Longevity and Mortality at Older Ages in the Czech Republic

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1 Longevity and Mortality at Older Ages in the Czech Republic Jana Langhamrová 1, Kornélia Cséfalvaiová 2, and Jitka Langhamrová 3 1 Department of Statistics and Probability, University of Economics, Prague, Czech Republic ( jana.langhamrova@vse.cz) 2 Department of Demography, University of Economics, Prague, Czech Republic ( kornelia.csefalvaiova@vse.cz) 3 Department of Demography, University of Economics, Prague, Czech Republic ( jitka.langhamrova@vse.cz) Abstract. In the past the level of development of countries was assessed by the level of infant mortality. The impact of advances in medicine and health care caused that the infant mortality rate in the Czech Republic over the past 60 years has greatly improved, and fell to very low values. In the current ageing society and due to the increased number of elderly persons in the population, we primary examine mortality particularly in older age intervals. Presented article examines the distribution of mortality in older age groups in the Czech Republic. The paper uses selected demographic, statistical and mathematical models and procedures. Furthermore, we focus on the trends in the variability of life expectancy of men and women at higher age in the Czech Republic. We describe current trends in longevity and mortality changes using the indicator of life expectancy and the modal age at death. Keywords: Longevity, Ageing, Life expectancy, Modal age at death. 1 Introduction During the last two centuries occurs in all developed populations a significant improvement in mortality rates. This results in a rapid growth of human life. In countries, where there is a relatively low level of mortality, life expectancy at birth for both sexes almost doubled in this period (Vienna Yearbook in article Modal age by Horiuchi, et al., 2013 and also Meslé and Vallin, 2011). Mankind has always been interested in the question of human life limits. We are interested in the increasing life expectancy and its upper limit. Talacko states that "under certain conditions we can express the life prolonging by a logistic curve and when we apply this logistic curve on Swedish, German and Czech data, there is a good correspondence between the theory and the observation" (Talacko, 1941, p. 6). Rychtaříková states that "the development of mortality in developed countries was neither linear nor logistic, but had more rapid or slower periods of decline, while some trends were unexpected" (Rychtaříková, 1992, p. 3). 16 th ASMDA Conference Proceedings, 30 June 4 July 2015, Piraeus, Greece 2015 ISAST

2 As life expectancy at birth is increasing, the share of the elderly is also higher in the population. In developed countries with low infant mortality, life expectancy is extending mainly due to better mortality rates of the elderly. For the evaluation of human life longevity the most commonly used indicator is the life expectancy at birth, which is an indicator of the type of average. Furthermore, we can use the probable age at death, which has a characteristic of the type of median and as a very suitable indicator appears in recent times the modal age at death, which is a characteristics of the type of mode and appears as more suitable for characterizing the longevity. Life expectancy has previously been heavily influenced by the level of infant mortality (Fig. 1), and from this reason commonly used indicators are modal age at death and median age at death. Modal age at death, which is the age at which people most often die, is nowadays frequently used. Modal age at death is considered to be a characteristic of longevity. Since the modal age at death is determined by adult mortality, we assume that it is a very good characteristic of longevity recently. 2 Development of Life Expectancies in the Czech Republic in In this article, due to the comparability of time series, we constructed our own mortality tables for the Czech Republic. To compare the development of mortality in the period , we used data of deaths by age and gender and the number of males and females for different ages from the Czech Statistical Office. First of all, from the number of deaths we calculated the mortality rates by age separately for men and women. Then, the mortality rates for ages 60 years and over were smoothed using the Gomperz-Makeham formula and were compared the real and smoothed data and tested using the Chi-square test of goodness of fit and independence. Mortality values were then again smoothed using moving averages. From these adjusted mortality rates we calculated life expectancies in the mortality tables for the years To calculate the modal age at death, we used the method as (Langhamrová, Cséfalvaiová, Langhamrová, 2014 and Fiala 2005). Life expectancy at birth is strongly influenced by the infant mortality (Fig. 1). Therefore, frequently used is the modal age at death (mode), or the probable age at death (median). Life expectancy is an aggregate indicator, but depends on the values of specific mortality by age and sex. Frequently followed is the life expectancy at the age of 20, 40, 60 or 80 years (Fig. 2). From the Figure 2 it is visible that the difference in life expectancy between men and women is decreasing as a higher age is achieved.

3 life expectancy (in years) life expectancy (in years) infant mortality coefficient (in ) Fig. 1. Development of life expectancy at birth (LE00) and infant mortality coefficient in the Czech Republic in the years LE00 infant mortality Source: CZSO, own calculation Fig. 2. Development of life expectancy at selected ages for males and females in the Czech Republic in the years LE20 men LE40 men LE60 men LE80 men LE20 women LE40 women LE60 women LE80 women Source: CZSO, own calculation The difference between the life expectancy of women and men evolves over the time (Fig. 3). The smallest differences can be observed at the beginning of the period. In 1920 we can see that the difference in the life expectancy at birth between females and males was approximately 2.5 years, and subsequently this

4 difference in life expectancies of men and women (in years) difference increased to the disadvantage of men. A period of mild growth or stagnation after 1990 has changed into decrease of the difference between women and men. It is due to the so-called male excess mortality. After 1990 began this phenomenon improve, in terms of faster improvement in the mortality rates of men and women and also due to significant improvement of mortality in middle and higher ages. Fig. 3. Development of difference in life expectancies between females and males at selected ages in the Czech Republic in the years years old 20 years old 40 years old 60 years old 80 years old Source: CZSO, own calculation Life expectancy includes features of average, with its advantages and disadvantages. From statistics we know that there are other measures of central tendency. These characteristics of lifespan do not have deficiencies like the average indicator. We can specify the so called modal age at death, which is the age at which most people die. It is the modal age at death in a stationary population. Modal age at death is often considered to be a characteristic of longevity. Appropriate additional indicator of longevity is the number of deaths in the mortality tables. It is also possible to determine how many people will survive this age. In earlier times, when the infant mortality was high, the mode equaled two for the series of deaths in the mortality table. When we are looking at the distribution of deaths by age in mortality tables for the Czech Republic in long time series (Fig. 4 and Fig. 5), it is bimodal, the first local maximum is at the age of 0 and the other local maximum is at the old age. In most of the developed countries, the mode at old age has been higher than the mode at the age of 0 during the last 50 years (Vienna Yearbook in article Modal age by Horiuchi, et al., 2013).

5 thousands of people thousands of people Fig. 4. Development of tabular number of deaths males at the completed age of x in the Czech Republic in the years age Source: CZSO, own calculation years Fig. 5. Development of tabular number of deaths females at the completed age of x in the Czech Republic in the years years 80 age Source: CZSO, own calculation It is visible, that in 1920, the number of deaths in their early life was still high. Modal age at death is considered to be valid for older ages. It is the age at which people are most likely to die. In this case, we are not interested in younger ages. In populations with low mortality the modal age at death is definitely in old age.

6 Modal age at death is increasing for both men and women, but ranges from 72 to less than 88 years during the whole time series. Unlike life expectancy at birth, the differences in values between the years 1920 and 2013 are not so significant. Fig. 6. Development of the number of males surviving to the exact age of x in the mortality tables in the Czech Republic in the years years age Source: CZSO, own calculation Fig. 7. Development of the number of females surviving to the exact age of x in the mortality tables in the Czech Republic in the years years 70 age Source: CZSO, own calculation Another used characteristic of the human lifespan is called the probable age at death of a person at the exact age x. Probable age at death is determined by the time at which exactly the half of the population will die in their age x. It is the median age of deaths older than x-years minus the age x. Probable age at death

7 differencies between females and males (in years) is also often considered as a half-life period of the x-year-olds, the time it takes for a given population to reduce its size to half keeping the given mortality rates. Probable age at death as a type of median is not dependent on the extremes. Development of the number of men and women surviving to the exact age x years in the mortality tables of the Czech Republic in the years is shown in the Fig. 6 and Fig. 7. In these figures we highlighted the half of the population (median). In 1920, the probable age at death for men was almost 56 years. During the followed period, probable age at deaths has increased up to almost 77 years in For women, the probable age at deaths was 59 years in 1920 and 83 years in Fig. 8. Development of difference in life expectancies (LE00), modal age at death (en) and probable age at death of 60 and 80-year-old (ep60, ep80) between females and males in the Czech Republic in the years LE00 en ep60 ep80 Source: CZSO, own calculation Fig. 8 shows the development of differences in life expectancies at birth, modal ages at death and probable age at death between women and men. It is obvious that with increasing age, the difference in the lengths of life for women and men decreases. Similar development such as life expectancy at birth passes also development of difference of modal age at death. It is seen that, so how improves mortality in older age groups, also differences of life expectancies, modal age at death and probable age at death for 60 years old are closer to each other. In the Fig. 9 and Fig. 10 is deliberately contrasted the modal age at death (the age at which people are most likely to die) with the probable age at death of a person at the exact age of 60 and 80 years in the period Probable age at death can be calculated from the life expectancy at the exact age of x plus already spent years. Age 60 was chosen to show, whether a person at the exact age of 60 by the current mortality level has the chance to survive to the modal

8 characteristics (in years) age at death. We calculated also the probable age at death of 80-year-old person. Modal age at death is marked as en, probable age at death of 60-year-old person ep60 a probable age at death of 80-year-old person ep80. The life expectancy of 60-year-old man has increased during the followed period, but not as significantly as the life expectancy at birth. As far as in 1920 a man lived 60 years, his probable age at death was similar to the modal age at death of a man in the same year. It is visible, that particularly after 1996 the difference between the modal age at death and probable age at death is increasing. It is caused by a change in mortality rates at higher ages and changes in the intensity of mortality by age. Today's man must survive at least the age of 70, in order his probable age at death was the same or higher than the modal age at death. Fig. 9. Development of modal age at death (en) and the probable age at death of 60 and 80-year-old (ep60, ep80) men in the Czech Republic in the years en ep60 ep80 Source: CZSO, own calculation Analogous figure for women (Fig. 10) shows that if a woman survived the age of 60 in 1920, she had the chance to survive the modal age at death. In the next period the situation is different. If she survives the age of 80, then recently ( ) she has the chance to survive the modal age at death. Modal age at death is marked as en, probable age at death of 60-year-old person ep60 a probable age at death of 80-year-old person ep80. Today's woman must survive at least the age of 79, in order her probable age at death was the same or higher than the modal age at death.

9 characteristics (in years) Fig. 10. Development of modal age at death (en) and the probable age at death of 60 and 80-year-old (ep60, ep80) women in the Czech Republic in the years en ep60 ep80 3 Conclusions Source: CZSO, own calculation In the Czech Republic, between years the life expectancy at birth for men increased from the value of years to years. It is a significant increase of 28 years. For women, the life expectancy increased from 49.6 years to 81.1, there is an increase of more than 31.5 years. This increase is significantly due to the reduction of infant mortality and improving mortality rates among middle-aged and elderly. When looking at the development of the modal age at death, it was years for men and for women in The growth has not been so fast in the studied period mainly due to old-age mortality. In 2013, it was 82.5 for men and 87.6 for women. Today's man must survive at least the age 70, in order his probable age at death was the same or higher than the modal age at death. Today's woman must survive at least the age of 79, in order her probable age at death was the same or higher than the modal age at death. With the improving mortality rates of the Czech population, the share of old people is increasing and it leads to the population ageing. We can assume that mortality will be declining and approaching the levels known from developed countries. In the future, it could be useful to observe not just traditional indicators such as life span indicators at exact age x, but also to follow the trend in the development of the modal age at death, which is considered to be a characteristic of longevity and currently seems to be an appropriate characteristic of mortality rates. Many people accepted the value of life expectancy as typical, most frequent. But when we look at life expectancy at birth (75.2 for males and 81.1 for females in 2013) is neglected fact that the typical age of death is 82.5 for males and 87.6 for females. And this is the reason for the increased use of modal age at death to characterize the mortality rate of the population.

10 Acknowledgments This article was supported by the Grant Agency of the Czech Republic, No. GA ČR S under the title Projection of the Czech Republic Population According to Educational Level and Marital Status. This article was supported by the Internal Grant Agency of the University of Economics, Prague No. 68/2014 under the title Economic and health connections of population ageing. References 1. Arltová, M., Langhamrová, J., Langhamrová, J. Development of life expectancy in the Czech Republic in years with an outlook to Prague Economic Papers, roč. 22, č. 1, s ISSN CZSO. Database of the Czech Statistical Office. Available on < 3. Fiala, T. Výpočty aktuárské demografie v tabulkovém procesoru. 1. vyd. Praha: Oeconomica, s. ISBN Horiuchi, Shiro, Nadine Ouellette, Siu Lan Karen Cheung and Jean-Marie Robine. Modal age at death : lifespan indicator in the era of longevity extension. Vienna Yearbook of Population Research 2013, vol. 11, pp 37-69, ISBN Langhamrová, J. Master thesis: Mortality in the Czech Republic in the years Prague: VŠE, Langhamrová, J., Miskolczi, M., Langhamrová, J. Life Expectancy Trends in CR and EU. Prague In: International Days of Statistics and Economics at VŠE, Prague. Prague : VŠE, 2011, s ISBN Langhamrová, Jana, Cséfalvaiová, Kornélia, Langhamrová, Jitka. Life Expectancy and Modal Age at Death in Selected European Countries in the Years In: SMTDA 2014 Stochastic Modeling Techniques and Data Analysis International Conference and Demographics Workshop. [online] Lisabon, Lisabon : University, 2014, s URL: L_SMTDA2014_Proceedings.pdf. 8. Meslé, F., and J. Vallin Historical Trends in Mortality. In International Handbook of Adult Mortality, ed. By R. G. Rogers and E. M. Crimmins, New York : Springer. 9. Miskolczi, M., Langhamrová, Jitka, Langhamrová, Jana. Trends in Life Expectancy Change in Central European Countries. Demografie, 2011, roč. 53, č. 4, s ISSN Rychtaříková, J. Dlouhodobé trendy úmrtnosti v Československu a Evropě, habilitační práce, Přírodovědecká fakulta Univerzity Karlovy, katedra demografie a geodemografie, 1992, 211 s. 11. Talacko, J. Dynamická pozorování ve statistice úmrtnosti. Praha: Ústřední statistický úřad, 1941.

11 Moment-matching multinomial lattices using Vandermonde matrices for option pricing Karl Lundengård 1, Carolyne Ogutu 2, Jonas Österberg1, Ying Ni 1, Sergei Silvestrov 1, and Patrick Weke 2 1 Division of Applied Mathematics, UKK, Mälardalen University, Högskoleplan 1, Box 883, Västerås, Sweden ( karl.lundengard@mdh.se; jonas.osterberg@mdh.se, ying.ni@mdh.se, sergei.silvestrov@mdh.se) 2 School of Mathematics, University of Nairobi, Box , Nairobi, Kenya ( cogutu@uonbi.ac.ke, pweke@uonbi.ac.ke) Abstract. Lattice models are discretization methods that divide the life of a financial option into time steps of equal length and model the underlying asset movement at each time step. A financial option of American or European style can be evaluated conveniently via backward induction using a lattice model. The most common lattice models are the well-known binomial- and trinomial lattice models, although several kinds of higher order models have also been examined in the literature. In the present paper we present an explicit scheme for creating a lattice model of arbitrary order and use the Vandermonde matrix to determine suitable parameters. Some selected models created using this scheme are examined with regard to their suitability for option pricing. Keywords: Lattice models, moment-matching, multinomial, Vandermonde matrix, option pricing. 1 Introduction In financial derivatives (e.g. an option), pricing, a lattice/tree model is used to discretize the life of a derivative into a number of time steps and model the underlying asset dynamics at each time step. A financial derivative can then be evaluated using backward induction. The advantage of a lattice model is its flexibility in dealing with nonstandard payoff functions and checking early exercising possibilities for American-style derivatives. Lattice models date back to the well-known, simple but powerful binomial lattice introduced by Cox et al.[4]. In a binomial lattice, the asset price S is assumed to either move up to Su or Sd at each time step with risk-neutral (jump) probabilities p and 1 p respectively. By a suitable choice of parameters u, d and p, the random walk in the binomial lattice converges to the continuous geometric Brownian motion under the celebrated Black-Scholes setting. As a natural generalization, a trinomial lattice where the asset price moves by up/middle/down factors u, m, d with probabilities p u, p m, p d = 1 p u p m during each time step, was developed by Boyle[1] in 1986 and Kamrad et al.[9] in Figlewski and Gao [5] proposed later a computationally more efficient version namely the adaptive mesh method. 16 th ASMDA Conference Proceedings, 30 June 4 July 2015, Piraeus, Greece 2015 ISAST

12 The parameterization of a binomial/trinomial is mainly based on a momentmatching technique. Usually the mean and variance of the asset return, under the normal distribution assumption in the risk-neutral world, during a time step are matched to their discrete counterparts. The condition that the probabilities must sum to one should also be imposed. Since the probabilities have to be positive and between zero and one, some conditions for the step size of the lattice and the moments of asset return must be satisfied to assure valid probabilities. Take the binomial lattice for instance, Cox et al.[4] proposed a parameterization under an additional constraint of d = 1/u, whereas Jarrow and Rudd [8] under an another additional constraint of p = 1/2. It is worth mentioning that in some other research work, higher-order moments of the binomial/trinomial distribution are also matched to their continuous counterparts for a better approximation (see for example Cho and Lee[3] and Tian[16]). Multinomial lattice models where asset price moves by more than 3 factors at each time step have also attracted attention of researchers. Some pioneer work are listed as follows. Boyle[2] in 1988 used a 3-dimensional multinomial lattice of order 5 for valuation of options on two underlying assets. Omberg[13] in the same year derived a family of n-order jump process (multinomial lattice) models for any n 2 by applying Gauss-Hermite quadrature. Madan et al.[11] developed a multinomial lattice option pricing model of an arbitrary order n for two limiting cases without and with jumps respectively. A more recent and more pedagogical description on the (central) moment-matching methodology for constructing a general n-order multinomial lattices under the geometric Brownian motion asset price process is given in Jabbour et al.[7]. In the present paper we study a symmetrically placed and evenly spaced lattice of order L which has a natural recombining property. We use a momentmatching technique where the first L raw moments of the asset return per time step which has an arbitrary (risk neutralized) infinitely divisible distribution with finite moments of order L (hence the logarithmic price process is Lévy), are matched to their discrete multinomial counterparts. In our lattice of order 2 for example, the logarithmic of asset price, denoted by X = ln S, moves either down to X + x 1 or up to X + x 2, x 1 = α, x 2 = α, with (risk-neutral) probabilities p 1, p 2, which reduces to the case in Cox et.al[4] under constraint d = 1/u with α = ln u. Matching exactly L moments in a L-order lattice, together with the condition that the probabilities must sum to zero, will give L + 1 equations, which is a natural way of determining L probabilities and the jump size parameter α. The scope of the paper however does not cover the straightforward procedure of derivative pricing on a lattice (refer to Hull[6]) and the convergence properties of the lattice (for further studies). The focus is on lattice-building itself, more specifically on determining analytical and explicit expressions for the probabilities and the jump size (Theorem 3,4). Probabilities for reaching final nodes at maturity (Theorem 1) and conditions under which valid parameters can be found are also presented (Theorem 5, 6). The novelty of the present paper is, for the first time to the authors knowledge, to apply some properties of Vandermonde matrices in determining the lattice parameters, building upon the results in Lundengård et al.[12]. The main contribution is to provide com-

13 w w w.tracker-software.co m w w w.tracker-software.co m plete and general results on analytical expressions/conditions for valid lattice parameters for the L-order lattice with arbitrary order L. For illustrative purpose we provide formulas for the parameters in a Quadrinomial/Pentanomial lattice (Section 4.1,4.2). PDF-XCHANGE Click to buy NOW! 2 Multinomial lattices In this section we will define a class of simple lattices that include those appearing in the binomial and trinomial models and examine some of their basic properties. Note that in the context of financial derivatives pricing, our lattice models the movement of logarithmic asset price X = ln S. Definition 1. Consider a lattice defined by an initial node (X 0, t 0 ) and the rule that for each node, (X, t) there is also a node at (X+x i, t+ t) for i = 1, 2,..., L which is connected to the previous node. The x i are all independent of t. If we have two nodes (X, t) and (Y, u) they are considered to be the same node when X = Y and t = u. Let the nodes of the lattice be defined by x i = (2i L 1)α, 1 i L (1) and some (s 0, t 0 ). Such a lattice will be called a lattice with symmetrically placed and evenly spaced nodes. The parameter α will be called the jump-size. Note that the distance between nodes is actually 2α. In figure 1 some lattices with symmetrically placed and evenly spaced nodes are illustrated. PDF-XCHANGE Click to buy NOW! Fig. 1: Examples of two lattices with symmetrically placed evenly spaced nodes with L = 4 and L = 5 respectively with total number of nodes at each value for t shown. For both lattices X 0 = 0, t 0, α = 1 and t = 1.

14 Theorem 1. Let N k be the number of nodes at t = t 0 + k t. If a lattice has symmetrically placed and evenly spaced nodes then N k = k(l 1) + 1. Proof. When there is an odd number of nodes there will always be a node at (X, t 0 + k t) for each node (X, t 0 + (k 1) t). There will also be L 1 2 nodes above the highest node and L 1 2 below the lowest node. Thus N k = N k 1 + L 1. Since N 0 = 1 and N k = k(l 1) + 1 = (k 1)(L 1) L 1 then theorem 1 can be proven for odd L using induction. If L is even and all nodes at t = t 0 + (k 1) t can be written (X 0 + mα, t 0 + (k 1) t) where m is an even number then all nodes at t = t 0 + k t can be written as (X 0 + nα, t 0 + (k 1) t) where n is an odd number. From this it follows that at t = t 0 + k t there will be N k 1 1 nodes between the highest and lowest node at t = t 0 + (k 1) t. There will be another L 2 nodes above and L 2 nodes below these nodes and thus N k = N k L. Since N 0 = 1 and N k = k(l 1) + 1 = (k 1)(L 1) L then theorem 1 can be proven for even L using induction. Theorem 2. Consider a lattice with evenly spaced nodes, x i+1 x i = x j+1 x j for all 1 j, j L. Denote the number of nodes in such a lattice at time t k = t 0 + k t with N k and label each node from the bottom up with (X m, t k ), m = 1, 2,..., N k. Order the different jump-sizes between nodes such that x 1 < x 2 <... < x L. The probability P (X m, t k ) equals the probability of following some path that start at (X 0, t 0 ) and ends at (X k, t 0 +k t). Consider a random walk on the lattice starting in (S 0, t 0 ). Suppose that at every step the probability of moving from (X, t) to (X + x i, t + t) is equal to p i. Then the probability of the random walk reaching node (X m, t k ) can be calculated by P (X m, t k ) = λ max L 1 λ L 1 =λ min L 1 λ max L 2 λ L 2 =λ min L 1 and the limits of the sums are defined by ( λ min L s = max 0, κ L s s 1 κ L s = k λ L i, i=1 ρl s L s 1 λ max 2 λ 2=λ min L 1 L! L i=1 p λi 1 i λ i 1!. (2) ), λ max L s = min (Λ L s, κ L s ), s 1 ρ L s = m λ L i (L i), i=1 ρ L s = (L s)λ L s + r L s, 0 r L s < L s. Whenever λ min L s > λmax L s that term in the sum is discarded. Note that we do not require the nodes to be symmetrically placed. Proof. Describe a path through the lattice that start at (X 0, t 0 ) and ends at (X m, t k ) with a sequence q = {i 1, i 2,..., i k } with i j {0, 1,..., L 1} where each i j corresponds to moving from (X, t) to (X + x ij, t + t). Let Q be the set of sequences corresponding to paths with steps of descending size. Denote the number of nodes in the lattice at time t k = t 0 + k t with N k and label each node from the bottom up with (X m, t k ), m = 1, 2,..., N k. If

15 we instead us the indices to denote the nodes, such that (m, k) corresponds to (X m, t k ), it is clear that any lattice with evenly spaced nodes is isomorphic to a lattice with (X 0, t 0 ) = (0, 0), x i = i 1, i = 1,..., L and t = 1. Then the probability of the random walk reaching (X m, t k ) via a path q Q, denoted P q (X m, t k ) is P q (X m, t k ) = j q p j+1. Since multiplication is commutative the order of the elements in a sequence corresponding to a path does not affect the probability of reaching the node (X m, t k ), only the number of times each p i appears in the sequence. In other words P q (X m, t k ) = L i=1 p λq(i 1) i where λ q (i) denotes the number of times i appears in q. This gives the following expression for the probability of the random walk reaching (S m, t k ) P (X m, t k ) = q Q n(q) L i=1 p λq(i 1) i (3) where n(q) = n(λ 1, λ 2,..., λ L 1 ) is the number of paths that has the same probability as the path represented by the path q. The set Q corresponds to the set of integer partitions with at most k parts where each part is smaller than or equal to L 1. Such an integer partition can be constructed by choosing a number, λ L 1, of parts of size L 1, and then consider the number integer partitions with at most k λ L 1 parts where each part is smaller than or equal to L 2 and then continue in the same manner for all possible part sizes. The minimum number of part of size L 1 can be found by assuming that all remaining parts are of size L 2 and that m can still be reached. Employing this method to construct Q gives a more explicit version of (3) P (X m, t k ) = λ max L 1 λ L 1 =λ min L 1 λ max L 2 λ L 2 =λ min L 2 λ max 1 λ 1=λ min L 1 Here the limits of the sums are defined by n(λ 1, λ 2,..., λ L 1 ) ρ L s = (L s)λ max L s + r L s, ( κ L s 1) (L s 1) < ρ L s λ min L s(l s) κ L s (L s 1) L i=0 p λi 1 i. (4) with κ L s and ρ L s defined as in theorem 2 and κ L s = κ L s λ min L s. Rewriting these expressions and ensuring that no λ L s are larger than L 1 or smaller than 0 are included gives the expressions for the limits given in theorem 2. The number of equivalent paths n(λ 1, λ 2,..., λ L 1 ) is the number of permutations of the k-multiset of L elements. By Stanley [15] this number is Thus we get expression (2). ( ) L n(q) = = λ 1, λ 2,..., λ L L! λ 1!λ 2!... λ L!.

16 3 Jump-sizes and probabilities on a moment-matching lattice Consider a lattice with symmetrically placed and evenly spaced nodes, that is a lattice with nodes defined by (1). The requirement that L-node lattice matches the k:th moment is L p i x k i α k = i=1 L p i (2i L 1) k α k = µ k (5) i=1 where µ k is the k:th moment of the actual (continuous) distribution of logarithmic asset return per time step, i.e. ln S t+ t S t or equivalently X t+ t Xt. We assume throughout the paper that the logarithmic asset return per time step has an infinitely divisible distribution with finite moments up to order L. So the actual logarithmic price follows a Léy process. We will also use the notation µ 0 = 1 so that matching to µ 0 is equivalent to the sum of all probabilities being equal to one. Matching the all moments µ k, k = 0, 1,..., L can be written as Mp = µ (6) where p is a column vector containing the probabilities, µ is a column vector containing the moments and A is the general lattice matrix with dimension L + 1 L whose elements are given by M m,n = (2n L 1)α m 1 (7) The probabilities can be found by matching µ 0, µ 1,..., µ L 2 and µ L 1. Note that if this is applied to a trinomial lattice the first three moments (mean, variance and skewness) will be matched but often (see for example Hull[6]) only the first two moments are matched. 3.1 Risk-neutral probabilities In this section we will examine the probabilities of moving from one node to another in such a way that (6) is satisfied. Theorem 3. The probability, p i, of going from (X, t) to (X + x i, t + t) is p i = 1 2 L 1 (i 1)!(L i)! L ( 1) j i α 1 j σ L j,i µ j 1 (8) where σ j,i is the jth elementary symmetric polynomial with x i set to zero j=1 σ j,i = 1 m 1<m 2<... <m j L n=1 j x mn (1 δ mn,i), δ a,b = { 1, a = b, 0, a b. (9)

17 Proof. The Vandermonde matrix is a wellknown type of matrix with elements defined by V i,j = x i 1 j. If the matrix is square and all x i are distinct then the matrix is invertible and the inverse is known to be 1 ( V 1 ) = L ij ( 1)j 1 σ L j,i (x k x i ), (10) k i with σ j,i as in theorem 3. This result is can be found in Macon and Spitzbart[10]. Let V L be an L L Vandermonde matrix with x i given by (1) and note that removing the last row from general lattice matrix given by (7) gives V L. The inverse of V L can be written ( V 1 L )ij = ( 1)j i 2 L 2 α j 1 σ L j,i (i 1)!(L i)!, (11) for details see Lundengård et al. [12]. Calculating V 1 l µ gives (8). The final row of the general lattice matrix will be used to find the jump-size. Theorem 4. The jump-size α is a positive real root to the following polynomial L 2 P L (α) = µ L c(l 2j + 1) µ L 2j α 2j, (12) c(j) = L i=1 j=1 ( 1) L j (2i L 1) L σ L j,i 2 L 1. (13) (i 1)!(L i)! Here denotes rounding down to the nearest integer. Proving this theorem requires some preliminary work. Lemma 1. Let A be a set of n distinct real values symmetrically distributed around zero. Denote the set of combinations of k elements from A with A k. Let π : A k R be the product of all elements in a given combination. If k is odd then π k (s) = 0 and if k is even then π k (s) = π k (s) 2 s A k s A k s à k 2 where à = {a A a > 0}, à k is the set of combinations of k elements from à and π : à k R is the product of the square of the elements in a combination multiplied by ( 1) k. Proof. When k is odd it is possible for all s A k to rewrite the product π k (s) = a l π k 1 (r) such that a l is not in r A k 1 for some a l A. If a l = 0 then π k (s) = 0 and for any other a l A there is a combination t s such that π k (t) = a l π k 1 (r) = π k (s) and thus s A k π k (s) = 0 + r A k 1 (a r a r )π k (s) = 0.

18 When k is even we can use an argument analogous to the odd case and conclude that for each combination s A k that can be rewritten such that π k (s) = a l π k 1 (r) where a l r for some a l A, r A k 1 there is also a combination that generate an annihilating term in the sum over all the products. Thus the only remaining terms in the sum over the products will contain both a l and a l as factors and thus any term can be written on the form π k (s) = a ( a) = ( 1) k 2 a 2. a s Lemma 2. The integer values given by (9) can be simplified in the following way: If j = 2k + 1 then σ j,i = x L i+1 π k (s), if j = 2k then σ j,i = π k (s). A useful consequence of this is s Ãk x i / s a s s Ãk x i / s σ j,i = ( 1) j σ j,l i+1. (14) Proof. With the notation used in Lemma 1 the expression in (9) can be rewritten as a sum of products of combinations of the elements in x, j σ j,i = (2n L 1)(1 δ mn,i) = π j (s) = 1 m 1<... <m j L s A j 1 x i / s, x i / s n=1 x i π j 1 (s) + s A j x i / s, x i / s π j (s), s A j i / s where A is the set formed by the values of the elements in x. Now Lemma 2 follows by directly applying Lemma 1. Proof. Follows immediately from lemma 2. Lemma 3. Let c(j) = L i=1 Then c(j) = 0 if L j is even and if L j is odd c(j) = L 2 i=1 Proof. Split the sum into two parts: L 2 c(j) = i=1 + ( 1) L j (2i L 1) L σ L j,i 2 L 1. (15) (i 1)!(L i)! ( 1) j i (2i L 1) L σ L j,i 2 L 2. (i 1)!(L i)! ( 1) j i (2i L 1) L σ L j,i 2 L 1 (i 1)!(L i)! L i= L 2 +1 ( 1) j i (2i L 1) L σ L j,i 2 L 1. (i 1)!(L i)!

19 Changing index in the second sum according to k = L i + 1 gives: L 2 c(j) = i=1 ( 1) j i (2i L 1) L σ L j,i 2 L 1 (i 1)!(L i)! L 2 +a + + a ( 1)j L+1 2 ( 1) j k L+1 (2k L + 1) L σ L j,l k+1 2 L 1 (n k)!(k 1)! ( 2 L+1 2 L 1 ) L σl j, L ( L 1 L )! ( ) L L+1. 2! where a = 1 when L is odd and a = 0 when L is even. Using 14 gives c(j) = ( 1 ( 1) L j) L 2 (2i L 1) L σ L j,i 2 L 1 (i 1)!(L i)!. i=1 Since the factor in front of the sum is zero when j is odd and two otherwise this concludes the proof. Proof (Theorem 4). Matching the L:th moment means satisfying µ L = L p i (2i L 1) L α L. (16) i=1 Using the explicit expressions for the p i given by (8) gives µ L L c(j)µ j 1 α L j+1 = 0 (17) j=1 with c(j) given by (13). The formula (8) follows from (17) and lemma 3. Note that there is no guarantee that there are any real positive roots to (12). The existence of one (or several) real positive roots does not guarantee that the probabilities given in theorem 3 will satisfy 0 p i 1. Some conditions that guarantee this is discussed in the next section. 4 Conditions for positive probabilities In this section p i refers to the probabilities given by theorem 3. Theorem 5. p i 0, i = 1, 2,..., L, L = 2k, k Z +, if and only if k ( 1) i 1 σ 2k 2j+1,i k α 2j 2 µ 2j 2 σ 2k 2j,i α 2j 1 µ 2j 1, i = 1, 2,..., k. (18) j=1 j=1

20 Proof. By letting L = 2k, k Z + and separating the terms in (8) containing even and odd moments into two separate sums into two sums the condition p i 0 is equivalent to k ( 1) i σ k 2k 2j,i α 2j 1 µ 2j 1 ( 1) i σ 2k 2j+1,i α 2j 2 µ 2j 2. (19) j=1 Considering p 2k i+1 0 and equation (14) an analogous argument gives k ( 1) i+1 σ 2k 2j,i µ 2j 1 j=1 α 2j 1 j=1 k ( 1) i σ 2k 2j+1,i α 2j 2 µ 2j 2. (20) j=1 Combining (19) and (20) gives expression (18). Theorem 6. p i 0, i = 1, 2,..., L, L = 2k + 1, k Z + if and only if k+1 ( 1) i 1 σ 2k 2j+1,i k α 2j 2 µ 2j 2 σ 2k 2j,i α 2j 1 µ 2j 1, i = 1, 2,..., k, (21) j=1 p k+1 = ( 1)k+1 2 2k (k!) 2 j=1 k j=0 σ 2k j+2,k+1 α 2j µ 2j 0. (22) Proof. Note that when L = 2k + 1 then x k+1 = 0 which means that σ L 1,i = 0, i k + 1. (23) From the relation described by equation (14) it follows that and thus σ L j+1,k+1 = ( 1) L j+1 σ L j+1,k+1 σ L j+1,k+1 = 0, when L j + 1 is odd. (24) Using (23) and (24) the expression for the probabilities, (8), can be written p i = ( 1) i 2 2k (i 1)!(2k + 1 i)! 2k j=1 p k+1 = ( 1)k+1 2 2k (k!) 2 ( 1) j+1 k j=0 Thus, analogously to the case with even L k+1 p i 0, i k + 1 ( 1) i σ 2k 2j,i µ 2j 1 j=1 α 2j 1 and p L i+1 0, i k + 1 is equivalent to k ( 1) i+1 σ 2k 2j,i µ 2j 1 j=1 α 2j 1 α j σ L j+1,i µ j, i k + 1, (25) σ 2k j+2,k+1 α 2j µ 2j. (26) k ( 1) i σ 2k 2j+1,i α 2j 2 µ 2j 2 (27) j=1 k ( 1) i σ 2k 2j+1,i α 2j 2 µ 2j 2. (28) Rewriting these expression using the relation σ j,i = ( 1) j σ j,l i+1 analogously to the case when L is even gives equation (21). j=1

21 4.1 Explicit expressions for quadrinomial and pentanomial lattices Quadrinomial lattice A quadrinomial lattice with α = 1, t = 1, and (X 0, t 0 ) = (0, 0) is shown in figure 1. If L = 4 then by theorem 4 the jump-size α must be a real positive root of the polynomial P 4 (α) = µ 4 10α 2 µ 2 + 9α 4. (29) The polynomial (29) has the following four roots α 1 = 1 5µ α 2 = 1 3 α 3,4 = 1 3 5µ 2 5µ 2 ± 25µ 2 2 9µ 4, (30) 25µ 2 2 9µ 4, (31) 25µ 2 2 9µ 4. It is clear that the only the roots given by two roots given by (30) and (31) can be real and positive. If and only if 25µ 2 2 9µ 4 > 0 (32) then α 1 will be real and positive. For α 2 to be real and positive the condition (32) will need to be satisfied as well as 5µ 2 25µ 2 2 9µ µ µ 2 2 9µ 4 0 9µ 4 and since all even moments are positive this condition is always satisfied and it is sufficient to satisfy (32) to guarantee α 1 R and α 2 R. The probability to jump to each node is given by these expressions p 1,4 = 1 ( 3 ± µ 1 48 α + 3µ 2 α 2 µ ) 3 α 3, (33) p 2,3 = 1 ( 27 27µ 1 48 α + 3µ 2 α 2 3µ ) 3 α 3. (34) Since the probabilities are already guaranteed to sum to one we only need to find conditions for p i 0. By theorem 5 p 2 0, p µ 2 α 2 p 1 0, p ( µ2 α 2 1 ) 1 α 1 9µ 1 µ 3, (35) α α 2 µ 1 µ 3 α 2. (36) Pentanomial lattice An illustration of the pentanomial lattice with (X 0, t 0 ) = (0, 0), α = 1, t = 1 can be seen in figure 1. This type of lattice with µ 1 = 0 has previously been examined in Primbs et al.[14].

22 If L = 5 then by theorem 4 the jump-size α must be a real positive root of the polynomial P 5 (α) = µ 5 20α 2 µ α 4 µ 1. (37) If µ 1 0 the polynomial (29) has the following four roots ( α 1 = 1 25µ µ 1 µ 5 + 5µ 3 4 2µ 1 ( 25µ µ 1 µ 5 5µ 3 α 2 = 1 4 α 3,4 = 1 4 2µ 1 ( 25µ µ 1 µ 5 ± 5µ 3 2µ 1 ) 1 2 ) 1 2 ) 1 2. (38) (39) If 25µ µ 1 µ 5, µ 1 > 0 and µ 3 > 0 then α 1 is real and non-negative. If 25µ µ 1 µ 5, µ 1 0, µ 3 < 0 and µ 5 < 0 then α 1 is real and non-negative. If 25µ µ 1 µ 5, µ 1 > 0 and µ 3 < 0 then α 2 is real and non-negative. If 25µ µ 1 µ 5, µ 1 0, µ 3 > 0 and µ 5 < 0 then α 2 is real and non-negative. If µ 1 0 and µ 5 0 then both (38) and (39) is real and non-negative. If µ 1 = 0 then (29) has two roots α 5,6 = ± 1 5 µ3 µ 5 10 µ 3 and only one of these roots will be positive at the time and only if µ 3 0 and µ 5 0 and µ 3 and µ 5 have opposite signs. The probability to jump to each node is given by these expressions p 1,5 = 1 ( ± 16µ α 4µ 2 α 2 4µ 3 α 3 + µ ) 4 α 4, (40) p 2,4 = 1 ( 32µ 1 96 α + 16µ 2 α 2 ± 2µ 3 α 3 µ ) 4 α 4, (41) p 3 = 1 64 ( 64 20µ 2 α 2 + µ 4 α 4 ). (42) Since the probabilities are already guaranteed to sum to one we only need to find conditions for p i 0. By theorem 6 p 2 0, p 4 0 p 1 0, p ( α 1 α 4µ 2 µ ) 4 α ( 2 16µ 2 µ ) 4 α 2 4 4µ 1 µ 3, (43) α µ 1 µ 3, (44) α 2 p µ 2 α 2 + µ 4 α 4 0 α4 5µ 2 16 α2 + µ 4 0. (45) 64

23 4.2 Explicit formulas for Geometrical Brownian Motion For the geometric Brownian motion the first five central moments are ) µ 1 = (r σ2 t, µ 2 = σ 2 t, µ 3 = 0, µ 4 = 3σ 4 t, µ 5 = 0. 2 Quadrinomial lattice Inserting the expression for the moments of the geometric Brownian motion into equation (29) gives the polynomial P 4 (α) = 3σ 2 t 10α 2 σ 2 t + 9α 4. (46) which will have two real roots if t 27 25σ 2, α 1,2 = σ 2 t ± σ 25 σ 2 t 2 27 t. (47) The probability to jump to each node is given by (33)-(34) and the conditions (35) and (36) becomes p 1 0, p 4 0 4σ 2 t ± σ 25σ 2 t 2 27 t 9α 1,2 r σ2 2 t, p 2 0, p 3 0 4σ 2 t ± σ 25σ 2 t 2 27 t α 1,2 r σ2 2 t. Pentanomial lattice Inserting the expression for the moments of the geometric Brownian motion into equation (37) gives the polynomial ) P 5 (α) = 64 (r σ2 t α 2. (48) 2 which only have one root, α = 0 which will give a degenerate lattice unless µ 1 = 0. The case when µ 1 = 0 is thoroughly examined in Primbs et al.[14]. 5 Conclusion In this paper we examined some basic properties of lattices with symmetrically placed and evenly spaced nodes and how they can be used to construct moment-matching lattices that can be used for financial derivatives pricing. The examination is limited to the case where the lattice is constructed from L nodes and matches L moments. Explicit expressions for probabilities are given in Theorem 3. We also show that the moment-matching places a restriction on the jump-sizes as they need to be a root of a particular polynomial, see Theorem 4. Furthermore we examine what moments that can give proper probabilities. Conditions for this is given by Theorems 5 and 6. Explicit conditions for L = 4 when the underlying asset is modelled by using geometric Brownian motion are given on page 13. The formulas for probabilities, jump-sizes and conditions for

24 positive probabilities for the quadrinomial (L = 4) as well as the pentanomial (L = 5) lattice are given in section 4.1. We also examine how the lattice method will behave with an underlying asset modelled by geometric Brownian motion for the quadrinomial and pentanomial case. For the pentanomial case we show that this particular lattice method is of limited use since it degenerates when µ 1 0, see page 13. Many of the results presented here are applicable when the lattice is used to approximate a Lévy process of the logarithmic asset price in the risk-neutral world. The asset return per time step, which under this setting has an infinitely divisible distribution, should have finite moments of order L. The approach presented here can be used to examine the properties of lattice methods with this family of node distributions. Further analysis of the convergence and numerical stability of the methods should be performed. References 1. P. Boyle Option valuation using a three-jump process. International Option Journal 3, 7 12, P. Boyle A lattice framework for option pricing with two state variables. Journal of Financial and Quantitative Analysis 23,No.1, 1 12, March H. Cho and K. Lee, An extension of the three jump process model for contigent claim valuation. The Journal of Derivatives 3, , J. C. Cox, S.A. Ross and M. Rubinstein Option pricing: A simplifed approach. Journal of Financial Economics 7, , October S. Figlewski and B. Gao The adaptive mesh model: a new approach to efficient option pricing Journal of Financial Economics 53, , J.C. Hull. Options, Futures and other Derivatives. Prentice Hall, 7th edition, G.M. Jabbour, M.V. Kramin, T.V. Kramin and S.D. Young. Multinomial Lattices and Derivatives Pricing. Advances in Quantitative Analysis of Finance and Accounting, 2, 1, 1 15, R. Jarrow and A. Rudd, Option Pricing, Irwin, B. Kamrad and P. Ritchken, Multinomial approximating models for options with k-state variables. Management science 37, No.12, , N. Macon and A. Spitzbart. Inverses of Vandermonde matrices. The American Mathematical Monthly, 65(2), , D.B. Madan, F. Milne and H. Shefrin The Multinomial option pricing model and its Brownian and Poisson limit. Review of Financial Studies 2, , K. Lundengård, C.Ogutu, S.Silvestrov and P.Weke. Asian Options, Jump- Diffusion Processes on a Lattice, and Vandermonde Matrices, in Modern Problems in Insurance Mathematics, Silvestrov, Dmitrii, Martin-Löf, Anders, Eds., Springer-Verlag, Berlin, (2014) 13. E. Omberg Efficient Discrete Time Jump Process Models in Option Pricing. Journal of Financial and Quantitative Analysis 23(2), , J.A. Primbs, M.Rathinam and Y.Yamada. Option Pricing with a Pentanomial Lattice Model that Incorporates Skewness and Kurtosis. Applied Mathematical Finance, 14:1, 1-17, R.P. Stanley. Enumerative Combinatorics. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2nd edition, Y. Tian A modified lattice approach to option pricing. Journal of Futures Markets 13, , 1993

25 Estimation of Parameters for the Multi-peaked AEF Current Functions Karl Lundengård 1, Milica Rančić 1, Vesna Javor 2, and Sergei Silvestrov 1 1 Division of Applied Mathematics, UKK, Mälardalen University, Högskoleplan 1, Box 883, Västerås, Sweden ( karl.lundengard@mdh.se; milica.rancic@mdh.se, sergei.silvestrov@mdh.se) 2 Department of Power Engineering, Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, Niš, Serbia ( vesna.javor@elfak.ni.ac.rs) Abstract. Examination of how the analytically extended function (AEF) can be used to approximate multi-peaked lightning current waveforms, is performed in the paper. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks is presented. This framework is used to find parameters for some common waveforms with a single peak, such as one-peaked Standard IEC lightning currents. Keywords: Analytically extended function, Electromagnetic compatibility, Electrostatic discharge current, Lightning discharge, Marquardt least-squares method. 1 Introduction Various single and multi-peaked functions are proposed in the literature for modelling lightning channel-base currents, Heidler[1], Heidler and Cvetic[2], Javor and Rancic[3], Javor[4]-[5]. For engineering and electromagnetic models, a general function that would be able to reproduce desired waveshapes is needed, such that analytical solutions for its derivatives, integrals, and integral transformations, exists. A multi-peaked channel-base current function has been proposed in Javor[4] as a generalization of the so-called TRF (two-rise front) function from Javor[5], which possesses such properties. In this paper we analyse a modification of such multi-peaked function, a socalled p -peak analytically extended function (AEF). Possibility of application of the AEF to modelling of various multi-peaked waveshapes is investigated. Estimation of its parameters has been performed using the Marquardt leastsquares method (MLSM), an efficient method for the estimation of non-linear function parameters, Marquardt[6]. It has been applied in many fields, including the lightning research for optimizing parameters of the Heidler function in Lovric et al.[7], or the Pulse function in Lundengård et al.[8]-[9]. Some numerical results are presented, including those for the Standard IEC 62305[10] currents of the first positive and first negative stokes, and an example of a fast-decaying lightning current waveform. Based on presented results, corresponding conclusions are made, and further research ideas are discussed. 16 th ASMDA Conference Proceedings, 30 June 4 July 2015, Piraeus, Greece 2015 ISAST

26 2 The p -peak analytically extended function Definition 1. Given I mq R, t mq R, q = 1, 2,..., p such that t m0 = 0 < t m1 t m2... t mp along with η q,k, β q,k R and 0 < n q Z for n q q = 1, 2,..., p + 1, k = 1, 2,..., n q such that η q,k = 1. The analytically extended function (AEF), i(t), with p peaks is defined as ( q 1 ) n q I mk + I mq η q,k x q (t) β2 q,k +1, t mq 1 t t mq, 1 q p, i(t) = ( p ) np+1 I mk η p+1,k x p+1 (t) β2 p+1,k, t mp t. where ( ) t t mq 1 tmq t exp t mq t mq x q (t) = ( t exp 1 t ) t mq t mq, 1 q p, q = p + 1 and t mq = t mq t mq 1. Sometimes the notation i(t; β, η) with β = [ ] [ ] β 1,1 β 1,2... β q,k... β p+1,np+1, η = η1,1 η 1,2... η q,k... η p+1,np+1 will be used to clarify what the particular parameters for a certain AEF are. Remark 1. The p -peak AEF can be written more compactly if we introduce the vectors η q = [η q,1 η q,2... η q,nq ], (1) [ x q (t) β2 q,1 +1 x q (t) β2 q, x q (t) +1] β2 q,nq, 1 q p x q (t) = [ ] (2) x q (t) β2 q,1 xq (t) β2 q,2... xq (t) β2 q,nq, q = p + 1 The more compact form is ( q 1 ) I mk + I mq η q x q (t), t mq 1 t t mq, 1 q p, i(t) = ( q ) I mk η q x q (t), t mq t, q = p + 1 (3) Since the AEF is a linear function of elementary functions its derivative can be found using standard methods.

27 Theorem 1. The derivative of the p -peak AEF is t mq t di(t) I mq t t mq 1 = dt x q (t) I mq t x q (t) t mq η q B q x q (t), t mq 1 t t mq, 1 q p, t mq t t mq η q B q x q (t), t mq t, q = p + 1 where β 2 p+1, β 2 0 βp+1,2 2 q, B p+1 =......, B 0 βq, q =......, βp+1,n 2 p βq,n 2 q + 1 for 1 q p. Lemma 1. The AEF is continuous and at each t mq the derivative is equal to zero. Proof. Within each interval t mq 1 t t mq the AEF is a linear combination of continuous functions and at each t mq the function will approach the same n q value from both directions unless all η q,k 0 but if η q,k 0 then η q,k 1. Noting that for any diagonal matrix B the expression n q η q B x q (t) = η q,k B kk x q (t) β2 q,k +1, 1 q p is well-defined and that the equivalent statement holds for q = p it is easy to see from (4) that the factor (t mq t) in the derivative ensures that the derivative is zero every time t = t mq. (4) Lemma 2. For any for t mq 1 t 0 t 1 t mq, 1 q p t1 ( x q (t) β dt = eβ 1 β β γ β + 1, β t 1 t mq, β t ) 0 t mq t mq t mq t 0 with t mq = t mq t mq 1 and (5) where γ(β, t 0, t 1 ) = γ (β + 1, βt 1 ) γ (β + 1, βt 0 ) γ(β, t) = t 0 τ β 1 e τ dτ is the lower incomplete Gamma function as in Abramowits and Stegun[11]. If t 0 = t mq 1 and t 1 = t mq then tmq x q (t) β dt = eβ 1 γ (β + 1, β). (6) t mq 1 ββ

28 Fig. 1. Illustration the AEF (solid line) and its derivative (dashed line) with the same I mq and t mq but different β q,k -parameters. (a) 0 < β q,k < 1, (b) 4 < β q,k < 5, (c) 12 < β q,k < 13, (d) a mixture of large and small β q,k -parameters. Proof. t1 t 0 x q (t) β dt = t1 t 0 = eβ 1 β β ( t tmq t mq t1 t 0 ( exp 1 t t m q ( β t t m q t mq t mq )) β dt ) β exp ( 1 β t t m q t mq ) dt. Changing variables according to τ = t tmq t mq gives t1 t 0 x q (t) β dt = eβ 1 β β τ1 τ 0 = eβ 1 β β γ When t 0 = t mq 1 and t 1 = t mq then τ β e 1 τ dt == eβ 1 β β (γ(β + 1, τ 1) γ(β + 1, τ 0 )) ( ). β + 1, β t 1 t mq, β t 0 t mq t mq t mq t1 and with γ(β + 1, 0) = 0 we get (6). t 0 x q (t) β dt = eβ 1 γ (β + 1, β) ββ

29 Lemma 3. For any for t mq 1 t 0 t 1 t mq, 1 q p ( t1 q 1 ) n q i(t) dt = (t 1 t 0 ) I mk + I mq η q,k g q (t 1, t 0 ), (7) t 0 g q (t 1, t 0 ) = e β2 q,k (β 2q,k + 1 ) β 2 q,k +1 γ ( βq,k 2 + 2, t 1 t mq 1, t 0 t mq 1 t mq t mq ) with γ(β, t 0, t 1 ) defined as in lemma 5. Proof. t1 t 0 i(t) dt = ( t1 q 1 t 0 = (t 1 t 0 ) = (t 1 t 0 ) ) n q I mk + I mq η q,k x q (t) β2 q,k +1 dt ( q 1 ) n q t1 I mk + I mq η q,k ( q 1 ) n q I mk + I mq η q,k g q (t 0, t 1 ) t 0 x q (t) β 2 q,k +1 dt Theorem 2. For t ma 1 t a t ma, t mb 1 t b t mb and 0 t a t b t mp ( tb a 1 ) n a i(t) dt = (t ma t a ) I mk + I ma η a,k g a (t a, t ma ) t a ( ( b 1 q 1 ) n q + t mq I mk + I mq η q,k ĝ ( βq,k )) q=a+1 + (t b t mb ) ( b 1 ) n b I mk + I mb η b,k g b (t mb, t b ) (8) where g q (t 0, t 1 ) is defined as in lemma 3 and ĝ(β) = eβ 1 γ (β + 1, β). ββ Proof. This theorem follows from integration being linear and lemma 3. Theorem 3. For t mp t 0 < t 1 < the integral of the AEF is ( t1 p ) np+1 i(t) dt = I mk η p+1,k g p+1 (t 1, t 0 ) (9) t 0 where g q (t 0, t 1 ) is defined as in lemma 3. When t 0 = t mp and t 1 the integral becomes ( p ) np+1 i(t) dt = I mk η p+1,k g ( β 2 ) p+1,k t mp (10)

30 with g(β) = eβ 1 (Γ (β + 1) γ (β + 1, β)), Γ (β) = ββ 0 t β 1 e t dt where Γ (β) is the Gamma function as in Abramowits and Stegun[11]. Proof. This theorem follows from integration being linear and lemma 3. In the next section we will estimate the parameters of the AEF that gives the best fit with respect to some data and for this the partial derivatives with respect to the β mq parameters will be useful. Since the AEF is a linear combination of elementary functions these partial derivatives can be found easily. Theorem 4. The partial derivatives of the p-peak AEF with respect to the β parameters are i 0 = 2 I β mq η q,k β q,k h q (t)x q (t) β2 q,k +1 q,k 0 { i 0 = β p+1,k I mp+1 η p+1,k β p+1,k h p+1 (t)x p+1 (t) β2 p+1,k, 0 t t mq 1,, t mq 1 t t mq, 1 q p,, t mq t,, 0 t t mp,, t mp t, (11) (12) where ( t tmq 1 ln t mq h q (t) = ( t ln t mq ) + t t m q 1 t mq + 1, 1 q p, ) + t t mq + 1, q = p Least-square fitting using MLSM Suppose that we have k q points (t q,k, i q,k ) such that t mq 1 < t q,1 < t q,2 <... < t q,kq < t mq and we wish to choose parameters η q,k and β q,k such that the sum of the squares of the residuals, k q S q = (i(t q,k ) i q,k ) 2, (13) is minimized. One way to estimate these parameters is to use the Marquardt least-square method. We will not discuss the details of the method here as the application of the method is analogous to the approach in Lundengård et al.[8]-[9]. Instead we will only give the necessary expressions for a matrix, J, containing the partial derivatives of the errors between the measured data and the AEF with respect to the different parameters.

31 3.1 Fitting the AEF In order to fit the AEF it is sufficient that k q n q. Suppose we have some estimate of the β-parameters which is collected in the vector b. It is then fairly simple to calculate an estimate for the η-parameters, see section 3.3, which we collect in h. We can then define a residual vector by (E) k = i(t q,k ; b, h) i q,k where i(t; b, h) is the AEF with the estimated parameters. The J matrix can in this case be described as J = i β q,1 t=tq,1 i β q,1 t=tq,2. i β q,1 t=tq,kq i i β q,2... β q,nq t=tq,1 t=tq,1 i i β q,2... β q,nq t=tq,2 t=tq, i i β q,2... β q,nq t=tq,kq t=tq,kq (14) where the partial derivatives are given by (11) and (12). 3.2 Fitting with data points as well as charge transfer and specific energy conditions By considering the charge transfer at the striking point, Q 0, and the specific energy, W 0, two further conditions need to be considered: First we will define Q(b, h) = Q 0 = W 0 = i(t) dt, (15) i(t) 2 dt. (16) i(t; b, h) dt and W (b, h) = These two quantities can be calculated as follows. 0 i(t; b, h) 2 dt. Theorem 5. Q(b, h) = ( ( p q 1 ) n q ) t mq I mk + I mq η q,k ĝ(βq,k 2 + 1) q=1 + ( p ) np+1 I mk η p+1,k g(β 2 p+1,k), (17)

32 W (b, h) = p q=1 ( q 1 ) 2 I mk + ( q 1 ) I mk n q + Im 2 q ηq,k 2 ĝ ( 2 βq,k ) n q Im 2 q n q r=1 s=r+1 I mq ( p ) 2 ( np + I mk ηp,k 2 g ( 2 βp,k 2 ) n p r=1 n p+1 s=r+1 n q η q,k ĝ(βq,k 2 + 1) η q,r η q,s ĝ ( βq,r 2 + βq,s ) where ĝ(β) and g(β) are defined in theorem 2 and 3. Proof. Formula (17) is found by combining (8) and (10). Formula (18) is found by noting that ( n ) 2 a k = n η p+1,r η p+1,s g ( β 2 p+1,r + β 2 p+1,s) ) (18) n 1 a 2 k + n r=1 s=r+1 a r a s and then reasoning analogously to the proofs for (8) and (10). We can calculate the charge transfer and specific energy given by the AEF with formula (17) and (18) respectively and get two additional residual terms E Q0 = Q(b, h) Q 0 and E W0 = W (b, h) W 0. Since these are global conditions this means that the parameters η and β no longer can be fitted separately in each interval. This means that we need to consider all data points simultaneously. The resulting J-matrix is J J = 0... J p+1 E Q0 E β 1,1... Q0 E β 1,n1... Q0 β p+1,1... E W0 β 1,1... E W 0 E β 1,n1... W0 β p+1,1... E Q0 β p+1,np+1 E W0 β p+1,np+1 (19) where J q = i β q,1 t=tq,1 i β q,1 t=tq,2. i β q,1 t=tq,kq i i β q,2... β q,nq t=tq,1 t=tq,1 i i β q,2... β q,nq t=tq,2 t=tq, i i β q,2... β q,nq t=tq,kq t=tq,kq

33 and the partial derivatives in the last two rows are given by dĝ Q 2 I mq η q,s β q,s dβ, 1 q p, β=β 2 = q,s +1 β q,s d g 2 I mp η p+1,s β p+1,s dβ, q = p + 1. β=β 2 p+1,s For 1 q p ( q 1 ) W = 2 I mk β q,s + 4 Im 2 dĝ q η q,s β q,s η q,s dβ dĝ I mq η q,s β q,s dβ n q + β=2β 2 q,s +2 k s η q,k β=β 2 q,s +1 dĝ dβ β=β 2 q,s +β 2 q,k +2 and ( p ) W = 4 I mk β p,s d g η p+1,s dβ η p+1,s β p+1,s n q + β=2β 2 p+1,s k s η p+1,k d g dβ β=β 2 p+1,s +β 2 p+1,k. The derivatives of ĝ(β) and g(β) are ) dĝ (1 dβ =1 + eβ ( ( ) ) Γ (β + 1) Ψ(β) ln(β) G(β) e β β, (20) ( ) d g e β dβ =1 e β β G(β) 1, (21) where Γ (β) is the Gamma function, Ψ(β) is the digamma function as in Abramowits and Stegun[11], and G(β) is a special case of the Meijer G-function and can be defined as ( ) G(β) = G 3,0 2,3 β 1, 1 0, 0, β + 1 using the notation from Prudnikov et al.[12]. When evaluating this function it might be more practical to rewrite G using other special functions ( G(β) = G 3,0 2,3 β 1, 1 0, 0, β + 1 ) = ββ+1 (β + 1) 2 2 F 2 (β + 1, β + 1; β + 2, β + 2; β) ( Ψ(β) + π cot(πβ) + ln(β) ) π csc (πβ) Γ ( β)

34 where 2F 2 (β + 1, β + 1; β + 2, β + 2; β) = ( 1) k β k (β + 1) 2 (β + k + 1) 2 is a special case of the hypergeometric function. These partial derivatives were found using Maple TM [13]. Note that all η-parameters must be recalculated for each step, how this is done is detailed in the section 3.3. k=0 3.3 Calculating the η-parameters when the β-parameters are known. Suppose that we have n q 1 points (t q,k, i q,k ) such that t mq 1 < t q,1 < t q,2 <... < t q,nq 1 < t mq. For an AEF that interpolates these points it must be true that q 1 I mk + I mq η q,s x q (t q,k ) βq,s = i q,k, k = 1, 2,..., n q 1. (22) n q s=1 Since η q,1 + η q, η q,nq = 1 equation (22) can be rewritten as n q 1 I mq s=1 η q,s ( xq (t q,k ) βq,s x q (t q,k ) βq,nq ) = iq,k x q (t q,k ) βq,nq for k = 1, 2,..., n q 1. This can be written as a matrix equation I ms (23) q 1 s=1 I mq Xq η q = ĩ q, (24) η q = [ ] ) q 1 η q,1 η q,2... η q,nq 1, (ĩq = i q,k x q (t q,k ) βq,nq I ms, k s=1 ( ) Xq = x q(k, s) = x q (t q,k ) βq,s x q (t q,k ) βq,nq, k,s and x q (t) given by (2). When all β q,k, k = 1, 2,..., n q are known then η q,k, k = 1, 2,..., n q 1 can be n q 1 found by solving equation (24) and η q,nq = 1 η q,k. If we have k q > n q 1 data points than the parameters can be estimated with the least-squares solution to (24), more specifically the solution to I 2 m q X q Xq η q = X q ĩq.

35 3.4 Explicit formulas for a single-peak AEF i(τ) I m1 = t t m1. Then the explicit Consider the case where p = 1, n 1 = n 2 = 2 and τ = formula for the AEF is { η 1,1 τ β2 1,1 +1 e (β2 1,1 +1)(1 τ) + η 1,2 τ β2 1,2 +1 e (β2 1,2 +1)(1 τ), 0 τ 1, η 2,1 τ β2 2,1 e β 2 2,1 (1 τ) + η 2,2 τ β2 2,2 e β 2 2,2 (1 τ), 1 τ. (25) Assume that four datapoints, (i k, τ k ), k = 1, 2, 3, 4, as well as the charge transfer and specific energy Q 0, W 0 are known. If we want to fit the AEF to this data using MLSM equation (19) gives f 1 (τ 1 ) f 2 (τ 1 ) 0 0 f 1 (τ 2 ) f 2 (τ 2 ) g 1 (τ 3 ) g 2 (τ 3 ) 0 0 g J = 1 (τ 4 ) g 2 (τ 4 ), Q(β, η) Q(β, η) Q(β, η) Q(β, η) β 1,1 β 1,2 β 2,1 β 2,2 β 1,1 W (β, η) β 1,2 W (β, η) β 2,1 W (β, η) β 2,2 W (β, η) f k (τ) = 2 η 1,k β 1,k τ β2 1,k +1 e (β2 1,k +1)(1 τ)( ln(τ) + 1 τ ), η 1,1 = i 1 I m1 τ β 2 1,2 1 e (β2 1,2 +1)(1 τ1), η 1,2 = 1 η 1,1, g k (τ) = 2 η 2,k β 2,k τ β2 2,k e β 2 2,k (1 τ)( ln(τ) + 1 τ ), η 2,1 = i 3 I m1 Q(β, η) I m1 = τ β 2 2,2 3 e β2 1,2 (1 τ3), η 2,2 = 1 η 2,1, β = [( β 2 1,1 + 1 ) ( β 2 1,2 + 1 ) β 2 2,1 β 2 2,2], η = [ ] η 1,1 η 1,2 η 2,1 η 2,2, 2 s=1 + η 1,s e β2 1,s ( β 2 1,s + 1 ) β 2 1,s +1 γ ( β 2 1,s + 2, β 2 2,s + 1 ) 2 e β2 2,s 1 η 2,s s=1 β 2β2 2,s 2,s ( ( Γ β 2 2,s + 1 ) γ ( β2,s 2 + 1, β2,s 2 )), dĝ Q 2 I m1 η 1,s β 1,s dβ, q = 1, β=β 2 = 1,s +1 β q,s d g 2 I mq η p,s β 2,s dβ, q = 2, β=β 2 2,s with derivatives of ĝ(β) and g(β) given by (20) and (21), β = [( β 2 1,1 + β 2 1,2 + 2 ) ( β 2 1,1 + β 2 1,2 + 2 ) (β 2 2,1 + β 2 2,2) (β 2 2,1 + β 2 2,2) ], η = [ η 2 1,1 η 2 1,2 η 2 2,1 η 2 2,2], η = [ (η1,1 η 1,2 ) (η 1,1 η 1,2 ) (η 2,1 η 2,2 ) (η 2,1 η 2,2 ) ], β q,s W (β, η) = 2 β q,s Q (2β, η) + β ) Q β q,s q, ((s 1) mod 2)+1 β q,s ( β, η ).

36 4 Results In this section some numerical results of fitting the AEF-function to some single-peaked waveshapes are presented and compared with the corresponding fitting of the Heidler function. The AEF given by (25) has been used to model few lightning current waveshapes whose parameters (rise/decay time ratio, T 1 /T 2, peak current value, I m1, time to peak current, t m1, charge transfer at the striking point, Q 0, specific energy, W 0, and time to 0.1I m1, t 1 ) are given in Table 1. Data points were chosen as follows: (i 1, τ 1 ) = (0.1 I m1, t 1 ), (i 3, τ 3 ) = (0.5 I m1, t h = t T 1 + T 2 ), (i 2, τ 2 ) = (0.9 I m1, t 2 = t T 1 ), (i 4, τ 4 ) = (i(1.5 t h ), 1.5 t h ). Fig. 2. First-positive stroke represented by the AEF function Fig. 3. First-negative stroke represented by the AEF function The AEF representation of the waveshape denoted as the first positive stroke current in IEC 62305[10], is shown in Fig. 2. Rising and decaying parts of the first negative stroke current from IEC 62305[10] are shown in Fig. 3 - left and right, respectively. β and η parameters of both waveshapes optimized by the MLSM are given in Table 1. We have also observed a so-called fast-decaying waveshape whose parameters are given in Table 1. It s representation using the AEF function is shown in Fig. 4, and corresponding β and η parameter values from Table 1.

37 Fig. 4. Fast-decaying waveshape represented by the AEF function Apart from the AEF function (solid line), the Heidler function representation of the same waveshapes (dashed line), and used data points (red solid circles) are also shown in the figures. First-positive stroke First-negative stroke Fast-decaying T 1/T 2 10/350 1/200 8/20 t m1 [µs] I m1 [ka] Q 0 [C] 100 / / W 0 [MJ/Ω] 10 / / t 1 [µs] β 1, β 1, β 2, β 2, η 1, η 1, η 2, η 2, Table 1. AEF function s parameters for some current waveshapes 5 Conclusion The paper deals with investigation of a possibility to approximate, in general, multi-peaked lightning currents using a modified AEF function. Furthermore, existance of the analytical solution for the derivative and the integral of such function has been proven, which is needed in order to perform lightning electromagnetic field (LEMF) calculations based on it. Two one-peaked Standard IEC waveforms, and a fast-decaying one have been represented using a variation of the proposed AEF function (25). The estimation of their parameters has been performed applying the MLS method

38 using two pairs of data points for each function part (rising and decaying). As it can be observed from the results, to improve the fitting, and therefore the representation of a certain waveshape, additional terms should be introduced, which is especially needed when the rising part is steep. Further research will focus on the application of the same optimization procedure to an AEF function with more terms in both the rising and the decaying part, and using more data points. Since the AEF function here is defined with an arbitrary number of peaks waveshapes with more than one peak are a natural subject of investigation. References 1. F. Heidler, Travelling current source model for LEMP calculation, in Proceedings of papers, 6th Int. Zurich Symp. EMC, Zurich, 1985, pp F. Heidler and J. Cvetic. A class of analytical functions to study the lightning effects associated with the current front, Transactions on Electrical Power, 12, 2, , V. Javor and P. D. Rancic. A channel-base current function for lightning returnstroke modeling, IEEE Transactions on EMC, 53, 1, , V. Javor. Multi-peaked functions for representation of lightning channel-base currents, Proceedings of papers, 2012 International Conference on Lightning Protection - ICLP, Vienna, Austria, 2012, pp V. Javor. New functions representing IEC standard and other typical lightning stroke currents, Journal of Lightning Research, 4, Suppl 2: M2, 50 59, D. W. Marquardt. An algorithm for least-squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics, 11, 2, , D. Lovric, S. Vujevic, and T. Modric. On the estimation of Heidler function parameters for reproduction of various standardized and recorded lightning current waveshapes, International Transactions on Electrical Energy Systems, 23, , K. Lundengård, M. Rančić, V. Javor, S. Silvestrov, Application of the Marquardt least-squares method to the estimation of Pulse function parameters in AIP Conference Proceedings 1637, ICNPAA, Narvik, Norway, 2014, pp K. Lundengård, M. Rančić, V. Javor, S. Silvestrov, Estimation of Pulse function parameters for approximating measured lightning currents using the Marquardt least-squares method in Conference Proceedings, EMC Europe, Gothenburg, Sweden, 2014, pp IEC Ed.2: Protection Against Lightning - Part I: General Principles, M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, A. P. Prudnikov, Yu Brychkov, and O. Marichev, Integrals and Series, Volume 3: More Special Functions. Gordon and Breach Science Publishers, New York, Maple Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.

39 Optimization of the determinant of the Vandermonde matrix and related matrices Karl Lundengård 1, Jonas Österberg1, and Sergei Silvestrov 1 Division of Applied Mathematics, UKK, Mälardalen University, Högskoleplan 1, Box 883, Västerås, Sweden ( karl.lundengard@mdh.se; karl.lundengard@mdh.se, jonas.osterberg@mdh.se, sergei.silvestrov@mdh.se) Abstract. The value of the Vandermonde determinant is optimized over various surfaces, including the sphere, ellipsoid and torus. Lagrange multipliers are used to find a system of polynomial equations which give the local extreme points in its solutions. Using Gröbner basis and other techniques the extreme points are given either explicitly or as roots of polynomials in one variable. The behavior of the Vandermonde determinant is also presented visually in some interesting cases. Keywords: Vandermonde determinant, optimization, Gröbner basis, orthogonal polynomials, ellipsoid, optimal experiment design, homogeneous polynomials. 1 Introduction In this paper we will consider the extreme points of the Vandermonde determinant on various surfaces. The examination is primarily motivated by mathematical curiosity but the techniques used here are likely to be extendable to some problems related to optimal experiment design for polynomial regression, see section 3. To the authors knowledge this problem has previously been examined on cubes (see section 3) and spheres, see Szegő [8]. Here we will consider some techniques to extend some of these results to other surfaces such as ellipsoids, cylinders, p-norm spheres and other surfaces defined by homogeneous polynomials. Our examination will mostly be restricted to three dimensions but many of the techniques can in principle be extended to higher dimensions. 2 The Vandermonde matrix A rectangular Vandermonde matrix of size m n is determined by n values x = (x 1,, x n ) and is defined by V mn (x) = [ x i 1 ] j = x 1 x 2 x n mn (1) x m 1 1 x m 1 2 x m 1 n Note that some authors use the transpose of this as the definition and possibly also let indices run from 0. All entries in the first row of Vandermonde matrices 16 th ASMDA Conference Proceedings, 30 June 4 July 2015, Piraeus, Greece 2015 ISAST

40 are ones and by considering 0 0 = 1 this is true even when some x j is zero. The determinant of the Vandermonde matrix is well known. Theorem 1. The determinant of (square) Vandermonde matrices has the form det V n (x) v n (x) = (x j x i ). (2) 1 i<j n This determinant is also simply referred to as the Vandermonde determinant or Vandermonde polynomial or Vandermondian[9]. In this paper we will use the method of Lagrange multipliers to optimize the Vandermonde determinant over a surface. For this purpose the following properties will be useful. Lemma 1. The Vandermonde determinant is a homogeneous polynomial of degree n(n 1) 2. Proof. Considering (2) the numbers of factor of v n (x) is Thus n i 1 = i=1 n(n 1). 2 v n (cx) = c n(n 1) 2 v n (x). (3) 3 Application to D-optimal experiment designs for polynomial regression with a cost-function Suppose an experiment is conducted where m data points from some compact interval, X R, i = 1, 2,..., m, are used to create a polynomial regression model of degree n 1. A vector containing the data points, x m = (x 1, x 2,..., x m ) X m, is called a design and a design is said to be D-optimal if det(m n (x m )) det(m n (y m )) for all y X m where m m M m (x) = i=1. m i=1 x i x n 1 i m x i... i=1 m i=1 x 2 i... m i=1 m i= m m x n i... i=1 x n 1 i x n i x 2n 2 i i=1 is the Fischer information matrix, see [5], [3]. Note that M m (x) = V n,m (x)v n,m (x) where V n,m (x) is an n m Vandermonde matrix. By the Cauchy-Binet formula the determinant of the information matrix M m can be decomposed into a sum of products of determinants of Vandermonde matrices det(m m (x)) = s S det(v n (s)) det(v n (s))

41 where S is the set of n-dimensional vectors corresponding to combinations of n elements from x, see [3] and [9]. If m = n then M m (x) is a product of two square Vandermonde matrices and thus maximizing det(m m (x)) = v n (x) 2 is equivalent to finding the extreme points of v n (x). Optimal design of experiments for polynomial regression models of various kinds have been examined, see [2] for an overview, but the examination have been restricted to x X n, which can be rewritten as x [ 1, 1] n without loss of generality. Here we consider other kinds of sets. Suppose there is a cost-function associated with the data such that the total cost of the experiment being below some threshold value, g(x) 1, defines some compact set, G = {x R m g(x) 1}, such that G X m. Since the Vandermonde determinant is a homogeneous polynomial for any c > 1 v n (x) > v n (cx) the extreme points will be on the surface of the compact set and thus it is enough to consider the set of points defined by g(x) = 1. 4 Optimization using Gröbner bases Gröbner bases together with algorithms to find them, and algorithms for solving a polynomial equation is an important tool that arises in many applications. One such application is the optimization of polynomials over affine varieties through the method of Lagrange multipliers. We will here give some main points and informal discussion on these methods as an introduction and to fix some notation. Definition 1. ([1]) Let f 1,, f m be polynomials in R[x 1,, x n ]. The affine variety V (f 1,, f m ) defined by f 1,, f m is the set of all points (x 1,, x n ) R n such that f i (x 1,, x n ) = 0 for all 1 i m. When n = 3 we will sometimes use the variables x, y, z instead of x 1, x 2, x 3. Affine varieties are this way the common zeros of a set of multivariate polynomials. Such sets of polynomials will generate a greater set of polynomials [1] by { m } f 1,, f m h i f i : h 1,, h m R[x 1,, x n ], i=1 and this larger set will define the same variety. But it will also define an ideal (a set of polynomials that contains the zero-polynomial and is closed under addition, and absorbs multiplication by any other polynomial) by I(f 1,, f m ) = f 1,, f m. A Gröbner basis for this ideal is then a finite set of polynomials {g 1,, g k } such that the ideal generated by the leading terms of the polynomials g 1,, g k is the same ideal as that generated by all the leading terms of polynomials in I = f 1,, f m. In this paper we consider the optimization of the Vandermonde determinant v n (x) over surfaces defined by a polynomial equation on the form s n (x 1,, x n ; p; a 1,, a n ) n a i x i p = 1, (4) i=1

42 where we will select the constants a i and p to get ellipsoids in three dimensions, cylinders in three dimensions, and spheres under the p-norm in n dimensions. The cases of the ellipsoids and the cylinders are suitable for solution by Gröbner basis methods, but due to the existing symmetries the spheres are more suitable for other methods, as provided in Section 8. From (3) and the convexity of the interior of the sets defined by (4), under a suitable choice of the constant p and non-negative a i, it is easy to see that the optimal value of v n on n i=1 a i x i p 1 will be attained on n i=1 a i x i p = 1. And so, by the method of Lagrange multipliers we have that the minimal/maximal values of v n (x 1,, x n ) on s n (x 1,, x n ) 1 will be attained at points such that v n / x i λ s n / x i = 0 for 1 i n and some constant λ and s n (x 1,, x n ) 1 = 0, [7]. For p = 2 the resulting set of equations will form a set of polynomials in λ, x 1,, x n. These polynomials will define an ideal over R[λ, x 1,, x n ], and by finding a Gröbner basis for this ideal we can use the especially nice properties of Gröbner bases to find analytical solutions to these problems, that is, to find roots for the polynomials in the computed basis. 5 Extreme points on the ellipsoid in three dimensions In this section we will find the extreme points of the Vandermonde determinant on the three dimensional ellipsoid given by ax 2 + by 2 + cz 2 = 1 (5) where a > 0, b > 0, c > 0. Using the method of Lagrange multipliers together with (5) and some rewriting gives that all stationary points of the Vandermonde determinant lie in the variety V = V ( ax 2 + by 2 + cz 2 1, ax + by + cz, ax(z x)(y x) by(z y)(y x) + cz(z y)(z x) ). Computing a Gröbner basis for V using the lexicographic order x > y > z give the following three basis polynomials: g 1 (z) =(a + b)(a b) 2 ( 4(a + b) 2 (a + c)(b + c) + 3c 2 (a 2 + ab + b 2 ) + 3c(a 3 + b 3 ) ) z 2 + 3c(a + b + c) ( 4(a + b)(a + c)(b + c) + (a 2 + b 2 )c + (a + b)c 2) z 4 c 2 (b + c)(a + c)(a + b + c) 2 z 6, (6) g 2 (y, z) = ( 2(a + b) 2 (a + c)(b + c) + c(a 2 + 2b 2 )(a + b + c) + 2bc 2 (a + b) ) z + q 1 z 5 q 2 z 3 b(a b)(a + b)(a + b + 3c)y, (7) g 3 (x, z) = ( 2(a + b) 2 (a + c)(b + c) + c(2a 2 + b 2 )(a + b + c) + 2ac 2 (a + b) ) z q 1 z 5 + q 2 z 3 a(a b)(a + b)(a + b + 3c)x, (8) q 1 = 9 c 2 (b + c)(a + c)(a + b + c) 2, q 2 = 3c(a + b + c)(3a 2 b + 4a 2 c + 3ab 2 + 6abc + 4ac 2 + 4b 2 c + 4bc 2 ).

43 The calculation of this basis was done using Maple TM [10]. Since g 1 only depends on z and g 2 and g 3 are first degree polynomial in y and x respectively the stationary points can be found by finding the roots of g 1 and then calculate the corresponding x and y coordinates. A general formula can be found in this case (since g 1 only contains even powers of z it can be treated as a third degree polynomial) but it is quite cumbersome and we will therefore not give it explicitly. Lemma 2. The extreme points of v 3 on an ellipsoid will have real coordinates. Proof. The discriminant is a useful tool for determining how many real roots low-level polynomials have. Following Irving [4] the discriminant, (p), of a third degree polynomial p(x) = c 0 + c 1 x + c 2 x 2 + c 3 x 3 is = 18c 1 c 2 c 3 c 4 4c 3 2c 4 + c 2 2c 2 3 4c 1 c c 2 1c 2 4 and if p(x) is non-negative then all roots will be real (but not necessarily distinct). Since the first basis polynomial g 1 only contains terms with even exponents and is of degree 6 the polynomial g 1 defined by g 1 (z 2 ) = g 1 (z) will be a polynomial of degree 3 whose roots are the square fo the roots of g 1. Calculating the discriminant of g 1 gives ( g 1 ) = 9(a b) 2 (a + b + 3c) 2 (a + b + c) 4 abc 3 ( 32(a 3 b 2 + a 3 c 2 + a 2 b 3 + a 2 c 3 + b 3 c 2 + b 2 c 3 ) + 61abc(a + b + c) 2). Since a, b and c are all positive numbers it is clear that (g 1 ) is non-negative. Furthermore, since a, b and c are positive numbers all terms in g 1 with odd powers have negative coefficients and all terms with even powers have positive coefficients. Thus if w < 0 then g 1 (w) > 0 and thus all roots must be positive. An illustration of an ellipsoid and the extreme points of the Vandermonde determinant on its surface is shown in Fig Extreme points on the cylinder in three dimensions In this section we will examine the local extreme points on an infinitely long in 3 dimensions cylinder aligned with the x-axis. In this case we do not need to use Gröbner basis techniques since the problem can be reduced to a one dimensional polynomial equation. The cylinder is defined by by 2 + cz 2 = 1, where b > 0, c > 0. (9) Using the method of Lagrange multipliers gives the equation system v 3 x = 0, v 3 y = 2λby, v 3 z = 2λcz.

44 Fig. 1: Illustration of the ellipsoid defined by x2 9 + y2 4 + z2 = 0 with the extreme points of the Vandermonde determinant marked. Displayed in Cartesian coordinates on the right and in ellipsoidal coordinates on the left. Taking the sum of each expression gives Combining (9) and (10) gives ( c ) b b + 1 cz 2 = 1 z = ± c by + cz = 0 y = c z. (10) b 1 c 1 y =. b + c b b + c Thus the plane defined by (10) intersects with the cylinder along the lines {( ) } c 1 b 1 l 1 = x,, x R = {(x, r, s) x R}, b b + c c b + c {( ) } c 1 b 1 l 2 = x,, x R = {(x, r, s) x R}. b b + c c b + c Finding the stationary points for v 3 along l 1 : ( ( v 3 (x, r, s) = x ) ) b c b + c c x + 1 (r + s), b b + c ( ( v 3 (x, r, s) = 2x + 1 )) b c x b + c c (r + s). b From this it follows that v 3 x (x, r, s) = 0 x = 1 2 b + c ( c b ) b. c Thus ( 1 1 x 1 = b + c 2 ( c b ) b c, c b, ) b c (11)

45 is the only stationary point on l 1. An analogous argument shows that x 2 = x 1 is the only stationary point on l 2. An example of where these points are placed on the cylinder is shown in Fig. 2. Fig. 2: Illustration of the cylinder defined by y z2 = 1 with the extreme points of the Vandermonde determinant marked. Displayed in Cartesian coordinates on the right and in cylindrical coordinates on the left. 7 Optimizing the Vandermonde determinant on a surface defined by a homogeneous polynomial When using Lagrange multipliers it can be desirable to not have to consider the λ-parameter (the scaling between the gradient and direction given by the constraint). we demonstrate a simple way to remove this parameter when the surface is defined by an homogeneous polynomial. Lemma 3. Let g : R R be a homogeneous polynomial such that g(cx) = c k g(x) with k n(n 1) 2. If g(x) = 1, x C n defines a continuous bounded surface then any point on the surface that is a stationary point for the Vandermonde determinant, z C n, can be written as z = cy where v n x i = g x=y x i, i 1, 2,..., n (12) x=y and c = g(y) 1 k. Proof. By the method of Lagrange multipliers the point y {x R n g(x) = 1} is a stationary point for the Vandermonde determinant if v n x k = λ g x=y x k, k 1, 2,..., n x=y for some λ R.

46 The stationary points on the surface given by g(cx) = c k will be given by c n(n 1) v n 2 x k = c k λ g x=y x k, k 1, 2,..., n x=y and if c is chosen such that λ = c n(1 n) 2 c k then the stationary points are defined by v n = g, k 1, 2,..., n. x k x k Suppose that y {x R n g(x) = c k } is a stationary point for v n then the point given by z = cy where c = g(y) 1 k will be a stationary point for the Vandermonde determinant and will lie on the surface defined by g(x) = 1. Lemma 4. If z is a stationary point for the Vandermonde determinant on the surface g(x) = 1 where g(x) is a homogeneous polynomial then z is either a stationary point or does not lie on the surface. Proof. Since g( x) = ( 1) k g(x) is either 1 or 1 then v n (x) = v n ( x) for any point, including z and the points in a neighbourhood around it which means that if g( x) = g(x) then the stationary points are preserved and otherwise the point will lie on the surface defined by g(x) = 1 instead of g(x) = 1. A well-known example of homogeneous polynomials are quadratic forms. If we let g(x) = x Sx then g(x) is a quadratic form which in turn is a homogeneous polynomial with k = 2. If S is a positive definite matrix then g(x) = 1 defines an ellipsoid. Here will will demonstrate the use of Lemma 3 to find the extreme points on a rotated ellipsoid. Consider the ellipsoid defined by 1 9 x y yz z2 = 1 (13) then by Lemma 2 we can instead consider the points in the variety V = V ( 2xy + 2xz + y 2 z x, Finding the Gröbner basis of V gives x 2 + 2xy 2yz + z y 3 4 z, 2xz y 2 + 2yz + x y 5 4 z). g 1 (z) = z(6z + 1)(260642z z + 697), g 2 (y, z) = z z z y, g 3 (x, z) = z z z x.

47 This system is not difficult to solve and the resulting points are: p 0 = (0, 0, 0), ( p 1 = 0, 1 6, 1 6 ( 45 2 p 2 = p 3 = ), 361, , ) 2, 722 ( , , ) The point p 0 is an artifact of the rewrite and does not lie on any ellipsoid and can therefore be discarded. By Lemma 4 there are also three more stationary points p 4 = p 1, p 5 = p 2 and p 6 = p 3. Rescaling each of these points according to Lemma 2 gives q i = g(p i ) which are all points on the ellipsoid defined by g(x) = 1. The result is illustrated in Fig. 3. Note that this example gives a simple case with a small Gröbner basis that is small and easy to find. Using this technique for other polynomials and in higher dimensions can require significant computational resources. Fig. 3: Illustration of the ellipsoid defined by (13) with the extreme points of the Vandermonde determinant marked. Displayed in Cartesian coordinates on the right and in ellipsoidal coordinates on the left. 8 The Vandermonde determinant on p-norm spheres The optimization of the Vandermonde determinant on the sphere (p = 2) and on the cube (p = ) lends themselves to methods in orthogonal polynomials. In fact, as shown by Stieltjes and recaptured by Szegő [8], and presented in more detail in and extended in [6] there is a fairly straightforward solution derived from electrostatic considerations, which implicitly deals with the Vandermonde determinant.

48 Consider the optimization of v n (x) over the sphere s n (x) = n x p i = 1, i=1 for suitable choices for p (even). Instead of optimizing v n we are free to optimize ln v n over the sphere to the same effect (v n (x) = 0 is not a solution, all x i are pair-wise distinct). This leaves us with the set of equations by Lagrange multipliers. ln v n x k = λ s n x k, s n = 1, (14) where the left-most equation holds for 1 k n. It is easy to show that the partial derivatives can be written ln v n x k = n i=1 i k 1 x k x i. (15) From this it is easy to show that (15) can be rewritten by introducing the univariate polynomial P n (x) = n i=1 (x x i) as ln v n = 1 P (x k ) x k 2 P (x k ). Now the leftmost equation of (14) can be written or more succinct 1 P (x k ) 2 P (x k ) = λ s n, x k P (x k ) 2λ s n x k P (x k ) = 0, (16) In the case p = 2 we are lucky since (16) becomes, by introducing the new multiplier ρ (P (x) + ρ n xp (x)) x=xk = 0, (17) and since the left part of this equation is a polynomial of degree n and has roots x 1,, x n we must have P (x) + ρ n xp (x) + σ n P (x) = 0, (18) for some ρ n, σ n that may depend on n. Now if choose P (x) to be monic, note that ρ n 0, and require us to be on the sphere we get P (x) = x n 1 2 xn 2 +, and by identifying coefficients we get ρ n and σ n : P (x) + n(1 n)xp (x) + n 2 (n 1)P (x) = 0, which is a nice and well known form of differential equation and defines a sequence of orthogonal polynomials that are rescaled Hermite polynomials [8], so we can find a recurrence relation for P n+1 in terms of P n and P n 1, and we

49 can, for a fixed n, construct the coefficients of P n recursively, without explicitly finding P 1,, P n 1. Now, this is for p = 2. For p = 4 we continue from (16) instead with ( P (x) + ρ n x 3 P (x) ) x=xk = 0. (19) Now the polynomial in x in the left part of this equation has shared roots with P (x) and so by the same method as for p = 2 we get: P (x) + ρ n x 3 P (x) + (σ n x 2 + τ n x + υ n )P (x) = 0. (20) It is easy to show for the sphere under any p-norm that the extreme points of ln v n (x) where x 1 < < x n are unique, see [8],[6], this coupled with the symmetry relation ln v n (x) = ln v n ( x), provides us with the property that the extreme points are symmetric in the sense that for all 1 i n we have that there exists a 1 j n such that x i = x j, for odd n we then have that x i = 0 for some i. We thus get polynomials P (x) on the form: P (x) = x n + c n 2 x n 2 + c n 4 x n 4 +, with every other coefficient zero, for even n we have only even powers, for odd n we have odd powers. By identifying powers in (20) we get that τ n xp (x) will not share any powers with any other part of the equation and so τ n = 0. We can also by identifying coefficients get nρ n + σ n = 0. We now have P (x) + ρ n x 3 P (x) + ( nρ n x 2 + υ n )P (x) = 0. (21) For n = 2 we get the specific system 2 + ρx 3 (2x) + ( 2ρx 2 + υ)(x 2 + c 0 ) = 0, but we actually don t need to calculate much here since it is easy do adapt the roots of x 2 + c 0 to the sphere with p = 4, we get: P 4 2 (x) = x The case n = 3 is also easy and by symmetry we get a zero coordinate: P 4 3 (x) = x x. The case n = 4 becomes a bit more interesting: (12x 2 + 2c 2 ) + ρx 3 (4x 3 + 2c 2 x) + ( 4ρx 2 + υ)(x 4 + c 2 x 2 + c 0 ) = 0, (υ 2ρc 2 )x 4 + (12 + υc 2 4ρc 0 )x 2 + (2c 2 + υc 0 ) = 0, This provides three equations. Now letting t = x 2 so that P (t) = t 2 + c 2 t + c 0 = (t t 1 )(t t 2 ) = t 2 (t 1 + t 2 )t + t 1 t 2, gives us the last equation x 4 i = 2 t 2 i = 2(c2 2 2c 0 ) = 1. Solving this gives us P4 4 (x) = x 4 2 x

50 9 Conclusion In this paper we have examined the extreme points of the Vandermonde determinant on various surfaces in three dimensions. Explicit expressions for the placement of the extreme points on an ellipsoid aligned with the axis and a cylinder aligned with the x-axis were found using Gröbner bases in section 5 and 6. A convenient way of rewriting the system of polynomial equations that the method of Lagrange multipliers gives was shown in section 7 and it was illustrated how this rewrite could be used to find the extreme points on an ellipsoid not aligned with the coordinate system. In section 8 a method for finding the extreme points on spheres with p- norm was discussed, specifically the case when p = 4 for two, three and four dimensions. Further work will be to extend the methods described here to higher dimensions and also examine matrices related to the Vandermonde matrices, for instance the matrices described in section 3. References 1. David Cox, John Little and Donal O Shea. Ideals, varieties, and algorithms. Springer, Holger Dette and Matthias Trampisch. A general approach to D-optimal designs for weighted univariate polynomial regression models. Journal of the Korean Statistical Society, 39, 1 26, Norbert Gaffke and Olaf Krafft. Exact D-Optimum Designs for Quadratic Regression. Journal of the Royal Statistical Society. Series B (Methodological), 44, 3, , Ronald S. Irving. Integers, Polynomials and Rings. Undergraduate Texts in Mathematics, Springer, New York, Jack C. Kiefer. Optimum Experimental Designs. Journal of the Royal Statistical Society. Series B (Methodological), 21, 2, , Karl Lundengård, Jonas Österberg and Sergei Silvestrov. Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant. arxiv, eprint arxiv: R. Tyrrell Rockafellar. Lagrange multipliers and optimality. SIAM Review, 35, 2, , Gabor Szegő, Orthogonal Polynomials. American Mathematics Society, Robert Vein and Paul Dale. Determinants and Their Applications in Mathematical Physics. Applied Mathematical Sciences, 134, Springer, New York, Maple Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.

51 An approximation of Ingredients Usage and the Effect on Dairy Milk Yield Francisco E. Martínez-Castañeda 1, Ana Lorga da Silva 2, Rodolfo R. Posadas- Domínguez 3 1 Instituto de Ciencias Agropecuarias y Rurales, Universidad Autónoma del Estado de México. Campus Universitario El Cerrillo Piedras Blancas, Toluca, 50100, México. femartinezc@uaemex.mx and 3 arielposadascueto@yahoo.com.mx 2 Departamento de Economia e Gestão, ECEO; CPES - ULHT, Campo Grande 376, Lisboa, Portugal, & CEDRIC-CNAM, Paris, France. ana.lorga@ulusofona.pt Abstract. In this work we collected data from fifteen small-scale dairy farms during thirteen months; In order to explain milk yield based on the usage of ingredients in dairy cows and on the number of the cows. Several panel data models have been tested such as models of fixed and random effects (Wooldridg[11]; Baltagi [1], based on animal feed, e.g. corn silage, green alfalfa, complement, protein, corn stover, etc., and the number of milking cows or the total of cows (the explanatory variables) to explain which and how they contribute positively and negatively to the milk yield. Milk yield is a natural process on female mammals, but the production level decrease after 8 to 10 weeks postpartum in cows. The effect on how ingredients are related with milk yield can contribute to more efficient decision-making. Based on the models we concluded that the total of cows does not be included as an explanatory variable, but only the total of milking cows. On the other hand it was taken in account that feed considered for the models is the total given to all the cows. The models and the influence of the explanatory variables on the production of the milk are presented. Key words: Panel Data, Regression Models, Livestock, Efficiency. 1 Introduction As known the term panel data refers to the pooling of observations on a crosssection of households, countries, firms, etc., over several time periods. This can be achieved by surveying a number of households or individuals and following them over time Baltagi[1]. In livestock management, feeding animals is the most important cost production. In small-scale dairy systems, they can represent around 55% of total cost Posadas-Domínguez et al.[8] but they are values registered between 52 to 16 th ASMDA Conference Proceedings, 30 June 4 July 2015, Piraeus, Greece 2015 ISAST

52 70% of total cost, of which purchased concentrates may represent up to 90 % of direct expenditures Espinoza-Ortega et al.[3]. According to Pica-Ciamarra and Otte[7], the best option to increase farmers income is through implementation of feeding strategies aimed at reducing costs, since world scenarios show that prices paid to farmers tend to remain static. Nutrition strategies and feed nutrition is calculated and implemented methodologically using a Standard value, that is Dry Matter, because all ingredients have different humidity. From a management point of view, the goal is not only to feed correctly the cows it is important to identify how inputs, in our case ingredients, are used and how they preformed in milk yield, because cost most be done in relation to inventories i.e. that the way they are used. The main objective of these study is to identify the usage of ingredients and with panel data analysis models, how ingredients explain milk yield in small scale dairy farms. 2 Methodology 2.1. Study Area The work took place in Texcoco, México, located in northeast of the State of Mexico and northeast of Mexico City. The State of Mexico is the seventh state in national milk production and the district of Texcoco occupies the second place in milk production in the state SIAP[10] and, historically, has been characterized by its dairy production, predominating at present SSDS based on crops of Lucerne (Medicago sativa) Zaragoza-Esparza et al.[12], maize (Zea mays) and oats (Avena sativa), with Holstein cattle as the main dairy breed (95.0 % of dairy cows). The main city of the district is Texcoco located at N and W, at an altitude of 2,250 m, a semi-dry temperate climate with a mean annual temperature of 15.9 C and a mean rainfall of 686 mm year-1 INEGI[4]. The study area selected was Texcoco s Mexico central Valley. Dairy small-scale sector represents 89% of all dairy systems in the area and contributed with 70% of milk produced. Farms size, were from 6 to 14 cows with four hectares land average per farm Data collection Monthly data of ingredients usage and milk yield from 15 small-scale dairy farms were collected. (April April 2013). Data were registered in production sheet formats and then captured and analyzed.

53 2.3. Models We are in a presence of panel data, 15 Farms were observed during 13 months (April April 2013). We intend to explain the Milk yield of small-scale farms, by the number of milking cows, and the ingredients (the quantity of each one) they eat, in each farm. Our first approach provides information about multicollinearity of some of explanatory variables. Using the Hausman test we verify that we should use a model with fixed effects (fe) (H = 33,9116; p-value=9,25658e-005). In this case the multicollinearity of the regressors is due to the characteristic of this sample. As known multicollinearity could lead to unreliable and unstable estimates of regression coefficients, but in fact as presented on Wooldridge[11], several experienced Econometricians conclude that multicollinearity is not a problem (we are not talking of course about perfect multicollinearity). In spite of we just mentioned about the presence of multicollinearity we constructed two indicators for models including Milking Cows and excluding Milking Cows (the variables that presents multicollinearity differs in each case), but the obtained models don t capture at all the effect of the others variables, so we decided to not present them in this work. Also approaches using 2SLS were tested, but the most obtained estimators were insignificant. We have estimated panel regression models, in which the variable we intend to explain is the number of Litters of milk monthly, which includes farm-fixed effects, explained by the ingredients that the Milking Cows eat each month by farm, and the total of Milking Cows in each farm in the observed months. We considered two main approaches, not including the variable Milking Cows and other including it, also both not including and including time dummy variables Models (presented in section 3) FE_1 and FE_3 includes the variable Milking Cows, and FE_2 and FE_4 they don t include the variable. Our general Fixed Effects (FE) models: y x x a u (1.1) it 0 1 it1... k itk i it y x... x t... t a u (1.2) it 0 1 it1 k itk k 1 1 k i it i=1,,13 the farms item t=1,,15 the period of time

54 k is the number of explanatory variables; k=8, if the variable milking cows is not included in the model, k=9 otherwise. t,..., t - time dummy variables 1 11 a is a farm fixed effect i u the error term it 2.4. Other analysis A descriptive statistical analysis of some variables was performed. 3 Results Average ingredient consumption per cow was kg. Normal consumption can be between 40 to 50 kg, depending in many variables like age, lactation period, environmental variables, etc. It is difficult to establish and accurate reference because related to how much food a cow eat, because specialized literature in dairy nutrition, refers every data to Dry Matter values. If so, we can identify an opportunity of action because producers are giving more ingredients that the cow can eat. Average milk yield per cow was daily. Martínez-García et al.[5] reported production yield between 10 to 15 litters in the northwest of State of Mexico. Posadas-Domínguez et al.[8] reported production averages, depending the size of the farms, between 17, 20 and 24 litters. It has been reported before the positive relation of farm size and productivity Carranza-Trinidad et al.[2]. According to Romo et al.[9] those small scale farms did attain or exceed the break-even point differed from unprofitable in their more efficient use of available resources. In this study, all farms produce more than 90% of their ingredients. The independence on external ingredients is a key strategy for small scale dairy producers. They gave them comparative advantages versus producers they have to buy their ingredients Posadas-Domínguez et al.[8] Table 1 resume the coefficient in milk yield of each Models tested.

55 Table 1. Ingredients usage coefficient in milk yield in small scale farms. April to April Corn silage FE_1 FE_2 FE_3 FE_4.360***.308***.300***.226*** (.080) (.053) (.087) (.056) Corn stover.254*** (.087) -.220*** (.058) -.281*** (.091) -.240*** (.058) Green alfafa.211*** (.032).047* (.024).199*** (.032).031* (.024) Corn.127 (1.092).823 (.730).479 (1.208) 2.209*** (.783) Other grains (2.177) *** (1.507) (2.288) *** (1.530) Complement *** *** Oat straw (0.938) (.640) (.944) (.622) -.793*** -.423* -.957*** -.649*** (.252) (.169) (.263) (.169) Protein *** (1.342) 2.778** (1.017) 9.713*** (1.377) 2.676** (.997) Milking Cows *** (30.660) *** (30.502) t (94.025) -113,846* (60.353) t (93.028) ** (60.694) t (94.468) * (62.045) t (91.195) (59.484) t (92.049) (59.221) t (93.143) (59.868) t (93.714) (60.468) t (88.548) (57.235) t ** (88.548) (57.272) t * (87.308) (56.201) t ** (87.095) ( ) _cons *** ( ) *** ( ) FE_1 F(14, 172) =19.16***, FE_2 F(14, 171) =44.65***, FE_3 F(14, 161) = 17.80***, FE_4 F(14, 160) = 45.46***. 2 FE _ FE _ FE _3.853 R, R, R, (Standard errors in parenthesis). *** p<0.01, ** p<0.05, * p<0.1 2 RFE _4.861 (56.193) *** ( )

56 FE_3 and FE_4, are models including time dummy variables. Figure 1 resume in percentage, the use of each ingredient. Fig. 1. Ingredient usage by farm. The most used ingredient was green alfalfa (50.1%), followed by corn silage (27.2%) and corn stover (12.9%) The coefficient associated to the variable green alfalfa becomes significant and positive in four models. If we consider the importance and the effect on milk yield and the percentage of use, it becomes the main strategy ingredient for these farms. In FE_2, the significance diminishes, and it is clear the effect that Milking cows in that model. Modest levels of alfalfa mixture fed improved forage energy and protein concentrations and tended to improve milk yield McCormic et al.[6] Corn silage has also in both models a positive contribution to the production of milk. Corn silage has become a technological strategy that producer have adopted many years ago and is a common practice. Martínez-García et al.[5] mentioned that quality forage and supplement intake were the variables that showed a significant (P<0.01) linear relationship with milk yield. In the groups of Quality forages they included: Fresh oat forage, Lucerne haylage, maize silage, Lucerne hay, oats hay and cut and carry pasture. But the ingredients corn stover, other grains and complement have different contributions in the absence or presence of milking cows.