On discrete least squares polynomial fit, linear spaces and data classification

François DUBEAU and Youness Mir

Département de mathématiques Université de Sherbrooke 2500 Boulevard de l’Université Sherbrooke (Qc), Canada, J1K 2R1

francois.dubeau@usherbrooke.ca youness.mir@usherbrooke.ca

Abstract : The best discrete least squares polynomial fit to a data set is revisited. We point out some properties related to the best polynomial and precise the dimension of vector spaces encountered to solve the problem. Finally, we suggest a classification of data sets based on their increasing or decreasing trend, and on their convexity or concavity form.

Key Words : Polynomial data fitting, weighted least squares, orthogonal polynomials, linear spaces, data classification.

AMS Subject Classification : 41A10, 62J02, 65D10, 62-07.

1. Introduction

Let {(ω_{i}, t_{i}, f_{i})}_{i=1}^{m} be a set of m data points where the t_{i}’s represent the distinct values of the independent variable, the f_{i}’s are the values of the measured function, and each ω_{i} is the weight associated to the data (t_{i}, f_{i}). The problem we consider is to find a polynomial p_{n} of degree at most n to fit the data. To measure how well the polynomial fit the data we use the weighted least squares deviation given by

(1.1)  F (p_{n}) = ∑_{i=1}^{m} ω_{i}(f_{i} − p_{n}(t_{i}))^{2}.

The best polynomial, called the weighted least squares estimate (WLSE), is given by

(1.2)  p_{n}^{∗} = argmin_{p_{n}∈P_{n}} F (p_{n}).

where P_{n} is the set of polynomials of degree at most n.

The goal of this paper is to clarify the dimension of some vector spaces encountered in solving this problem, establish a property useful for proving the existence of a WLSE for exponential models [2], and suggest a way to classify data using the best polynomial fits. For a standard presentation of the theory related to best (polynomial) least squares fit see [1, 3, 6, 7].

The best polynomial fit problem can be solved by considering an orthogonal projection onto P_{n} or, equivalently, by considering an orthogonal projection onto a subspace of ℝ^{m}. In Section 2 we briefly review the solution of the problem in P_{n} and specify the dimension of subspaces of polynomials. In the first part of the Section 3 we consider the subspaces of ℝ^{m} that play a role in solving the problem in ℝ^{m}. In the second part of this Section 3 we solve the problem using a projection onto a subspace of ℝ^{m}. Finally in Section 4 we suggest a way to classify data which will be useful in the problem of finding existence results for weighted least squares estimator [2].

2. Polynomial weighted least squares fitting in P_{n}

In the first part of this section we present the underlying subspaces of P = Lin{ t^{j} | j = 0, 1, 2, … } related to the polynomial weighted least squares problem. In the second part we solve the problem using a projection onto a subspace of P.

2.1. Vector spaces. Let us recall that P_{n} = Lin{ t^{j} | j = 0, 1, ..., n }. We consider also the following two other polynomial subspaces

(2.1)  PV_{k}^{+} = Lin{ v_{k,i}^{+}(t) = (t + t_{i})^{k} | i = 1, ..., m } ⊆ P_{k},
(2.2)  PV_{k}^{−} = Lin{ v_{k,i}^{−}(t) = (t − t_{i})^{k} | i = 1, ..., m } ⊆ P_{k},

for any nonnegative integer k = 0, 1, 2, … The next two results specify the dimension of these subspaces.

Theorem 2.1. Let P_{n} = Lin{t^{j} | j = 0, ..., n } ⊆ P, then dim P_{n} = n + 1. □

Theorem 2.2. Let k be any nonnegative integer and let PV_{k}^{+} and PV_{k}^{−} be defined by (2.1) and (2.2).

(a) If k ≤ m − 1 then PV_{k}^{+} = P_{k} = PV_{k}^{−}, and dim PV_{k}^{+} = k + 1 = dim PV_{k}^{−}.

(b) If k ≥ m then PV_{k}^{+} ⊊ P_{k}, PV_{k}^{−} ⊊ P_{k}, and dim PV_{k}^{+} = m = dim PV_{k}^{−}.

Proof. We prove the result for PV_{k}^{+} only, the proof for PV_{k}^{−} is identical. Since

∑_{i=1}^{m} μ_{i}v_{k,i}^{+}(t) = ∑_{i=1}^{m} μ_{i}( ∑_{j=0}^{k} ( \begin{matrix}k \\ j \end{matrix} ) t_{i}^{j}t^{k−j}) = ∑_{j=0}^{k} ( \begin{matrix}k \\ j \end{matrix} ) ( ∑_{i=1}^{m} μ_{i}t_{i}^{j})t^{k−j},

then ∑_{i=1}^{m} μ_{i}v_{k,i}^{+}(t) = 0 if and only if ∑_{j=0}^{k} ( \begin{matrix}k \\ j \end{matrix} ) ( ∑_{i=1}^{m} μ_{i}t_{i}^{j})t^{k−j} = 0. From

Theorem 2.1, the set {t^{j}}_{j=0}^{k} is linearly independent, it follows that ∑_{i=1}^{m} μ_{i}t_{i}^{j} = 0 for j = 0, ..., k. The matrix associated to this system is a Vandermonde type matrix. The rank of this matrix in min{k + 1, m} and the result follows. □

2.2. Polynomial weighted least squares fitting. Under the condition that n < m, we introduce the scalar product on P_{n} defined by

⟨p, q⟩ = ∑_{i=1}^{m} ω_{i}p(t_{i})q(t_{i})

for any pair of polynomials p and q in P_{n}. In this case (1.1) becomes

F (p_{n}) = ∥f − p_{n}∥^{2}

where ∥ · ∥ is the norm on P_{n} induced by the scalar product. For the f_{i}’s we use the notation f_{i} = f(t_{i}) (i = 1, ..., m). It is well known that p_{n}^{∗} is unique and is characterized by the normal equations

⟨f − p_{n}^{∗}, p_{n}⟩ = 0

for all p_{n} ∈ P_{n}.

In this setting, to simplify the computation of p_{n}^{∗}, we can find a sequence of orthogonal polynomials by applying the Gram-Schmidt orthogonalization process to the standard basis {1, t, t^{2}, ⋯, t^{n}} of P_{n}. These orthogonal polynomials are given by

q_{0}(t) = 1, q_{1}(t) = t − α_{1},

and for j = 2, ..., n,

q_{j}(t) = (t − α_{j})q_{j−1}(t) − β_{j}q_{j−2}(t)

where

α_{j} = \frac{⟨tq_{j−1}, q_{j−1}⟩}{⟨q_{j−1}, q_{j−1}⟩} (j = 1, 2, ..., n),

and

β_{j} = \frac{⟨tq_{j−1}, q_{j−2}⟩}{⟨q_{j−2}, q_{j−2}⟩} (j = 2, 3, ..., n).

Hence the best n-degree least squares polynomial p_{n}^{∗} can be written as

(2.3)  p_{n}^{∗}(t) = ∑_{j=0}^{n} γ_{j}^{∗}q_{j}(t)

where

γ_{j}^{∗} = \frac{⟨f, q_{j}⟩}{⟨q_{j}, q_{j}⟩} (j = 0, 1, …, n).

The next two results will be useful for finding sufficient conditions for the existence of the WLSE for a 3-parametric exponential model [2].

Theorem 2.3. ⟨f − p_{n−1}^{∗}, t^{n}⟩ = γ_{n}^{∗}∥q_{n}∥^{2} for n = 0, …, m − 1 .

Proof. For n = 0 it is obvious because p_{n−1}^{∗} = 0. For n > 0, since q_{n}(t) = t^{n} + p_{n−1}(t) where p_{n−1}(t) is a polynomial of degree ≤ n − 1, and

p_{n}^{∗}(t) = γ_{n}^{∗}q_{n}(t) + p_{n−1}^{∗}(t),

we have

γ_{n}^{∗}∥q_{n}∥^{2} = ⟨γ_{n}^{∗}q_{n}, q_{n}⟩
= ⟨p_{n}^{∗} − p_{n−1}^{∗}, q_{n}⟩
= ⟨p_{n}^{∗} − f, q_{n}⟩ + ⟨f − p_{n−1}^{∗}, q_{n}⟩
= ⟨f − p_{n−1}^{∗}, t^{n} + p_{n−1}⟩
= ⟨f − p_{n−1}^{∗}, t^{n}⟩.  □

Theorem 2.4. If the q_{j}’s are the orthogonal polynomials associated to {(ω_{i}, t_{i})}_{i=1}^{m}, the orthogonal polynomials q̃_{j}’s associated to {(ω_{i}, t̃_{i} = −t_{i})}_{i=1}^{m} are given by

q̃_{j}(t) = (−1)^{j}q_{j}(−t). □

3. Polynomial weighted least squares fitting in ℝ^{m}

In the first part of this section we present the underlying subspaces of ℝ^{m} related to the polynomial weighted least squares problem. In the second part we solve the problem using a projection onto a subspace of ℝ^{m}.

3.1. Vector spaces. Let {t_{i}}_{i=1}^{m} be a set of m distinct real numbers. For any positive integer j let us define the vectors \overrightarrow{t}^{j} ∈ ℝ^{m} by

\overrightarrow{t}^{j} = ( t_{1}^{j} ; t_{2}^{j} ; ⋮ ; t_{m}^{j} ) ∈ ℝ^{m}.

For any positive integer k, we also define the vectors

\overrightarrow{v}_{k,i}^{+} = (\overrightarrow{t} + t_{i}\overrightarrow{1})^{k} = ∑_{j=0}^{k} ( \begin{matrix}k \\ j \end{matrix} ) t_{i}^{j}\overrightarrow{t}^{k−j}  for  i = 1, ..., m,

and

\overrightarrow{v}_{k,i}^{−} = (\overrightarrow{t} − t_{i}\overrightarrow{1})^{k} = ∑_{j=0}^{k}(−1)^{j} ( \begin{matrix}k \\ j \end{matrix} ) t_{i}^{j}\overrightarrow{t}^{k−j}  for  i = 1, ..., m.

In this section we clarify the properties of the following vector spaces, in particular the dimension of the vector spaces,

(3.1)  T^{n} = Lin\{ \overrightarrow{t}^{j} | j = 0, ..., n \},
(3.2)  V_{k}^{+} = Lin\{ \overrightarrow{v}_{k,i}^{+} | i = 1, ..., m \},
(3.3)  V_{k}^{−} = Lin\{ \overrightarrow{v}_{k,i}^{−} | i = 1, ..., m \},

for any integers n and k such that n ≥ 0 and 0 ≤ k ≤ m − 1.

Theorem 3.1. Let T^{n} = Lin\{\overrightarrow{t}^{j} | j = 0, ..., n \} ⊆ ℝ^{m}.

(a) If n < m, the set \{\overrightarrow{t}^{j}\}_{j=0}^{n} is linearly independent and dim T^{n} = n + 1 .

(b) If n ≥ m, the set \{\overrightarrow{t}^{j}\}_{j=0}^{n} is linearly dependent and dim T^{n} = m.

Proof. We consider ∑_{j=0}^{n} λ_{j}\overrightarrow{t}^{j} = 0. But the Vandermonde matrix

A_{m,n+1} = ( \overrightarrow{t}^{0} \overrightarrow{t}^{1} ⋯ \overrightarrow{t}^{n} )

is of rank n + 1 as long as n < m, and hence λ_{j} = 0 for j = 0, ..., n. If n ≥ m its rank is m and there exits non zero solutions to the system. Hence the result follows because T^{n} ⊆ ℝ^{m}. □

Remark 3.2. For any positive integer l, since \overrightarrow{t}^{m+l} ∈ T^{m−1} = ℝ^{m}, we have

\overrightarrow{t}^{m+l} = ∑_{j=0}^{m−1} λ_{j}(l)\overrightarrow{t}^{j},

where

\overrightarrow{λ}(l) = ( λ_{0}(l) ; λ_{1}(l) ; ⋮ ; λ_{m−1}(l) ) = A_{m,m}^{−1}\overrightarrow{t}^{m+l} = A_{m,m}^{−1}diag(\overrightarrow{t}^{m})\overrightarrow{t}^{l},

and

diag(\overrightarrow{t}^{m}) = ( \begin{matrix}t_{1}^{m} & 0 & ⋯ & 0 \\ 0 & t_{2}^{m} & ... & ⋮ \\ ⋮ & ... & ... & 0 \\ 0 & ⋯ & 0 & t_{m}^{m} \end{matrix} ) . □

Theorem 3.3. Let k be any integer such that 0 ≤ k ≤ m − 1, and let V_{k}^{+} and V_{k}^{−} be defined by (3.2) and (3.3), then

V_{k}^{+} = T^{k} = V_{k}^{−}

and

dim V_{k}^{+} = k + 1 = dim V_{k}^{−}.

Proof. We prove the result for V_{k}^{+} only, the proof for V_{k}^{−} is identical. Since

∑_{i=1}^{m} μ_{i}\overrightarrow{v}_{k,i}^{+} = ∑_{i=1}^{m} μ_{i}( ∑_{j=0}^{k} ( \begin{matrix}k \\ j \end{matrix} ) t_{i}^{j}\overrightarrow{t}^{k−j})

= ∑_{j=0}^{k} ( \begin{matrix}k \\ j \end{matrix} ) ( ∑_{i=1}^{m} μ_{i}t_{i}^{j})\overrightarrow{t}^{k−j},

then ∑_{i=1}^{m} μ_{i}\overrightarrow{v}_{k,i}^{+} = 0 if and only if ∑_{j=0}^{k} ( \begin{matrix}k \\ j \end{matrix} ) ( ∑_{i=1}^{m} μ_{i}t_{i}^{j})\overrightarrow{t}^{k−j} = 0. From

Theorem 3.1, the set \{\overrightarrow{t}^{k−j}\}_{j=0}^{k} is linearly independent for k < m , it follows that

∑_{i=1}^{m} μ_{i}t_{i}^{j} = 0 for j = 0, ..., k. But this system of k + 1 equations and m unknowns has a unique solution only for k = m − 1. Moreover the matrix associated to this system is A_{m,k+1}^{T} is of rank k + 1 for k < m. Hence

dim V_{k}^{+} = k + 1. □

For k ≥ m we have no clear result about the dimension of V_{k}^{−} and V_{k}^{+} as illustrated by the following example for m = 3.

Example 3.4. Let m = 3.

(a) For V_{k}^{−}, since we have

Det(\overrightarrow{v}_{k,1}^{−},\overrightarrow{v}_{k,2}^{−},\overrightarrow{v}_{k,3}^{−}) = | \begin{matrix}0 & (t_{1} − t_{2})^{k} & (t_{1} − t_{3})^{k} \\ (t_{2} − t_{1})^{k} & 0 & (t_{2} − t_{3})^{k} \\ (t_{3} − t_{1})^{k} & (t_{3} − t_{2})^{k} & 0 \end{matrix} |

= [1 + (−1)^{k}](t_{1} − t_{2})^{k}(t_{2} − t_{3})^{k}(t_{3} − t_{1})^{k}

= \{ \begin{matrix}0 & if k is odd, \\ 2(t_{1} − t_{2})^{k}(t_{2} − t_{3})^{k}(t_{3} − t_{1})^{k} & if k is even, \end{matrix}

it follows that

dim V_{k}^{−} = \{ \begin{matrix}2 & if k is odd, \\ 3 & if k is even. \end{matrix}

(b) For V_{k}^{+}, we have

Det(\overrightarrow{v}_{k,1}^{+},\overrightarrow{v}_{k,2}^{+},\overrightarrow{v}_{k,3}^{+}) = | \begin{matrix}(2t_{1})^{k} & (t_{1} + t_{2})^{k} & (t_{1} + t_{3})^{k} \\ (t_{2} + t_{1})^{k} & (2t_{2})^{k} & (t_{2} + t_{3})^{k} \\ (t_{3} + t_{1})^{k} & (t_{3} + t_{2})^{k} & (2t_{3})^{k} \end{matrix} |

= (8t_{1}t_{2}t_{3})^{k} + 2(t_{1} + t_{2})^{k}(t_{2} + t_{3})^{k}(t_{3} + t_{1})^{k}
−2^{k}[t_{1}^{k}(t_{2} + t_{3})^{2k} + t_{2}^{k}(t_{3} + t_{1})^{2k} + t_{3}^{k}(t_{1} + t_{2})^{2k}].

This determinant can be 0. Indeed for t_{1} + t_{3} = 0 and t_{2} = 0 the determinant is 0 for odd k. It follows that dim V_{k}^{+} is 2 or 3 depending on the values of t_{1}, t_{2} and t_{3}. □

Remark 3.5. In [4] it is asserted that V_{2}^{−} is of dimension m which is clearly false except for m = 3. As a consequence the proof given in [4] for the existence of a WLSE for a 3-parametric exponential function is not correct. There are also errors in the proof of the existence of a WLSE in [5]. □

3.2. Polynomial weighted least squares fitting. We introduce the scalar product on ℝ^{m} defined by

⟨\overrightarrow{u}, \overrightarrow{v}⟩ = ∑_{i=1}^{m} ω_{i}u_{i}v_{i} ,

for any pair of vectors \overrightarrow{u} and \overrightarrow{v} in ℝ^{m}

\overrightarrow{u} = ( u_{1} ; u_{2} ; ⋮ ; u_{m} )  and  \overrightarrow{v} = ( v_{1} ; v_{2} ; ⋮ ; v_{m} ) .

The norm on ℝ^{m} induced by the scalar product is ∥\overrightarrow{u}∥ = ⟨\overrightarrow{u}, \overrightarrow{u}⟩^{1/2}. Then (1.1) becomes

F (p_{n}) = ∥\overrightarrow{f} − \overrightarrow{p}_{n}∥^{2},

where

\overrightarrow{p}_{n} = ∑_{j=0}^{n} α_{j}\overrightarrow{t}^{j},  \overrightarrow{t}^{j} = ( t_{1}^{j} ; t_{2}^{j} ; ⋮ ; t_{m}^{j} ) ,  and  \overrightarrow{f} = ( f_{1} ; f_{2} ; ⋮ ; f_{m} ) .

The problem is to find the orthogonal projection of \overrightarrow{f} on T^{n}. This projection is completely characterized by the normal equations

⟨\overrightarrow{f} − \overrightarrow{p}_{n}^{∗}, \overrightarrow{p}_{n}⟩ = 0

for all \overrightarrow{p}_{n} ∈ T^{n}.

Again, to simplify the computation of p_{n}^{∗}, we can determine an orthogonal basis \{\overrightarrow{q}_{j}\}_{j=0}^{n} for T^{n} by applying the Gram-Schmidt process to its basis \{\overrightarrow{t}^{j}\}_{j=0}^{n}. We obtain

\overrightarrow{q}_{0} = \overrightarrow{1}, \overrightarrow{q}_{1} = \overrightarrow{t} − α_{1}\overrightarrow{1},

and for j = 2, ..., n,

\overrightarrow{q}_{j} = (\overrightarrow{t} − α_{j}\overrightarrow{1}) · \overrightarrow{q}_{j−1} − β_{j}\overrightarrow{q}_{j−2}

where

α_{j} = \frac{⟨\overrightarrow{t} · \overrightarrow{q}_{j−1}, \overrightarrow{q}_{j−1}⟩}{⟨\overrightarrow{q}_{j−1}, \overrightarrow{q}_{j−1}⟩} (j = 1, 2, 3, …),

and

β_{j} = \frac{⟨\overrightarrow{t} · \overrightarrow{q}_{j−1}, \overrightarrow{q}_{j−2}⟩}{⟨\overrightarrow{q}_{j−2}, \overrightarrow{q}_{j−2}⟩} (j = 2, 3, 4, …).

In these identities, \overrightarrow{u} · \overrightarrow{v} is the coordinatewise multiplication of two vectors of ℝ^{m} defined by

\overrightarrow{u} · \overrightarrow{v} = ( u_{1}v_{1} ; u_{2}v_{2} ; ⋮ ; u_{m}v_{m} ) .

Let us observe that \overrightarrow{q}_{j} ∈ T^{j} for j = 0, ..., n.

It follows that the projection is given by

(3.4)  \overrightarrow{p}_{n}^{∗} = ∑_{j=0}^{n} γ_{j}^{∗}\overrightarrow{q}_{j}

where

γ_{j}^{∗} = \frac{⟨\overrightarrow{f}, \overrightarrow{q}_{j}⟩}{⟨\overrightarrow{q}_{j}, \overrightarrow{q}_{j}⟩} (j = 0, 1, …, n).

The next theorem is equivalent to Theorem 2.3.

Theorem 3.6. ⟨\overrightarrow{f} − \overrightarrow{p}_{n−1}^{∗}, \overrightarrow{t}^{n}⟩ = γ_{n}^{∗}∥\overrightarrow{q}_{n}∥^{2} for n = 0, …, m − 1 .

Proof. For n = 0 we have \overrightarrow{p}_{n−1}^{∗} = \overrightarrow{0} and the result follows. For n > 0, since \overrightarrow{q}_{n} = \overrightarrow{t}^{n} + \overrightarrow{p}_{n−1} where \overrightarrow{p}_{n−1} is a vector in T^{n−1}, and

\overrightarrow{p}_{n}^{∗} = γ_{n}^{∗}\overrightarrow{q}_{n} + \overrightarrow{p}_{n−1}^{∗},

we have

γ_{n}^{∗}∥\overrightarrow{q}_{n}∥^{2} = ⟨γ_{n}^{∗}\overrightarrow{q}_{n}, \overrightarrow{q}_{n}⟩
= ⟨\overrightarrow{p}_{n}^{∗} − \overrightarrow{p}_{n−1}^{∗}, \overrightarrow{q}_{n}⟩
= ⟨\overrightarrow{p}_{n}^{∗} − \overrightarrow{f}, \overrightarrow{q}_{n}⟩ + ⟨\overrightarrow{f} − \overrightarrow{p}_{n−1}^{∗}, \overrightarrow{q}_{n}⟩
= ⟨\overrightarrow{f} − \overrightarrow{p}_{n−1}^{∗}, \overrightarrow{t}^{n} + \overrightarrow{p}_{n−1}⟩
= ⟨\overrightarrow{f} − \overrightarrow{p}_{n−1}^{∗}, \overrightarrow{t}^{n}⟩.  □

4. Classification of data

Let {(ω_{i}, t_{i}, f_{i})}_{i=1}^{m} be a set of m data points. If we use a discrete least squares polynomial to fit the data with the orthogonal basis {q_{j}}_{j=0}^{n}, the coefficients of p_{n}^{∗} with respect to its expansion (2.3) or (3.4) suggest the following classification of the data.

Definition 4.1. The data {(ω_{i}, t_{i}, f_{i})}_{i=1}^{m} are said to be :

(i) essentially stationary if γ_{1}^{∗} = 0; (ii) essentially increasing, respectively decreasing, if γ_{1}^{∗} > 0, respectively γ_{1}^{∗} < 0; (iii) essentially linear if γ_{2}^{∗} = 0; (iv) essentially convex, respectively concave, if γ_{2}^{∗} > 0, respectively γ_{2}^{∗} < 0. □

Let us note that we could continue the classification with the higher order coefficients γ_{n}^{∗} for n = 3, ..., m − 1.

Finally if we apply symmetric transformations to the data we obtain the following result.

Theorem 4.2. Effect of symmetric transformations on the data.

(a) If the data {(ω_{i}, t_{i}, f_{i})}_{i=1}^{m} are essentially increasing, resp. decreasing, then the data {(ω_{i}, −t_{i}, f_{i})}_{i=1}^{m} are essentially decreasing, resp. increasing. The stationarity, linearity, and concavity or convexity properties are not modified by this transform.

(b) If the data {(ω_{i}, t_{i}, f_{i})}_{i=1}^{m} are essentially increasing, resp. decreasing, and essentially convex, resp. concave, then the data {(ω_{i}, t_{i}, −f_{i})}_{i=1}^{m} are essentially decreasing, resp. increasing, and essentially concave, resp. convex. The stationarity and linearity properties are not modified by this transform. □

5. Conclusion

We have revisited the polynomial weighted least squares analysis. Doing so we have specified the dimension of three vector subspaces of P (Theorem 2.1 and Theorem 2.2) and of ℝ^{m} (Theorem 3.1 and Theorem 3.3) used for solving this problem . We also have established a property (Theorem 2.3 and Theorem 3.6) and suggested a classification of data (Definition 4.1) which will play a role in finding sufficient conditions for the existence of a WLSE for a 3-parametric exponential model [2].

6. Acknowledgments

This work has been supported by a NSERC (Natural Sciences and Engineering Research Council of Canada) individual discovery grant for the first author.

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