91Ó°ÊÓ

The sample data \(x_{1}, x_{2}, \ldots, x_{n}\) sometimes represents a time series, where \(x_{t}=\) the observed value of a response variable \(x\) at time \(t\). Often the observed series shows a great deal of random variation, which makes it difficult to study longerterm behavior. In such situations, it is desirable to produce a smoothed version of the series. One technique for doing so involves exponential smoothing. The value of a smoothing constant \(\alpha\) is chosen \((0<\alpha<1)\). Then with \(\bar{x}_{t}=\) smoothed value at time \(t\), we set \(\bar{x}_{1}=x_{1}\), and for \(t=2\), \(3, \ldots, n, \bar{x}_{t}=\alpha x_{t}+(1-\alpha) \bar{x}_{t-1}\). a. Consider the following time series in which \(x_{t}=\) temperature \(\left({ }^{\circ} \mathrm{F}\right)\) of effluent at a sewage treatment plant on day \(t: 47,54,53,50,46,46,47,50,51,50,46,52\), 50,50 . Plot each \(x_{t}\) against \(t\) on a two-dimensional coordinate system (a time-series plot). Does there appear to be any pattern? b. Calculate the \(\bar{x}_{t} \mathrm{~s}\) using \(\alpha=.1\). Repeat using \(\alpha=.5\). Which value of \(\alpha\) gives a smoother \(\bar{x}_{t}\) series? c. Substitute \(\bar{x}_{t-1}=\alpha x_{t-1}+(1-\alpha) \bar{x}_{t-2}\) on the right-hand side of the expression for \(\bar{x}_{t}\), then substitute \(\bar{x}_{t-2}\) in terms of \(x_{t-2}\) and \(\bar{x}_{t-3}\), and so on. On how many of the values \(x_{t}, x_{t-1}, \ldots, x_{1}\) does \(\bar{x}_{t}\) depend? What happens to the coefficient on \(x_{t-k}\) as \(k\) increases? d. Refer to part (c). If \(t\) is large, how sensitive is \(\bar{x}_{t}\) to the initialization \(\bar{x}_{1}=x_{1}\) ? Explain.

Step by step solution

01

Understanding the Problem

We are given a time series representing temperature observations over different days and asked to conduct exponential smoothing with two different values of \( \alpha \). Additionally, we are to explore theoretical questions about the characteristics of exponential smoothing.

02

Recognize Parts of the Time Series

Part (a) of the problem involves plotting the given temperatures against the corresponding days. These values are: 47, 54, 53, 50, 46, 46, 47, 50, 51, 50, 46, 52, 50, and 50. By plotting these points on a time-series plot, we can visually inspect for any patterns such as trends or cycles.

03

Calculate Smoothed Values for \( \alpha = 0.1 \)

To perform exponential smoothing with \( \alpha = 0.1 \), first set \( \bar{x}_1 = 47 \). Each subsequent \( \bar{x}_t \) can be calculated with the formula \( \bar{x}_t = 0.1 x_t + 0.9 \bar{x}_{t-1} \). Use this iterative formula to calculate all \( \bar{x}_t \) values up to day 14.

04

Calculate Smoothed Values for \( \alpha = 0.5 \)

Repeat the process with \( \alpha = 0.5 \). Set \( \bar{x}_1 = 47 \) again, and use \( \bar{x}_t = 0.5 x_t + 0.5 \bar{x}_{t-1} \) to calculate the smoothed values \( \bar{x}_t \). Observe which \( \alpha \) value generates a smoother series.

05

Substitute Smoothed Equation to Uncover Dependence

For part (c), substitute \( \bar{x}_{t-1} = \alpha x_{t-1} + (1-\alpha) \bar{x}_{t-2} \) into \( \bar{x}_t \) and continue substituting \( \bar{x}_{t-k} \) in terms of previous \( x_{t-k} \) and \( \bar{x}_{t-k-1} \). This shows \( \bar{x}_t \) depends on all past \( x \) values.

06

Understand Coefficient Behavior as \( k \) Increases

Observe that the coefficient on \( x_{t-k} \) is \( \alpha (1 - \alpha)^k \). As \( k \) increases, \( (1 - \alpha)^k \) approaches zero, making the influence of earlier \( x \) values diminish exponentially, especially if \( \alpha \) is not close to zero.

07

Sensitivity to Initialization

In part (d), for large \( t \), the influence of the initial value \( \bar{x}_1 = x_1 \) diminishes because the weighting from past data exponentially decays. Thus, \( \bar{x}_t \) becomes less sensitive to how \( \bar{x}_1 \) was initialized, more accurately reflecting recent data trends.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Time Series Analysis

Time series analysis is a vital statistical approach that deals with data points collected or recorded at time intervals. It helps us understand underlying patterns in the data over time. In the given exercise, temperatures from a sewage treatment plant are observed over several days, forming a time series. By plotting these temperatures against the corresponding days, we can visually assess whether any trends, cycles, or random variations exist.

Time series analysis can often reveal seasonal trends or cyclical patterns, which are essential in forecasting future values or understanding the past behaviors of the data. However, sometimes such data may show random variations, making it challenging to detect significant trends or patterns. This leads us to use techniques like exponential smoothing to create a smoother dataset, reducing noise and highlighting the actual trend.

Ultimately, studying time series data helps in various fields like economics, financial markets, and environmental studies, providing insights for decision-making and predicting future trends.

Smoothing Constant

In exponential smoothing methods, the smoothing constant, represented by \( \alpha \), is a crucial parameter that defines the level of smoothing applied to the time series data. Ranging between 0 and 1, it determines the weight given to the most recent observation compared to the past observations.

A smaller \( \alpha \) means that more weight is given to historical data, producing a smoother series with less sensitivity to recent fluctuations. Contrarily, a larger \( \alpha \) gives more weight to the latest data point, which can reflect recent changes more quickly but may also introduce more variability into the smoothed data.

In the exercise, calculations with \( \alpha = 0.1 \) and \( \alpha = 0.5 \) illustrate how different values influence the smoothness of the resultant series. A smaller \( \alpha \) yields a smoother curve that better absorbs the shock of random errors, while a larger \( \alpha \) is faster in responding to changes in the data.

Dependency on Past Values

The dependency on past values in exponential smoothing is pivotal because each smoothed value \( \bar{x}_t \) builds upon a combination of the current observed value \( x_t \) and the previously smoothed value \( \bar{x}_{t-1} \). The formula \( \bar{x}_t = \alpha x_t + (1 - \alpha) \bar{x}_{t-1} \) demonstrates this relationship.

To delve deeper, by recursively substituting smoothed values back into this equation, \( \bar{x}_t \) can be expressed as dependent on all previous \( x \) values. This expansion allows us to see the proportion of each past observation's contribution to the current smoothed value. The coefficient on each past observation declines exponentially, indicated by the factor \( (1 - \alpha)^{k} \), where \( k \) represents the time lag from the current point.

This structure implies that older observations contribute less to the smoothed value, with their influence reducing quickly as time progresses. This decrement is especially true when \( \alpha \) is not close to zero, reflecting a balance between honoring historical data and emphasizing recent observations.

Initialization Sensitivity

Initialization sensitivity refers to how the initial smoothed value \( \bar{x}_1 \) affects the subsequent smoothed values in exponential smoothing. In practice, \( \bar{x}_1 \) is often set equal to the first observation \( x_1 \).

As the time series progresses, the impact of this initial setting diminishes due to the nature of the smoothing process. Each subsequent \( \bar{x}_t \) becomes progressively less dependent on \( \bar{x}_1 \) as older observations' contributions fade, particularly when the smoothing constant \( \alpha \) allows a stronger focus on more recent data.

Thus, for larger \( t \), the initialization has a negligible influence on \( \bar{x}_t \). This characteristic enhances the reliability of the method, ensuring that \( \bar{x}_t \) more accurately reflects current patterns and reduces concern over the arbitrary choice of \( \bar{x}_1 \). This diminishing influence assures robust forecasts and trend analysis in long datasets, providing confidence in the method's stability over time.