Time Series Analysis of Wind Speed Using VAR and the Generalized Impulse Response Technique Bradley T. Ewing
Rawls Professor of Operations Management Area of Information Systems & Quantitative Sciences
Rawls College of Business
& Wind Science and Engineering Research Center Texas Tech University
Lubbock, TX 79409-2101 USA
[email protected] Jamie Brown Kruse
Professor of Economics and Director Center for Natural Hazard Research
East Carolina University Greenville, NC
USA John L. Schroeder Assistant Professor Department of Geosciences
& Wind Science and Engineering Research Center Texas Tech University
Lubbock, TX USA Douglas A. Smith Associate Professor Department of Civil Engineering
& Wind Science and Engineering Research Center Texas Tech University
Lubbock, TX USA
Revised March 2006
Abstract
This research examines the interdependence in time series wind speed data measured in the same location at four different heights. A multiple-equation system known as a vector autoregression is proposed for characterizing the time series dynamics of wind. Additionally, the recently developed method of generalized impulse response analysis provides insight into the cross-effects of the wind series and their responses to shocks. Findings are based on analysis of contemporaneous wind speed time histories taken at 13, 33, 70 and 160 feet above ground level with a sampling rate of 10 Hz. The results indicate that wind speeds measured at 70 feet was the most variable. Further, the turbulence persisted longer at the 70 foot measurement than at the other heights. The greatest interdependence is observed at 13 feet. Gusts at 160 feet led to the greatest persistence to an “own” shock and led to greatest persistence in the responses of the other wind series.
Time Series Analysis of Wind Speed Using VAR and the Generalized Impulse Response Technique I. Introduction
Understanding the time series dynamics of wind speed is an essential element in many types of applied research. For example, the design and construction of wind resistant structures requires the characterization of several wind processes including wind speed. Models of wind speed also play important roles in the operation of wind power plants and applied atmospheric sciences. For example, the characteristics of wind speed are important factors in the determination of the cut-in and cut-out wind speeds of wind turbines and partially govern the placement of mesoscale networks of meteorological observations stations. Wind speed models will likely become an important factor in financial markets given the growing popularity of weather derivatives and the need to manage weather-related risks, including wind risk.1 This study models wind speed measured at the same location at various heights over
a period of time. The purpose of this research is to characterize cross-variable dynamics in these wind speeds.
Time domain models for predicting wind speeds at a specific point in space exist but accounting for the effects of the surrounding wind environment is new. Simultaneous wind speed prediction at multiple points is the first step toward improving the analytical and computational techniques for predicting wind pressures or loads on ground-based structures. To date, most attempts to characterize wind speed using time series analysis have concentrated on variations of single equation models [e.g., Huang and Chalabi (1995), Sfetsos (2000, 2002), More and Deo (2003), Poggi et al. (2003), and Hussain et al. (2004)]. Studies that empirically estimate multiple-equation systems are lacking because wind speed data measured at multiple heights over a single site have not been readily available in the time domain. A multiple-equation system or vector autoregression (VAR) is capable of capturing the underlying cross-variable dynamics among wind at various heights. Thus, this paper addresses the concern of Kestens and Teugels (2002) that the quality of wind speed data and analysis might be improved by the “inclusion of measurable covariates while developing models.” (p. 821) Further, wind speed interaction at various heights has yet to be examined using the statistical time series technique called innovation accounting. This technique is a particularly fruitful and appropriate avenue of research as it captures the vertical relationships in a wind
profile in addition to interdependencies in the time domain. Wind gusts represent shocks to the system that can transmit up or down over time. This type of information should be useful to engineers who often rely on wind data gathered at a particular height while designing structures to withstand wind occurring at a variety of heights and speeds. Similarly, those involved in wind power generation and energy, and potentially other fields as well, may benefit from our results.
II. Wind Speed Data
The wind data used in this study was acquired from the Wind Engineering Research Field Laboratory (WERFL).2 The facility consists of a 30 ft x 45 ft x 13 ft high (9.14m x 13.7m x 3.96m high)
metal building and a 160 ft (48.77m) high meteorological tower that is instrumented at 5 heights. The WERFL site is in flat open terrain. The roughness length for the site (as established from field measurements) ranges between 0.002 and 0.288 ft ( 0.0006 to 0.088m) with an average of 0.044 ft (0.013m). The roughness length for the 15-minute duration record analyzed herein is 0.029 ft (0.009m).
The data used in this investigation consist of sequences of observations in the time domain. Wind speed is measured at four different heights: 13, 33, 70, and 160 feet. The wind speed time histories consist of 9000 observations within a 15-minute interval (sampling rate of 10 Hz) acquired from each height. In what follows we let Xn (n = 13, 33, 70, 160) denote wind speed at the four respective heights.
Table 1 provides descriptive statistics for the four wind speed series. Mean wind speed increases with height of measurement. Wind speed turbulence, as indicated by standard deviation, is highest for X70
followed by X33, X13, and X160, respectively. Figure 1 shows plots of the four wind speed series. A
casual review of the plots suggests that the four wind speeds series move together over time, although the series are not perfectly correlated. Certainly no definitive statement about the response of wind speed at one height to changes in wind speed at other heights should be made based on a cursory review of these plots.
III. Wind Speed in a Vector Autoregression (VAR) Framework
2 References in Zhou, Smith, and Metha (2003) provide detailed information on the collection of the
We are interested in how wind speed at one height responds to gusts or shocks in wind speed measured at other heights, as well as identifying interdependencies and cross-variable dynamics. For this purpose we estimate a multiple-equation, vector autoregressive (VAR) model and utilize the corresponding innovation accounting method known as impulse response analysis [Mills (1999)]. The VAR methodology makes minimal theoretical demands on the structure of the underlying model.3
The nonstructural VAR model is estimated on a set of (endogenously determined) variables. In our case, these endogenous variables are wind speeds measured at heights of 13, 33, 70, and 160 feet. While the VAR and impulse response analysis are particularly well-suited for dynamic analysis, the conventional impulse response method has been criticized because results are subject to the “orthogonality assumption.” If two (or more) of the error terms in the various equations contained in the VAR system are contemporaneously correlated, then the impulse responses are not robust to the ordering of the variables. In fact, the impulse responses may display noticeably different patterns [Lutkenpohl (1991)]. The conventional orthogonalized impulse response employs a Cholesky decomposition of the positive definite covariance matrix of the shocks [see Mills (1999), Enders (2004)]. This restriction forces a shock to (at least) one series to have no contemporaneous effect on some other series. Often, this imposes an unrealistic assumption, especially when there is some degree of contemporaneous interdependence among the series within the period of the shock. This is likely to be the case in a study of wind speed since there is no a priori reason to assume that wind gusts (shocks) at one height are independent of wind speed at other
heights either in the current period or in future periods. Generalized Impulse Analysis does not impose this
restriction. Recent developments in VAR and impulse response analysis have focused on responses that are not sensitive to the ordering of the variables in the VAR and thus provide more robust results. This paper employs the generalized impulse response function proposed by Pesaran and Shin (1998) and Koop, Pesaran, and Potter (1996).
The standard structure of the n-equation VAR(m) model can be written as (1) Xt = A0 + A1Xt-1 + A2Xt-2 +...+ AmXt-m + vt
where Xt is the n×1 vector of variables to be examined, A0 is a n×1 constant term or intercept vector, A1
3 In weather-related research, Campbell and Diebold (2003) forecast average daily temperatures using
heating and cooling degree days data and show that nonstructural time series models perform on par with, and sometimes better than, many structural models.
through Am are n×n coefficient matrices, and vt is a the corresponding n×1 disturbance vector. Now define
LτX
t equal to Xt-τ, for τ=1,2,…,m. Let d(L) = d0+d1L+d2L2+…+dpLp so that β(L)= 1-β1L-β2L2-…-βmLm.
Therefore, L simply denotes a polynomial in the lag operator. Thus, the four-equation VAR(m) model is
written compactly as: (1) β(L)Xt = A0 + vt
where m represents the number of lags or (vector) autoregressive terms, Xt is the 4×1 vector of wind
speeds, A0 is the constant term vector, and vt represents the corresponding disturbance vector (i.e., vit are
the shocks to wind speed measured at the four different heights). Note that current period wind speed values are expressed solely in terms of past wind speed values, as well as error terms. Further, the VAR may be estimated using ordinary least squares regression which produces unbiased and asymptotically efficient estimates since all the regressors are predetermined and provided that the error term is serially uncorrelated. The latter property may be assured by the choice of m. Unlike many “short run” forecasting models, the VAR is well-suited for determining the inter-relationships among the n-variables in the system.
Corresponding to the estimation of the VAR is the technique of generalized impulse response analysis. Generally speaking, this technique allows for the simulation of how wind speed at one height responds to a shock or wind gust at another height over a future time horizon. Unexpected changes in a
variable are called innovations or shocks in time series analysis [Harvey (1994)]. An innovation is also called an impulse, primarily to reflect the notion of a one-time shock occurring at some point in time. The value of the impulse, which is often chosen to be one standard error, is nonzero in the initial impact period (t=1) and zero elsewhere (t≠1). The dynamic nature of the relationship between wind speeds at various heights would likely lead to the impulse having an impact in future periods as well as the period of the shock.
Unexpected changes in wind speed (gusts) may arise for a number of reasons, such as mechanical shearing and the associated breakdown of the mean flow, differential heating and thermal convection within the lower atmosphere, thunderstorm downdrafts and their associated outflows, cold fronts, drylines and other organized boundaries. Impulse response analysis provides useful information as to how wind speed at a particular height is likely to respond to various wind gusts. Consider now the following moving average representation of the multiple-equation, VAR(m) model where the constant terms may be ignored:
(2) Xt = Ψ(L)vt
Let E(vtv′t) = Σv such that shocks are contemporaneously correlated. The generalized impulse response
function of Xi to a unit (one standard deviation) shock in Xj is given by:
(3) Ψij,h = (σii)-1/2 (e'
jΣvei)
where σii is the ith diagonal element of Σv, ei is a selection vector with the ith element equal to one and all
other elements equal to zero, and h is the horizon.
The impulse responses computed using the generalized method are invariant to any re-ordering of the variables in the VAR. Additionally, because orthogonality is not imposed, they allow for meaningful interpretations of the initial impact response of each variable to shocks to any other variable. The generalized method is more robust than the orthogonalized method. In the study of wind in which events may occur that simultaneously or contemporaneously affect wind speed at various heights, the ability to capture these immediate responses of endogenous variables to shocks is especially appealing.
IV. Discussion of Results
A vector autoregression was estimated where the four equations corresponded to the wind speed at 13, 33, 70, and 160 feet. A constant term was included in each equation. The order of the VAR is the number of lags used in the estimation process and was determined to be three based on Schwartz Bayesian criterion (SBC), as well as visual inspection of the autocorrelation and partial autocorrelation functions.4
The SBC is a popular method for choosing the lag specification in VAR models. Harvey (1994) provides more details on fitting or specifying the VAR model as well as the selection process used.
The VAR(3) results are provided in Table 2. It is informative to first consider the autoregressive properties inherent in the four wind speed series. Note that the AR(3) terms in each series are statistically significant and thus indicate a high degree of persistence in wind speed. The autoregressive component in each equation exhibits a similar pattern regardless of height. That is, the AR(1) term is positive and greater than one and the AR(2) and/or AR(3) term is negative. The AR(2) and AR(3) terms are all much smaller in magnitude than the AR(1) term in each of the respective equations. This pattern reflects a series which
4 Zhou, Smith, and Mehta (2003) also determined that an AR(3) single-equation model was appropriate for
depends strongly on the most recent past observation such that the current observation of the series appears to gain momentum from the last observation. However, the momentum effect is followed by a period of cycling and reversal driven by the negative coefficients on the AR(2) and/or AR(3) terms. This latter feature is sometimes referred to in time series analysis as overshooting as it suggests a series does not
immediately return to its unconditional mean value following a positive (negative) shock but instead overshoots (i.e., goes below/above) its baseline value before the shock dissipates.
Univariate AR(3) models show similar characteristics as the VAR(3) model. All three terms are statistically significant with the first AR term being in the range of 1.2 to 1.8. The second term is typically negative and the third term either positive of negative. Similar to the VAR model, the univariate model overshoots its baseline value. The three AR coefficients in the model are highly linearly correlated with each other thus knowing the AR(1) term allows complete specification of the AR(3) model. The first term in the AR(3) model is functionally related to the physical parameters of roughness length and height above ground and the white noise variance term is related to the physical parameter shear velocity.
The VAR results presented in Table 2 also reveal insight into the interdependencies of wind speed measured at the four heights. Consider first the last column corresponding to the X13 equation. The
current value of X13 depends on past values of each of the other variables. In particular, X13 depends
positively and significantly on last period’s value of X160 (note the estimated coefficient on X160(t-1)
equals 0.020 with corresponding t-statistic of 1.76). In contrast, we find that X13 depends negatively and
significantly on last period’s value of X70 (estimated coefficient = -0.03, t-statistic = -1.72). There is a
much longer delay in the response of X13 to X33 as the estimated coefficient on X33(t-3) is positive and
significant (coefficient = 0.03, t-statistic = 3.35). X70 is highly dependent on X160 with all three lagged
values of wind at 160 feet entering significantly into the X70 equation. The first and third lag of X160 are
found to be positive and statistically significant (estimated coefficients = 0.02 and 0.02, respectively, with corresponding t-statistics = 2.21 and 2.36). The second lag of X160 is found to be negative and statistically
significant (estimated coefficient = -0.03, t-statistic = -2.76). This high degree of dependence on X160,
which has the highest mean value, together with our finding that X70 has the largest AR(1) term is
consistent with X70 exhibiting the highest measured standard deviation. Finally, the VAR indicates that
wind speeds. Generally speaking, the standard interpretation of the VAR suggests that wind speed at 70 feet is the most volatile and persistent of the group. However, wind speed measured at 13 feet is the most interdependent, while wind speeds at 33 and 160 feet are the most independent.
While the statistical significance of a particular past value (or values) is informative, it does not address whether or not a series responds to shocks or gusts that appear in another series. Impulse response analysis allows for examination of responses to shocks. The generalized responses are shown in Figure 2. The “cross-effects” are given in the off-diagonal plots and show the response of Xi to a shock to Xj (for
i≠j). The “own-effects” are shown along the diagonal going from the upper left corner to the bottom right corner. The generalized impulse response functions are plotted out to four hundred ninety-nine periods after the shock (h=500). Significance of the impulse response is determined by the use of confidence intervals representing plus/minus two standard deviations [Runkle (1987)]. At points where the confidence bands do not straddle the line at zero, the impulse response is considered to be statistically different from zero at the 5 percent level of significance or less (p-value ≤ 0.05).
Consider first the response of the wind speed series to X13 shocks (see column at far right). The
responses to X13 shocks are all positive and they are larger and more pronounced going from X160 to X70
to X33. In fact, the response of X160 to the X13 shock is statistically insignificant at all horizons. X33 also
exhibits slightly greater persistence to the X13 shock than does X70, with the effect of the shock at h=1
dissipating completely (i.e., becoming statistically insignificant) around h=200 for the case of X33 and
around h=180 in the case of X70.
The responses of X13, X70, and X160 to an X33 shock are shown in the next to last column. X13
responds positively and significantly to the X33 shock from the time of the shock (h=1) until about h=375.
Similarly, X70 responds positively and significantly with the effect lasting to around h=325, though the
magnitude of the X70 response is less than X13 response. X160 does not exhibit a response to the X13
shock until about h=50, and the effect is only marginally significant and exhibits far less persistence than the other wind speeds.
Examining the responses to an X70 shock we find that the largest effect (in terms of magnitude)
occurs in X33, then in X13, and X160, respectively. However, the persistence of the X70 shock is only
The responses of X70, X33, and X13 to a shock to X160 exhibit roughly the same degree of
persistence. The magnitude of the response is largest in X70, followed by X33 and then X13.
Generally speaking, a comparison of the “cross effects” indicates several interesting conclusions. First, shocks to X13 are associated with responses that are the least persistent and smallest in magnitude.
Physically this makes sense, since mechanical breakdown of the flow yields relatively small eddies (wind gusts), which are most prevalent near the source of friction (the earth’s surface) [Holmes (2001)]. Second, the largest and most persistent responses are usually at heights closest to where the shock occurred, that is, the transmission of wind shocks is strongest in the wind series closest to where the shock emanated. Similarly, the transmission of shocks weakens the farther away the responding wind series is from where the shock was initiated. This result also makes sense since the vertical scale of a gust is limited and has a finite value. Hence, a shock seen at one height is most likely to be seen at an adjacent height. Moreover, examining the lag to peak response in the impulse response functions that relate two different height levels, it appears that the peaks of the response functions between two levels are related to the distances between the levels and to the shear velocity derived from the velocity profile. This may imply that the persistence is related to the rate of momentum transfer downward from higher level winds. Third, shocks to X160 have
the greatest persistence in the responses of the other wind series.
Finally, the “own-effects” of shocks are shown to have a similar decaying pattern regardless of the height in which wind speed was measured. However, the generalized impulse responses shown along the diagonal indicate that the greatest degree of persistence to an “own” shock is for X160, followed by X70, X33, and X13, respectively, indicating a vertical dependency of the associated wind gust scale generally
increases with height as one moves upward.
V. Concluding Remarks
Time series methods were used to examine the interdependence in wind speed data measured in the same location at four different heights (13, 33, 70, and 160 feet). The time series dynamics of wind were characterized using a multiple-equation system known as a vector autoregression and corresponding innovation accounting technique called generalized impulse response analysis. The recently developed method of generalized impulse response analysis provided insight into the cross-effects of the wind series
and their responses to shocks. The results from the VAR suggested that wind speed at 70 feet is the most volatile and persistent of the group. However, wind speed measured at 13 feet is the most interdependent, while wind speeds at 33 and 160 feet are the most independent. The impulse responses indicated that shocks to X13 were associated with responses that were the least persistent and smallest in magnitude. It is
not surprising that the largest and most persistent responses were at heights closest to where the shock occurred. Shocks to X160 led to the greatest persistence in the responses of the other wind series. Further,
the greatest degree of persistence to an “own” shock is for X160, followed by X70, X33, and X13,
respectively. Finally, it is acknowledged that the results presented here are applicable to the examined data. Results based on the analysis of additional data sets are the focus of future research.
Acknowledgement: This work was performed under the Department of Commerce NIST/TTU Cooperative Agreement Award 70NANB8H0059
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Table 1. Descriptive Statistics X13 X33 X70 X160 Mean 16.15 18.05 20.15 22.61 Maximum 24.30 26.74 28.89 29.92 Minimum 7.25 9.30 11.77 15.84 Std. Dev. 2.88 2.95 3.05 2.50
Notes: Xn (n = 13, 33, 70, 160) denotes the wind speed measures at the respective heights (reported in
Table 2. VAR (Vector AutoRegression) Results X160 X70 X33 X13 X160(t-1) 1.078284 0.017995 0.016823 0.020148 [ 104.379]a [ 2.21091]b [ 1.29208] [ 1.75574]c X160(t-2) 0.117444 -0.033306 -0.031607 -0.020156 [ 7.65997]a [-2.75708]a [-1.63560] [-1.18343] X160(t-3) -0.205523 0.019226 0.018248 0.001494 [-19.8859]a [ 2.36101]a [ 1.40091] [ 0.13013] X70(t-1) 0.017040 1.541258 0.023704 -0.025577 [ 1.27569] [ 146.443]a [ 1.40795] [-1.72365]c X70(t-2) -0.018603 -0.481821 -0.011313 0.037466 [-0.78391] [-25.7693]a [-0.37822] [ 1.42122] X70(t-3) 0.005182 -0.071391 -0.000246 -0.013507 [ 0.38745] [-6.77483]a [-0.01458] [-0.90913] X33(t-1) 0.000850 -0.003032 1.149955 0.009131 [ 0.10405] [-0.47135] [ 111.755]a [ 1.00679] X33(t-2) -0.008851 0.004607 0.045540 -0.020397 [-0.70159] [ 0.46348] [ 2.86398]a [-1.45540] X33(t-3) 0.009165 0.004450 -0.221164 0.030514 [ 1.11881] [ 0.68950] [-21.4209]a [ 3.35326]a X13(t-1) 0.000927 -0.005108 0.017549 1.332736 [ 0.09954] [-0.69588] [ 1.49439] [ 128.768]a X13(t-2) -0.010935 0.011235 -0.009873 -0.164440 [-0.69933] [ 0.91198] [-0.50098] [-9.46734]a X13(t-3) 0.009208 -0.005150 0.000678 -0.192497 [ 0.99348] [-0.70523] [ 0.05807] [-18.6957]a Constant 0.140019 0.027393 0.004871 0.042203 [ 5.60808]a [ 1.39251] [ 0.15478] [ 1.52163] R-squared 0.990143 0.995877 0.988767 0.990806 Adj. R-squared 0.990130 0.995872 0.988752 0.990794 F-statistic 75205.15a 180849.6a 65899.64a 80684.99a
Notes: Xn (n = 13, 33, 70, 160) denotes the wind speed measures at the respective heights. There are 8997
usable observations after adjusting endpoints to account for three lags. Actual t-statistics are reported in brackets. Superscripts a, b, c indicate significance at the 1%, 5%, and 10% levels, respectively. F-statistic test the null hypothesis that the regressors in the respective equations are jointly equal to zero.
Figure 1. Wind Speed at Heights of 13, 33, 70, and 160 Feet 8 12 16 20 24 28 32 2500 5000 7500 X13 8 12 16 20 24 28 32 2500 5000 7500 X33 8 12 16 20 24 28 32 2500 5000 7500 X70 8 12 16 20 24 28 32 2500 5000 7500 X160
Notes: Xn (n = 13, 33, 70, 160) denotes the wind speed measures at the respective heights. There are 9000
Figure 2. Generalized Impulse Response Functions -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X160 to X160 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X160 to X70 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X160 to X33 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X160 to X13 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X70 to X160 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X70 to X70 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X70 to X33 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X70 to X13 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X33 to X160 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X33 to X70 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X33 to X33 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X33 to X13 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X13 to X160 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X13 to X70 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X13 to X33 -.1 .0 .1 .2 .3 .4 .5 .6 100 200 300 400 500 Response of X13 to X13
Notes: Xn (n = 13, 33, 70, 160) denotes the wind speed measures at the respective heights. The forecast
horizon (h) is given on the horizontal axis. The vertical axis measures the magnitude of the response to the impulse, scaled such that 1.0 equals one standard deviation. Confidence bands, used to determine statistical significance (at p-value ≤ .05) of an impulse response at horizon h, where h = 1,2 ,…500, are shown as dashed (----) lines and represent ±2 standard errors.
