小波频谱分析（代码完整，数据齐全）资源-CSDN文库

共14个文件

m：10个

docx：1个

ps：1个

版权申诉

matlab

74 浏览量 2023-12-29 17:44:04 上传评论收藏 1.89MB RAR 举报

资源推荐

资源详情

资源评论

收起资源包目录

wavelet.rar （14个子文件）

fmin.m 3KB

小波分析实用教程.pdf 1.53MB

wavelvbo.m 571B

wave_signif.m 7KB

小波分析.docx 374KB

aqi_daily_贵阳2.txt 321B

wavetest.m 6KB

lvbo.m 1KB

SST_NINO3.ps 247KB

chisquare_inv.m 1KB

wavelet.m 6KB

ws.m 6KB

wave_bases.m 3KB

chisquare_solve.m 686B

Bulletin of the American Meteorological Society

1. Introduction

Wavelet analysis is becoming a common tool for

analyzing localized variations of power within a time

series. By decomposing a time series into time–fre-

quency space, one is able to determine both the domi-

nant modes of variability and how those modes vary

in time. The wavelet transform has been used for nu-

merous studies in geophysics, including tropical con-

vection (Weng and Lau 1994), the El Niño–Southern

Oscillation (ENSO; Gu and Philander 1995; Wang and

Wang 1996), atmospheric cold fronts (Gamage and

Blumen 1993), central England temperature (Baliunas

et al. 1997), the dispersion of ocean waves (Meyers et

al. 1993), wave growth and breaking (Liu 1994), and

coherent structures in turbulent flows (Farge 1992). A

complete description of geophysical applications can

be found in Foufoula-Georgiou and Kumar (1995),

while a theoretical treatment of wavelet analysis is

given in Daubechies (1992).

Unfortunately, many studies using wavelet analy-

sis have suffered from an apparent lack of quantita-

tive results. The wavelet transform has been regarded

by many as an interesting diversion that produces col-

orful pictures, yet purely qualitative results. This mis-

conception is in some sense the fault of wavelet analy-

sis itself, as it involves a transform from a one-dimen-

sional time series (or frequency spectrum) to a diffuse

two-dimensional time–frequency image. This diffuse-

ness has been exacerbated by the use of arbitrary nor-

malizations and the lack of statistical significance tests.

In Lau and Weng (1995), an excellent introduction

to wavelet analysis is provided. Their paper, however,

did not provide all of the essential details necessary

for wavelet analysis and avoided the issue of statisti-

cal significance.

The purpose of this paper is to provide an easy-to-

use wavelet analysis toolkit, including statistical sig-

nificance testing. The consistent use of examples of

A Practical Guide to

Wavelet Analysis

Christopher Torrence and Gilbert P. Compo

Program in Atmospheric and Oceanic Sciences, University of Colorado, Boulder, Colorado

ABSTRACT

A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Niño–

Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an

appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet

scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theo-

retical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence

intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum.

Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to

wavelet analysis such as filtering, the power Hovmöller, cross-wavelet spectra, and coherence are described.

The statistical significance tests are used to give a quantitative measure of changes in ENSO variance on interdecadal

timescales. Using new datasets that extend back to 1871, the Niño3 sea surface temperature and the Southern Oscilla-

tion index show significantly higher power during 1880–1920 and 1960–90, and lower power during 1920–60, as well

as a possible 15-yr modulation of variance. The power Hovmöller of sea level pressure shows significant variations in

2–8-yr wavelet power in both longitude and time.

Corresponding author address: Dr. Christopher Torrence, Ad-

vanced Study Program, National Center for Atmospheric Re-

search, P.O. Box 3000, Boulder, CO 80307-3000.

E-mail: torrence@ucar.edu

In final form 20 October 1997.

Vol. 79, No. 1, January 1998

ENSO provides a substantive addition to

the ENSO literature. In particular, the

statistical significance testing allows

greater confidence in the previous wave-

let-based ENSO results of Wang and

Wang (1996). The use of new datasets

with longer time series permits a more

robust classification of interdecadal

changes in ENSO variance.

The first section describes the datasets

used for the examples. Section 3 de-

scribes the method of wavelet analysis

using discrete notation. This includes a

discussion of the inherent limitations of

the windowed Fourier transform (WFT),

the definition of the wavelet transform,

the choice of a wavelet basis function,

edge effects due to finite-length time se-

ries, the relationship between wavelet

scale and Fourier period, and time series

reconstruction. Section 4 presents the

theoretical wavelet spectra for both

white-noise and red-noise processes.

These theoretical spectra are compared to

Monte Carlo results and are used to es-

tablish significance levels and confi-

dence intervals for the wavelet power

spectrum. Section 5 describes time or

scale averaging to increase significance

levels and confidence intervals. Section

6 describes other wavelet applications

such as filtering, the power Hovmöller,

cross-wavelet spectra, and wavelet co-

herence. The summary contains a step-

by-step guide to wavelet analysis.

2. Data

Several time series will be used for examples of

wavelet analysis. These include the Niño3 sea surface

temperature (SST) used as a measure of the amplitude

of the El Niño–Southern Oscillation (ENSO). The

Niño3 SST index is defined as the seasonal SST av-

eraged over the central Pacific (5°S–5°N, 90°–

150°W). Data for 1871–1996 are from an area aver-

age of the U.K. Meteorological Office GISST2.3

(Rayner et al. 1996), while data for January–June 1997

are from the Climate Prediction Center (CPC) opti-

mally interpolated Niño3 SST index (courtesy of D.

Garrett at CPC, NOAA). The seasonal means for the

entire record have been removed to define an anomaly

time series. The Niño3 SST is shown in the top plot

of Fig. 1a.

Gridded sea level pressure (SLP) data is from the

UKMO/CSIRO historical GMSLP2.1f (courtesy of D.

Parker and T. Basnett, Hadley Centre for Climate Pre-

diction and Research, UKMO). The data is on a 5°

global grid, with monthly resolution from January

1871 to December 1994. Anomaly time series have

been constructed by removing the first three harmon-

ics of the annual cycle (periods of 365.25, 182.625, and

121.75 days) using a least-squares fit.

The Southern Oscillation index is derived from the

GMSLP2.1f and is defined as the seasonally averaged

pressure difference between the eastern Pacific (20°S,

150°W) and the western Pacific (10°S, 130°E).

1880 1900 1920 1940 1960 1980 2000

-2

-1

(

a. NINO3 SST

1880 1900 1920 1940 1960 1980 2000

Time (year)

Period (years)

64.0

32.0

16.0

8.0

4.0

2.0

1.0

0.5

Scale (years)

b. Morlet

1880 1900 1920 1940 1960 1980 2000

Time

ear

)

Period (years)

16.000

8.000

4.000

2.000

1.000

0.500

0.250

0.125

Scale

ears

)

c. DOG

FIG. 1. (a) The Niño3 SST time series used for the wavelet analysis. (b) The

local wavelet power spectrum of (a) using the Morlet wavelet, normalized by 1/

(

= 0.54°C

). The left axis is the Fourier period (in yr) corresponding to the

wavelet scale on the right axis. The bottom axis is time (yr). The shaded contours

are at normalized variances of 1, 2, 5, and 10. The thick contour encloses regions

of greater than 95% confidence for a red-noise process with a lag-1 coefficient of

0.72. Cross-hatched regions on either end indicate the “cone of influence,” where

edge effects become important. (c) Same as (b) but using the real-valued Mexican

hat wavelet (derivative of a Gaussian; DOG m = 2). The shaded contour is at

normalized variance of 2.0.

Bulletin of the American Meteorological Society

3. Wavelet analysis

This section describes the method of wavelet analy-

sis, includes a discussion of different wavelet func-

tions, and gives details for the analysis of the wavelet

power spectrum. Results in this section are adapted to

discrete notation from the continuous formulas given

in Daubechies (1990). Practical details in applying

wavelet analysis are taken from Farge (1992), Weng

and Lau (1994), and Meyers et al. (1993). Each sec-

tion is illustrated with examples using the Niño3 SST.

a. Windowed Fourier transform

The WFT represents one analysis tool for extract-

ing local-frequency information from a signal. The

Fourier transform is performed on a sliding segment

of length T from a time series of time step

t and total

length N

t, thus returning frequencies from T

−1

at each time step. The segments can be win-

dowed with an arbitrary function such as a boxcar (no

smoothing) or a Gaussian window (Kaiser 1994).

As discussed by Kaiser (1994), the WFT represents

an inaccurate and inefficient method of time–fre-

quency localization, as it imposes a scale or “response

interval” T into the analysis. The inaccuracy arises

from the aliasing of high- and low-frequency compo-

nents that do not fall within the frequency range of the

window. The inefficiency comes from the T/(2

t) fre-

quencies, which must be analyzed at each time step,

regardless of the window size or the dominant frequen-

cies present. In addition, several window lengths must

usually be analyzed to determine the most appropri-

ate choice. For analyses where a predetermined scal-

ing may not be appropriate because of a wide range

of dominant frequencies, a method of time–frequency

localization that is scale independent, such as wave-

let analysis, should be employed.

b. Wavelet transform

The wavelet transform can be used to analyze time

series that contain nonstationary power at many dif-

ferent frequencies (Daubechies 1990). Assume that

one has a time series, x

, with equal time spacing

and n = 0 … N − 1. Also assume that one has a wave-

let function,

(

), that depends on a nondimensional

“time” parameter

. To be “admissible” as a wavelet,

this function must have zero mean and be localized in

both time and frequency space (Farge 1992). An ex-

ample is the Morlet wavelet, consisting of a plane

wave modulated by a Gaussian:

ηπ

ωη

14 2

(

)

−−

, (1)

where

is the nondimensional frequency, here taken

to be 6 to satisfy the admissibility condition (Farge

1992). This wavelet is shown in Fig. 2a.

The term “wavelet function” is used generically to

refer to either orthogonal or nonorthogonal wavelets.

The term “wavelet basis” refers only to an orthogo-

nal set of functions. The use of an orthogonal basis

implies the use of the discrete wavelet transform,

while a nonorthogonal wavelet function can be used

-4 -2 0 2 4

-0.3

0.0

0.3

ψ (t / s)

a. Morlet

-2 -1 0 1 2

(s ω)

-4 -2 0 2 4

-0.3

0.0

0.3

b. Paul (m=4)

-2 -1 0 1 2

-4 -2 0 2 4

-0.3

0.0

0.3

c. DOG (m=2)

-2 -1 0 1 2

-4 -2 0 2 4

t / s

-0.3

0.0

0.3

d. DOG (m=6)

-2 -1 0 1 2

s ω / (2π)

FIG. 2. Four different wavelet bases, from Table 1. The plots

on the left give the real part (solid) and imaginary part (dashed)

for the wavelets in the time domain. The plots on the right give

the corresponding wavelets in the frequency domain. For plotting

purposes, the scale was chosen to be s = 10

t. (a) Morlet, (b) Paul

(m = 4), (c) Mexican hat (DOG m = 2), and (d) DOG (m = 6).

Vol. 79, No. 1, January 1998

with either the discrete or the continuous wavelet

transform (Farge 1992). In this paper, only the con-

tinuous transform is used, although all of the results

for significance testing, smoothing in time and scale,

and cross wavelets are applicable to the discrete wave-

let transform.

The continuous wavelet transform of a discrete se-

quence x

is defined as the convolution of x

with a

scaled and translated version of

(

Ws x

nnt

(

)

=∗

′

−

(

)













′

−

∑

, (2)

where the (*) indicates the complex conjugate. By

varying the wavelet scale s and translating along the

localized time index n, one can construct a picture

showing both the amplitude of any features versus the

scale and how this amplitude varies with time. The

subscript 0 on

has been dropped to indicate that this

has also been normalized (see next section). Al-

though it is possible to calculate the wavelet transform

using (2), it is considerably faster to do the calcula-

tions in Fourier space.

To approximate the continuous wavelet transform,

the convolution (2) should be done N times for each

scale, where N is the number of points in the time se-

ries (Kaiser 1994). (The choice of doing all N convo-

lutions is arbitrary, and one could choose a smaller

number, say by skipping every other point in n.) By

choosing N points, the convolution theorem allows us

do all N convolutions simultaneously in Fourier space

using a discrete Fourier transform (DFT). The DFT

of x

ikn N

−

∑

, (3)

where k = 0 … N − 1 is the frequency index. In the

continuous limit, the Fourier transform of a function

(t/s) is given by

). By the convolution theorem,

the wavelet transform is the inverse Fourier transform

of the product:

Ws x s e

int

(

)

=∗

(

)

−

∑

ψω

ωδ

, (4)

where the angular frequency is defined as

≤

−>











. (5)

Using (4) and a standard Fourier transform routine, one

can calculate the continuous wavelet transform (for a

given s) at all n simultaneously and efficiently.

c. Normalization

To ensure that the wavelet transforms (4) at each

scale s are directly comparable to each other and to the

transforms of other time series, the wavelet function

at each scale s is normalized to have unit energy:

ˆˆ

ψω

(

)













(

)

. (6)

Examples of different wavelet functions are given in

Table 1 and illustrated in Fig. 2. Each of the unscaled

are defined in Table 1 to have

;

ψω ω

′

(

)

′

−∞

+∞

∫

that is, they have been normalized to have unit energy.

Using these normalizations, at each scale s one has

ψω

(

)

−

∑

, (7)

where N is the number of points. Thus, the wavelet

transform is weighted only by the amplitude of the

Fourier coefficients x$

and not by the wavelet function.

If one is using the convolution formula (2), the nor-

malization is

′

−

(

)

























′

−

(

)













nnt

, (8)

where

(

) is normalized to have unit energy.

d. Wavelet power spectrum

Because the wavelet function

(

) is in general

complex, the wavelet transform W

(s) is also complex.

The transform can then be divided into the real part,

Bulletin of the American Meteorological Society

ℜ{W

(s)}, and imaginary part, ℑ{W

(s)}, or ampli-

tude, |W

(s)|, and phase, tan

−1

[ℑ{W

(s)}/ℜ{W

(s)}]. Fi-

nally, one can define the wavelet power spectrum as

(s)|

. For real-valued wavelet functions such as the

DOGs (derivatives of a Gaussian) the imaginary part

is zero and the phase is undefined.

To make it easier to compare different wavelet

power spectra, it is desirable to find a common nor-

malization for the wavelet spectrum. Using the nor-

malization in (6), and referring to (4), the expectation

value for |W

(s)|

is equal to N times the expectation

value for |x$

. For a white-noise time series, this ex-

pectation value is

/N, where

is the variance. Thus,

for a white-noise process, the expectation value for the

wavelet transform is |W

(s)|

at all n and s.

Figure 1b shows the normalized wavelet power

spectrum, |W

(s)|

, for the Niño3 SST time series.

The normalization by 1/

gives a measure of the

power relative to white noise. In Fig. 1b, most of the

power is concentrated within the ENSO band of 2–8

yr, although there is appreciable power at longer peri-

ods. The 2–8-yr band for ENSO agrees with other stud-

ies (Trenberth 1976) and is also seen in the Fourier

spectrum in Fig. 3. With wavelet analysis, one can see

variations in the frequency of occurrence and ampli-

tude of El Niño (warm) and La Niña (cold) events.

During 1875–1920 and 1960–90 there were many

warm and cold events of large amplitude, while dur-

ing 1920–60 there were few events (Torrence and

Webster 1997). From 1875–1910, there was a slight

shift from a period near 4 yr to a period closer to 2 yr,

while from 1960–90 the shift is from shorter to longer

periods.

These results are similar to those of Wang and

Wang (1996), who used both wavelet and waveform

analysis on ENSO indices derived from the Compre-

hensive Ocean–Atmosphere Data Set (COADS)

dataset. Wang and Wang’s analysis showed reduced

wavelet power before 1950, especially 1875–1920. The

reduced power is possibly due to the sparseness and de-

creased reliability of the pre-1950 COADS data (Folland

et al. 1984). With the GISST2.3 data, the wavelet trans-

form of Niño3 SST in Fig. 1b shows that the pre-1920

period has equal power to the post-1960 period.

TABLE 1. Three wavelet basis functions and their properties. Constant factors for

and

ensure a total energy of unity.

e-folding Fourier

Name

(

)

∧

) time

wavelength

Morlet

ωη

−−14 2

πω

ωω

−

−−

()

(

)

ωω

(

= frequency)

Paul

(

)

−

(

)

−+

()

Hse

−

(

)

(

)

(

)

−

ωω

(m = order)

DOG

−

(

)









(

)

−

Γ+









(

)

−

()

(m = derivative)

) = Heaviside step function, H(

) = 1 if

ω > 0

, H(

) = 0 otherwise.

DOG = derivative of a Gaussian; m = 2 is the Marr or Mexican hat wavelet.

评论收藏

内容反馈

版权申诉

神经网络机器学习智能算法画图绘图

粉丝: 2373
资源: 569

小波频谱分析（代码完整，数据齐全）

频谱分析代码

音乐频谱分析 相当齐全

基于小波的多尺度分析（代码完整，数据齐全）

数据进行频谱分析

小波分解，ceemd分解（代码完整，数据齐全）

复小波频谱分析在地震数据处理中的应用.pdf

正弦信号和三角波频谱分析显示频谱图

频谱分析c++源代码

vc++源代码 频谱分析器

频谱分析代码 哈工大

小波阈值去噪（代码完整，数据齐全）

小波分解重构的MATALB代码（代码完整，数据齐全）

频谱分析上位机代码.zip

arm1138频谱分析代码

信号频谱分析代码--MATLAB

基于小波频谱分析的滚动轴承故障诊断研究

离散信号频谱分析（matlab源码）

语音信号的采集与频谱分析(附代码).pdf

倒频谱.rar_倒谱分析_倒频谱_倒频谱 matlab_频谱分析_齿轮数据

kmeans聚类分析的MATLAB代码，（代码完整，数据齐全）

fft_fft_数据频谱分析_

暗通道去雾，改进的暗通道去雾（代码完整，数据齐全）

基于遗传算法的药品配送（代码完整，数据齐全）

最完整的python教学资料代码和文档很齐全

我爱北京三角波频谱图_频谱图_频谱分析_

频谱仪的代码.

(完整word)如何使用matlab对一段离散数据进行频谱分析.doc

MATLAB小波分析与应用30个案例分析

光学滤波与频谱分析的研究

最新资源

音乐频谱分析相当齐全

vc++源代码频谱分析器

频谱分析代码哈工大