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ESTIMATING DAILY MEAN TEMPERATURE Marie Novak Harry Podschwit Aaron Zimmerman

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Questions If daily temperatures were described by a sine curve, the average daily temperature would indeed be the average of min and max. How well is daily temperature described by a sine curve? What is the effect on bias and variability of different observational schemes?

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The Data Times and locations January - Visby Island, Sweden June - Red Oak, Iowa, USA Temperature measurements taken every minute Red Oak data more variable than the January data Variance(Red Oak) ≈ 6.05°C 2 Variance(Visby Island) ≈ 3.56°C 2

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The Models

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Two-Stage Cosine Model Fit the sunlight portion of the day with a cosine model Use NLS: temp = A*cos(2 π*B*time + C) + D + error Add straight line segment between sunset and sunrise Integrate over the piecewise function and average

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Trends by Pressure

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Good Fit Bad Fit

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Bias and Variability Model error Error between different models Measurement error Error in observation times Error in linear combination models Error in different linear combination schemes

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Model Error Investigated the tendency of 5 different models to over/under-estimate the daily mean temperature Iceland model Edlund model Ekholm model U.S. model 2-Stage Cosine model (Aaron’s model)

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Just How Accurate Are These Models? Visby Island, Sweden ModelRMSEMean error 95% CI Iceland0.4827 0.0015-0.1785, 0.1815 Edlund0.8406 0.0598-0.2528, 0.3725 Ekholm0.2851-0.0665-0.1699, 0.0368 Min- Max 0.4287-0.0938-0.2485, 0.0637 2-Stage Cosine 0.4007 0.0635-0.0840, 0.2110 Red Oak, IA ModelRMSEMean error 95% CI Iceland0.6620 0.1572-0.0870, 0.4014 Edlund1.200 0.56050.1576, 0.9636 Ekholm0.5442-0.2089-0.3998, 0.0181 Min- Max 0.8380-0.2040-0.5126, 0.1048 2-Stage Cosine 0.4340-0.1555-0.3118, 0.0008

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What Are the Consequences of Errors in Measurement Time? Recalculate the error of each model for all of the temperature values from the bottom to the top of the hour How does the error change if you were 1 minute late in taking your measurements? 5 minutes? 59 minutes?

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Visby Island, SwedenRed Oak, IA

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Observation Error Results Visby Island, Sweden Min. RMSE Max. RMSE Mean RMSE Iceland0.47440.76940.5530 Edlund0.73820.88240.8308 Ekholm0.26880.31930.2902 Red Oak, IA Min. RMSE Max. RMSE Mean RMSE Iceland0.63851.4770.8314 Edlund0.42500.48290.4546 Ekholm0.53090.69880.6292 Min. error Max. error Mean error Iceland-0.05440.24190.1518 Edlund0.13340.59370.3411 Ekholm-0.1520-0.0781-0.1163 Min. error Max. error Mean error Iceland-0.05440.0939-0.0034 Edlund-0.04740.09200.0276 Ekholm-0.0876-0.0306-0.0585

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What If You Were Really, Really Bad at Taking Measurements? Simulated error in observation times by randomly sampling data points within the hour Simulation repeated 10,000 times and RMSE of daily mean temperature over the month calculated

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Visby Island, SwedenRed Oak, IA

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Observation Error Results Visby Island, Sweden Min. RMSE Max. RMSE Mean RMSE Iceland0.43730.92880.5648 Edlund0.69270.93200.8464 Ekholm0.30660.50430.4018 Red Oak, IA Min. RMSE Max. RMSE Mean RMSE Iceland0.61111.58580.8475 Edlund1.02561.43871.2454 Ekholm0.31710.60310.4670 Min. error Max. error Mean error Iceland-0.23060.39900.1920 Edlund 0.1531 0.4167 Ekholm-0.24780.0124-0.1171 Min. error Max. error Mean error Iceland-0.0902 0.1601-0.0025 Edlund-0.1127 0.1209 0.0316 Ekholm-0.2387-0.0816-0.1568

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What Kind of Biases Are Possible From Linear Combinations of Temperature Data? Performed a Monte Carlo simulation in which the daily mean temperature was calculated with a random linear combination of the temperature data points taken at every hour Dot product of random weighting and hourly temperature readings

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Visby, SwedenRed Oak, IA Pearson correlation coefficient: 0.373 Spearman correlation coefficient: 0.358 Pearson correlation coefficient: 0.583 Spearman correlation coefficient: 0.562

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Visby Island, Sweden Before noon After noon Positive error 4323822187 Negative error 295804995 Red Oak, IA Before noon After noon Positive error 2585020773 Negative error 469206457 The contingency table of the simulated data.[X 2 =4330.182, p-value < 2.2 * 10 - 16 ] ⱷ=0.208 The contingency table of the simulated data.[X 2 =13231.4, p-value < 2.2 * 10 -16 ] ⱷ=0.364

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Conclusions For Visby Island, little inter-hour variation For Red Oak, enough inter-hour variation to make meaningful changes to model given error in measurement times Linear combinations of temperature data tended to underestimate DMT when more weight was put on temperatures early in the day. Similarly, the models tended to overestimate when more weight was put on temperatures later in the day.

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Conclusions There was no one “best” model Geographic/seasonal factors Edlund model lowest RMSE for Visby Island but not for Red Oak, IA Iceland model lowest mean error for Visby Island, highly variable The Ekholm and Min-Max model tended to underestimate for both data sets but not significantly so For Red Oak data, the 2-stage cosine model tended to underestimate; the Iceland and Edlund models tended to overestimate (although Iceland not significantly) Implications for worldwide standardized method of measurement?

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