ESTIMATING DAILY MEAN TEMPERATUREMarie NovakHarry PodschwitAaron Zimmerman

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?

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°C2

• Variance(Visby Island) ≈ 3.56°C2

The Models

• Iceland

• Edlund model (Sweden)

• Ekholm model (Sweden)

• a,b,c,d and e are specific to the time of year

• Min-Max model (U.S. and others)

Two-Stage Cosine Model

• Fit the sunlight portion of the day with a cosine model

• Use NLS: temp = A*cos(2π*B*time + C) + D + error• Add straight line segment between sunset and sunrise

• Integrate over the piecewise function and average

Trends by Pressure

Good Fit Bad Fit

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

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)

Just How Accurate Are These Models?Visby Island, Sweden

Model RMSE Mean error

95% CI

Iceland 0.4827 0.0015 -0.1785, 0.1815

Edlund 0.8406 0.0598 -0.2528, 0.3725

Ekholm 0.2851 -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

Model RMSE Mean error

95% CI

Iceland 0.6620 0.1572 -0.0870, 0.4014

Edlund 1.200 0.5605 0.1576, 0.9636

Ekholm 0.5442 -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

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?

Visby Island, Sweden Red Oak, IA

Observation Error Results

Visby Island, Sweden

Min. RMSE

Max.RMSE

MeanRMSE

Iceland 0.4744 0.7694 0.5530

Edlund 0.7382 0.8824 0.8308

Ekholm 0.2688 0.3193 0.2902

Red Oak, IA

Min. RMSE

Max.RMSE

MeanRMSE

Iceland 0.6385 1.477 0.8314

Edlund 0.4250 0.4829 0.4546

Ekholm 0.5309 0.6988 0.6292

Min. error

Max.error

Meanerror

Iceland -0.0544 0.2419 0.1518

Edlund 0.1334 0.5937 0.3411

Ekholm -0.1520 -0.0781 -0.1163

Min. error

Max.error

Meanerror

Iceland -0.0544 0.0939 -0.0034

Edlund -0.0474 0.0920 0.0276

Ekholm -0.0876 -0.0306 -0.0585

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

Visby Island, Sweden Red Oak, IA

Observation Error Results

Visby Island, Sweden

Min. RMSE

Max.RMSE

MeanRMSE

Iceland 0.4373 0.9288 0.5648

Edlund 0.6927 0.9320 0.8464

Ekholm 0.3066 0.5043 0.4018

Red Oak, IA

Min. RMSE

Max.RMSE

MeanRMSE

Iceland 0.6111 1.5858 0.8475

Edlund 1.0256 1.4387 1.2454

Ekholm 0.3171 0.6031 0.4670

Min. error

Max.error

Meanerror

Iceland -0.2306 0.3990 0.1920

Edlund 0.1531 0.1531 0.4167

Ekholm -0.2478 0.0124 -0.1171

Min. error

Max.error

Meanerror

Iceland -0.0902 0.1601 -0.0025

Edlund -0.1127 0.1209 0.0316

Ekholm -0.2387 -0.0816 -0.1568

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

Visby, Sweden Red Oak, IA

Pearson correlation coefficient: 0.373Spearman correlation coefficient: 0.358

Pearson correlation coefficient: 0.583Spearman correlation coefficient: 0.562

Visby Island, Sweden

Before noon

After noon

Positive error

43238 22187

Negative error

29580 4995

Red Oak, IA

Before noon

After noon

Positive error

25850 20773

Negative error

46920 6457

The contingency table of the simulated data.[X2=4330.182, p-value < 2.2 * 10-

16]=0.208ⱷ

The contingency table of the simulated data.[X2=13231.4, p-value < 2.2 * 10-16]

=0.364ⱷ

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.

Conclusions• There was no one “best” model

• Geographic/seasonal factors• Edlund modellowest 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?