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This general pattern is well illustrated in a "classic" Series G data set (Box and to identify trend components in the time series data; however, as long as the trend be visually identified in the series as a pattern that repeats every k elements. . Two autoregressive (p) parameters: ACF - a sine-wave shape pattern or a set.

**Table of contents**

- General Introduction
- Persistence of Storms and Calms
- How To Identify Patterns in Time Series Data: Time Series Analysis
- Global Wave Statistics Online
- Wave Parameters: Wavelength, Amplitude, Period, Frequency & Speed

Again such a cyclical component can effect sales in an additive manner or multiplicative manner. First a moving average is computed for the series, with the moving average window width equal to the length of one season. If the length of the season is even, then the user can choose to use either equal weights for the moving average or unequal weights can be used, where the first and last observation in the moving average window are averaged. In the moving average series, all seasonal within-season variability will be eliminated; thus, the differences in additive models or ratios in multiplicative models of the observed and smoothed series will isolate the seasonal component plus irregular component.

Specifically, the moving average is subtracted from the observed series for additive models or the observed series is divided by the moving average values for multiplicative models.

### General Introduction

The seasonal component is then computed as the average for additive models or medial average for multiplicative models for each point in the season. The medial average of a set of values is the mean after the smallest and largest values are excluded. The resulting values represent the average seasonal component of the series. The original series can be adjusted by subtracting from it additive models or dividing it by multiplicative models the seasonal component.

The resulting series is the seasonally adjusted series i. Remember that the cyclical component is different from the seasonal component in that it is usually longer than one season, and different cycles can be of different lengths. The combined trend and cyclical component can be approximated by applying to the seasonally adjusted series a 5 point centered weighed moving average smoothing transformation with the weights of 1, 2, 3, 2, 1.

Random or irregular component. Finally, the random or irregular error component can be isolated by subtracting from the seasonally adjusted series additive models or dividing the adjusted series by multiplicative models the trend-cycle component. The general ideas of seasonal decomposition and adjustment are discussed in the context of the Census I seasonal adjustment method Seasonal Decomposition Census I.

The Census method II 2 is an extension and refinement of the simple adjustment method. Subsequently, the term X has become synonymous with this refined version of the Census method II. In addition to the documentation that can be obtained from the Census Bureau, a detailed summary of this method is also provided in Makridakis, Wheelwright, and McGee and Makridakis and Wheelwright If you plot those data, it is apparent that 1 there appears to be an upwards linear trend in the passenger loads over the years, and 2 there is a recurring pattern or seasonality within each year i.

The purpose of seasonal decomposition and adjustment is to isolate those components, that is, to de-compose the series into the trend effect, seasonal effects, and remaining variability. The "classic" technique designed to accomplish this decomposition was developed in the 's and is also known as the Census I method see the Census I overview section.

This technique is also described and discussed in detail in Makridakis, Wheelwright, and McGee , and Makridakis and Wheelwright The difference between a cyclical and a seasonal component is that the latter occurs at regular seasonal intervals, while cyclical factors usually have a longer duration that varies from cycle to cycle. The trend and cyclical components are customarily combined into a trend-cycle component TC t.

X t represents the observed value of the time series at time t. Consider the difference between an additive and multiplicative seasonal component in an example: Thus, you could add to your forecasts for every December the amount of 3 million to account for this seasonal fluctuation. The previous example can be extended to illustrate the additive and multiplicative trend-cycle components. In terms of the toy example, a "fashion" trend may produce a steady increase in sales e. In addition, cyclical components may impact sales. To reiterate, a cyclical component is different from a seasonal component in that it usually is of longer duration, and that it occurs at irregular intervals.

The basic method for seasonal decomposition and adjustment outlined in the Basic Ideas and Terms topic can be refined in several ways. In fact, unlike many other time-series modeling techniques e. Some of the major refinements are listed below. Different months have different numbers of days, and different numbers of trading-days i. When analyzing, for example, monthly revenue figures for an amusement park, the fluctuation in the different numbers of Saturdays and Sundays peak days in the different months will surely contribute significantly to the variability in monthly revenues.

The X variant of the Census II method allows the user to test whether such trading-day variability exists in the series, and, if so, to adjust the series accordingly. Most real-world time series contain outliers, that is, extreme fluctuations due to rare events. For example, a strike may affect production in a particular month of one year.

Such extreme outliers may bias the estimates of the seasonal and trend components. The X procedure includes provisions to deal with extreme values through the use of "statistical control principles," that is, values that are above or below a certain range expressed in terms of multiples of sigma , the standard deviation can be modified or dropped before final estimates for the seasonality are computed.

The refinement for outliers, extreme values, and different numbers of trading-days can be applied more than once, in order to obtain successively improved estimates of the components. The X method applies a series of successive refinements of the estimates to arrive at the final trend-cycle, seasonal, and irregular components, and the seasonally adjusted series. Tests and summary statistics.

### Persistence of Storms and Calms

In addition to estimating the major components of the series, various summary statistics can be computed. For example, analysis of variance tables can be prepared to test the significance of seasonal variability and trading-day variability see above in the series; the X procedure will also compute the percentage change from month to month in the random and trend-cycle components. As the duration or span in terms of months or quarters for quarterly X increases, the change in the trend-cycle component will likely also increase, while the change in the random component should remain about the same.

The width of the average span at which the changes in the random component are about equal to the changes in the trend-cycle component is called the month quarter for cyclical dominance , or MCD QCD for short. These and various other results are discussed in greater detail below. The computations performed by the X procedure are best discussed in the context of the results tables that are reported.

The adjustment process is divided into seven major steps, which are customarily labeled with consecutive letters A through G. Prior adjustment monthly seasonal adjustment only. Before any seasonal adjustment is performed on the monthly time series, various prior user- defined adjustments can be incorporated. The user can specify a second series that contains prior adjustment factors; the values in that series will either be subtracted additive model from the original series, or the original series will be divided by these values multiplicative model.

For multiplicative models, user-specified trading-day adjustment weights can also be specified. These weights will be used to adjust the monthly observations depending on the number of respective trading-days represented by the observation. Preliminary estimation of trading-day variation monthly X and weights. Next, preliminary trading-day adjustment factors monthly X only and weights for reducing the effect of extreme observations are computed.

Final estimation of trading-day variation and irregular weights monthly X- The adjustments and weights computed in B above are then used to derive improved trend-cycle and seasonal estimates. These improved estimates are used to compute the final trading-day factors monthly X only and weights.

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Final estimation of seasonal factors, trend-cycle, irregular, and seasonally adjusted series. The final trading-day factors and weights computed in C above are used to compute the final estimates of the components. Modified original, seasonally adjusted, and irregular series. The original and final seasonally adjusted series, and the irregular component are modified for extremes. The resulting modified series allow the user to examine the stability of the seasonal adjustment. In this part of the computations, various summary measures see below are computed to allow the user to examine the relative importance of the different components, the average fluctuation from month-to-month quarter-to-quarter , the average number of consecutive changes in the same direction average number of runs , etc.

Finally, you will compute various charts graphs to summarize the results. For example, the final seasonally adjusted series will be plotted, in chronological order, or by month see below. Customarily, these tables are numbered, and also identified by a letter to indicate the respective part of the analysis. For example, table B 11 shows the initial seasonally adjusted series; C 11 is the refined seasonally adjusted series, and D 11 is the final seasonally adjusted series. Shown below is a list of all available tables. Also, for quarterly adjustment, some of the computations outlined below are slightly different; for example instead of a term [monthly] moving average, a 4-term [quarterly] moving average is applied to compute the seasonal factors; the initial trend-cycle estimate is computed via a centered 4-term moving average, the final trend-cycle estimate in each part is computed by a 5-term Henderson average.

Following the convention of the Bureau of the Census version of the X method, three levels of printout detail are offered: Standard 17 to 27 tables , Long 27 to 39 tables , and Full 44 to 59 tables. For the charts, two levels of detail are available: Distributed lags analysis is a specialized technique for examining the relationships between variables that involve some delay.

For example, suppose that you are a manufacturer of computer software, and you want to determine the relationship between the number of inquiries that are received, and the number of orders that are placed by your customers. You could record those numbers monthly for a one-year period, and then correlate the two variables. Put another way, there will be a time lagged correlation between the number of inquiries and the number of orders that are received. Time-lagged correlations are particularly common in econometrics.

For example, the benefits of investments in new machinery usually only become evident after some time. Higher income will change people's choice of rental apartments, however, this relationship will be lagged because it will take some time for people to terminate their current leases, find new apartments, and move. In general, the relationship between capital appropriations and capital expenditures will be lagged, because it will require some time before investment decisions are actually acted upon.

In all of these cases, we have an independent or explanatory variable that affects the dependent variables with some lag. The distributed lags method allows you to investigate those lags. Detailed discussions of distributed lags correlation can be found in most econometrics textbooks, for example, in Judge, Griffith, Hill, Luetkepohl, and Lee , Maddala , and Fomby, Hill, and Johnson In the following paragraphs we will present a brief description of these methods.

We will assume that you are familiar with the concept of correlation see Basic Statistics , and the basic ideas of multiple regression see Multiple Regression. Suppose we have a dependent variable y and an independent or explanatory variable x which are both measured repeatedly over time. In some textbooks, the dependent variable is also referred to as the endogenous variable, and the independent or explanatory variable the exogenous variable.

The simplest way to describe the relationship between the two would be in a simple linear relationship:. In this equation, the value of the dependent variable at time t is expressed as a linear function of x measured at times t , t-1 , t-2 , etc. Thus, the dependent variable is a linear function of x , and x is lagged by 1, 2, etc. The beta weights i can be considered slope parameters in this equation. If the weights for the lagged time periods are statistically significant, we can conclude that the y variable is predicted or explained with the respective lag.

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## How To Identify Patterns in Time Series Data: Time Series Analysis

A common problem that often arises when computing the weights for the multiple linear regression model shown above is that the values of adjacent in time values in the x variable are highly correlated. In extreme cases, their independent contributions to the prediction of y may become so redundant that the correlation matrix of measures can no longer be inverted, and thus, the beta weights cannot be computed.

In less extreme cases, the computation of the beta weights and their standard errors can become very imprecise, due to round-off error. In the context of Multiple Regression this general computational problem is discussed as the multicollinearity or matrix ill-conditioning issue. Almon proposed a procedure that will reduce the multicollinearity in this case. Specifically, suppose we express each weight in the linear regression equation in the following manner:.

Almon could show that in many cases it is easier i. Note that with this method, the precision of the beta weight estimates is dependent on the degree or order of the polynomial approximation. A general problem with this technique is that, of course, the lag length and correct polynomial degree are not known a priori. The effects of misspecifications of these parameters are potentially serious in terms of biased estimation. This issue is discussed in greater detail in Frost , Schmidt and Waud , Schmidt and Sickles , and Trivedi and Pagan Spectrum analysis is concerned with the exploration of cyclical patterns of data.

The purpose of the analysis is to decompose a complex time series with cyclical components into a few underlying sinusoidal sine and cosine functions of particular wavelengths. The term "spectrum" provides an appropriate metaphor for the nature of this analysis: Suppose you study a beam of white sun light, which at first looks like a random white noise accumulation of light of different wavelengths. However, when put through a prism, we can separate the different wave lengths or cyclical components that make up white sun light.

In fact, via this technique we can now identify and distinguish between different sources of light. Thus, by identifying the important underlying cyclical components, we have learned something about the phenomenon of interest. In essence, performing spectrum analysis on a time series is like putting the series through a prism in order to identify the wave lengths and importance of underlying cyclical components.

A much cited example for spectrum analysis is the cyclical nature of sun spot activity e. It turns out that sun spot activity varies over 11 year cycles.

Other examples of celestial phenomena, weather patterns, fluctuations in commodity prices, economic activity, etc. To contrast this technique with ARIMA or Exponential Smoothing , the purpose of spectrum analysis is to identify the seasonal fluctuations of different lengths, while in the former types of analysis, the length of the seasonal component is usually known or guessed a priori and then included in some theoretical model of moving averages or autocorrelations.

The classic text on spectrum analysis is Bloomfield ; however, other detailed discussions can be found in Jenkins and Watts , Brillinger , Brigham , Elliott and Rao , Priestley , Shumway , or Wei For more information, see Time Series Analysis - Index and the following topics:. Cross-spectrum analysis is an extension of Single Spectrum Fourier Analysis to the simultaneous analysis of two series.

In the following paragraphs, we will assume that you have already read the introduction to single spectrum analysis. Detailed discussions of this technique can be found in Bloomfield , Jenkins and Watts , Brillinger , Brigham , Elliott and Rao , Priestley , Shumway , or Wei Strong periodicity in the series at the respective frequency. The purpose of cross-spectrum analysis is to uncover the correlations between two series at different frequencies. For example, sun spot activity may be related to weather phenomena here on earth.

If so, then if we were to record those phenomena e. That is, we may find a periodicity in the weather data that is "in-sync" with the sun spot cycles. We can easily think of other areas of research where such knowledge could be very useful; for example, various economic indicators may show similar correlated cyclical behavior; various physiological measures likely will also display "coordinated" i. At first sight it is not easy to see the relationship between the two series.

However, as shown below the series were created so that they would contain two strong correlated periodicities. Shown below are parts of the summary from the cross-spectrum analysis the spectral estimates were smoothed with a Parzen window of width 3. The complete summary contains all spectrum statistics computed for each variable, as described in the Single Spectrum Fourier Analysis overview section. Looking at the results shown above, it is clear that both variables show strong periodicities at the frequencies.

Analogous to the results for the single variables, the complete summary will also display periodogram values for the cross periodogram. However, the cross-spectrum consists of complex numbers that can be divided into a real and an imaginary part. These can be smoothed to obtain the cross-density and quadrature density quad density for short estimates, respectively. The reasons for smoothing, and the different common weight functions for smoothing are discussed in the Single Spectrum Fourier Analysis. The square root of the sum of the squared cross-density and quad-density values is called the cross- amplitude.

The cross-amplitude can be interpreted as a measure of covariance between the respective frequency components in the two series. Thus we can conclude from the results shown in the table above that the. You can standardize the cross-amplitude values by squaring them and dividing by the product of the spectrum density estimates for each series. The result is called the squared coherency , which can be interpreted similar to the squared correlation coefficient see Correlations - Overview , that is, the coherency value is the squared correlation between the cyclical components in the two series at the respective frequency.

However, the coherency values should not be interpreted by themselves; for example, when the spectral density estimates in both series are very small, large coherency values may result the divisor in the computation of the coherency values will be very small , even though there are no strong cyclical components in either series at the respective frequencies. The gain value is computed by dividing the cross-amplitude value by the spectrum density estimates for one of the two series in the analysis. Consequently, two gain values are computed, which can be interpreted as the standard least squares regression coefficients for the respective frequencies.

## Global Wave Statistics Online

The phase shift estimates usually denoted by the Greek letter are measures of the extent to which each frequency component of one series leads the other. Now, let's return to the example data set presented above. In fact, the two series were created as:. Indeed, the analysis presented in this overview reproduced the periodicity "inserted" into the data very well. The "wave length" of a sine or cosine function is typically expressed in terms of the number of cycles per unit time Frequency , often denoted by the Greek letter nu ; some textbooks also use f.

For example, the number of letters handled in a post office may show 12 cycles per year: On the first of every month a large amount of mail is sent many bills come due on the first of the month , then the amount of mail decreases in the middle of the month, then it increases again towards the end of the month.

Therefore, every month the fluctuation in the amount of mail handled by the post office will go through a full cycle. Thus, if the unit of analysis is one year, then n would be equal to 12, as there would be 12 cycles per year. Of course, there will likely be other cycles with different frequencies. The period T of a sine or cosine function is defined as the length of time required for one full cycle. Thus, it is the reciprocal of the frequency, or: Put into words, there is a period in the series of length 0.

As mentioned before, the purpose of spectrum analysis is to decompose the original series into underlying sine and cosine functions of different frequencies, in order to determine those that appear particularly strong or important. One way to do so would be to cast the issue as a linear Multiple Regression problem, where the dependent variable is the observed time series, and the independent variables are the sine functions of all possible discrete frequencies.

Following the common notation from classical harmonic analysis, in this equation lambda is the frequency expressed in terms of radians per unit time, that is: What is important here is to recognize that the computational problem of fitting sine and cosine functions of different lengths to the data can be considered in terms of multiple linear regression. Note that the cosine parameters a k and sine parameters b k are regression coefficients that tell us the degree to which the respective functions are correlated with the data.

Overall there are q different sine and cosine functions; intuitively as also discussed in Multiple Regression , it should be clear that we cannot have more sine and cosine functions than there are data points in the series. In other words, there will be as many different sinusoidal waves as there are data points, and we will be able to completely reproduce the series from the underlying functions.

Note that if the number of cases in the series is odd, then the last data point will usually be ignored; in order for a sinusoidal function to be identified, you need at least two points: To summarize, spectrum analysis will identify the correlation of sine and cosine functions of different frequency with the observed data.

Complex numbers real and imaginary numbers. In many textbooks on spectrum analysis, the structural model shown above is presented in terms of complex numbers, that is, the parameter estimation process is described in terms of the Fourier transform of a series into real and imaginary parts. Complex numbers are the superset that includes all real and imaginary numbers. Imaginary numbers, by definition, are numbers that are multiplied by the constant i , where i is defined as the square root of Obviously, the square root of -1 does not exist, hence the term imaginary number; however, meaningful arithmetic operations on imaginary numbers can still be performed e.

It is useful to think of real and imaginary numbers as forming a two dimensional plane, where the horizontal or X -axis represents all real numbers, and the vertical or Y -axis represents all imaginary numbers. Complex numbers can then be represented as points in the two- dimensional plane. In fact, in this manner the mathematical discussion and required computations are often more elegant and easier to perform; which is why many textbooks prefer the presentation of spectrum analysis in terms of complex numbers.

Shumway presents a simple example to clarify the underlying "mechanics" of spectrum analysis. Let's create a series with 16 cases following the equation shown above, and then see how we may "extract" the information that was put in it. First, create a variable and define it as:. This variable is made up of two underlying periodicities: The cosine coefficient 1.

The spectrum analysis summary is shown below. The sine and cosine functions are mutually independent or orthogonal ; thus we may sum the squared coefficients for each frequency to obtain the periodogram. Specifically, the periodogram values above are computed as:. The periodogram values can be interpreted in terms of variance sums of squares of the data at the respective frequency or period. Customarily, the periodogram values are plotted against the frequencies or periods. In the example above, a sine function with a frequency of 0.

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## Wave Parameters: Wavelength, Amplitude, Period, Frequency & Speed

The speed of a wave v is related to the frequency , wave period , and wavelength by the following simple equations:. For example, the musical note "A" is a sound wave with a frequency of Hz. The wavelength of the wave is What is the speed of the sound wave? To determine the speed of the wave, we can use equation 3 and substitute the given values for wavelength and frequency , making sure we are using the standard units. Interestingly, the speed of sound in air depends on temperature and pressure.

A musician who plays a wind instrument, such as a trumpet, could tune her trumpet at the base of a mountain, hike up the mountain to where the air pressure is lower, and find that her trumpet is no longer in tune. Similarly, a change in air temperature could also change the tuning of the instrument. As the example above illustrates, waves are all around us in everyday life. The Ancient Greeks began their study of waves by thinking about music, but now almost every branch of physics involves waves in one way or another.

Waves have been of interest to philosophers and scientists alike for thousands of years. This module introduces the history of wave theory and offers basic explanations of longitudinal and transverse waves. Wave periods are described in terms of amplitude and length. Wave motion and the concepts of wave speed and frequency are also explored. The study of waves dates back to the ancient Greeks who observed how the vibrating strings of musical instruments would generate sound.

While there are two fundamental types of waves - longitudinal and transverse - waves can take many forms e. Waves can be described by their exhibited properties: Bookmark Glossary Terms On July 17, , three huge waves — "tsunamis" — up to 15 meters high struck the north coast of Papua New Guinea, killing at least 2, people. Comprehension Checkpoint Our understanding of wave motion began with the study of a. Illustration of a longitudinal wave. Illustration of a transverse wave. Comprehension Checkpoint The height of a wave is know as its a. Comprehension Checkpoint A frequency of 0.