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 \(t=2,3, \ldots, n, \bar{x}_{t}=\alpha x_{t}+(1-\alpha) \bar{x}_{t-1} .\) c. Substitute \(\bar{x}_{r-1}=\alpha x_{r-1}+(1-\alpha) \bar{x}_{r-2}\) on the right-hand side of the expression for \(\bar{x}_{n}\) then substitute \(\bar{x}_{t-2}\) in terms of \(x_{1-2}\) and \(\bar{x}_{1-3}\), and so on. On how many of the values \(x_{r}, x_{t-1}, \ldots, x_{1}\) does \(\bar{x}_{1}\) 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. [Note: A relevant reference is the article "Simple Statistics for Interpreting Environmental Data," Water Pollution Control Fed. J., 1981: 167-175.] 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 \(\bar{x}_{t}\) 's using \(\alpha=.1\). Repeat using \(\alpha=.5\). Which value of \(\alpha\) gives a smoother \(\bar{x}_{t}\) series?

Step by step solution

01

Understand the Formula

The exponential smoothing formula is described as follows: For the initial time, \( \bar{x}_{1} = x_{1} \) For subsequent times, \( \bar{x}_{t} = \alpha x_{t} + (1-\alpha) \bar{x}_{t-1} \) where \( 0 < \alpha < 1 \). This formula helps smooth out the time series data by giving more weight to recent observations.

02

Decomposition of \( \bar{x}_{t} \)

Substitute \( \bar{x}_{r-1} = \alpha x_{r-1} + (1-\alpha) \bar{x}_{r-2} \) into \( \bar{x}_{t} \): \[ \bar{x}_{t} = \alpha x_{t} + (1-\alpha)(\alpha x_{t-1} + (1-\alpha)\bar{x}_{t-2}) \] Simplifying further, we continue substituting backward to express \( \bar{x}_{t} \) in terms of the initial value \( \bar{x}_{1} \), until all terms involve \( x\) values and weights of \((1-\alpha)^k \). As \( k \) increases, coefficients of \( x_{t-k} \) decrease exponentially due to \((1-\alpha)^k \).

03

Sensitivity of \( \bar{x}_{t} \) to the Initial Value

With large \( t \), the terms involving \( x_{1} \) become less significant since the coefficient \((1-\alpha)^t\) becomes very small. Therefore, \( \bar{x}_{t} \) becomes less sensitive to the initial condition as \( t \) grows.

04

Time Series Plot

Plot the given temperatures against the days to see if there's any apparent pattern. This plot helps visualize fluctuations or trends over time. Given the data \[ 47,54,53,50,46,46,47,50,51,50,46,52,50,50 \], plot each \( x_{t} \) against its respective \( t \). Any cycles, trends or random behaviors should be assessed through this plot.

05

Calculate Smoothed Values \( \bar{x}_{t} \) for \( \alpha = 0.1 \)

Start with \( \bar{x}_{1} = x_{1} = 47 \). Use \( \alpha = 0.1 \) to calculate each subsequent \( \bar{x}_{t} \):- \( \bar{x}_{2} = 0.1 \times 54 + 0.9 \times 47 = 47.7 \)- Continue the process up to \( \bar{x}_{14} \). This process gradually smoothens the data.

06

Calculate Smoothed Values \( \bar{x}_{t} \) for \( \alpha = 0.5 \)

Again, start with \( \bar{x}_{1} = x_{1} = 47 \). Use \( \alpha = 0.5 \):- \( \bar{x}_{2} = 0.5 \times 54 + 0.5 \times 47 = 50.5 \)- Repeat up to \( \bar{x}_{14} \). This approach gives different smoothing due to the greater weight on current values.

07

Compare Smoothing Effects

Compare the smoothing results from \( \alpha = 0.1 \) and \( \alpha = 0.5 \). The smaller \( \alpha \) often creates a smoother curve as it places less weight on the most recent value and more on the "lagged" data. Larger \( \alpha \) is more sensitive to recent changes, resulting in less smooth series.

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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 statistical technique used to analyze a sequence of data points collected or recorded at specific time intervals. The goal is to identify trends, patterns, and other characteristics within the data.
In many practical situations, such as weather prediction or stock market analysis, time series help us understand past patterns to predict future behavior.

• Components: Generally, a time series consists of several components, such as trend (long-term direction), seasonal (systematic variations), and noise (random variations).
• Application: Time series can help in various fields like finance, economics, and environmental science.

Studying time series involves recognizing these components to make informed decisions and forecasts.

Smoothing Techniques

Smoothing techniques are methods used to remove noise from data, making it easier to observe underlying trends and patterns. These techniques are crucial in time series analysis as they help in reducing the effect of random fluctuations.
Different types of smoothing techniques include:

• Moving Average: Averages data at different time intervals to smooth out short-term fluctuations.
• Weighted Moving Average: Similar to moving average, but assigns different weights to each data point.
• Exponential Smoothing: Provides more weight to recent observations, decreasing the weights exponentially for older data points.

The choice of technique depends on the specific characteristics of the data and the objectives of the analysis.

Smoothing Constant Alpha

The smoothing constant, denoted as \( \alpha \), is a crucial parameter in exponential smoothing methods. It determines the weight given to the most recent observation compared to past smoothed values.

• Range: \( \alpha \) ranges from 0 to 1, where a value close to 1 means more emphasis on recent data, while a value close to 0 gives more weight to past values.
• Impact: A smaller \( \alpha \) results in a smoother series, because it averages out the random noise over a longer period. Conversely, a larger \( \alpha \) makes the series sensitive to recent changes, reflecting them quickly in the smoothed series.
• Selection: Choosing the correct \( \alpha \) can significantly affect the outcome of the analysis and is often selected based on the objective of the smoothing and the desired level of responsiveness.

Therefore, \( \alpha \) is key in balancing the trade-off between smoothing the data and reacting to latest changes.

Time Series Plot

A time series plot is a graphical representation of data points in a time series set against time. It helps visually assess any trends, patterns, seasonality, or anomalies.
The x-axis represents time, while the y-axis represents the observed values. This kind of plot is essential for:

• Trend Identification: Observing increasing or decreasing trends over time.
• Seasonality Detection: Identifying periodic patterns that repeat over known intervals.
• Spotting Outliers: Recognizing any anomalies or unusual events in the data set.

Creating a time series plot is usually the first step in time series analysis, as it gives a quick overview of the data's natural behavior.

Statistical Methods

Statistical methods are vital in time series analysis to make sense of the data. They provide frameworks for drawing conclusions and making predictions.
Tools and techniques applied in analyzing time series include:

• Autoregressive Integrated Moving Average (ARIMA): A sophisticated method that combines autoregression and moving averages to forecast future points.
• Decomposition: Breaking down a series into trend, seasonal, and noise components to analyze each separately.
• Covariance and Correlation: Measures to assess the relationship between different time series.

Statistical methods not only offer ways to model data but also validate the assumptions made about the data, ensuring robust conclusions.