Determination Of Trend By Weighted Moving Average Process

1. Core Concept Overview & Intuition

1.1 Introduction

A time series is a sequence of observations recorded in chronological order. When such data are plotted against time, the underlying movement is rarely a straight line; it typically exhibits a long-term drift together with shorter oscillations. The long-term drift is the trend, and the central task of this unit is to estimate it objectively rather than by visual inspection alone.

The weighted moving average process is a smoothing technique for estimating trend. Its logic is as follows. Suppose we have already decided on the degree of a polynomial that we believe describes the trend adequately. Rather than fitting that polynomial to the entire series at once, we fit it to a small, convenient block of consecutive observations and then slide the block forward one observation at a time. At each position of the block, the fitted value at the middle point of the block is recorded as the trend value for that time point. Because the fitted middle value turns out to be a fixed linear combination of the observations in the block, the procedure reduces to multiplying each observation by a prescribed weight and summing.

The essential intuition is that of a local polynomial fit. A low-degree polynomial fitted to a short window captures the local level of the series while suppressing short-term irregular fluctuations. The weights are symmetric about the centre of the window and sum to unity, so the resulting average has the same units as the original data and does not distort the overall level of the series.

1.2 Learning Outcomes

By the end of this unit the learner will be able to:

1. Explain the distinction between fitting a polynomial to the entire series and fitting it locally to a moving block of consecutive points.
2. Derive the weights of a weighted moving average from the normal equations of a least-squares polynomial fit.
3. State and apply the weight formula for a cubic polynomial fitted to five consecutive points.
4. Compute trend values for a given series using the weighted moving average weights, correctly identifying the time point to which each trend value corresponds.
5. Recognise the symmetry and unit-sum properties of the weights and use them as checks on derived weights.

1.3 Definition of a Time Series

> Definition (Time Series). A time series is a set of observations $X_t$ arranged in chronological order, where the subscript $t$ indexes the time period at which the observation is recorded. The time index may be a year, a quarter, a month, a week, a day, or an hour, depending on the frequency of recording.

The defining feature of a time series is that the ordering of the observations carries information. Unlike a random sample, where the order of the values is irrelevant, in a time series the position of each observation in the sequence is essential to its interpretation.

1.4 Types of Time Series

Time series are classified according to the nature of the variable being recorded and the mechanism by which it is observed.

• Univariate and multivariate series. A univariate series records a single variable over time, for example the quarterly profit of a firm. A multivariate series records two or more variables simultaneously, for example price and quantity sold in each period.
• Continuous and discrete series. A continuous series is recorded without interruption, such as a temperature trace from a recording instrument. A discrete series is recorded at separated time points, such as annual census populations or quarterly sales figures. All series treated in this unit are discrete.
• Series classified by frequency of recording. Data recorded annually, quarterly, monthly, weekly, daily, or hourly. The frequency determines which smoothing windows are natural: a four-point window aligns with quarterly data and a twelve-point window with monthly data.
• Series classified by the components they exhibit. A series may display trend only, trend with seasonal variation, or trend with cyclical and irregular movement. The weighted moving average is used to isolate the trend component.

1.5 Some Terminologies

• Trend ($T_t$). The long-term general direction of the series, whether increasing, decreasing, or stable, after short-term fluctuations have been removed.
• Moving average. A sequence of averages, each computed from a fixed number of consecutive observations, with the block advanced by one observation at each step.
• Extent (order) of the moving average. The number of consecutive observations included in each block. An extent of five means each average uses five successive values.
• Weights. The fixed numerical coefficients attached to the observations within a block when the local fitted value is expressed as a linear combination of those observations.
• Centred moving average. A moving average whose value is attributed to the middle observation of its block. When the extent is odd, the middle observation is uniquely defined; when the extent is even, centring requires a further averaging step.
• Smoothing. The operation of replacing each observation by a local average so that irregular fluctuations are attenuated and the trend is made visible.
• Fitted polynomial. A polynomial $X_t=a_0+a_1t+a_2t^2+a_3t^3+\cdots$ used to represent the trend, where $a_0,a_1,a_2,a_3$ are constants estimated from the data.
• Normal equations. The system of linear equations obtained by minimising the sum of squared deviations between the observations and the fitted polynomial.

1.6 Objectives of Time Series Analysis

The analysis of a time series is undertaken for several connected purposes:

1. Description. To summarise the historical behaviour of the variable in terms of its components, principally the trend.
2. Estimation of trend. To obtain a smooth representation of the long-term movement, which is precisely the objective served by the weighted moving average process.
3. Isolation of components. To separate the trend from the other components so that each may be studied on its own.
4. Forecasting. To project the estimated trend forward in order to anticipate future values of the variable.
5. Comparison and control. To compare the performance of different periods against the estimated trend, and to detect departures from expected behaviour.

1.7 Introduction to the Weighted Moving Average Process

Suppose the degree of the polynomial to be fitted has been decided. Instead of fitting the polynomial to the entire data set, we may fit it to a smaller convenient set of points and keep moving the set along the series to determine the trend. For example, if the trend curve is cubic, we could fit it to a set of four or five consecutive points and keep moving to determine the trend. This method of determining trend is called the weighted moving average method.

The procedure of determining trend is equivalent to taking linear combinations of the observations with coefficients that can be worked out from the least-squares fit. The coefficients are the weights, and because they are derived once for a given polynomial degree and window length, they may be applied mechanically to every block of the series.

1.8 Learning Outcomes for the Weighted Moving Average Process

By the end of this lesson the learner will be able to determine trend values using the method of weighted moving average process, including the derivation of the weights from the normal equations and the correct alignment of each trend value with its time point.

2. Mathematical Formalization & Core Definitions

2.1 Local Polynomial Representation

Let a block of consecutive observations be indexed symmetrically about a central point. For a block of five consecutive points we write the observations as

$$
X_{-2},\; X_{-1},\; X_{0},\; X_{1},\; X_{2}
$$

where the subscript denotes the position relative to the centre, so that $X_0$ is the middle observation of the block. The cubic polynomial fitted to this block is

$$
X_t=a_0+a_1t+a_2t^2+a_3t^3
$$

where $t\in{-2,-1,0,1,2}$ and $a_0,a_1,a_2,a_3$ are the coefficients to be estimated. The trend value at the centre of the block is the fitted value at $t=0$, which is

$$
X_0=a_0
$$

Thus the problem of obtaining the trend value at the middle point reduces to the problem of estimating the constant term $a_0$ of the local polynomial.

2.2 The Least-Squares Criterion and the Normal Equations

The coefficients are chosen to minimise the sum of squared deviations between the observed values and the fitted polynomial over the block. The normal equations for the cubic fit to five points are

$$
\sum_{t=-2}^{2}X_t=5a_0+a_1\sum t+a_2\sum t^2+a_3\sum t^3 \qquad (i)
$$

$$
\sum_{t=-2}^{2}tX_t=a_0\sum t+a_1\sum t^2+a_2\sum t^3+a_3\sum t^4 \qquad (ii)
$$

$$
\sum_{t=-2}^{2}t^2X_t=a_0\sum t^2+a_1\sum t^3+a_2\sum t^4+a_3\sum t^5 \qquad (iii)
$$

$$
\sum_{t=-2}^{2}t^3X_t=a_0\sum t^3+a_1\sum t^4+a_2\sum t^5+a_3\sum t^6 \qquad (iv)
$$

where every summation on $t$ runs from $-2$ to $2$.

2.3 Symmetry of the Power Sums

Because the index set ${-2,-1,0,1,2}$ is symmetric about zero, every odd power sum vanishes:

$$
\sum t=0,\qquad \sum t^3=0,\qquad \sum t^5=0
$$

The even power sums are

$$
\sum t^2=10,\qquad \sum t^4=34,\qquad \sum t^6=130
$$

and the count of points is $5$. Substituting these values into the normal equations eliminates the odd-order coefficients from equations $(i)$ and $(iii)$, leaving the reduced pair

$$
\sum X_t=5a_0+10a_2
$$

$$
\sum t^2X_t=10a_0+34a_2
$$

This reduction is the key algebraic step: the symmetry of the index set decouples the even and odd coefficients, so that $a_0$ and $a_2$ may be found from two equations alone.

2.4 Solving for the Central Value

Solving the reduced pair simultaneously for $a_0$ gives

$$
a_0=\frac{1}{35}\left(12\sum_{t=-2}^{2}t^2X_t-17\sum_{t=-2}^{2}X_t\right)
$$

Expanding the two sums over the five points and collecting coefficients of each observation yields the weight form

$$
a_0=\frac{1}{35}\left(-3X_{-2}+12X_{-1}+17X_0+12X_1-3X_2\right)
$$

Since the trend value at the centre is $a_0$, this expression is the weighted moving average of extent five appropriate to a cubic polynomial.

> Definition (Weights of a Weighted Moving Average). The weights of a weighted moving average of extent $n$ are the fixed coefficients $w_{-m},\dots,w_0,\dots,w_m$, where $n=2m+1$, such that the trend value at the centre of each block is

$$
> \hat{T}0=\sumw_jX_j}^{m
$$

> with $\sum_{j=-m}^{m}w_j=1$ and $w_{-j}=w_j$ for every $j$.

For the cubic fit to five points the weights are

$$
w_{-2}=-\frac{3}{35},\quad w_{-1}=\frac{12}{35},\quad w_0=\frac{17}{35},\quad w_1=\frac{12}{35},\quad w_2=-\frac{3}{35}
$$

These numbers are symmetric about the centre and sum to unity:

$$
-\frac{3}{35}+\frac{12}{35}+\frac{17}{35}+\frac{12}{35}-\frac{3}{35}=\frac{35}{35}=1
$$

2.5 Alignment of Trend Values with Time

The first trend value obtainable from a block of five points is located at the middle of the first five observations, that is at $t=3$ when the series is indexed $t=1,2,\dots,12$. The second trend value is located at $t=4$, and so on. In general, a weighted moving average of extent $n=2m+1$ yields trend values for the time points $t=m+1,\,m+2,\dots,N-m$, where $N$ is the total number of observations. The first $m$ and the last $m$ observations therefore receive no trend value from this procedure.

2.6 Properties of the Weights

The weights derived above possess three structural properties that follow directly from their construction and serve as checks on any derived set.

• Symmetry. $w_{-j}=w_j$ for all $j$. This follows from the symmetry of the index set about the centre.
• Unit sum. $\sum_j w_j=1$. This follows because a constant series must be reproduced exactly by the local fit, so the fitted central value of a constant series equals that constant.
• Zero sum of weighted odd powers. The weights annihilate linear and cubic components in the sense that the local fit reproduces polynomials of degree at most three exactly at the centre.