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Mississippi State University Snow Depth Trends-A Single Series 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

Mississippi State University Snow Depth Trends-A Single Series 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 {X1, X2, . . .}. 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

Mississippi State University Snow Depth Trends-A Single Series 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

Mississippi State University Snow Depth Trends-A Single Series 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

Mississippi State University Snow Depth Trends-A Single Series 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

Mississippi State University Snow Depth Trends-A Single Series 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

• f post office

7/8/1948 Station moved to another part Gladys Peterson New thermometer support

• f 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

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Consider the Snowdepth Data of Napolean, ND

1901 1911 1921 1931 1941 1951 1961 1971 1981 1991 2001 20 40 60 80 100 120

Year Snow Depth (Centimeters)

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Winter 1977-78

Oct Nov Dec Jan Feb Mar Apr May Jun 20 40 60 80 100 120

Time of Year Snow Depth (Centimeters)

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

The Model

Our model is based on the storage balance equation Xt = max{Xt−1 + Zt, 0}. We assume that {Zt} is white noise, independent of {Xt−1, Xt−2, . . . , X1}, with periodic dynamics: E[Zt] = mt and Var(Zt) = w2

t . We assume that Zt is normally distributed.

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Model Parameterization

For seasonal dynamics, we assume that {mt} and {wt} are periodic in time with period T = 365 days. Let mt = Pt

• A + B cos

2π(t − ρ) T

• + δt + αt
• ,

with δt =          ∆1, 1 ≤ t < τ1 ∆2, τ1 ≤ t < τ2 . . . . . . ∆k, τk−1 ≤ t ≤ N .

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series 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 S(θ) =

N

• t=1

(Xt − ˆ Xt)2 σ2

t

=

d−1

• n=0

T

• ν=1

(XnT+ν − ˆ XnT+ν)2 σ2

ν

, where σ2

t = E[(Xt − ˆ

Xt)2|Xt−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

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

One-step Ahead Predictors

Since {Xt} is a periodic Markov chain, ˆ Xt = E[Xt|Xt−1]. An explicit form is needed for the conditional mean is E[Xt+1|Xt] = {Xt + mt+1}

• 1 − Φ

−Xt − mt+1 wt+1

• +

wt+1φ Xt + mt+1 wt+1

• .

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Smoothed Daily Standard Deviations

Oct 17 Nov 1 Dec 1 Jan 1 Feb 1 March 1 April 1 May 1 5 10 15 20 25 30 35 40 45 Day of Year Sample Stndard Deviation Smoothed Daily Standard Deviations Sample Standard Deviation

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series 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) ˆ θ

D

− → 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: F/d ≈ ∂2S(θ) ∂θ∂θ′ −1

• θ=ˆ

θ .

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series 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

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

The Mean Daily Change in Snow Depth Function mt

1901 1911 1921 1931 1941 1951 1961 1971 1981 1991 2001 −0.5 0.5 1

Year centimeters M(t)−5 breakpoints

1901 1911 1921 1931 1941 1951 1961 1971 1981 1991 2001 −0.5 0.5 1

M(t)− 17 breakpoints Year Centimeters

1901 1911 1921 1931 1941 1951 1961 1971 1981 1991 2001 −0.5 0.5 1

Year Centimeters M(t)− Zero Breakpoints

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Snow Depths with One-day Ahead Preditions

Oct 17 Nov 1 Dec 1 Jan 1 Feb 1 March 1 Apr 1 May 1 5 10 15 20 25 30 35 40 45 50 Time of Year Snow Depth (Centimeters) Predicted Snow Depth Actual Snow Depth Upper 95 Percent Confidence Bound Lower 95 Percent Confidence Bound

We see the the model with the conditional least squares estimates tracks the daily data tightly.

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Residual Analysis

−5 −4 −3 −2 −1 1 2 3 4 5 −10 −5 5 10 Standard Normal QQ Plot of Residuals Standard Normal Quantile Quantile of Residual −10 −8 −6 −4 −2 2 4 6 8 10 2000 4000 6000 Histogram of Residuals Residuals Frequency −8 −6 −4 −2 2 4 6 8 10 −10 −5 5 10 Lag One Plot of Residuals Residual at Time t Residual at Time t−1

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Snow depth Predictions for the Final 10 Years of Data

94 95 96 97 98 99 00 01 02 03 10 20 30 40 50 60 70 80 Year Snow Depth (Centimeters) Actual and Predicted Snow Depths for Last 10 Years of Data

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

North American Snow Depth Trends

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

The Data

The creation of 1◦ × 1◦ daily snow depth grids from station level data from 1900-2000 is discussed by Dyer and Mote (2006). Kluver et al. (2016) update the creation and validation procedures introduced in Dyer and Mote (2006), and extend the time record to 1900-2009. The grids considered in the data examined here extend from 25◦ N to 71◦ N latitude and from 53◦ W to 168◦ W longitude.

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

The Data Continued...

Station data are interpolated to 1◦ × 1◦ grids via the Spheremap spatial interpretation program by Willmott et

• al. (1984).

Kluver et al. (2016) report station density and found that the number of stations in a grid cell reporting snow depths varies widely with time (see Figure 2 of Kluver et al. 2016), with a large increase in the number of reporting stations beginning in 1948 due to the full digitization of Cooperative Observation Program (COOP) station network. Kluver et al. (2016) caution against using pre-1948 data in estimating trends, since station density is sparse before this time.

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Let {Xt}N

t=1 denote the snow depth series at a fixed grid for

days t = 1, . . . , N At a given grid, define Yd = #(W)−1

ν∈W XdT+ν as the

average winter snow depth for WCY d ∈ {0, . . . , n − 1}, where #(A) is the number of elements in the set A and W denotes the snow season for this grid. Our basic model for the yearly average snow depths {Yd}n−1

d=0

is the time series regression Yd = α + γd + µd + ǫd. (1)

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Here, α models the overall mean γ is a linear trend parameter µd is a changepoint effect {ǫd}n−1

d=0 is a zero-mean first order autoregressive (AR(1))

error process that induces temporal correlation into the yearly averages. The changepoint effect µd, with k mean shifts at times τ1 < τ2 < . . . < τk, respectively, is µd =          ∆0 = 0, τ0 < d < τ1 ∆1, τ1 ≤ d < τ2 . . . . . . ∆k, τk ≤ d < τk+1 , where τ0 = 0 and τk+1 = n by convention. Here, ∆ℓ is interpreted as the changepoint effect of the ℓth regime.

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

While the parameter γ controls the trend and is the object of

• ur study, it is not easily interpretable within the storage

equation For example, because of the maximum in the storage equation, ˆ γ = 50 does not imply a depth change of 50 units per time. A quantity that does have interpretable units of depth change per time is ˆ β = k

r=1

• n∈Hr

T

ν=1

• XnT+ν − ¯

Xr(ν)

• (nT + ν − ¯

tr(ν)) k

r=1

• n∈Hr

T

ν=1 (nT + ν − ¯

tr(ν))2 , where Hr = {t : τr−1 ≤ t < τr} denotes the set of times in the rth time regime, ¯ Xr(ν) denotes the average snow depth on day ν, and ¯ tr(ν) denotes the average time during the rth regime, respectively.

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Changepoint details and Diagnostic at Warm Lake

1950 1960 1970 1980 1990 2000 100 200 300 Warm Lake, ID: Daily Snow Depths CM 1950 1960 1970 1980 1990 2000 50 100 150 Count Changepoints by Year: Entire Study Region 1950 1960 1970 1980 1990 2000 100 200 300 CM Warm Lake, ID: Simulated Daily Snow Depths 1950 1960 1970 1980 1990 2000 50 100 150 200 Warm Lake, ID: Average Yearly Snow and Model Fits CM MDL Fit Average Yearly Snow Depth Simple Linear Trend Fit

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Simulated Data ˆ β Estimates: No Changepoints

< -10

• 10 - -9
• 9 - -8
• 8 - -7
• 7 - -6
• 6 - -5
• 5 - -4
• 4 - -3
• 3 - -2
• 2 - -1
• 1 - 0

0 - 1 1 - 2 2 - 3 3 - 4 4 - 5 5 - 6 6 - 7 7 - 8 8 - 9 9 - 10 10 <

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Simulatedˆ β Estimates: Changepoints Included

< -10

• 10 - -9
• 9 - -8
• 8 - -7
• 7 - -6
• 6 - -5
• 5 - -4
• 4 - -3
• 3 - -2
• 2 - -1
• 1 - 0

0 - 1 1 - 2 2 - 3 3 - 4 4 - 5 5 - 6 6 - 7 7 - 8 8 - 9 9 - 10 10 <

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Simulated Z Scores: No Change Points

< -5.0

• 5.0 - -4.5
• 4.5 - -4.0
• 4.0 - -3.5
• 3.5 - -3.0
• 3.0 - -2.5
• 2.5 - -2.0
• 2.0 - -1.5
• 1.5 - -1.0
• 1.0 - -0.5
• 0.5 - 0.0

0.0 - 0.5 0.5 - 1.0 1.0 - 1.5 1.5 - 2.0 2.0 - 2.5 2.5 - 3.0 3.0 - 3.5 3.5 - 4.0 4.0 - 4.5 4.5 - 5.0 5.0 <

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Simulated Z Scores: Changepoints Included

< -5.0

• 5.0 - -4.5
• 4.5 - -4.0
• 4.0 - -3.5
• 3.5 - -3.0
• 3.0 - -2.5
• 2.5 - -2.0
• 2.0 - -1.5
• 1.5 - -1.0
• 1.0 - -0.5
• 0.5 - 0.0

0.0 - 0.5 0.5 - 1.0 1.0 - 1.5 1.5 - 2.0 2.0 - 2.5 2.5 - 3.0 3.0 - 3.5 3.5 - 4.0 4.0 - 4.5 4.5 - 5.0 5.0 <

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America

Mississippi State University Snow Depth Trends-A Single Series North American Snow Depth Trends

Thank You!!

Jonathan Woody Mississppi State University Statistics A Statistical Analysis of Snow Depth Trends in North America