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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.

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 [

There are many practical techniques for smoothing a time series [

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 [

This paper is organized as follows. Section

The equation for the general model for the approximation of the time series

The partition of the domain of the time series

An approximation

An iteration over

The measure of smoothness of the iterations at time

For most of the examples presented in this paper,

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 (

The magnitude

Given the iterates

If

An iterative process is used to obtain

Similarly, if

For a time series

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 (

It is convenient to introduce another notation to represent a partition:

The upper graphs include

The graph of the trend

The upper graphs represent

The upper graph is the approximation

The trend

Two time series, provided by the Ontario Ministry of Transportation ([

In Figure

In Figure

In Figure

The subdomains that are the same in the two partitions

For a step level change between

A simple example indicates the approach for a series that has a missing value or a possible outlier at

Graphs for the example are shown:

The point of this exercise is to determine the

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

24 | 0.986 (0.013) | [0.94, 1.01] |

12 | 0.967 (0.018) | [0.91, 1.00] |

6 | 0.930 (0.023) | [0.87, 0.98] |

4 | 0.888 (0.027) | [0.81, 0.96] |

3 | 0.843 (0.035) | [0.76, 0.92] |

The terms in the equation (

To describe the properties of

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.