}{u \cdot u}u)^2}}$$

$$\sqrt{\frac{(v - \frac{u \cdot v}{u \cdot u}u)^2}{(\frac{u \cdot v}{u \cdot u}u)^2}}$$

$$\sqrt{\frac{{v^2 - 2(v\frac{u \cdot v}{u \cdot u}u)} + (\frac{u \cdot v}{u \cdot u}u)^2}{(\frac{u \cdot v}{u \cdot u}u)^2}}$$

# Simplify⌗

Alright, so now we know that the simplest option is to use Cos because it is the only option that exclusively uses the adjacent and hypotenuse sides. When using projection, we can only calculate the opposite side after we know the hypotenuse and adjacent sides, so it makes sense to use those sides directly.

$$\text{angle} = \arccos{\bigg(\sqrt{\frac{(\frac{u \cdot v}{u \cdot u}u)^2}{v \cdot v}}\bigg)}$$

This is the simplest that we can make the equation - it’s our final answer!

# Wait a second…⌗

We spotted a problem in our last blog post regarding obtuse angles. That also happens with this method.

Since the projection can face the opposite direction of the initial line, our $\theta$ will always be acute. We can detect and fix this by calculating if $b$ and $b_{proj}$ face the same way. If they aren’t, then the actual angle is $180 - \theta$.

# <- Up Next⌗

We just derived an alternative way to use the dot product to calculate the angle between two vectors - using projection as a starting point!

We’ve now covered 3 different methods of calculating the angle formed by two vectors:

1. Standard (Law of Cosines)
2. Visual (Change of Basis)
3. Alternative (Projection)

Up next, we’re going to benchmark these different methods and see which one is fastest on a computer.

1. If the vector is already perpendicular with the line/plane/whatever, then the projected vector will have a length of 0 and the angle will be undefined. Additionally, if the vector already resides in the subspace, the difference between the original and projected vector will be zero, and a triangle will not be formed. ↩

2. Projection matricies are so cool. You just define what each component (unit vector) maps to in your subspace. Then, when you have a new vector, you just feed each of its components into the mappings, and you get out the projection. Another way of looking at it is that the matrix collapses the output space to only be that subspace, and the mappings ensure that we get as close as possible to the input vector. ↩


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# Permutation and Combination

### 1.1

1.

Soln:

A man can enter the stadium in 4 ways. Again the man can leave the stadium in 9 ways.

So, total no.of ways with which a man enters and then leaves the stadium = 4 * 9 = 36ways.

2.

Soln:

There are 6 choices for a student to enter the hostel. There are 5 choices for a student to leave the hostel as different door is to be used.

So, total no.of ways = 6 * 5 = 30.

3.

Soln:

There are 7 choices for 1st son, 6 choices for 2nd son and 5 choices for 3rd son.

Now, by the basic principle of counting, the total number of ways of choice = 7 * 6 * 5 = 210.

4.

Soln:

A man can go from city A to city B in 5 ways. As he has to return by a different road, so he can return from city B to city A in 4 ways.

So, total no.of ways by which a man can go from city A to city B and returns by a different road = 5 * 4 = 20 ways.

5.

Soln:

A person can go from city A to city B in 5 ways. Again, he can go from city B to city C in 4 ways. So, a person can go from city A to city C in 5 * 4 = 20ways. The person has to return from C to A without driving on the same road twice, So, he can return from city C to city B in 3 ways and from city B to city A in 4 ways.

So, he can return from city C to city A in 3 * 4 = 12 ways.

So, Total no.of ways by which a person can go from city A to city C and return from city C to city A = 20 * 12 = 240 ways.

6.

Soln:

Numbers formed should be of at least 3 digits means they may be of 3 digits, 4 digits, 5 digits or 6 digits.

There are 6 choices for digit in the units place. There are 5 and 4 choices for digits in ten and hundred’s place respectively.

So, total number of ways by which 3 digits numbers can be formed = 6.5.4 = 120

Similarly, the total no.of ways by which 4 digits numbers can be formed = 6.5.4.3 = 360.

the total no. of ways by which 5 digits numbers can be formed = 6.5.4.3.2 = 720.

The total no.of ways by which 4 digits numbers can be formed = 6.5.4.3.2.1 = 720.

So, total no.of ways by which the numbers of at least 3 digits can be formed = 120 + 360 + 720 + 720 = 1920.

7.

Soln:

The numbers formed must be of three digits and less than 500, so the digit in the hundred’s place should be 1,2,3 or 4. So, there are 4 choices for the digit in the hundred’s place. There are 5 choice for the digit in the ten’s place. There are 4 choices for the digit in the unit’s place.

So, no of ways by which 3 digits numbers les than 500 can be formed = 4.5.4 = 80.

8.

Soln:

The numbers formed should be even. So, the digit in the unit’s place must be 2 or 4. So, the digit in unit’s place must be 2 or 4. So, for the digit in unit’s place, there are 2 choices. So, after fixing the digit in the unit’s place, remaining 4 figures can be arranged in P(4,4) ways.

Ie. $\frac{{\left( 4 \right)!}}{{\left( {4 - 4} \right)!}}$ = $\frac{{4!}}{{0!}}$ = $\frac{{4{\rm{*}}3{\rm{*}}2{\rm{*}}1}}{1}$ = 24 ways.

So, total no.of ways by which 5 even numbers can be formed = 2 * 24 = 48.

9.

Soln:

The numbers formed must be of 4 digits. The digit in the thousand’s place must always be 4. For this, there is only one choice. After that, n = 6 – 1 = 5, r = 4 – 1 = 3. Then remaining 5 figures can be placed in remaining 3 places in:

Or, P(5,3) ways = $\frac{{5!}}{{\left( {5 - 3} \right)!}}$ = $\frac{{5!}}{{2!}}$ = $\frac{{5{\rm{*}}4{\rm{*}}3{\rm{*}}2{\rm{*}}1}}{{2{\rm{*}}1}}$ = 60 ways.

So, Total no.of ways by which 4 digits numbers between 4,000 and 5,000 can be formed = 1 * 60 = 60.

10.

Soln:

For the three digits numbers, there are 5 ways to fill in the 1st place, there are 4 ways to fill in the 2nd place and there are 3 ways to fill in the 3rd place. By the basic principle of counting, number of three digits numbers = 5 * 4 * 3 = 60.

Again, for three digit numbers which are divisible by 5, the number in the unit place must be 5. So, the unit place can be filled up in 1 way. After filling up the unit place 4 numbers are left. Ten’s place can be filled up in 4 ways and hundredths place can be filled up in 3 ways. Then by the basic principle of counting, no.of 3 digits numbers which are divisible by 5 = 1 * 4 * 3 = 12.

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Loan Repayment / Amortization

Question:

A loan of $12,500 is made at an effective interest rate of 8.5%. Payments are made at the end of each interest period. Each payment equals twice the interest due untilthe borrower pays off the outstanding debt with a final payment of, at most,$1,800. Find the number of payments n and the amount of the final payment.
Suppose in a purely competitive market that American firms consider labor costs to be mostly variable while Japanese firms consider labor costs to be mostly fixed. What implication would this have for the viability of firms in each country if they compete
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When all the conditions are identical, in the case of flow through pipes with heat transfer,...
When all the conditions are identical, in the case of flow through pipes with heat transfer, the velocity profiles will be identical for liquid heating and gas cooling a liquid heating and liquid cooling gas heating and gas cooling heating and cooling of any fluid 92-...
Find the critical values χ2L and χ2R for the given confidence level c and sample size...
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2. Solve problem 9.3. Mma.-20625 lb.ft, Vmax-2500 lb 9.3. A single-bay frame of the type generally...
2. Solve problem 9.3. Mma.-20625 lb.ft, Vmax-2500 lb 9.3. A single-bay frame of the type generally illustrated in Figure 9.5 carries a horizontal load of 5000 lb acting at the upper-left joint. Assume that h 15 ft and L 25 ft. Draw shear and moment diagrams. Indicate numerical values. Use an approxi...
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Stuutuu Resources Hint mestion 19 of 37 > Write the equilibrium-constant expression for the reaction shown...
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# SSAT Upper Level Math : How to find the probability of an outcome

## Example Questions

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### Example Question #1681 : Ssat Upper Level Quantitative (Math)

Eight standard decks of 52 cards are shuffled together; four jokers are shuffled in as well. A card is selected at random from this new deck. What is the probability that a spade will be selected?

Explanation:

Eight decks of 52 cards, with 4 jokers added, comprise a total of

cards. A standard deck of 52 cards has 13 spades, so the eight decks together have a total of

spades. This makes the probability of a random draw resulting in a spade

### Example Question #1681 : Ssat Upper Level Quantitative (Math)

The above set of balls is placed in a box. Give the probability that, if a ball is drawn at random, it will be a gray ball marked with an odd number.

Explanation:

As seen in the diagram below, nine of the twenty-five balls in the box are both gray and marked with odd numbers.

This makes the probability that a random selection will yield one of these balls  .

### Example Question #31 : Outcomes

The above set of balls is placed in a box. Give the probability that, if a ball is drawn at random, it will be a white ball marked with a prime number.

Explanation:

Out of the whole numbers from 1 through 9, there are four prime numbers - 2, 3, 5, and 7. As seen in the diagram below, there are five out of twenty-five balls that are both white and marked with a prime number.

This makes the probability of drawing one of these balls .

### Example Question #55 : Data Analysis / Probablility

Mark is playing a board game in which spaces represent property that can be "bought" and "sold"; the above diagram represents part of the board. Mark is represented by the red triangle, and he is moving in a left-to-right direction.

Mark is about to roll a pair of fair six-sided dice. If he lands on one of the yellow spaces, he will have to pay a huge fee to his opponent, Jennifer, who "owns" the properties represented by the spaces; if this happens, he will be out of the game.

What are the odds of this happening?

11 to 7 against

2 to 1 against

5 to 4 against

Even odds

11 to 7 against

Explanation:

The undesirable rolls are 5, 6, and 8. As can be seen in the diagram below, out of thirty-six equally probable outcomes possible with the roll of two fair six-sided dice, fourteen are undesirable, and twenty-two are desirable.

The odds against an undesirable roll are

- that is, 11 to 7.

### Example Question #31 : Probability

The above set of balls is placed in a box. Give the probability that, if a ball is drawn at random, it will be a gray ball marked with a number that is a multiple of 3.

Explanation:

Of the twenty-five balls in the box, five are both gray and marked with a number that is a multiple of 3 (3, 6, 90), as seen in the figure below:

This makes the probability of drawing one of these balls

.

### Example Question #32 : Probability

Above is a part of the board for a popular board game. Terry is moving clockwise on the board; his playing piece is represented by the black triangle. He is about to roll a pair of fair six-sided dice. He collects $10,000 every time he lands on or passes the space marked "Onward!" What are the odds in favor of, or against, him collecting$10,000 on this turn?

11 to 1 in favor

11 to 1 against

5 to 1 against

5 to 1 in favor

11 to 1 in favor

Explanation:

Terry will collect $10,000 if he rolls a four or higher. As can be seen in the diagram below, out of thirty-six equally probable outcomes possible with the roll of two fair six-sided dice, there are only three rolls unfavorable to this outcome, and thirty-three favorable: This makes the odds , or 11 to 1, in favor of$10,000 being collected on this turn.

### Example Question #32 : Probability

In the game of Crazy Eights, which is played with an ordinary deck of 52 playing cards, each player lays down a card from his hand on top of a discard pile according to the card that has previously been played. The card he is allowed to play must be a card of the same rank or suit, or it must be an eight of any suit.

At the beginning of the hand, the seven of diamonds was turned over; Kenny, who was first, played a six of diamonds on top of it. Dick, who plays next, has the following cards in his hand:

As can be seen, Dick does not have a playable card. By the rules, he must draw a card, and, if he can play it, he must. What is the probability that the card he draws will be playable?

Explanation:

Dick knows 10 of the cards the deck - the eight in his hand, and the two in the discard pile. Of the other  cards, the following are playable:

The four eights;

The other three sixes (hearts, clubs, spades); and,

The other ten diamonds (the six and the seven have been played, and the eight of diamonds has already been counted).

This makes

of the 42 cards playable, so the probability that a playable card will be drawn is .

### Example Question #33 : Probability

In the game of Crazy Eights, which is played with an ordinary deck of 52 playing cards, each player lays down a card from his hand on top of a discard pile according to the card that has previously been played. The card he is allowed to play must be a card of the same rank or suit, or it must be an eight of any suit.

At the beginning of the hand, the two of clubs has been turned over. Brenda has in her hand eight cards, as follows:

As can be seen, Brenda cannot play a card. By the rules, she must draw a card, and, if she can play it, she must. What is the probability that the card she draws will be playable?

Explanation:

Brenda knows nine of the cards - the one in the discard pile and the eight in her hand. Therefore, the next card can be one of  different cards. The playable cards are:

The four eights;

The other three twos (hearts, spades, diamonds); and,

The other eleven clubs (the two has been played, and the eight of diamonds has already been counted).

This makes

of the 42 cards playable, so the probability that a playable card will be drawn is .

### Example Question #1691 : Ssat Upper Level Quantitative (Math)

In a jar, there are 3 blue marbles, 5 red marbles, 8 green marbles. If Bob reaches his hand in a jar, and grabs one marble, what is the likelihood he will pick up a blue marble?

Explanation:

First, calculate the total number of marbles in the jar, which is . Because 3 of the marbles are blue, the chances of picking a blue marble .

### Example Question #1696 : Ssat Upper Level Quantitative (Math)

A fair six-sided die has on its faces the numbers 1, 1, 1, 2, 2, 3; a fair eight-sided die has on its faces the numbers 1, 1, 1, 1, 2, 2, 3, 4. If both dice are rolled, what is the probability that the numbers add up to 3 or less?

Explanation:

In order for the roll to add up to 3 or less, neither die can come up 3 or 4. Therefore, we look at the probabilities that each die can come up 1 and 2.

The six-sided die comes up 1 with probability ; it comes up 2 with probability .

The six-sided die comes up 1 with probability ; it comes up 2 with probability .

A roll of 3 or less can happen three ways:

Case 1: Both dice come up 1. The probability of this, by the Multiplication Principle, is

Case 2: The six-sided die comes up 1 and the eight-sided die comes up 2. The probability of this, by the Multiplication Principle, is

Case 3: The six-sided die comes up 2 and the eight-sided die comes up 1. The probability of this, by the Multiplication Principle, is

These are mutually exclusive events, so add these probabilities:

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# How do you find the derivative of y=x^4-3x^2+7x-2?

Aug 5, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = 4 {x}^{3} - 6 x + 7$

#### Explanation:

To find derivative of a polynomial, one can use $\frac{d}{\mathrm{dx}} {x}^{n} = n {x}^{n - 1}$.

Hence, as $y = {x}^{4} - 3 {x}^{2} + 7 x - 2$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 4 {x}^{4 - 1} - 3 \times 2 {x}^{2 - 1} + 7$ or

$\frac{\mathrm{dy}}{\mathrm{dx}} = 4 {x}^{3} - 6 x + 7$


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An acid-base indicator has K_b of 1.0 × 10^(-5). The acid form of the indicator is red and the basic form is blue.

### Question Asked by a Student from EXXAMM.com Team

Q 1189334217.     An acid-base indicator has K_b of 1.0 × 10^(-5). The acid form of the indicator is red and the basic form is blue. Then:
A

pH is 8.4 when indicator is 80% red

B

pH is 9.6 when indicator is 80% blue

C

pH is 8.4 when indicator is 80% blue

D

pH is 9.6 when indicator is 80% red

#### HINT

(Provided By a Student and Checked/Corrected by EXXAMM.com Team)

#### Access free resources including

• 100% free video lectures with detailed notes and examples
• Previous Year Papers
• Mock Tests
• Practices question categorized in topics and 4 levels with detailed solutions
• Syllabus & Pattern Analysis


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# The top view diagram at right illustrates two point sources, S_1, and S_2. 1. On the diagram,...

## Question:

The top view diagram at right illustrates two point sources, {eq}S_1 {/eq}, and {eq}S_2 {/eq}.

1. On the diagram, indicate points for which the value of {eq}\Delta D {/eq} is:

i) largest and

ii) smallest. ({eq}\Delta D {/eq} is the difference in distances to the sources.)

2. What are the largest and smallest values of {eq}\Delta D {/eq} for this situation? Explain your reasoning.

## Distance:

Distance is a physical variable. The measurement that used to find that how a point is far from a fixed point is known as distance. The distance between two points is calculated or measured in meters.

(a)

Given Data

• The difference in distances to the sources is {eq}\Delta D {/eq}.
• The distance between two points sources {eq}S_1 {/eq} and {eq}S_2 {/eq} is {eq}d {/eq}.

Let a point {eq}Q {/eq} is a point on the line extended {eq}{S_1}{S_2} {/eq} for which the distance between the {eq}{S_1}Q {/eq} and {eq}{S_2}Q {/eq} is maximum.

So,

{eq}\begin{align*} \Delta {D_{\max }} &= {S_1}Q - {S_2}Q\\ &= d \end{align*} {/eq}

Thus, point {eq}Q {/eq} is the point for which the values of {eq}\Delta D {/eq} is largest.

Let {eq}PM {/eq} is the perpendicular bisector of the line {eq}{S_1}{S_2} {/eq}.

{eq}{S_1}P {/eq} and {eq}{S_2}P {/eq} is the distance of point {eq}P {/eq} from the source. As point {eq}P {/eq} is on the perpendicular of distance {eq}d {/eq}, the distances {eq}{S_1}P {/eq} and {eq}{S_2}P {/eq} will be equal.

So,

{eq}\begin{align*} {{S_1}P} &= {{S_2}P}\\ {{S_1}P} - {{S_2}P} &= 0\\ \Delta{D_{min}} &= 0 \end{align*} {/eq}

Thus, point {eq}P {/eq} is the point for which the values of {eq}\Delta D {/eq} is smallest.

(b)

As the distance of sources from point on the perpendicular bisector of {eq}{S_1}{S_2} {/eq} is equal, the value of {eq}\Delta D {/eq} is 0. For any point which is on line extended from line {eq}{S_1}{S_2} {/eq}, the value of {eq}\Delta D {/eq} is d which is maximum. Thus, from above calculation and discussion the largest and smallest values of {eq}\Delta D {/eq} is d and 0 respectively.