Mathematics and Statistics 4(1): 40-45, 2016 DOI: 10.13189/ms.2016.040105

http://www.hrpub.org

The Constructive Implicit Function Theorem and Proof in Logistic Mixtures Xiao Liu Methods in Empirical Educational Research, TUM School of Education and Centre for International Student Assessment (ZIB), TU M¨unchen, Arcisstr. 21, 80333 Munich, Germany

c Copyright ⃝2016 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License

Abstract There is the work by Bridges et al (1999) on the key features of a constructive proof of the implicit function theorem, including some applications to physics and mechanics. For mixtures of logistic distributions such information is lacking, although a special instance of the implicit function theorem prevails therein. The theorem is needed to see that the ridgeline function, which carries information about the topography and critical points of a general logistic mixture problem, is well-defined [2]. In this paper, we express the implicit function theorem and related constructive techniques in their multivariate extension and propose analogs of Bridges and colleagues’ results for the multivariate logistic mixture setting. In particular, the techniques such as the inverse of Lagrange’s mean value theorem [4] allow to prove that the key concept of a logistic ridgeline function is well-defined in proper vicinities of its arguments.

Keywords Constructive Implicit Function Theorem, Logistic Mixture, Lagrange Mean Value Theorem, Ridgeline

1 Introduction In this paper, we focuse on constructive techniques that are based on expressing the proof of implicit function theorem in their multivariate extension and propose analogs of Bridges and colleagues’ results for the multivariate logistic mixture setting. According to [2], applying the implicit function theorem, we can prove that a unique explicit formula for the ridgeline function is possible locally in Theorem 1; In this paper, we propose analogs of Bridges et al.’s results (see [1]) for the multivariate logistic distribution. Applying Lagrange’s mean value theorem (see [4]), we can get the first lemma. Moreover, we are interested in uniform differentiability of 2 variables on Lemma 3 due to [5], which carries important information about the proof of our goal.

2 Applications to Topography Theorem 1 later frequently allows us to show some ridgeline and contour plots, where the ridgeline function satisfies that the left side of formula (1) is equal to null (see [3]). The following example is in the case of two dimensions and three components. Example 1. The mixture logistic density with D ( ) 0 µ1 = , 0 ( ) 1 µ2 = , 1 ( ) 2 µ3 = , 2

= 2 and K = 3, and the parameters ( ) 1 s1 = , 0.07 ( ) 0.07 s2 = , 1 ( ) 1 1 s3 = , π1 = π2 = π3 = . 0.07 3

Figure 1 shows the contours of the density given in Example 1.

Mathematics and Statistics 4(1): 40-45, 2016

41

Figure 1. Density contour plot for the three component mixture density of Example 1.

3

The Constructive Implicit Function Theorem For the most part we confine our attention to the following special case of the Implicit Function Theorem.

Theorem 1. Let ψ(α, x) :=

K ∑

αi

i=1

D ∑ k=1

( ) D+1 s−1 1 − ik Eik

(1)

x −µ xk −µik xk −µik ∑ − js ij D ij where Eik = 1 + e sik + e sik e , j̸=k be a differentiable mapping of a neighbourhood of (α0 , x0 ) ∈ RK ×RD into Rp , let ψ(α0 , x0 ) = 0, and let det(D2 ψ(α0 , x0 )) ̸= ¯ 0 , r) ⊂ RK → RD such that for each α ∈ B(α ¯ 0 , r), 0. Then there exists r > 0 and a differentiable function f : B(α (α, f (α)) is the unique solution x of the equation ψ(α, x) = 0 in some neighbourhood of (α0 , x0 ).

Where we use standard modern notations for derivatives, such as D for the derivative itself, and Dl for the lth partial derivative (l = 1, 2), of a mapping from a subset of RK × RD to Rp . This result, given here for the logistic mixture case, has obvious generalizations to any general implicit function as following corollary. To keep the presentation short we refrain from presenting them here, as their treatment follows the same strategy analogously to [1]. Corollary 1. Let ψ be a differentiable mapping of a neighbourhood of (x0 , y0 ) ∈ RK × RD into Rp , let ψ(x0 , y0 ) = 0, and ¯ 0 , r) ⊂ RK → RD such that for let det(D2 ψ(x0 , y0 )) ̸= 0. Then there exists r > 0 and a differentiable function f : B(x ¯ each x ∈ B(x0 , r), (x, f (x)) is the unique solution y of the equation ψ(x, y) = 0 in some neighbourhood of (x0 , y0 ). In next section, we will give the proof of Theorem 1. At last, we will show some necessary lemmata for preparing the proof of our main theorem in appendix.

4

A New Proof of The Constructive Implicit Function Theorem in The Case of Logistic Mixtures

4.1

Proof of Theorem 1

Proof. (i) Suppose that the assumption of Theorem 1 are satisfied, and choose r, s > 0 as in Lemma 2. Assume J := B × C = {(α, x) : |α − α0 | ≤ r, |x − x0 | ≤ s}, which is a compact set. Fix ξ with |ξ − α0 | ≤ r, and there ∃ ε > 0 such that 0 ψ 2 (ξ, x0 − m); this is contradictory due to (12) in view of the choice of s in Lemma 2. 1 Therefore γ = 0, then we can choose xn for each n such that |xn − x0 | ≤ s and |ψ(ξ, xn )| < 2 . Now the work in part (i) n 2 of this proof shows that ∃ N , such that when nj , nl > N , we can get |xnj − xnl | < ; then the sequence {xn } is a Cauchy n sequence, and hence converges to a limit x∞ on the segment C = [x0 − m, x0 + m]. The same argument shows that x∞ is also the unique solution x on C of the equation ψ(ξ, x) = 0, thus we can define a function f : [α0 − n, α0 + n] → C due to f (ξ) = x∞ . On account of Lemma 3, we complete the proof.

A Definition of Uniform Differentiability in 2 Variables Definition 1. Let f : Rm ×Rn → R be differentiable and such that ∇f is uniformly continuous. We define that f is uniformly differentiable, i.e., for any ε > 0, there is a δ > 0 such that for all a, x ∈ Rm and b, y ∈ Rn , we have |f (a, b) − f (x, y) − D1 f (x, y)(a − x) − D2 f (x, y)(b − y)| then

⟨

b − a,

inf |Df (x)| x∈[a,b]

Thus

⟩

≤ f (b) − f (a) .

inf |Df (x)| ≤ |b − a|−1 |f (b) − f (a)|. x∈[a,b]

(7) (8)

(9)

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The following example of Lemma 1 can be applied for calculating the average velocity of non-uniform motion in kinematics. Example 2. Let f (x) = ax2 + bx

(a ̸= 0)

then Df (x) = 2ax + b. So we have a(a + b − a)2 + b(a + b − a) − aa2 − ba } { 1 = (b − a) 2a[a + (b − a)] + b . 2 The core to our mathematical expression of the existence of an implicit function in the logistic mixtures case is provided by the following lemma. Lemma 2. Under the hypotheses of Theorem 1, there exist m ∈ RD , |m| = s, n ∈ RK , |n| = r, and r, s > 0 such that 2 s|D2 ψ(α0 , x0 )|, 3

(10)

|D2 ψ(α0 + h, x0 + g)| > 0,

(11)

|ψ(α0 , x0 ± m)| ≥ for ∀ h ∈ RK , g ∈ RD inf

|h|≤r,|g|≤s

and inf |ψ(α0 + h, x0 ± m)| ≥

|h|≤r

1 s|D2 ψ(α0 , x0 )| > sup |ψ(α0 + h, x0 )|. 2 |h|≤r

(12)

Proof. Choose an open ball B, with centre (α0 , x0 ) and radius R. Following formula (1), obviously D2 ψ(α, x) ∈ C 0 (B). Then we can get 1 |D2 ψ(α, x)| > |D2 ψ(α0 , x0 )| (13) 2 for all (α, x) ∈ B. According to that ψ is differentiable at (α0 , x0 ), there exists s ∈ (0, R) such that if |x − x0 | ≤ s, we obtain ψ(α0 , x) − ψ(α0 , x0 ) 4 ≤ |D2 ψ(α0 , x0 )|, 3 x − x0 then |ψ(α0 , x) − D2 ψ(α0 , x0 )(x − x0 )| ≤

1 |D2 ψ(α0 , x0 )(x − x0 )| 3

(14)

and therefore

2 |D2 ψ(α0 , x0 )(x − x0 )|. (15) 3 In particular, choose x = x0 ± m, we obtain inequality (10). Since ψ(α0 , x0 ) = 0 and ψ is continuous, we can now choose r ∈ (0, R) such that inequality in (12) hold. According to r < R, our choice of R ensures that we can obtain inequality (11). |ψ(α0 , x)| ≥

Next we separate out the proof of the differentiability of the implicit function. It will be convenient to establish the existence of Theorem 1 before. Lemma 3. Let B be a compact ball in RK , C a compact domain in RD , and ψ : B × C → Rp be a uniformly differentiable function such that 0 < m := inf |D2 ψ|. (16) B×C

Suppose that there exists a function f : B → C such that ψ(α, x) = 0 for ∀ α ∈ B, x := f (α) ∈ C. Then f is uniformly differentiable on B, and D1 ψ(ξ, f (ξ)) (17) f ′ (ξ) = − D2 ψ(ξ, f (ξ)) for any ξ ∈ B.

44

The Constructive Implicit Function Theorem and Proof in Logistic Mixtures

Proof. Let 0 < ε < 17 m, and let α(1) , α(2) be points of B, we define √ (1) (2) 2 (1) (2) ∥α − α ∥ := ΣK i=1 (αi − αi ) .

(18)

According to Definition 1, we have the definition of uniform differentiability in 2 variables (also see [5]), so we can get ( ψ α(1) , f (α(1) )) − ψ (α(2) , f (α(2) )) − D ψ (α(2) , f (α(2) )) (α(1) − α(2) ) 1 (19) ( )( ) −D2 ψ α(2) , f (α(2) ) f (α(1) ) − f (α(2) ) √ ∥α(1) − α(2) ∥2 + ∥f (α(1) ) − f (α(2) )∥2 ≤ ε. Then

≤ ≤

( ) ( ) ψ α(1) , f (α(1) ) − ψ α(2) , f (α(2) ) ( )( ) −D1 ψ α(2) , f (α(2) ) α(1) − α(2) ( )( ) −D2 ψ α(2) , f (α(2) ) f (α(1) ) − f (α(2) ) √ ε ∥α(1) − α(2) ∥2 + ∥f (α(1) ) − f (α(2) )∥2



) (



ε α(1) − α(2) + f (α(1) ) − f (α(2) ) .

( ) ( ) ( ) Otherwise ψ α(1) , f (α(1) ) = ψ α(2) , f (α(2) ) = 0 and D2 ψ α(2) , f (α(2) ) ≥ m due to formula (16), so D ψ (α(2) , f (α(2) )) ( ) 1 ( (1) (2) (1) (2) ) α −α + f (α ) − f (α ) D2 ψ α(2) , f (α(2) )



) (



≤ m−1 ε α(1) − α(2) + f (α(1) ) − f (α(2) )



) 1 (

(1)



≤

α − α(2) + f (α(1) ) − f (α(2) ) . 7 Therefore

(21)



D ψ (α(2) , f (α(2) )) ( ) 1

(1) (2) ( ) α(1) − α(2) + − α

α

.

f (α(1) ) − f (α(2) ) ≤ 7 D2 ψ α(2) , f (α(2) )

Choosing a bound M for |D1 ψ| on the compact set B × C, we see that

(

)



f (α(1) ) − f (α(2) ) ≤ 7M m−1 + 1 α(1) − α(2) . It follows from formula (21) that

≤

D ψ (α(2) , f (α(2) )) ( ) 1 ( ) α(1) − α(2) + f (α(1) ) − f (α(2) ) D2 ψ α(2) , f (α(2) )

( )

m−1 7M m−1 + 2 ε α(1) − α(2) → 0

when ε → 0. Thus f is uniformly differentiable on B, with f ′ (ξ) = − for ∀ ξ ∈ B.

(20)

D1 ψ(ξ, f (ξ)) D2 ψ(ξ, f (ξ))

(22)

(23)

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REFERENCES [1] D. Bridges, C. Calude, B. Pavlov, D. Stefanescu. The Constructive Implicit Function Theorem and Applications in Mechanics, Chaos Solitons and Fractals, Vol.10, 927-934, 1999. ¨ u. Multivariate Logistic Mixtures, European Conference on Data Analysis (ECDA) , The University of [2] X. Liu, A. Unl¨ Bremen, 105, 2014. [3] X. Liu. Multivariate Logistic Mixtures, Universal Journal of Applied Mathematics, Vol.3, No.4, 77-87, 2015. [4] P. K. Sahoo, T. Riedel. Mean Value Theorems and Functional Equations, World Scientific Press, New Jersey, 1998. [5] E. M. Stein. Singular Integrals and Differentiability Properties of Functions, Princeton University Press, New Jersey, 1970.