Approximations of Time Series

Abstract

A method is proposed to approximate the main features or patterns including interventions that may occur in a time series. Collision data from the Ontario Ministry of Transportation illustrate the approach using monthly collision counts from police reports over a 10-year period from 1990 to 1999. The domain of the time series is partitioned into nonoverlapping subdomains. The major condition on the approximation requires that the series and the approximation have the same average value over each subdomain. To obtain a smooth approximation, based on the second difference of the series, a few iterations are necessary since an iteration over one subdomain is affected by the previous iteration over the adjacent subdomains.

1. Introduction

Graduated licensing system (GLS) is a method of gradual exposure of young novice drivers into the driving environment, allowing them to obtain initial experience with driving under supervision, followed by more independent driving under higher-risk circumstances [1]. This model was widely incorporated into driver licensing programs across the US and Canada as well as other countries over the 1990s. Most of these programs have incorporated similar restrictions into their initial phases [2]. These include driving with supervision, restricted driving at night, limited teenage passengers, and zero blood alcohol level while driving. This method has had limited long-term evaluation in North America, but long-term followup in New Zealand suggested a reduced but persistent long-term reduction in young driver collisions as a result of its implementation [3]. The collision data for Ontario drivers, around the time of the introduction of the GLS ([2], p. 126), illustrate the variety of approximations of time series that are possible.

There are many practical techniques for smoothing a time series [4]. The smoothed value at a point is a weighted average involving the elements in the series that are within a local window about the point. One way to generate the weights in a moving average involves a local polynomial approximation of order 3 (or 5) where the window includes 5 (or 7 et cetera) points. The weights are then defined by regression. Another approach defines the weights in terms of an appropriate kernel, and this method applies more generally to bivariate data [5]. The advantages of local estimates compared with global estimates are discussed in [6].

The first step in the computational process involves the partition of the domain of the time series into subdomains. The subdomains are then labelled as odd-numbered or even-numbered. Iterations are then performed over the odd numbered subdomains followed by the iterations over the even-numbered subdomains. This process is numerically efficient since the iterations over one set of subdomains update the boundary conditions for the iterations over the remaining subdomains [7]. To determine a smooth and accurate approximation, this set of iterations is repeated a few times.

This paper is organized as follows. Section 2 describes the form of the approximations, the partition of the time series, and the minimization over the subdomains of the partition. In Section 3, the equations for the approximation are derived, and some computational details are given in Section 4. Under certain assumptions, approximations of time series with variable spacing are possible. The time series and their approximations are presented in Section 5, and an example outlines the approach for step level changes, missing data, and outliers in Section 6. In Section 7, the approximation over a subdomain is determined by a fourth-order polynomial and a straight line. Finally, guidelines for the application of the proposed approach to a time series are outlined in “Concluding remarks.”

2. The General Model

The equation for the general model for the approximation of the time series {𝑍_𝑡∣𝑡=1,…,𝑁} is𝑍_𝑡=^𝑛−1_𝑘=0𝑂^𝑘𝑡+𝑄^𝑘𝑡+𝑅^𝑛𝑡,(2.1) where 𝑄^0𝑡=𝑇_𝑡+𝑀_𝑡 and 𝑅^𝑛𝑡=𝐴^𝑛𝑡𝑊^𝑛𝑡. In these equations, 𝑂^𝑘𝑡 is the term for outliers (if present); 𝑄^𝑘𝑡 is the 𝑘th approximation; 𝑇_𝑡 is a trend and includes the level changes; 𝑀_𝑡 is a nonperiodic oscillatory function; 𝑅^𝑛𝑡 is the remainder; 𝐴^𝑛𝑡 is a measure of the variation of the remainder. A restriction on the approximations requires that the root-mean-square (RMS) value of the remainder is a decreasing sequence with increasing 𝑛. The form of (2.1) is similar to the asymptotic expansion of a function that contains a small positive parameter ([8], p. 1–4).

The partition of the domain of the time series {𝑍_𝑡}, henceforth denoted by 𝑍_𝑡, is chosen in order to accurately approximate possible patterns in the time series. Let 𝑃={𝐸_𝑘∣𝑘=1,…,𝑀} be a partition of [1,𝑁] where the nonoverlapping subdomains are 𝐸_𝑘={𝑡_𝑘−𝑛_𝑘+1,…,𝑡_𝑘}, 𝑛_𝑘≥1, where 𝑛_𝑘 is the number of elements in the 𝑘th subdomain 𝐸_𝑘. The overlapping subdomains, over which the iterations are computed, are defined as 𝐸^𝑜𝑘={𝑡_𝑘−𝑛_𝑘,𝐸_𝑘,𝑡_𝑘+1}. For 𝑘=1, 𝐸^𝑜1={0,𝐸₁,𝑛₁+1} and for 𝑘=𝑀, 𝐸^𝑜𝑀={𝑁−𝑛_𝑁,𝐸_𝑁,𝑁+1} so that the overlapping subdomains are defined over the interval [0,𝑁+1].

An approximation 𝑄_𝑡 of a time series 𝑍_𝑡, where 𝑍_𝑡=𝑄_𝑡+𝑅_𝑡, is determined by a few iterations 𝐼^𝑛𝑡 starting with 𝐼^0𝑡=𝑍_𝑡. Once the desired accuracy is obtained, the last iteration is defined as 𝑄_𝑡. All iterations and the approximation 𝑄_𝑡 along with the remainder 𝑅_𝑡 satisfy the following properties.(1)An iteration over 𝐸_𝑘 has the same average value 𝑎_𝑘 as the time series _𝑡_∈𝐸𝑘𝑍_𝑡=_{𝑡∈𝐸𝑘}𝐼^𝑛𝑡=𝑎_𝑘,(2.2) and, hence, the average value of the remainder 𝑅_𝑡 is zero over this subdomain. If 𝑍_𝑡 is a measure of “energy” in the process, then 𝑄_𝑡 conserves energy over each subdomain in the partition. In the particular case 𝑛_𝑘=1, 𝐸_𝑘={𝑡_𝑘} and 𝑡=𝑡_𝑘 is a fixed point for the approximation so that 𝑄_𝑡∶=𝑍_𝑡 at 𝑡=𝑡_𝑘.(2)The measure of smoothness of the iterations at time 𝑡 for the 𝑛th iteration is defined by 𝛿^𝑛𝑡=𝐼^𝑛𝑡+1+𝐼^𝑛𝑡−1−2𝐼^𝑛𝑡 which is the second difference of 𝐼^𝑛𝑡 at time 𝑡. The norm on 𝐸^𝑜𝑘 is defined as the RMS value Δ^𝑛𝑘=⎛⎜⎜⎝∑^{𝑡∈𝐸𝑘}𝛿^𝑛𝑡²𝑛_𝑘⎞⎟⎟⎠^1/2.(2.3) Provided that the number of elements in 𝐸_𝑘 is greater than 1, then the condition that Δ^𝑛𝑘 has a minimum value is imposed. 𝐼^𝑛𝑡 is required for 𝑡=𝑡_𝑘−𝑛_𝑘 and 𝑡=𝑡_𝑘+1 in 𝐸^𝑜𝑘 to determine 𝛿^𝑛𝑡 at the endpoints of 𝐸_𝑘. These two values of 𝐼^𝑛𝑡 are the boundary conditions for the minimization on 𝐸_𝑘.(3)For most of the examples presented in this paper, 𝑡=1 and 𝑡=𝑁 are fixed points so that 𝑄₁=𝑍₁ (𝑛₁=1) and 𝑄_𝑀=𝑍_𝑁 (𝑛_𝑀=1). These values provide the boundary conditions for the minimization over the adjacent subdomain. For the general case where 𝑛₁>1 in 𝐸₁ (𝑛_𝑀>1 in 𝐸_𝑀), one of the boundary conditions is missing in 𝐸^𝑜1 (𝐸^𝑜𝑀) so that an external boundary condition is required as described in the last paragraph of Section 3.(4)In some cases there are two or more approximations over one or more subdomains and a criterion is required to choose the best approximation. From (2.1), 𝑅^𝑘𝑡=𝑄^𝑘𝑡+𝑅^𝑘+1_t(𝑅^0𝑡∶=𝑍_𝑡) where 𝑄^𝑘𝑡 is determined from 𝑅^𝑘𝑡 over a partition 𝑃_𝑘. For the example in Section 5, the simplest case occurs when 𝑃_𝑘 is a refinement of 𝑃_𝑘−1; that is, 𝑃_𝑘=∪𝑃^ℓ𝑘 where 𝑃^ℓ𝑘 covers 𝐸_ℓ in 𝑃_𝑘−1. Then an approximation over 𝐸_ℓ is 𝑄^𝑘𝑡=0 and the other is defined by 𝑃^ℓ𝑘. Let the RMS value of the remainder 𝑅^𝑡𝑘+1 over 𝑃^ℓ𝑘 be denoted by 𝑆^𝑘+1, and 𝑆^𝑘 is the RMS value over 𝐸_ℓ in 𝑅^𝑘𝑡. The approximation defined by 𝑃^ℓ𝑘 is a significant improvement if the ratio 𝑆^𝑘+1/𝑆^𝑘≤𝜖 for a chosen value of 𝜖. As shown in Section 7, an upper bound for 𝜖 takes on values between 0.75 and 0.9. For the example in Section 6 involving an outlier, there are two approximations for 𝑄^0𝑡.(5)The magnitude 𝐴^𝑛𝑡 of the remainder 𝑅^𝑛𝑡 is defined to be the RMS value of the remainder over each subdomain in a partition, and this definition implies that the RMS value of the series 𝑊^𝑛𝑡 in (2.1) is equal to 1. For the example presented in Section 5, the subdomains are uniform with 12 elements.

3. Mathematical Details

Given the iterates 𝐼^𝑡𝑛−1 and 𝛿^𝑡𝑛−1 on 𝐸^𝑜𝑘, the iterates 𝐼^𝑛𝑡 for 𝑡∈𝐸_𝑘 are computed such that the sum of squares of 𝛿^𝑛𝑡 has a minimum value. For the moment, the first and last interval 𝐸₁ and 𝐸_𝑀 are excluded. The following variables are required to set up the equations for the minimization over the subdomain:𝑋^𝑛𝑘=𝛿𝐵^𝑛𝑡_{𝑘−𝑛𝑘+1𝑛𝑘},…,𝛿^𝑛𝑡_𝑘^,𝑌^𝑛𝑘=𝐼^𝑛𝑡_{𝑘−𝑛𝑘+1},…,𝐼^𝑛𝑡_𝑘^,=𝐼^𝑛𝑡_{𝑘−𝑛𝑘},0,…,0,𝐼^𝑛𝑡_𝑘+1^(3.1) are 𝑛_𝑘×1 matrices and a prime on a matrix indicates the transposed matrix. From the definition of 𝛿^𝑛𝑡 in Section 2, these matrices are related by 𝑋^𝑛𝑘−𝐴𝑌^𝑛𝑘=𝐵^𝑛𝑘, where 𝐴 is a 𝑛_𝑘×𝑛_𝑘 tridiagonal symmetric matrix with elements{(−2,1);(1,−2,1);…;(1,−2,1);(1,−2)},(3.2) where −2 is on the main diagonal. The equations for the iterations are obtained by replacing 𝐵^𝑛𝑘 with 𝐵^𝑘𝑛−1. Since the sum of 𝐼^𝑛𝑡 for 𝑡∈𝐸_𝑘 is a constant for all 𝑛, then 𝐻^𝑌^𝑛𝑘=_{𝑡∈𝐸𝑘}𝑍_𝑡=𝑎_𝑘𝑛_𝑘,(3.3) where 𝐻=[1,1,…,1]^, and 𝑎_𝑘 is the average value of 𝑍_𝑡 over 𝐸_𝑘. The condition on the sum in terms of 𝑋^𝑛𝑘 is 𝐸^(𝑋^𝑛𝑘−𝐵^𝑘𝑛−1)=𝑎_𝑘𝑛_𝑘, where 𝐸 is the solution of 𝐴𝐸=𝐻. Thus, 𝐸′𝑋^𝑛𝑘=𝑛_𝑘𝜌^𝑘𝑛−1 where𝜌^𝑘𝑛−1=𝑎_𝑘−𝐼^{𝑡𝑛−1𝑘−1}+𝐼^𝑡𝑛−1_𝑘+12,(3.4) and 𝐸[1]=𝐸[𝑛_𝑘]=−𝑛_𝑘/2 (Section 7). The solution for 𝑋^𝑛𝑘 such that (𝑋^𝑛𝑘)^𝑋^𝑛𝑘 has a minimum is 𝑋^𝑛𝑘=𝜌^𝑘𝑛−1𝐺²(𝑛_𝑘)𝐸, where 𝐺²(𝑛_𝑘)=𝑛_𝑘/(𝐸′𝐸). Finally, Δ^𝑛𝑘=|𝜌^𝑘𝑛−1|𝐺(𝑛_𝑘), and the solution 𝑌^𝑛𝑘 is𝑉𝑌^𝑛𝑘=𝐼^𝑡𝑛−1_{𝑘−𝑛𝑘}𝑉₁+𝐼^𝑡𝑛−1_𝑘+1𝑉₂+𝜌^𝑘𝑛−1𝑉₃,(3.5)₁[𝑖]=1−𝑖𝑛_𝑘+1,𝑉²[𝑖]=𝑖𝑛_𝑘+1,𝐴𝑉³=𝐺²𝑛_𝑘.(3.6) The iteration 𝐼^𝑛𝑡=𝑌^𝑛𝑘[𝑖], where 𝑖=1,…,𝑛_𝑘 and 𝑡=𝑡_𝑘−𝑛_𝑘+1,…,𝑡_𝑘 in 𝐸_𝑘, respectively. For a given 𝑛_𝑘, 𝐸 and then 𝑉₃ are uniquely determined, and 𝑉₃ is computed in advance for all of the subdomains that occur in a time series.

3.1. External Boundary Conditions

If 𝑡=1 is not a fixed point, then the subdomain 𝐸^𝑜1 requires a boundary condition. Here are three possible external boundary conditions to impose at 𝑡=0 that can be used to reflect the possible behavior of the time series near the endpoint. (1)𝐼^0𝑛−1=𝐼^1𝑛−1. The slope of the tangent is zero at 𝑡=1. (2)𝐼^0𝑛−1 is defined so that 𝜌^1𝑛−1=0 in (3.4). This condition implies that the approximation over 𝐸₁ is a segment of a straight line. (3)An iterative process is used to obtain 𝑍₀ such that the RMS value of the remainder over the adjacent subdomain(s) has a minimum value.

Similarly, if 𝑡=𝑁 is not a fixed point, then the external boundary conditions are obtained by replacing 𝐼^0𝑛−1, 𝐼^1𝑛−1, 𝜌^1𝑛−1, 𝑍₀ with 𝐼^{𝑛−1𝑁+1}, 𝐼^𝑁𝑛−1, 𝜌^𝑀𝑛−1, 𝑍_𝑁+1, respectively.

4. Computational Aspects

For a time series 𝑍_𝑡 and a partition 𝑃, there is a related series defined by Z_𝑡=𝑎_𝑘 for 𝑡∈𝐸_𝑘, where 𝑎_𝑘 is the average value of 𝑍_𝑡 for 𝑡∈𝐸_𝑘. This property holds for all of the approximations in this paper,

The approximations for a time series 𝑍_𝑡 and the averaged time series Z_𝑡 are the same to the desired accuracy provided that the same partition and the same external boundary conditions (if any) are applied.

Consequently, any time series with variable spacing can be approximated provided that the estimates of the average values of the time series over the subdomains are adequate.

The approximation for the averaged series is employed especially for larger subdomains (𝑛_𝑘≈12 or more). The efficiency of the computations is increased if the boundary conditions in the first set of iterations (even and odd) are the average of the four values of the series that straddle the subdomains 𝐸_𝑘−1 and 𝐸_𝑘. The averaged series was used in all computations, although the approximation obtained from 𝑍_𝑡 may be more efficient in special cases.

It is convenient to introduce another notation to represent a partition: 𝑃 = {𝑛₁,𝑛₂,…;…;…,𝑛_𝑀}, where the number of elements in the subdomains in the first block is {𝑛₁,𝑛₂,…} and in the last block by {…,𝑛_𝑀}. These blocks are a convenient way to separate the seasons or a set of months. Also, the approximation 𝑄^𝑘𝑡 obtained by iterating the time series 𝑛 times, using the partition 𝑃_𝑘, is denoted by 𝒫^𝑛𝑘{𝑅^𝑘𝑡} (𝑅^0𝑡∶=𝑍_𝑡). The number of iterations 𝑛 is determined from the difference𝐷^𝑛𝑡=𝒫^𝑛𝑘𝑅^𝑘𝑡−𝒫^𝑘2𝑛𝑅^𝑘𝑡(4.1) by imposing the condition that max(|𝐷^𝑛𝑡|)<𝐿. 𝐿=1 in Figures 1 and 2, 𝐿=0.04 in Figures 3 and 4, and 𝐿=0.01 in Figure 5. All calculations in this paper were performed using Maple software [9].

Figure 1 

The upper graphs include 𝑍𝑡 (thin line), the accidents per month for young novice drivers, and the approximation 𝑄0𝑡 (thick line). The lower graphs represent the remainder 𝑅1𝑡=𝑍𝑡−𝑄0𝑡 (thin line) and the amplitude of the remainder 𝐴1𝑡 (thick line). The RMS value of the remainder is 25.

Figure 2 

The graph of the trend 𝑇𝑡 (thick line) and the approximation 𝑄0𝑡 (thin line) from Figure 1 for young novice drivers are shown.

Figure 3 

The upper graphs represent 𝑍𝑡 (thin line), the accidents (in thousands) per month for all drivers, and the approximation 𝑄0𝑡 (thick line). The RMS value of the remainder 𝑅1𝑡 is 1.83. The lower graphs include the second approximation 𝑄1𝑡 (thick line) and the remainder 𝑅1𝑡 (thin line).

Figure 4 

The upper graph is the approximation 𝑄0𝑡+𝑄1𝑡 (in thousands) for all drivers. The lower graph is the remainder 𝑅2𝑡; the RMS value is 1.29.

Figure 5 

The trend 𝑇𝑡 (thin line) of two segments of a series 𝑍𝑡 is 𝑇𝑡=0 for 𝑡≤60 and 𝑇𝑡=1 for 𝑡≥61; whereas, 𝑄𝑡 (thick line) is the trend of the series over [1,120].

5. Applications

Two time series, provided by the Ontario Ministry of Transportation ([2], p. 126), illustrate the approximations. The graph of the time series for the monthly accidents for young novice drivers is given in Figure 1 where the main feature here is the intervention that occurs at 52 months owing to the introduction of the GLS on April 1, 1994. The corresponding graph for all drivers is shown in Figure 3 where the sharp drop in the graph from the maximum in December/January to April, except for the last 2 years, is a strong feature of the series.

In Figure 1, the uniform partition 𝑃₀ = {1,3;4;…;4;3,1}, except for the first and last blocks, provides a smooth approximation 𝑄^0𝑡=𝒫⁴⁰{𝑍_𝑡} and captures the intervention well. In this case, the first and last elements are fixed (𝑄⁰¹=𝑍₁ and 𝑄⁰¹²⁰=𝑍₁₂₀). Since the sum of the elements in the remainder over 𝐸_𝑘 is zero, then 𝒫₀{𝑅^1𝑡}={0}. Other uniform partitions are possible where there are 3 or 6 elements in each subdomain. The approximation in the former case is slightly less smooth than 𝑄^0𝑡 in Figure 1, and the RMS value of the remainder is 26. For the case of 6 elements, the RMS value of the remainder is 30. A more accurate approximation is obtained if the subdomains have two elements; however, this approximation has an angular appearance since it more closely approximates the time series.

In Figure 2, the approximation 𝑄^0𝑡 of Figure 1 is expressed as a sum of a trend and an oscillatory series. The partition for the trend 𝑇_𝑡=𝒫^6𝑇{𝑄^0𝑡} is 𝑃_𝑇 = {12,12,12,12;6,6,6,6;12,12,12,12}. The external boundary condition implies that the tangent is horizontal at the endpoints of the series. The trend in this example is defined as a seasonal approximation of the time series where the subdomains contain 6 elements over the domain of the intervention. The remainder is the oscillatory series 𝑀_𝑡=𝑄^0𝑡−𝑇_𝑡.

In Figure 3, the points for January or December (plus one November) and April are fixed points for the approximation 𝑄^0𝑡=𝒫³⁰{𝑍_𝑡}. The partition is 𝑃₀={1,2,1,7,1;3,1,7,1;3,1,7,1;3,1,8;1,2,1,7,1;3,1,6,2;1,2,1,8;1,2,1,6,1;2,2,1,8;1,2,1,7,1}. The second approximation captures the increase in the number of accidents that occur in the summer months by approximating the remainder 𝑅^1𝑡 in 𝑍_𝑡=𝑄^0𝑡+𝑅^1𝑡 to obtain 𝑅^1𝑡=𝑄^1𝑡+𝑅^2𝑡 where 𝑄^1𝑡=𝒫³¹{𝑅^1𝑡}. The partition 𝑃₁ is a refinement of 𝑃₀ where 7 is replaced with 3,4;6 with 3,3; 8 with 3,5. Consequently, the approximations 𝑄^0𝑡 and 𝑄^0𝑡+𝑄^1𝑡 have the same average value over the subdomains of 𝑃₀.

The subdomains that are the same in the two partitions 𝑃₀ and 𝑃₁ are indicated by the intervals over which the approximation is zero in Figure 3. For the remaining intervals, the ratio of the RMS value of the remainder 𝑅^2𝑡 in Figure 4 to the RMS value of 𝑅^1𝑡 is equal to 0.49 so that this second approximation is significant. Furthermore, each of the ten segments of 𝑄^1𝑡, excluding the segments in which the approximation is identically equal to zero, has a value between 0.40 and 0.52. The approximation 𝑄^0𝑡+𝑄^1𝑡 of 𝑍_𝑡 and the remainder 𝑅^2𝑡 appear in Figure 4.

6. Level Changes, Missing Data, and Outliers

For a step level change between 𝑡=𝜏 and 𝑡=𝜏+1, an approximation may not provide an adequate approximation for the time series in the subdomains on both sides of the step. For 𝑡>𝜏, an external boundary condition is applied at 𝑡=𝜏 such that the remainder of the approximation has a minimum RMS value. The same approach is applied to the series for 𝑡<𝜏+1. These ideas are illustrated in Figure 5 where the partition for the approximation 𝑄_𝑡=𝒫⁶{𝑇_𝑡} over [1,120] is 𝑃 = {1,11;12;…;12;11,1}. The approximation exhibits a phenomenon that is similar to a Fourier series near a discontinuity in that the approximation overshoots on the right and undershoots to the left of the jump. The maximum and minimum of 𝑄_𝑡 are 1.176 and −0.173. Moving away from the jump in either direction, the oscillations of 𝑄_𝑡−𝑇_𝑡 occur with rapidly decreasing amplitude. The details in item 4 of Section 2 are applied over the subdomains adjacent to 𝑡=60 to choose between the two approximations of the trend. Interventions and level shifts, from an autoregressive moving average point of view, are presented in [10] and [11].

A simple example indicates the approach for a series that has a missing value or a possible outlier at 𝑡=6. The series is {𝑍_𝑡}= {0.6,−0.3,−0.5,−0.2,0.4,𝑍₆,1,0.8,0.9,1.1,1.0,1.2} where 𝑍₆ is not defined in the case of a missing value. For both cases, the approximation 𝑄^0𝑡 is determined for the partition 𝑃= {1;4;1;5;1}, where the value of the series is 𝑋 at 𝑡=6, such that the RMS value of the remainder is a minimum. A good initial estimate for 𝑋 is the average value of the time series in a window about 𝑡=6 [5]. Then an iterative process is started to obtain 𝑋≈1.25, as shown in Figure 6, and the RMS value of the remainder is 0.104. The smoothest approximation over 𝑃 occurs for 𝑋≈0.12, where ∑²¹¹𝛿^2𝑡 has a minimum value. For the case of a possible outlier, 𝑂^0𝑡=0 for 𝑡≠6 in (2.1) and 𝑂⁰⁶=𝑍₆−1.25 provided that the ratio of the RMS value of the approximation with 𝑋=1.25 to the RMS value of the approximation under the assumption that 𝑍₆ is not an outlier satisfies the condition in item 4 of Section 2.

Figure 6 

Graphs for the example are shown: {𝑍𝑡} (thin line); approximation (thick line), 𝑋 = 1.25 (large dot) at 𝑡=6; smooth approximation, 𝑋=0.12 at 𝑡=6.

7. Properties of Approximations

Approximations of Random Samples
The point of this exercise is to determine the 𝜖 in item 4 of Section 2 such that the only reasonable approximation for a series of random samples is 𝑄^0𝑡 equal to the mean of the series. 12,000 random samples from the normal distribution with a mean of 0 and a standard deviation of 1 were generated using Maple to form 100 time series with 120 elements in each series. For each series, five approximations were determined where the subdomains of the uniform partition contained 3, 4, 6, 12, and 24 elements. The external boundary condition for the approximation is the condition of zero slope of the tangent. For each series, the ratio of the RMS value of the remainder for the approximation to the RMS value of the series were calculated, and the results are given in Table 1. The approximations corresponding to 24 and 12 elements are smooth and appear to reflect an underlying pattern in the series; whereas, for the cases 3 and 4, the approximations are contorted. An upper bound for 𝜖 is less than the minimum values in the range.

Table 1 

The number of elements in the subdomains along with the mean, standard error (StE), and range of the ratios is shown.

Quartic Polynomial
The terms in the equation (3.5) for the approximation over 𝐸^𝑜𝑘 have a simple interpretation. For the first two terms in (3.5), (𝑖,𝑉₁[𝑖]) and (𝑖,𝑉₂[𝑖]) are points on straight lines. For the last term, Maple solves 𝐴𝐸=𝐻 for 𝐸 and (3.6) for 𝑉₃ exactly; consequently, an accurate computation shows that (𝑖,𝐸[𝑖]) and (𝑖,𝑉₃[𝑖]) are points on the graph of a quadratic and a quartic polynomial, respectively.
To describe the properties of 𝐸 and 𝑉3, it is necessary to change variables from 𝑡 to 𝑠 where 𝑠=𝑡−[𝑡_𝑘−(𝑛_𝑘−1)/2] and 𝑠=0 is the central point of the interval 𝐸^𝑜𝑘. In the 𝑠 variable, the boundary conditions are applied at the points 𝑠=±𝑠_𝑜, where 𝑠_𝑜=(𝑛_𝑘+1)/2. The equation of the quadratic polynomial 𝑣(𝑠,𝑛_𝑘) for 𝐸 is defined by 𝑑²𝑣/𝑑𝑠²=1 and 𝑣=0 at 𝑠=±𝑠_𝑜 so that 𝑣=(𝑠²−𝑠^2𝑜)/2. For any integer 𝑖, 𝐸[𝑖] is equal to 𝑣 at the corresponding value for 𝑠, and 𝐸′𝐸=((2𝑠_𝑜)⁵−2𝑠_𝑜)/120. The equation for the quartic polynomial 𝑢(𝑠,𝑛_𝑘) for 𝑉₃, provided 𝑛_𝑘≥3, is determined by 𝑑⁴𝑢/𝑑𝑠⁴=𝐺²(𝑛_𝑘), where the roots of the equation for 𝑢 are 𝑠=±𝑠_𝑜 and 𝑠²=5𝑠^2𝑜+1. Thus, 𝑢=𝐺²(𝑛_𝑘)(𝑠²−𝑠^2𝑜)(𝑠²−5𝑠^2𝑜−1)/24. 𝐺(𝑛_𝑘) is a measure of the smoothness of the approximation over 𝐸^𝑜𝑘: G(2) = 1.0, G(3) = 0.59, G(4) = 0.39, G(6) = 0.21, G(12) = 0.062, and G(24) = 0.017.

Concluding Remarks
The major input for the approximation of a time series involves the partition of the domain. Initially a uniform partition is chosen and, if seasonal behavior is present in the series, a subset of the partitions cover the domain for the seasons. In general, as the length of the subintervals decreases, the approximation is less smooth and the accuracy of the approximation increases. The best approximation occurs at the point at which the approximation is acceptably smooth. The subintervals can be enlarged to determine a much smoother approximation that can be labelled as a trend while still respecting the seasonal aspects of the series; however, if an intervention is present, then some adjustment of the partition may be necessary in the region of the intervention. For time series with a well-defined local maximum or minimum, the approximation can be assigned the same value as the series by taking the partition to be a single point of the domain. For series with jumps and other complexities, examples are provided to suggest how to proceed in these cases.
An approach in the literature, as indicated in the introduction, defines the approximation at a point as a weighted average of the values of the values of the time series in a window about the point. This approach may smooth out interesting features in the time series and, if applied over a smaller intervals, the approximation will not be smooth. Since the proposed model is not based on regression, a comparison of the two approaches has not been considered.