Summary Nonstationary Time Series Multitude of Representations Possibilities from Applied Computational Harmonic Analysis Tests of Stationarity
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1 Nonstationary Time Series, Priestley s Evolutionary Spectra and Wavelets Guy Nason, School of Mathematics, University of Bristol Summary Nonstationary Time Series Multitude of Representations Possibilities from Applied Computational Harmonic Analysis Tests of Stationarity Nonstationary Time Series. c U. Bristol 1 / 32
2 Nonstationary Time Series, Priestley s Evolutionary Spectra and Wavelets Guy Nason, School of Mathematics, University of Bristol Summary Nonstationary Time Series Multitude of Representations Possibilities from Applied Computational Harmonic Analysis Tests of Stationarity Nonstationary Time Series. c U. Bristol 1 / 32
3 Nonstationary Time Series, Priestley s Evolutionary Spectra and Wavelets Guy Nason, School of Mathematics, University of Bristol Summary Nonstationary Time Series Multitude of Representations Possibilities from Applied Computational Harmonic Analysis Tests of Stationarity Nonstationary Time Series. c U. Bristol 1 / 32
4 Nonstationary Time Series, Priestley s Evolutionary Spectra and Wavelets Guy Nason, School of Mathematics, University of Bristol Summary Nonstationary Time Series Multitude of Representations Possibilities from Applied Computational Harmonic Analysis Tests of Stationarity Nonstationary Time Series. c U. Bristol 1 / 32
5 Nonstationary Time Series, Priestley s Evolutionary Spectra and Wavelets Guy Nason, School of Mathematics, University of Bristol Summary Nonstationary Time Series Multitude of Representations Possibilities from Applied Computational Harmonic Analysis Tests of Stationarity Nonstationary Time Series. c U. Bristol 1 / 32
6 Nonstationary Time Series, Priestley s Evolutionary Spectra and Wavelets Guy Nason, School of Mathematics, University of Bristol Summary Nonstationary Time Series Multitude of Representations Possibilities from Applied Computational Harmonic Analysis Tests of Stationarity Nonstationary Time Series. c U. Bristol 1 / 32
7 Time Series Time series are everywhere. We want to model them, estimate parameters, check model fit, and try other models. forecast future values. We need models! Or change the data, or the sampling mechanism: another story. Nonstationary Time Series. c U. Bristol 2 / 32
8 Time Series Time series are everywhere. We want to model them, estimate parameters, check model fit, and try other models. forecast future values. We need models! Or change the data, or the sampling mechanism: another story. Nonstationary Time Series. c U. Bristol 2 / 32
9 Time Series Time series are everywhere. We want to model them, estimate parameters, check model fit, and try other models. forecast future values. We need models! Or change the data, or the sampling mechanism: another story. Nonstationary Time Series. c U. Bristol 2 / 32
10 Time Series Time series are everywhere. We want to model them, estimate parameters, check model fit, and try other models. forecast future values. We need models! Or change the data, or the sampling mechanism: another story. Nonstationary Time Series. c U. Bristol 2 / 32
11 Time Series Time series are everywhere. We want to model them, estimate parameters, check model fit, and try other models. forecast future values. We need models! Or change the data, or the sampling mechanism: another story. Nonstationary Time Series. c U. Bristol 2 / 32
12 Time Series Time series are everywhere. We want to model them, estimate parameters, check model fit, and try other models. forecast future values. We need models! Or change the data, or the sampling mechanism: another story. Nonstationary Time Series. c U. Bristol 2 / 32
13 Time Series Time series are everywhere. We want to model them, estimate parameters, check model fit, and try other models. forecast future values. We need models! Or change the data, or the sampling mechanism: another story. Nonstationary Time Series. c U. Bristol 2 / 32
14 Time Series Time series are everywhere. We want to model them, estimate parameters, check model fit, and try other models. forecast future values. We need models! Or change the data, or the sampling mechanism: another story. Nonstationary Time Series. c U. Bristol 2 / 32
15 Example Time Series: Data Source MIDAS: Met Office Integrated Data Archive System Originates from the UK Meteorological Office Distributed through the NERC British Atmospheric Data Centre Acknowledgement and thanks is due. Nonstationary Time Series. c U. Bristol 3 / 32
16 Example Time Series: Data Source MIDAS: Met Office Integrated Data Archive System Originates from the UK Meteorological Office Distributed through the NERC British Atmospheric Data Centre Acknowledgement and thanks is due. Nonstationary Time Series. c U. Bristol 3 / 32
17 Example Time Series: Data Source MIDAS: Met Office Integrated Data Archive System Originates from the UK Meteorological Office Distributed through the NERC British Atmospheric Data Centre Acknowledgement and thanks is due. Nonstationary Time Series. c U. Bristol 3 / 32
18 Example Time Series: Data Source MIDAS: Met Office Integrated Data Archive System Originates from the UK Meteorological Office Distributed through the NERC British Atmospheric Data Centre Acknowledgement and thanks is due. Nonstationary Time Series. c U. Bristol 3 / 32
19 Hourly Wind Speeds at Cardinham, Bodmin, Cornwall Wind Speed at Cardinham (knots) Hour (Days since 1st January 2009) Nonstationary Time Series. c U. Bristol 4 / 32
20 First Differences of Wind Speed First Difference Wind Speed at Cardinham (knots) Hour (Days since 1st January 2009) Nonstationary Time Series. c U. Bristol 5 / 32
21 Time Series Models The classical methods of time series analysis... are all based on two crucial assumptions, namely that: (a) all series are stationary (at least to order 2), or can be reduced to stationarity... (b) all models are linear,... Priestley (1981), page 816. Nonstationary Time Series. c U. Bristol 6 / 32
22 Nonstationary Series and However, stationarity and linearity are... approximations to the real situation.... first establish some method of characterizing... non-stationary processes,... we describe... non-stationary processes based on the theory of evolutionary spectra. This approach was developed by Priestley (1965b, 1966, 1967) Priestley (1981), page 816. Nonstationary Time Series. c U. Bristol 7 / 32
23 Nonstationary Series and However, stationarity and linearity are... approximations to the real situation.... first establish some method of characterizing... non-stationary processes,... we describe... non-stationary processes based on the theory of evolutionary spectra. This approach was developed by Priestley (1965b, 1966, 1967) Priestley (1981), page 816. Nonstationary Time Series. c U. Bristol 7 / 32
24 Oscillatory Function (Priestley, 1981, Def ) The function of t, φ t (ω) will be said to be an oscillatory function if, for some (necessarily unique) θ(ω), it may be written in the form where A t (ω) is of the form φ t (ω) = A t (ω)e iθ(ω)t, A t (ω) = e itu dk ω (u), with dk ω (u) having an absolute maximum at u = 0. Nonstationary Time Series. c U. Bristol 8 / 32
25 Oscillatory Process (Priestley, 1981, Def ) If there exists a family of oscillatory functions {φ t (ω)} in terms of which the process {X (t)} has a representation of the form X (t) = φ t (ω)dz(ω), where Z(ω) is an orthogonal process with E[ dz(ω) 2 ] = dµ(ω), then {X (t)} will be termed an oscillatory process. Nonstationary Time Series. c U. Bristol 9 / 32
26 Evolutionary Power Spectra (Priestley, 1981, Def ) We define the evolutionary power spectrum at time t dh t (ω) by dh t (ω) = A t (ω) 2 dµ(ω). When X (t) is stationary and θ(ω) = ω then dh t (ω) reduces to the regular spectrum, h(ω). H t (ω) is the integrated time-frequency spectrum. Assuming smoothness the evolutionary spectral density function is h t (ω) = H t(ω). Nonstationary Time Series. c U. Bristol 10 / 32
27 Evolutionary Power Spectra (Priestley, 1981, Def ) We define the evolutionary power spectrum at time t dh t (ω) by dh t (ω) = A t (ω) 2 dµ(ω). When X (t) is stationary and θ(ω) = ω then dh t (ω) reduces to the regular spectrum, h(ω). H t (ω) is the integrated time-frequency spectrum. Assuming smoothness the evolutionary spectral density function is h t (ω) = H t(ω). Nonstationary Time Series. c U. Bristol 10 / 32
28 Evolutionary Power Spectra (Priestley, 1981, Def ) We define the evolutionary power spectrum at time t dh t (ω) by dh t (ω) = A t (ω) 2 dµ(ω). When X (t) is stationary and θ(ω) = ω then dh t (ω) reduces to the regular spectrum, h(ω). H t (ω) is the integrated time-frequency spectrum. Assuming smoothness the evolutionary spectral density function is h t (ω) = H t(ω). Nonstationary Time Series. c U. Bristol 10 / 32
29 Evolutionary Power Spectra (Priestley, 1981, Def ) We define the evolutionary power spectrum at time t dh t (ω) by dh t (ω) = A t (ω) 2 dµ(ω). When X (t) is stationary and θ(ω) = ω then dh t (ω) reduces to the regular spectrum, h(ω). H t (ω) is the integrated time-frequency spectrum. Assuming smoothness the evolutionary spectral density function is h t (ω) = H t(ω). Nonstationary Time Series. c U. Bristol 10 / 32
30 Stationarity Tests (Priestley & Subba Rao, 1969, JRSS-B) 1 Use Priestley s (1965) double-window estimator: ĥ t (ω). 2 Define Y (t, ω) = log ĥ t (ω). 3 Then, approximately, E{Y (t, ω)} = log h t (ω), 4 And, crucially, var{y (t, ω)} = σ 2. In other words Y (t, ω) = log h t (ω) + ɛ(t, ω), which we can discretize over a set of times t 1,..., t I and frequencies ω 1,..., ω J to get the nice linear model: in an obvious way. H : Y ij = µ + α i + β j + γ ij + ɛ ij. Approximately ɛ ij iid N(0, σ 2 ) if t i, ω j spaced out enough. Nonstationary Time Series. c U. Bristol 11 / 32
31 Stationarity Tests (Priestley & Subba Rao, 1969, JRSS-B) 1 Use Priestley s (1965) double-window estimator: ĥ t (ω). 2 Define Y (t, ω) = log ĥ t (ω). 3 Then, approximately, E{Y (t, ω)} = log h t (ω), 4 And, crucially, var{y (t, ω)} = σ 2. In other words Y (t, ω) = log h t (ω) + ɛ(t, ω), which we can discretize over a set of times t 1,..., t I and frequencies ω 1,..., ω J to get the nice linear model: in an obvious way. H : Y ij = µ + α i + β j + γ ij + ɛ ij. Approximately ɛ ij iid N(0, σ 2 ) if t i, ω j spaced out enough. Nonstationary Time Series. c U. Bristol 11 / 32
32 Stationarity Tests (Priestley & Subba Rao, 1969, JRSS-B) 1 Use Priestley s (1965) double-window estimator: ĥ t (ω). 2 Define Y (t, ω) = log ĥ t (ω). 3 Then, approximately, E{Y (t, ω)} = log h t (ω), 4 And, crucially, var{y (t, ω)} = σ 2. In other words Y (t, ω) = log h t (ω) + ɛ(t, ω), which we can discretize over a set of times t 1,..., t I and frequencies ω 1,..., ω J to get the nice linear model: in an obvious way. H : Y ij = µ + α i + β j + γ ij + ɛ ij. Approximately ɛ ij iid N(0, σ 2 ) if t i, ω j spaced out enough. Nonstationary Time Series. c U. Bristol 11 / 32
33 Stationarity Tests (Priestley & Subba Rao, 1969, JRSS-B) 1 Use Priestley s (1965) double-window estimator: ĥ t (ω). 2 Define Y (t, ω) = log ĥ t (ω). 3 Then, approximately, E{Y (t, ω)} = log h t (ω), 4 And, crucially, var{y (t, ω)} = σ 2. In other words Y (t, ω) = log h t (ω) + ɛ(t, ω), which we can discretize over a set of times t 1,..., t I and frequencies ω 1,..., ω J to get the nice linear model: in an obvious way. H : Y ij = µ + α i + β j + γ ij + ɛ ij. Approximately ɛ ij iid N(0, σ 2 ) if t i, ω j spaced out enough. Nonstationary Time Series. c U. Bristol 11 / 32
34 Stationarity Tests (Priestley & Subba Rao, 1969, JRSS-B) 1 Use Priestley s (1965) double-window estimator: ĥ t (ω). 2 Define Y (t, ω) = log ĥ t (ω). 3 Then, approximately, E{Y (t, ω)} = log h t (ω), 4 And, crucially, var{y (t, ω)} = σ 2. In other words Y (t, ω) = log h t (ω) + ɛ(t, ω), which we can discretize over a set of times t 1,..., t I and frequencies ω 1,..., ω J to get the nice linear model: in an obvious way. H : Y ij = µ + α i + β j + γ ij + ɛ ij. Approximately ɛ ij iid N(0, σ 2 ) if t i, ω j spaced out enough. Nonstationary Time Series. c U. Bristol 11 / 32
35 Stationarity Tests (Priestley & Subba Rao, 1969, JRSS-B) 1 Use Priestley s (1965) double-window estimator: ĥ t (ω). 2 Define Y (t, ω) = log ĥ t (ω). 3 Then, approximately, E{Y (t, ω)} = log h t (ω), 4 And, crucially, var{y (t, ω)} = σ 2. In other words Y (t, ω) = log h t (ω) + ɛ(t, ω), which we can discretize over a set of times t 1,..., t I and frequencies ω 1,..., ω J to get the nice linear model: in an obvious way. H : Y ij = µ + α i + β j + γ ij + ɛ ij. Approximately ɛ ij iid N(0, σ 2 ) if t i, ω j spaced out enough. Nonstationary Time Series. c U. Bristol 11 / 32
36 Stationarity Tests (Priestley & Subba Rao, 1969, JRSS-B) 1 Use Priestley s (1965) double-window estimator: ĥ t (ω). 2 Define Y (t, ω) = log ĥ t (ω). 3 Then, approximately, E{Y (t, ω)} = log h t (ω), 4 And, crucially, var{y (t, ω)} = σ 2. In other words Y (t, ω) = log h t (ω) + ɛ(t, ω), which we can discretize over a set of times t 1,..., t I and frequencies ω 1,..., ω J to get the nice linear model: in an obvious way. H : Y ij = µ + α i + β j + γ ij + ɛ ij. Approximately ɛ ij iid N(0, σ 2 ) if t i, ω j spaced out enough. Nonstationary Time Series. c U. Bristol 11 / 32
37 Test Procedure and Modern Implementation Test stationarity by inferring whether α i = 0 and γ ij = 0, i, j. Implementation: stationarity() in fractal R package. Uses improved multitaper estimate: reduces bias. fractal posted in By Bill Constantine & Donald Percival of the Applied Physics Laboratory, U of Washington, USA. Thirty-eight years after the Priestley and Subba Rao paper! Way ahead of their time! p-value for Cardinham data using stationarity() is Nonstationary Time Series. c U. Bristol 12 / 32
38 Test Procedure and Modern Implementation Test stationarity by inferring whether α i = 0 and γ ij = 0, i, j. Implementation: stationarity() in fractal R package. Uses improved multitaper estimate: reduces bias. fractal posted in By Bill Constantine & Donald Percival of the Applied Physics Laboratory, U of Washington, USA. Thirty-eight years after the Priestley and Subba Rao paper! Way ahead of their time! p-value for Cardinham data using stationarity() is Nonstationary Time Series. c U. Bristol 12 / 32
39 Test Procedure and Modern Implementation Test stationarity by inferring whether α i = 0 and γ ij = 0, i, j. Implementation: stationarity() in fractal R package. Uses improved multitaper estimate: reduces bias. fractal posted in By Bill Constantine & Donald Percival of the Applied Physics Laboratory, U of Washington, USA. Thirty-eight years after the Priestley and Subba Rao paper! Way ahead of their time! p-value for Cardinham data using stationarity() is Nonstationary Time Series. c U. Bristol 12 / 32
40 Test Procedure and Modern Implementation Test stationarity by inferring whether α i = 0 and γ ij = 0, i, j. Implementation: stationarity() in fractal R package. Uses improved multitaper estimate: reduces bias. fractal posted in By Bill Constantine & Donald Percival of the Applied Physics Laboratory, U of Washington, USA. Thirty-eight years after the Priestley and Subba Rao paper! Way ahead of their time! p-value for Cardinham data using stationarity() is Nonstationary Time Series. c U. Bristol 12 / 32
41 Test Procedure and Modern Implementation Test stationarity by inferring whether α i = 0 and γ ij = 0, i, j. Implementation: stationarity() in fractal R package. Uses improved multitaper estimate: reduces bias. fractal posted in By Bill Constantine & Donald Percival of the Applied Physics Laboratory, U of Washington, USA. Thirty-eight years after the Priestley and Subba Rao paper! Way ahead of their time! p-value for Cardinham data using stationarity() is Nonstationary Time Series. c U. Bristol 12 / 32
42 Test Procedure and Modern Implementation Test stationarity by inferring whether α i = 0 and γ ij = 0, i, j. Implementation: stationarity() in fractal R package. Uses improved multitaper estimate: reduces bias. fractal posted in By Bill Constantine & Donald Percival of the Applied Physics Laboratory, U of Washington, USA. Thirty-eight years after the Priestley and Subba Rao paper! Way ahead of their time! p-value for Cardinham data using stationarity() is Nonstationary Time Series. c U. Bristol 12 / 32
43 Test Procedure and Modern Implementation Test stationarity by inferring whether α i = 0 and γ ij = 0, i, j. Implementation: stationarity() in fractal R package. Uses improved multitaper estimate: reduces bias. fractal posted in By Bill Constantine & Donald Percival of the Applied Physics Laboratory, U of Washington, USA. Thirty-eight years after the Priestley and Subba Rao paper! Way ahead of their time! p-value for Cardinham data using stationarity() is Nonstationary Time Series. c U. Bristol 12 / 32
44 c Nonstationary Time Series. U. Bristol 13 / 32
45 Parzen and Priestley s Vision Parzen (1959) has pointed out that if there exists a representation X (t) = φ t (ω)dz(ω), then there is a multitude of different representations of the process, each representation based on a different family of functions. The situation is in some ways similar to the selection of a basis for a vector space. However, if the process is non-stationary this choice [complex exponential family] of family of functions is no longer valid. Priestley (1981) p Nonstationary Time Series. c U. Bristol 14 / 32
46 Parzen and Priestley s Vision Parzen (1959) has pointed out that if there exists a representation X (t) = φ t (ω)dz(ω), then there is a multitude of different representations of the process, each representation based on a different family of functions. The situation is in some ways similar to the selection of a basis for a vector space. However, if the process is non-stationary this choice [complex exponential family] of family of functions is no longer valid. Priestley (1981) p Nonstationary Time Series. c U. Bristol 14 / 32
47 Parzen and Priestley s Vision Parzen (1959) has pointed out that if there exists a representation X (t) = φ t (ω)dz(ω), then there is a multitude of different representations of the process, each representation based on a different family of functions. The situation is in some ways similar to the selection of a basis for a vector space. However, if the process is non-stationary this choice [complex exponential family] of family of functions is no longer valid. Priestley (1981) p Nonstationary Time Series. c U. Bristol 14 / 32
48 Priestley s Oscillatory Functions From p. 824 Priestley settles on θ(ω) = ω and φ t (ω) = A t (ω)e iωt. This captures uniformly modulated processes A t (ω) = C(t). Nonstationary Time Series. c U. Bristol 15 / 32
49 Priestley s Oscillatory Functions From p. 824 Priestley settles on θ(ω) = ω and φ t (ω) = A t (ω)e iωt. This captures uniformly modulated processes A t (ω) = C(t). Nonstationary Time Series. c U. Bristol 15 / 32
50 Semi-Stationary Processes Consider linear filter with frequency response function Γ(ω). Stationary processes have useful property that h (Y ) (ω 1 ) is unaffected by ω ω 1, i.e. h (Y ) (ω 1 ) = Γ(ω 1 ) 2 h (X ) (ω 1 ),... Priestley mimics stationary case and approx. useful property. He achieves this by A t (ω) slowly evolving fn. of t = semi-stationary processes. Today, related to locally stationary processes; also has the advantage of permitting estimation. Nonstationary Time Series. c U. Bristol 16 / 32
51 Semi-Stationary Processes Consider linear filter with frequency response function Γ(ω). Stationary processes have useful property that h (Y ) (ω 1 ) is unaffected by ω ω 1, i.e. h (Y ) (ω 1 ) = Γ(ω 1 ) 2 h (X ) (ω 1 ),... Priestley mimics stationary case and approx. useful property. He achieves this by A t (ω) slowly evolving fn. of t = semi-stationary processes. Today, related to locally stationary processes; also has the advantage of permitting estimation. Nonstationary Time Series. c U. Bristol 16 / 32
52 Semi-Stationary Processes Consider linear filter with frequency response function Γ(ω). Stationary processes have useful property that h (Y ) (ω 1 ) is unaffected by ω ω 1, i.e. h (Y ) (ω 1 ) = Γ(ω 1 ) 2 h (X ) (ω 1 ),... Priestley mimics stationary case and approx. useful property. He achieves this by A t (ω) slowly evolving fn. of t = semi-stationary processes. Today, related to locally stationary processes; also has the advantage of permitting estimation. Nonstationary Time Series. c U. Bristol 16 / 32
53 Semi-Stationary Processes Consider linear filter with frequency response function Γ(ω). Stationary processes have useful property that h (Y ) (ω 1 ) is unaffected by ω ω 1, i.e. h (Y ) (ω 1 ) = Γ(ω 1 ) 2 h (X ) (ω 1 ),... Priestley mimics stationary case and approx. useful property. He achieves this by A t (ω) slowly evolving fn. of t = semi-stationary processes. Today, related to locally stationary processes; also has the advantage of permitting estimation. Nonstationary Time Series. c U. Bristol 16 / 32
54 Semi-Stationary Processes Consider linear filter with frequency response function Γ(ω). Stationary processes have useful property that h (Y ) (ω 1 ) is unaffected by ω ω 1, i.e. h (Y ) (ω 1 ) = Γ(ω 1 ) 2 h (X ) (ω 1 ),... Priestley mimics stationary case and approx. useful property. He achieves this by A t (ω) slowly evolving fn. of t = semi-stationary processes. Today, related to locally stationary processes; also has the advantage of permitting estimation. Nonstationary Time Series. c U. Bristol 16 / 32
55 Heisenberg-Gabor uncertainty principle more accurately we estimate h t (ω) as a function of time the less accurately we can determine it as a function of frequency, Priestley, 1981, p. 835 (Daniells, 1965 and Tjøstheim, 1976). To estimate time-varying behaviour, we will necessarily have to sacrifice some frequency resolution. In some situations local Fourier highly inappropriate Nonstationary Time Series. c U. Bristol 17 / 32
56 Heisenberg-Gabor uncertainty principle more accurately we estimate h t (ω) as a function of time the less accurately we can determine it as a function of frequency, Priestley, 1981, p. 835 (Daniells, 1965 and Tjøstheim, 1976). To estimate time-varying behaviour, we will necessarily have to sacrifice some frequency resolution. In some situations local Fourier highly inappropriate Nonstationary Time Series. c U. Bristol 17 / 32
57 Heisenberg-Gabor uncertainty principle more accurately we estimate h t (ω) as a function of time the less accurately we can determine it as a function of frequency, Priestley, 1981, p. 835 (Daniells, 1965 and Tjøstheim, 1976). To estimate time-varying behaviour, we will necessarily have to sacrifice some frequency resolution. In some situations local Fourier highly inappropriate Nonstationary Time Series. c U. Bristol 17 / 32
58 Other Possibilities Single Bases: Fourier Nonstationary Time Series. c U. Bristol 18 / 32
59 Other Possibilities Single Bases: Fourier (of course). Nonstationary Time Series. c U. Bristol 18 / 32
60 Other Possibilities Single Bases: Fourier (of course). T /2 frequencies, T time steps Nonstationary Time Series. c U. Bristol 18 / 32
61 Other Possibilities Single Bases: Fourier (of course). T /2 frequencies, T time steps Wavelets, O(N). Nonstationary Time Series. c U. Bristol 18 / 32
62 Other Possibilities Single Bases: Fourier (of course). T /2 frequencies, T time steps Wavelets, O(N). log T frequency bands, T time steps Nonstationary Time Series. c U. Bristol 18 / 32
63 Other Possibilities Single Bases: Fourier (of course). T /2 frequencies, T time steps Wavelets, O(N). log T frequency bands, T time steps Walsh series Nonstationary Time Series. c U. Bristol 18 / 32
64 Other Possibilities Single Bases: Fourier (of course). T /2 frequencies, T time steps Wavelets, O(N). log T frequency bands, T time steps Walsh series Libraries or Dictionaries of Bases (richer signal analysis) Nonstationary Time Series. c U. Bristol 18 / 32
65 Other Possibilities Single Bases: Fourier (of course). T /2 frequencies, T time steps Wavelets, O(N). log T frequency bands, T time steps Walsh series Libraries or Dictionaries of Bases (richer signal analysis) Local Cosine or Sine bases Nonstationary Time Series. c U. Bristol 18 / 32
66 Other Possibilities Single Bases: Fourier (of course). T /2 frequencies, T time steps Wavelets, O(N). log T frequency bands, T time steps Walsh series Libraries or Dictionaries of Bases (richer signal analysis) Local Cosine or Sine bases Wavelet packet library Nonstationary Time Series. c U. Bristol 18 / 32
67 Other Possibilities Single Bases: Fourier (of course). T /2 frequencies, T time steps Wavelets, O(N). log T frequency bands, T time steps Walsh series Libraries or Dictionaries of Bases (richer signal analysis) Local Cosine or Sine bases Wavelet packet library Adapted waveform analysis of Coifman, fast O(N log N) transforms. Nonstationary Time Series. c U. Bristol 18 / 32
68 Bell Function for Local Cosine Bases Figure 7.4 A bell used for a local cosine basis 37 Nonstationary Time Series. c U. Bristol 19 / 32
69 Local Cosine Basis Let {a k } be a sequence of real numbers and {ɛ k } of positive numbers such that a ±k ± and a k + ɛ k < a k+1 ɛ k+1 ; let b k (x) be the (ɛ k, ɛ k+1 ) bell over [a k, a k+1 ]; then {u k,j } where { } u k,j (x) = {2/(a k+1 a k )} 1/2 (2j + 1)π(x ak ) b k (x) cos, 2(a k+1 a k ) k Z, j N is an orthonormal basis of L 2 (R). Walter and Shen (2001) Theorem 7.3 (Originally Coifman and Meyer (1991)). Nonstationary Time Series. c U. Bristol 20 / 32
70 Local Cosine Basis Let {a k } be a sequence of real numbers and {ɛ k } of positive numbers such that a ±k ± and a k + ɛ k < a k+1 ɛ k+1 ; let b k (x) be the (ɛ k, ɛ k+1 ) bell over [a k, a k+1 ]; then {u k,j } where u k,j (x) = { 2/(a k+1 a k ) } { } 1/2 (2j + 1)π(x ak ) bk (x) cos, 2(a k+1 a k ) k Z, j N is an orthonormal basis of L 2 (R). Walter and Shen (2001) Theorem 7.3 Nonstationary Time Series. c U. Bristol 21 / 32
71 Local Cosine Basis Let {a k } be a sequence of real numbers and {ɛ k } of positive numbers such that a ±k ± and a k + ɛ k < a k+1 ɛ k+1 ; let b k (x) be the (ɛ k, ɛ k+1 ) bell over [a k, a k+1 ]; then {u k,j } where { } (2j + 1)π(x u k,j (x) = {2/(a k+1 a k )} 1/2 ak ) b k (x) cos, 2(a k+1 a k ) k Z, j N is an orthonormal basis of L 2 (R). Walter and Shen (2001) Theorem 7.3 Nonstationary Time Series. c U. Bristol 22 / 32
72 Figure 7.5 Three elements in the local cosine basis with bell of Figure Nonstationary Time Series. c U. Bristol 23 / 32
73 Figure 7.6 Two additional elements of the local cosine basis showing the bell 39 Nonstationary Time Series. c U. Bristol 24 / 32
74 Different Families... Back to Priestley: there is a multitude of different representations of the process, each representation based on a different family of functions. So, there are many things we might try. Not all of them are oscillatory functions, or we don t know E.g. wavelets, Locally Stationary Wavelet Processes: X t = w j,k ψ j,k t ξ j,k, j=1 k Z w j,k = amplitude, ψ = oscillation, ξ = randomness. Nonstationary Time Series. c U. Bristol 25 / 32
75 Different Families... Back to Priestley: there is a multitude of different representations of the process, each representation based on a different family of functions. So, there are many things we might try. Not all of them are oscillatory functions, or we don t know E.g. wavelets, Locally Stationary Wavelet Processes: X t = w j,k ψ j,k t ξ j,k, j=1 k Z w j,k = amplitude, ψ = oscillation, ξ = randomness. Nonstationary Time Series. c U. Bristol 25 / 32
76 Different Families... Back to Priestley: there is a multitude of different representations of the process, each representation based on a different family of functions. So, there are many things we might try. Not all of them are oscillatory functions, or we don t know E.g. wavelets, Locally Stationary Wavelet Processes: X t = w j,k ψ j,k t ξ j,k, j=1 k Z w j,k = amplitude, ψ = oscillation, ξ = randomness. Nonstationary Time Series. c U. Bristol 25 / 32
77 Different Families... Back to Priestley: there is a multitude of different representations of the process, each representation based on a different family of functions. So, there are many things we might try. Not all of them are oscillatory functions, or we don t know E.g. wavelets, Locally Stationary Wavelet Processes: X t = w j,k ψ j,k t ξ j,k, j=1 k Z w j,k = amplitude, ψ = oscillation, ξ = randomness. Nonstationary Time Series. c U. Bristol 25 / 32
78 A Wavelet Based Test for Stationarity (Nason 2013) Use raw wavelet periodogram, I j,k = d 2 j,k where d j,k = t X tψ j,k t Time-scale analogue of regular periodogram. Define β j (z) = EI j,k, where z = k/t Under stationarity H 0 function β j (z) is constant. In mind locally stationary wavelet process alternative, but not necessary Nonstationary Time Series. c U. Bristol 26 / 32
79 A Wavelet Based Test for Stationarity (Nason 2013) Use raw wavelet periodogram, I j,k = d 2 j,k where d j,k = t X tψ j,k t Time-scale analogue of regular periodogram. Define β j (z) = EI j,k, where z = k/t Under stationarity H 0 function β j (z) is constant. In mind locally stationary wavelet process alternative, but not necessary Nonstationary Time Series. c U. Bristol 26 / 32
80 A Wavelet Based Test for Stationarity (Nason 2013) Use raw wavelet periodogram, I j,k = d 2 j,k where d j,k = t X tψ j,k t Time-scale analogue of regular periodogram. Define β j (z) = EI j,k, where z = k/t Under stationarity H 0 function β j (z) is constant. In mind locally stationary wavelet process alternative, but not necessary Nonstationary Time Series. c U. Bristol 26 / 32
81 A Wavelet Based Test for Stationarity (Nason 2013) Use raw wavelet periodogram, I j,k = d 2 j,k where d j,k = t X tψ j,k t Time-scale analogue of regular periodogram. Define β j (z) = EI j,k, where z = k/t Under stationarity H 0 function β j (z) is constant. In mind locally stationary wavelet process alternative, but not necessary Nonstationary Time Series. c U. Bristol 26 / 32
82 A Wavelet Based Test for Stationarity (Nason 2013) Use raw wavelet periodogram, I j,k = d 2 j,k where d j,k = t X tψ j,k t Time-scale analogue of regular periodogram. Define β j (z) = EI j,k, where z = k/t Under stationarity H 0 function β j (z) is constant. In mind locally stationary wavelet process alternative, but not necessary Nonstationary Time Series. c U. Bristol 26 / 32
83 A Wavelet Based Test for Stationarity (Nason 2013) Use raw wavelet periodogram, I j,k = d 2 j,k where d j,k = t X tψ j,k t Time-scale analogue of regular periodogram. Define β j (z) = EI j,k, where z = k/t Under stationarity H 0 function β j (z) is constant. In mind locally stationary wavelet process alternative, but not necessary Nonstationary Time Series. c U. Bristol 26 / 32
84 A Wavelet Based Test for Stationarity, 2 Use Neumann and von Sachs (2000) to test constancy of β j (z) Uses Haar wavelet coefficients of I j,k as fn. of k, which are ˆv l,m Test H 0 : v l,m = 0 for all l, m, asymptotic Gaussian theory Use multiple test control, Bonferroni, FDR R package locits contains the software Nonstationary Time Series. c U. Bristol 27 / 32
85 A Wavelet Based Test for Stationarity, 2 Use Neumann and von Sachs (2000) to test constancy of β j (z) Uses Haar wavelet coefficients of I j,k as fn. of k, which are ˆv l,m Test H 0 : v l,m = 0 for all l, m, asymptotic Gaussian theory Use multiple test control, Bonferroni, FDR R package locits contains the software Nonstationary Time Series. c U. Bristol 27 / 32
86 A Wavelet Based Test for Stationarity, 2 Use Neumann and von Sachs (2000) to test constancy of β j (z) Uses Haar wavelet coefficients of I j,k as fn. of k, which are ˆv l,m Test H 0 : v l,m = 0 for all l, m, asymptotic Gaussian theory Use multiple test control, Bonferroni, FDR R package locits contains the software Nonstationary Time Series. c U. Bristol 27 / 32
87 A Wavelet Based Test for Stationarity, 2 Use Neumann and von Sachs (2000) to test constancy of β j (z) Uses Haar wavelet coefficients of I j,k as fn. of k, which are ˆv l,m Test H 0 : v l,m = 0 for all l, m, asymptotic Gaussian theory Use multiple test control, Bonferroni, FDR R package locits contains the software Nonstationary Time Series. c U. Bristol 27 / 32
88 A Wavelet Based Test for Stationarity, 2 Use Neumann and von Sachs (2000) to test constancy of β j (z) Uses Haar wavelet coefficients of I j,k as fn. of k, which are ˆv l,m Test H 0 : v l,m = 0 for all l, m, asymptotic Gaussian theory Use multiple test control, Bonferroni, FDR R package locits contains the software Nonstationary Time Series. c U. Bristol 27 / 32
89 Wavelet Test of Stationarity on Cardinham 1st Diffs First Difference Wind Speed at Cardinham (knots) Hour (Days since 1st January 2009) MC Type: FDR. 7 rejected. Nonstationary Time Series. c U. Bristol 28 / 32
90 Cardinham Localized Autocovariance Localized Autocovariance of Cardinham Wind First Differences Hour (Days since 1st January 2009) Nonstationary Time Series. c U. Bristol 29 / 32
91 Localized Autocovariance for Cardinham: 4 days 0400 Local Autocovariance of Cardinham Wind First Differences Lag c('4 days 04:00', lag) Nonstationary Time Series. c U. Bristol 30 / 32
92 Localized Autocovariance for Cardinham: 16 days 1600 Local Autocovariance of Cardinham Wind First Differences Lag c('16 days 16:00', lag) Nonstationary Time Series. c U. Bristol 31 / 32
93 Summary Nonstationary time series models Oscillatory & Semi-Stationary processes Multitude of representations, which one? Wavelets, Computational Harmonic Analysis Essential for picking up alternatives. Priestley: major contributions to statistics and time series. Nonstationary Time Series. c U. Bristol 32 / 32
94 Summary Nonstationary time series models Oscillatory & Semi-Stationary processes Multitude of representations, which one? Wavelets, Computational Harmonic Analysis Essential for picking up alternatives. Priestley: major contributions to statistics and time series. Nonstationary Time Series. c U. Bristol 32 / 32
95 Summary Nonstationary time series models Oscillatory & Semi-Stationary processes Multitude of representations, which one? Wavelets, Computational Harmonic Analysis Essential for picking up alternatives. Priestley: major contributions to statistics and time series. Nonstationary Time Series. c U. Bristol 32 / 32
96 Summary Nonstationary time series models Oscillatory & Semi-Stationary processes Multitude of representations, which one? Wavelets, Computational Harmonic Analysis Essential for picking up alternatives. Priestley: major contributions to statistics and time series. Nonstationary Time Series. c U. Bristol 32 / 32
97 Summary Nonstationary time series models Oscillatory & Semi-Stationary processes Multitude of representations, which one? Wavelets, Computational Harmonic Analysis Essential for picking up alternatives. Priestley: major contributions to statistics and time series. Nonstationary Time Series. c U. Bristol 32 / 32
98 Summary Nonstationary time series models Oscillatory & Semi-Stationary processes Multitude of representations, which one? Wavelets, Computational Harmonic Analysis Essential for picking up alternatives. Priestley: major contributions to statistics and time series. Nonstationary Time Series. c U. Bristol 32 / 32
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