Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends A Statistical Analysis of Snow Depth Trends in North America Jonathan Woody Mississppi State University Statistics June 13, 2017 Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends What is a Changepoint? A changepoint is a time of discontinuity in the structure of a time series of data { X 1 , X 2 , . . . } . A series could experience a change in first moment, variance, or even in distribution. Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends Changepoints Wreak Havoc on Inferences:Tuscaloosa, Al Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends A Stochastic Storage Model for Assessment of Daily Snow Depth Trends Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends Napolean Data We use the Napoleon, North Dakota daily snow depth records from January 1, 1901 to December 31, 2003. The data consists of 37,595 daily measurements. Approximately 3% of the daiy measurements are missing. The missing data occur approximately uniformly over seasons when snowfall is possible. Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends Napoleon Meta-data Table: Napoleon Meta-data Date # Time Change Observer Note 8/19/1939 18:00 Site was already established CJ Hoof Station began 4/1/1889 1.3 mi SE of PO 6/20/1946 7:45 New observer, no move Gladys Peterson Station 1.5 mi SE of post office 7/8/1948 Station moved to another part Gladys Peterson New thermometer support of farm 11/11/1949 18:30 Moved 70 feet SE Gladys Peterson 1.3 mi SE of post office to improve exposure 3/17/1954 1 18:00 Moved to 1.5 mi NW of Napoleon Ted Frank 0.3 mi N of post office 4/18/1956 1A New observer, no move Alvin Schuchard 2/19/1957 1B New observer, no move Warren Wentz 5/8/1957 2 18:00 New observer, moved Gladys Peterson At ice cream store 3 blocks east of PO, 3.5 blocks SE of old location 7/1/1958 2A Recording rain gauge removed 8/28/1958 3 18:00 Equipment moved 0.6 mi W Gladys Peterson Moved to 0.3 mi W of PO to observer’s house 9/30/1965 4 18:00 No move, update form Gladys Peterson 9/10/1968 4A New observer, no move Warren Wentz 8/18/1969 5 7:00 Moved to Soo Depot, 0.5 mi E Warren Wentz Moved to more convenient location, station at 0.1 mi NE of PO 12/1/1973 5A New observer, no move Terry Wentz No move 6/14/1976 6 7:00 Moved to mother’s house Warren Wentz Station at 0.5 mi NE 0.4 mi NE 6A Address correction 7/11/1985 6B 7:00 Moved across street Warren Wentz MMTS installed to observer’s house 12/23/1987 7 7:00 No move, update form 10/20/1992 8 8:00 Moved 0.1 mi SW Bruce Wentz Son of previous observer, to new residence now 0.4 mi NE of PO Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends Consider the Snowdepth Data of Napolean, ND 120 100 Snow Depth (Centimeters) 80 60 40 20 0 1901 1911 1921 1931 1941 1951 1961 1971 1981 1991 2001 Year Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends Winter 1977-78 120 100 Snow Depth (Centimeters) 80 60 40 20 0 Oct Nov Dec Jan Feb Mar Apr May Jun Time of Year Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends The Model Our model is based on the storage balance equation X t = max { X t − 1 + Z t , 0 } . We assume that { Z t } is white noise, independent of { X t − 1 , X t − 2 , . . . , X 1 } , with periodic dynamics: E [ Z t ] = m t and Var( Z t ) = w 2 t . We assume that Z t is normally distributed. Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends Model Parameterization For seasonal dynamics, we assume that { m t } and { w t } are periodic in time with period T = 365 days. Let � � 2 π ( t − ρ ) � � m t = P t A + B cos + δ t + α t , T with ∆ 1 , 1 ≤ t < τ 1 ∆ 2 , τ 1 ≤ t < τ 2 δ t = . . . . . . . ∆ k , τ k − 1 ≤ t ≤ N Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends Model Estimation Let θ = ( A , B , ρ, α, ∆ 2 , . . . , ∆ k ) ′ be a vector containing all model parameters. The sum of squares function that we will minimize is N d − 1 T ( X t − ˆ ( X nT + ν − ˆ X t ) 2 X nT + ν ) 2 � � � S ( θ ) = = , σ 2 σ 2 t ν t =1 n =0 ν =1 t = E [( X t − ˆ where σ 2 X t ) 2 | X t − 1 ] and d = N / T = 103 is the number of years of data. Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends One-step Ahead Predictors Since { X t } is a periodic Markov chain, ˆ X t = E [ X t | X t − 1 ]. An explicit form is needed for the conditional mean is � � − X t − m t +1 �� E [ X t +1 | X t ] = { X t + m t +1 } 1 − Φ w t +1 � X t + m t +1 � + w t +1 φ . w t +1 Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends Smoothed Daily Standard Deviations 45 Smoothed Daily Standard Deviations 40 Sample Standard Deviation 35 Sample Stndard Deviation 30 25 20 15 10 5 0 Oct 17 Nov 1 Dec 1 Jan 1 Feb 1 March 1 April 1 May 1 Day of Year Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends Asymptotic Properties It can be shown that the estimator θ that minimizes S ( θ ) is consistent and asymptotically normal in that the distributional convergence (Kimko and Nelson, 1978) D ˆ θ − → N ( θ , F / d ) , as d → ∞ is achieved. F / d can be approximated by the inverse of the second derivative matrix of S ( θ ) evaluated at θ = ˆ θ : � − 1 � � ∂ 2 S ( θ ) � F / d ≈ . � ∂ θ ∂ θ ′ � θ =ˆ θ Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
Snow Depth Trends-A Single Series Mississippi State University North American Snow Depth Trends Table: Summary of Model Parameter Estimates Parameters 17 Breakpoints 5 Breakpoints No Breakpoints A -2.5982 (0.0755) -2.5951 (0.0742) -2.7224 (0.0678) B 2.9144 (0.0707) 2.9062 (0.0700) 2.9002 (0.0700) ρ 3.8374 (0.6811) 3.7964 (0.6823) 3.8986 (0.6810) α -0.5004 (0.1927) -0.4748 (0.1803) 0.2250 (0.0466) ∆ 1 (8 / 19 / 1939) 0.2558 (0.0728) 0.2263 (0.0629) - ∆ 2 (6 / 20 / 1946) 0.1269 (0.1114) - - ∆ 3 (7 / 8 / 1948) 0.4639 (0.1421) - - ∆ 4 (11 / 11 / 1949) 0.2419 (0.0911) - - ∆ 5 (3 / 17 / 1954) 0.1924 (0.1196) - - ∆ 6 (4 / 18 / 1956) -0.0360 (0.1949) 0.0074 (0.0960) - ∆ 7 (2 / 19 / 1957) 0.3707 (0.5387) - - ∆ 8 (5 / 8 / 1957) -0.2248 (0.1905) - - ∆ 9 (7 / 1 / 1958) - - - ∆ 10 (8 / 28 / 1958) 0.0626 (0.0990) - - ∆ 11 (9 / 30 / 1965) -0.0381 (0.1327) - - ∆ 12 (9 / 10 / 1968) 0.5633 (0.2352) 0.5626 (0.1118) - ∆ 13 (8 / 18 / 1969) 0.4975 (0.1225) - - ∆ 14 (12 / 1 / 1973) 0.7380 (0.1559) - - ∆ 15 (6 / 14 / 1976) 0.5899 (0.1285) - - ∆ 16 (7 / 11 / 1985) 0.2021 (0.1685) 0.2835 (0.1442) - ∆ 17 (12 / 23 / 1987) 0.3923 (0.1685) - - ∆ 18 (10 / 20 / 1992) 0.6332 (0.1565) 0.6123 (0.1473) - Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America
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