Technical report from Automatic Control at Linköpings universitet
Mean and covariance matrix of a
multivariate normal distribution with one
doubly-truncated component
Henri Nurminen, Rafael Rui, Tohid Ardeshiri, Alexandre
Bazanella, Fredrik Gustafsson
Division of Automatic Control
E-mail: henri.nurminen@tut.fi, rafael.rui@ufrgs.br,
tohid@isy.liu.se, bazanella@ufrgs.br, fredrik@isy.liu.se
7th July 2016
Report no.: LiTH-ISY-R-3092
Address:
Department of Electrical Engineering Linköpings universitet
SE-581 83 Linköping, Sweden
WWW: http://www.control.isy.liu.se
AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET
Technical reports from the Automatic Control group in Linköping are available from http://www.control.isy.liu.se/publications.
Abstract
This technical report gives analytical formulas for the mean and covariance matrix of a multivariate normal distribution with one component truncated from both below and above.
Keywords: doubly-truncated multivariate normal distribution, mean, co-variance matrix
Mean and covariance matrix of a multivariate
normal distribution with one doubly-truncated
component
Henri Nurminen
∗, Rafael Rui
†, Tohid Ardeshiri,
Alexandre Bazanella
†, and Fredrik Gustafsson
2016-09-02
Abstract
This technical report gives analytical formulas for the mean and co-variance matrix of a multivariate normal distribution with one component truncated from both below and above. The result is used in the compu-tation of the moments of a mixture of such distributions in [1].
1 Introduction
In this technical report doubly-truncated multivariate normal distribution (DTMND) is a multivariate normal distribution, where one component is truncated from both below and above. In this report we present and derive the analytical for-mulas for the mean and covariance matrix of a DTMND. The forfor-mulas are used in [1], where the piecewise a ne dynamical model results in the posterior dis-tribution being a mixture of DTMNDs. Computing the moments of a mixture of DTMNDs is straightforward given the moments of the mixture components. Without loss of generality, we assume that the double truncation is applied to the rst component of the random vector. For numerical methods, evaluating the presented formulas requires evaluation of the Cholesky decomposition [2, Ch. 2.2.2] as well as the probability density function (PDF) and cumulative density function (CDF) of the univariate standard normal distribution.
Notations: The functions φ and Φ are the PDF and the CDF of the uni-variate standard normal distribution, and the notation [a]idenotes the ith com-ponent of the vector a and the notation [A](i,j)denotes the element (i, j) of the matrix A. Ip is the p × p identity matrix, 0p the p-dimensional column vector
∗H. Nurminen is with the Department of Automation Science and Engineering,
Tam-pere University of Technology (TUT), PO Box 692, 33101 TamTam-pere, Finland (e-mail: henri.nurminen@tut. ). H. Nurminen receives funding from TUT Graduate School, the Foun-dation of Nokia Corporation, and Tekniikan edistämissäätiö.
†R. Rui and A. Bazanella are with Department of Electrical Engineering,
Universi-dade Federal do Rio Grande do Sul, Porto Alegre 90040-060, Brazil (email: rafael.rui, bazanella@ufrgs.br) and are supported by Conselho Nacional de Desenvolvimento Cientí co e Tecnològico (CNPq).
of zeros, and 1A(x)the indicator function
1A(x) =
1, x ∈ A 0, x /∈ A .
2 Formulas for mean and covariance matrix
Let x ∈ Rn be a random variable of the DTMND with the PDF
p(x) ∝ N (x; µ, Σ) · 1[l1,l2]([x]1), (1)
where µ ∈ Rnis the location parameter vector, Σ ∈ Rn×nis the positive de nite squared-scale matrix, and l1, l2 ∈ R are the truncation limits. Further, let Λ be the lower triangular matrix for which Σ = ΛΛT and whose diagonal entries are strictly positive. This type of square-root matrix can be obtained using the Cholesky decomposition [2, Ch. 2.2.2].
Then, the expectation value and covariance matrix of x are
E[x] = Λ m∗ 0n−1 + µ (2) V[x] = Λ s∗ 0T n−1 0n−1 In−1 ΛT (3) where m∗=φ(λ1) − φ(λ2) Z , (4) s∗= 1 +λ1φ(λ1) − λ2φ(λ2) Z − (m ∗)2, (5) with λ1= l1− [µ]1 [Λ](1,1) , λ2= l2− [µ]1 [Λ](1,1) , Z = Φ(λ2) − Φ(λ1).
3 Derivation
Let y ∈ Rn be a DTMND with the PDF
py(y) ∝ N (y; 0, In) · 1[λ1,λ2]([y]1).
The components of y are independent, so the moments of y are obtained using the formula for the doubly-truncated univariate normal random variable [3, Ch. 10.1]. The mean and the covariance matrix are thus
E[y] = m∗ 0n−1 , (6) V[y] = s∗ 0T n−1 0n−1 In−1 , (7)
where m∗and s∗ are those in (4) and (5). Let now z = Λy + µ. The PDF of z is then
As Λ is a lower triangular matrix, [Λ−1] (1,1:n)= h 1 [Λ](1,1) 0 T n−1i, so [y]1= ([z]1− [µ]1)/[Λ](1,1). Thus, (8) becomes
pz(z) ∝ N (Λ−1(z − µ); 0, I) · 1[λ1,λ2] [z]1− [µ]1 [Λ](1,1) · det(Λ)−1 (9) = N (z; µ, Σ) · 1[l1,l2]([z]1) , (10) because ΛΛT = Σ, l
i = [Λ](1,1)λi+ [µ]1 for i ∈ {1, 2} and [Λ](1,1) is positive. That is, z has the same distribution as x, so the expected value and covariance matrix of x are
E[x] = E[z] = Λ E[y] + µ (11)
V[x] = V[z] = ΛV[y]ΛT. (12)
By substituting (6) and (7) to (11) and (12), respectively, we get the formulas (2) and (3).
References
[1] R. Rui, T. Ardeshiri, H. Nurminen, A. Bazanella, and F. Gustafsson, State estimation for piecewise a ne state-space models, 2016. [Online]. Available: http://arxiv.org/abs/1609.00365v1
[2] Å. Björck, Numerical Methods for Least Squares Problems. SIAM, 1996. [3] N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate
Dis-tributions, Vol. 1, 2nd ed. John Wiley & Sons, Inc., November 1994.
Avdelning, Institution Division, Department
Division of Automatic Control Department of Electrical Engineering
Datum Date 2016-07-07 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport
URL för elektronisk version http://www.control.isy.liu.se
ISBN ISRN
Serietitel och serienummer
Title of series, numbering ISSN1400-3902
LiTH-ISY-R-3092
Titel
Title Mean and covariance matrix of a multivariate normal distribution with one doubly-truncatedcomponent
Författare
Author Henri Nurminen, Rafael Rui, Tohid Ardeshiri, Alexandre Bazanella, Fredrik Gustafsson
Sammanfattning Abstract
This technical report gives analytical formulas for the mean and covariance matrix of a multivariate normal distribution with one component truncated from both below and above.
Nyckelord
