Your student project may fit ARIMA processes to daily temperature or rainfall.

Take heed: This project template discussed daily temperature. Rainfall is more complex, since the error term is not normally distributed. We explain some rainfall items, but you must do independent work. Keep in mind two things:

The project template on daily temperature is for candidates who need more guidance and

have difficulty designing their own projects. We expect candidates who choose this project template to provide complete write-ups of the statistical reasoning for ARIMA modeling.

Global warming is a much discussed topic. Many web sites analyze temperature changes and other historical weather statistics. We encourage you to explore other ideas for student projects besides ARIMA modeling of daily temperature.

Take heed: The weather service provides data bases for Fortran and SAS applications. We have placed time series data on the NEAS web site in Excel (CDS) formats. Download a compressed file, extract one time series, open it in Excel, then save it in Excel format.

Weather data are well suited to ARIMA modeling, since the patterns are seasonal with strong autoregressive properties. Several candidates have modeled daily temperature or daily rainfall.

Illustration: Ambitious candidates may do a project using hourly temperature, combining daily seasonality with yearly seasonality, strong autoregression, and cycles from warm/cold front movements. We provide data from one weather station in text fromat.

Daily temperature is seasonal: high in the summer and low in the winter.

De-seasonalize the data and fit an ARIMA process.

We use an ARIMA process with a 12 month autoregressive parameter.

The volatility of daily temperature overwhelms the seasonal pattern. The first differences may be positive 55% of the time from mid-January to mid-July and negative the other 45%. The time series is a jagged curve: high stochastic variation overlaid on a smooth cycle.

To smooth the seasonal pattern, use a multi-year centered moving average.

Illustration: For the expected temperature on January 10, we might use a fifty year average of the average daily temperature on January 3 through January 17. The average of 50 × 15 days = 750 days gives a smooth annual cycle.

Take heed: The type of average depends on the length of the time series, such as

You may adjust for long-term trends and cycles in daily temperature.

Take heed: For rainfall, use a longer-term average, such as 100 years with 25 days in each year.

Compare your averages with those published by the weather service. Weather.com shows daily averages for every location in the U.S.; see their monthly forecasts.)

Several candidates have done student projects on daily temperatures. A few of them used monthly averages, relying on our recommendations for interest rates. For weather studies, use daily figures, not monthly figures.

For good ARIMA models, we work with many years, giving thousands of data points. Excel handles all the data crunching for you.

De-seasonalizing the daily temperature for a single year removes the trend as well (if any). If the time series has long-term trends or cycles, use separate periods or detrend the data.

Daily rainfall (precipitation) is an interesting time series project template. Candidates who live in rainy climes or who want a project with different properties may use daily rainfall.

(1) The daily rainfall time series is not a differentiable function, since many days have zero rainfall. We can model the time series with an ARIMA process, but the error terms are not normally distributed. Tests of significance assume a normal distribution, so be careful in your interpretation of the results. Bartlett’s test is not appropriate for this time series.

Recommendation: We can use time series models, but hypothesis testing is harder. An ARIMA process does not model the time series adequately. Your student project may form an ARIMA process and show that the residuals do not have a normal distribution.

If rain is infrequent in a location, do not use daily rainfall. You can not properly evaluate the ARIMA models, and no model fits well.

(2) Rainfall is seasonal, but the seasonality is complex and varies by location. Use a longer time period and more smoothing to get the mean daily rainfall.

The number of days for the long-term average depends on the percentage of days with rainfall. A hundred days with rainfall is sufficient for the student project. If rain occurs one day in four, use 400 days. For 25 years of data, use a 17 day centered moving average.

If the average daily rainfall is not smooth, use a longer centered moving average, such as 25 days or 31 days. But be careful: a wider moving average loses information. If average rainfall differ between May and June and we use a 90 day moving average, we get smooth curves that have less information. If we use a 365 day moving average, we get even smoother curves but we lose all the seasonality.

Take heed: Examine the graphs of different moving averages. Choose an average that smooths stochasticity but retains seasonality.

The weather service provides mean daily rainfall for the month. Do not use this monthly mean for each day of the month, as it distorts the autocorrelations.

Illustration: Suppose the monthly means of daily rainfall are 4 for May and 2 for June. May 31 and June 1 might have the same rainfall of 3. But the seasonally adjusted figures are –1 for May 31 and +1 for June 1. This gives a negative sample autocorrelation for consecutive days with the same rainfall – an illogical result.

Determine the average daily rainfall for each day. Verify your averages with the monthly means of the weather service.

Illustration: Suppose the monthly means of daily rainfall are 4 for May and 2 for June. Your analysis gives an overall average of 6 for May and 8 for June. You have probably made an error. But if your overall averages are 3.6 for May and 1.8 for June, your results are close enough to proceed with the analysis.

(3) Rainfall and daily temperature are related. We can construct a simple regression model using daily temperature and daily rainfall and fit an ARIMA process to the residuals.

A change in the daily temperature often heralds rainfall, as a warm front pushes against a cold front. Your student project can regress daily rainfall on the first differences of the daily temperature and model the residuals with an ARIMA process.

The student project does not construct a real weather forecast. We don’t actually forecast daily rainfall by an ARIMA process on the residuals of a regression on daily temperature. We encourage structural models (residuals from a regression analysis) for the student project because they allow us to apply more techniques from the on-line courses and they often produce better fitting ARIMA models.

The NEAS web site has files with high and low daily temperatures in degrees Fahrenheit and daily rainfall in hundredths of an inch for 1,221 national weather service locations in the continental U.S. The weather service collects daily temperatures from over 3,000 locations, and it publishes long-term historical file for these 1,221 locations.

The starting and ending dates vary by location. Most files begin before 1900 and end either at 2000 or at 2005.

To choose a location, start with the file Station Names.

To pick a particular station, highlight Column D (the state abbreviations) and find (control F) the state you want, such as "NY." Select one of the stations, such as New York Central Park or Rochester Airport. Each station has a code: 305801 = New York Central Park and 307167 = Rochester Airport.

The weather station are not in numerical order. Check which WinZip file has the weather station you want. The table showing the zip file for each weather station may not yet be posted on the discussion forum.

If you are not choosy, use a station with many years of data and few missing records.

The data files are in 14 WinZip files, each of which contains up 120 station files.

Save the file in Excel format, so that your formulas, charts, and graphs are saved. CSV is a compressed format, so we can store records for 1,221 stations. For a student project with one or two stations, use standard Excel format.

Take heed: If you can’t open a CSV file or save it in Excel format, post a question on the discussion forum. Most candidates can un-zip compressed files, but some machines lack the required software. We will place several plain Excel files on the discussion forum.

Some observations are missing in almost every file. These records show a high or low daily temperature of –999 or 0. Correct the data one of two ways:

Both corrections slightly distort the ARIMA model. With 100 years of data, a few missing values does not have a material distortion.

Take heed: Inspect the data to ensure its integrity. If you overlook missing values, your time series analysis may be invalid, and you will have wasted several hours.

Illustration: Station 280325

Station 280325 (Atlantic City State Marina, New Jersey) has 48,212 records from January 1, 1874, to December 31, 2005. December 31, 1894, is missing data:

Interpolate the high daily temperature as 31 and the low daily temperature as 16.5.

Most missing values are at a single date. Some values are missing for several days.

Illustration: Station 280325 is missing data for 5/1/1987-5/4/1987. We interpolate for the missing values as

If values are missing for a long sequence of days, delete the rows.

Values are missing for 10/1/1998 – 6/30/1990. Delete these rows from your work-sheet. The one day-to-day change from 9/30/1998 to 7/1/1990 slightly distorts the ARIMA model, but it does not have a material effect on a time series of 48,000 values.

Take heed: The denominator for the daily average over all years may depend on the day. For Station 280325, the denominator is either 130 or 131. We use the countif and sumif built-in functions to form averages.

Take heed: If you delete rows, a 365 day seasonal autoregressive parameter is distorted. For daily temperature, we de-seasonalize the data; we do not use seasonal autoregressive parameters. See the discussion forum posting on seasonality.

This step is geared to candidates with no knowledge of Excel. If you use dates in Excel, you can perform the items here more efficiently.

The observation day is in column A in MM/DD/YYYY format (month/day/year format). You must extract the month and day to compute averages for seasonal adjustments.

The illustrative worksheet determines a "day of the year index" as month × 100 + day.

This MMDD format is easy to understand. We use this number for the countif and sumif built-in functions.

Excel has a built-in function to convert dates to an index starting at January 1, 1900. If you are familiar with Excel’s date indices, they may simplify your code. The instructions here are geared to new Excel users; they are simple, but not efficient.

Caution: Take care to form monthly averages correctly. Not all days of the year have the same number of observations if you eliminate some records.

Alternatively, use data from 1/1/1874 to 12/31/1997 for your student project. You must still correct a few missing numbers, but the interpolations should take a few minutes.

To analyze seasonality, form the month, day, and year of each observation.

Illustration: For the date 3/18/1926 in cell A19070:

For the date 3/18/1874 in cell A78, the date functions return value!. We manually extract the month, day, and year.

In the daily temperature files, dates before 1/1/1990, have the format MM/DD/YYYY.

The Excel string function return strings, not numbers. To convert them to numbers, use the Excel value built-in function.

To compute average daily temperature for a given day, use the Excel sumif and countif built-in functions.

Take heed: If the date strings in Column A did not have ten characters each, we would search for the slashes in the string: =search("/",A78,1).

To de-seasonalize the daily temperatures, we assign each day of the year a unique value. For simplicity, we use the value month × 100 + day

We use this value for the countif and sumif functions.

Seasonality has several forms. The discussion forum posting on seasonality explains the types of seasonality and the methods used to adjust for them.

The daily temperature depends on the time of year, not on the value at the same date one year ago. We de-seasonalize the data; we do not use a 365 day autoregressive parameter.

Take heed: The simplest way to code the average daily temperature for each record in the file is to copy these formulas to Cells M1347585. For each row, we compute the number of times the day occurs in the time series, the sum of the daily high temperatures for that day, and the average daily high temperature for that day.

But countif and sumif are memory intensive functions. Excel searches through 47,574 records for the day and then adds the high temperature to the sum. Excel does 2 × 47,574² = 4,526,570,952 operations, which can bring your machine to a halt. We use a less memory intensive procedure to compute the average daily temperatures.

We create a V-Look-Up Table for the day index and the average daily temperature.

First we add a row for February 29, which does not occur in 1874. Enter the date 229 in Cell M11, and copy Cells M1212 to Cells M1111. Note that February 29 occurs 32 times in this file, not 130 times.

Select the cells L11376. We need columns L and O (not Columns M and N), but the extra columns in the V-Look-Up Table don’t cause a problem.

Choose insert from the menu bar; choose names > create. Enter the name VlookupTable for the cells L11376.

Place the formula vlookup(L377,VlookupTable,4) in Cell O377. The formula says:

Look up the value in Cell L377 in the first column of the VlookupTable (Cells L11:L376). From the matching row of the table, take the value in the fourth column (Column O), which is the average daily temperature.

Copy the formula in Cell O377 to Cells O37847585. Each row has the average daily temperature for its day.

Take heed: The instructions here mix simplicity and efficiency. The ARIMA modeling can be done more efficiently in VBA, in "R," in SAS, or in MINITAB. If you use VBA for the average temperatures, your spread-sheet will run faster. For daily correlograms of a 100+ year time series, use the VBA macro in the illustrative work-sheet.

We want the expected temperature to determine the seasonally adjusted temperature.

Recommendation: Verify this relation. Compute the standard deviation of daily temperature in January (using Excel’s built-in function) and the standard deviation of the average daily temperature in January. The relation will not be precise, because the expected temperature is not the same on each day of January.

The average daily temperature should form a smooth sine-shaped curve, with a peak in July and a nadir (low point) in January.

Form a line chart using the chart wizard.

Take heed: The illustrative worksheets do not graph the data. Graphs are formed easily with the chart wizard. If you use line charts, set the chart wizard default to line charts, not bar charts, to save a step. Be sure to label the axes and add titles and legends before copying the charts to Word (or whatever text file you use).

The averages are not perfectly smooth, which distorts the fitted ARIMA model.

Take heed: Several types of centered moving averages can be used. The simplest method is an N day centered moving average. Pick an N that smooths the random fluctuations but keeps the true differences among days.

Try 7, 15, and 31 day centered moving averages. Graph the results for one year. Each year is the same, so your graph needs just 365 days.

Some statisticians use weighted centered moving averages. For five points, you might use weights of 15%, 20%, 30%, 20%, 15%. Some statisticians use complex weighting systems.

You may vary the number of points over the year.

Recommendation: No specific average is correct. Your student project should explain what you did, such as "A 50 year average leaves a somewhat jagged curve. I used centered moving averages of 3, 5, 7, … days. The curve becomes smoother until about 21 days, so I used a 21 day centered moving average." We check if you understand the logic of moving averages, not if your choice is the same as the course instructor’s.

The illustrative worksheet shows a 31 day centered moving average.

Extend the centered moving average to the rest of the rows. Copy the formula to all cells in Column P except the first 15 cells and the last 15 cells. Excel’s average built-in function is efficient, so we do not use the vlookup function. We make several manual adjustments.

If you deleted rows from your time series because of missing values, replace the 15 centered moving averages right before and after the missing values.

Illustration: The illustrative work-sheet has missing values for 10/1/1998 – 6/30/1990. We replace the centered moving averages for 9/16/1998–9/30/1993 and 7/1/1990– 7/15/1990 with the correct centered moving averages for those days.

The centered moving average works well except at the apex and nadir of the curve.

The graph of the smoothed average daily temperature is too flat at the apex and nadir. You may fit the graph to a smooth curve or judgmentally adjust the averages.

Take heed: The illustrative work-sheet does not make these corrections. Examine the graphs of your time series and judge if a correction is needed. Do not get bogged down in second decimal place accuracy if the effect is not material. You might find that a 25 day weighted and centered moving average works fine, with no correction needed.

Take heed: If you have deleted many rows with missing values, it may be easier to use the vlookup built-in function as we did for each day’s 130 year average.

Seasonal adjustments can be additive or multiplicative. Multiplicative models are common for other time series (not daily temperature).

Illustration: If sales in December 20X8 are $100,000, and December sales are 25% higher than those of the average month, the seasonally adjusted sales for December 20X8 are $100,000 / 1.25 = $80,000.

For daily temperatures, multiplicative vs additive models mean:

The illustrative work-sheet uses an additive model.

Take heed: If the temperature scale is arbitrary, the variance of the error term is unrelated to the mean temperature. The Fahrenheit scale for daily temperature is arbitrary.

White noise: If the time series is a white noise process,

The mean seasonally adjusted daily temperature for each day is zero. The observations are also the residuals if this is a white noise process.

Graph the time series to see these patterns.

Take heed: If you graph all 47,574 days with one line chart, you won’t see the patterns. Select half a year (182 days) and examine the chart.

Take heed: The correlogram for the first year on the illustrative work-sheet suggests an ARMA(1,1) process. The daily temperatures for 1874 alone do not give the optimal model for the next 130 years. Be sure that your model is robust.

A robust model does not depend on the period. Suppose that

We infer that one year does not give a robust model, but 65 years gives a robust model.

Take heed: Every weather station differs. The time series for another weather station may differ in two time periods.

Take heed: Forming a correlogram from 47,574 observations is memory intensive. The response time varies with the memory and power of your computer.

If you code the subproduct built-in function for the first cell with the offset feature and copy it to the other cells on a slow machine, you may wait an hour for the response.

The illustrative work-sheets show several sample autocorrelation functions. For your student project, use the chart wizard to form correlograms.

The first sample autocorrelation function uses one year of daily temperatures, to verify that the cell formulas are correct. Examine the following items:

The annual seasonality and stochasticity are obvious in the data. Long-term trends and cycles are not easy to see. We expect a stationary time series, with rapidly declining sample autocorrelations. From the tenth lag onwards, the sample autocorrelations should have about equal numbers of positive and negative figures.

Recommendation: The sample autocorrelation of lag 1 may range from 20% to 70%, depending on the location. Your student project may compare two weather stations:

Form a correlogram from two or three years of observations, to see how quickly your machine computes the sample autocorrelations. The response time depends on your machine’s memory and CPU speed.

As you add years, the response time increases exponentially. If a correlogram for two years takes a minute, a correlogram for 50 years may take several hours.

Take heed: These warnings about response time are for older machines. A high powered laptop bought after 2006 will handle the computations.

Important: Unless your machine is powerful, set calculation to manual and remove the check-mark from re-calculate when saving.

Take heed: The illustrative work-sheet has sample autocorrelations that decline rapidly to zero for the 365 day correlogram and sample autocorrelations that stay above zero for about thousands of lags for the 130 year correlogram. You can distinguish a stationary from a non-stationary time series with 40 or 50 lags.

For the Box-Pierce Q statistic, use 40 or 50 lags, unless your computer is powerful. Use more lags if the result is indeterminate.

If you have daily temperature for 120 years, use three periods of 40 years each. If the three periods gives the same (or similar) ARIMA processes, your model is robust.

Take heed: We don’t expect a ARIMA process more complex than ARMA(1,1) for daily temperature, and we presume the ARIMA process is the same for all the years. Long-term trends and cycles may causes non-stationary processes and different ARIMA parameters. Compute just the statistics you need for the student project. The illustrative worksheet uses the Atlantic City data to show the complications from non-stationary time series.

The illustrative work-sheet shows sample autocorrelations for one year of data: January 16, 1874, to January 15, 1875. We use temperature deviations, having an average of zero for all years. This simplifies your computations.

We explain the cell formula for the offset and sumproduct built-in functions.

OFFSET(R$27,0,0,365-S27,1) means an array of cells:

The SUMPRODUCT(OFFSET(R$27,0,0,365-S27,1),R28:R$391) formula specifies the second array of figures as ending in R391 and beginning 364 rows earlier. When we copy the formula to Cell Q28, the second array of figures becomes R29:R$391.

Take heed: Once you grasp the logic, use the VBA macro for sample autocorrelations.

Take heed: This sample autocorrelation function uses cell formulas, so that you can copy the formulas to your own student project. We recommend that you use the VBA macro instead, which is more efficient, easier to use, and less likely to freeze your worksheet. But some candidates do not want to use VBA, so we show a cell formula solution as well.

Column P has 31 day centered moving averages of the daily average high temperatures.

A year and a half have missing values in this time series. The centered moving averages for the 15 days right before and right after the missing data are not correct. Instead, copy the centered moving averages from a year with no missing data to these rows.

Take heed: The VBA macro adjusts for the non-zero average. The computations here are needed only for the cell formulas.

We adjust for the non-zero average.

Autofill column S with lags ranging from 1 in row 12 to 47574 in row 47585. A simple way to autofill is to place a 1 in Cell S12 and a 2 in Cell S13. Select both cells and drag the autofill handle on the lower right corner of cell S13 to Cell S47585.

Place the formula "=SUMPRODUCT(OFFSET(R$12,0,0,47574-S12,1),R13:R$47585)" in Cell T12. Be sure to use the correct figures for your time series.

See how much time Excel needs to compute the sumproduct. Depending on the response time, copy the formula to the next ten cells or the next 100 cells.

When Calculation is Manual, press F9 to re-calculate.

The sample autocorrelations for Atlantic City are positive for the first 6,345 lags. They are negative for the next few lags, and then turn positive again. After 9,000 lags, the sample autocorrelations are equally likely to be positive as negative.

The time series of daily temperatures is not stationary over the full 130½ year period: 132 years from 1/1/1874 to 12/31/2005 minus 1½ years of missing data.

Your student project should identify why the time series is not stationary and make the needed adjustments to fit an ARIMA process.

A time series may be non-stationary for several reasons: random walks, trends, cycles.

Actual changes in the daily temperature may occur for several reasons:

These items are disputed, and we do not know the causes of daily temperature changes over time. Your student project will not decide the disputes, but it shows the sensitivity of ARIMA modeling to slight changes in trends or means. Use the following steps.

The illustrative work-sheet shows four sample autocorrelation functions.

One correlogram uses all 47,574 observations and cell formulas, not the VBA macro. The next three correlograms uses the VBA macro.

The sample autocorrelations are 0.51442 for lag 1 and 0.26526 for lag 2:

For the first group of 10,000 observations, the sample autocorrelations are 0.50885 for lag 1 and 0.26648 for lag 2.

For a middle group of 10,000 observations, the sample autocorrelations are 0.49628 for lag 1 and 0.24502 for lag 2.

For the last group of 10,000 observations, the sample autocorrelations are 0.54844 for lag 1 and 0.27875 for lag 2.

Determining if the time series is stationary is not easy.

But the time series is not a random walk and the high volatility of the daily temperature makes it hard to identify a trend. The AR(1) model is robust:

First differences

We examine the first differences when the time series is a random walk or has a trend. Taking first differences of a stationary autoregressive process does not improve the model.

Excel is ideal for medium size operations, such as fitting exponential trends to average claim severities. Many users set calculation to automatic: Excel recalculates the cell formulas whenever an input cell changes or when the work-sheet is saved.

A daily temperature time series of 135 years has 136 × 365.25 = 49,674 observations. If your student project examines the ARIMA processes for high vs low temperatures, you may have two time series of 50,000 observations each.

The sumproduct formula for the first lag of a time series with 50,000 values adds 50,000 products, requiring 100,000 operations. The column of sumproducts needs ½ × 50,000 × 100,000 = 2,500,000,000 operations. Two columns use 5 billion operations. Calculating these columns takes much time: between ten minutes and several hours, depending on the power of your computer. To work efficiently

You can work on your spread-sheet and save frequently without freezing your computer.

The time series analysis is a series of steps. You compute sample autocorrelations from raw temperature readings and from seasonally adjusted figures, using several ways to adjust for seasonality. You compute correlograms for the entire period or for sub-periods. You may use high or low temperatures.

Excel is so powerful that one tends to leave all the computations in a single work-sheet. But a work-sheet with billions of computations is likely to freeze, and you may lose hours of work. The recommendations below should speed your work and prevent loss of material.

We give a VBA macro to compute the sample autocorrelations. The macro runs quickly and does not leave cell formulas in the worksheet.

If you use cell formulas for the sample autocorrelations, use separate workbooks or worksheets for each part of your student project. Complete each part and save it, so you may come back to your saved version if the next part of your student project is in error.

Dates: convert Excel dates or string dates to a date index and the year. Correct errors and missing values in the temperature readings. Create a worksheet or workbook with the clean data and a date index for each row.

Illustration: Suppose the original weather data runs from 1/1/1870 to 12/31/2000. Convert a date of November 15, 1878, to a date index of 1115 and a year of 1878. Inspect the data for errors and missing values. Interpolate or delete rows, as appropriate. Keep a record of the data you change.

Seasonal adjustments: convert raw temperature readings to seasonally adjusted figures. Use separate worksheets for high vs low temperatures. Examine the adjusted time series for trend and cycles with graphs and moving averages. Form time series of detrended, seasonally adjusted values.

The write-up explains the seasonal adjustment and any observed trends or cycles.

Correlograms: Form correlograms separately for each time series and period you examine. Try various adjustments for trend or cycles, and examine each period whose mean differs.

ARIMA fitting: Use linear regression or Yule-Walker equations for the autoregressive and moving average parameters. Form residuals for each model and test them using the Box-Pierce Q statistic and Bartlett’s test.

The student project shows if you can interpret statistical techniques. Correlograms are hard to interpret if a time series is not stationary or has few observations

We analyze the 365 day correlogram of daily temperature for 1/16/1874 to 1/15/1875.

The sample autocorrelations have two components:

A 95% confidence interval uses ± 1.96 standard deviations.

A common rule of thumb is to replace each figure by a range that is ±1 standard deviation about the observed value. The sample autocorrelations for the first lag or two have higher error terms, and we might use a range of ± 2 × standard deviation.

Take heed: An effect that dissipates in N days does not mean an autoregressive process with N terms. An AR(1) process has an infinite memory: the effect continues indefinitely, though it may be overwhelmed by the random fluctuation after a few lags.

The sample autocorrelation of lag 1 may reflect an autoregressive or a moving average process. In your project write-up, explain what the ARIMA process implies.

Illustration: We contrast AR(1) and MA(1) processes for daily temperature.

The MA(1) process dies out in a day; the AR(1) process dies out gradually.

The MA(1) process says that if it was colder than expected one day, a cold front is moving in, and we should expect colder weather the next day.

Take heed: A student project on daily temperature may compare AR(1), AR(2), & ARMA(1,1) processes. For the AR(1) and AR(2) processes, the linear regression gives the autoregressive parameters.

For other student projects, fluctuations complicate the choice of ARIMA process. The 130+ years of daily temperature observations smooth the random fluctuations in the correlogram. Although the Yule-Walker equations do not give optimal parameters for short time series, they give reasonably exact parameters for a time series of 50,000 observations.

Take heed: Some of the sample autocorrelation reflects the long-term trend or cycle, or the joining of distinct time series. Your fitted model is more accurate if you eliminate the patterns in the time series.

The illustrative work-sheet shows AR(1) models for several time series:

The 365 day AR(1) model is fit with a regression analysis of 364 pairs of figures.

Take heed: The one period lag converts 365 values to 364 pairs of values. The last day (day #365) is not used as an independent variable, since the one year time series has no day #366 as a dependent variable.

The sample autocorrelation has one more observation.

Copy the seasonally adjusted temperatures from Cells P1247585 to Cells W13:W47586.

Use Excel’s regression add-in or the slope and intercept built-in functions to estimate the AR(1) model.

Begin with a regression on two or three years of observations to be sure you use the Excel functions correctly.

We don’t expect a consistent trend in daily temperature. The average daily temperature hasn’t changed much over the past hundred million years. Estimates of global warming or cooling over the past hundred years suggest a change of 1 or 2 degrees Fahrenheit.

But daily temperature has strong medium term cycles. Meteorologist believe the earth has been slowly warming since the last Ice Age. Every century, the earth may warm or cool a few degrees. Cities (with smog) and coastal regions may have other cycles or trends.

Short term cycles from ocean currents and short term trends from urbanization and smog may have a strong effect on daily temperatures.

Illustration: The build-up in auto exhaust during the week in metropolitan areas may raise rainfall on weekends, as water vapor condenses on exhaust particles. The change in daily temperature is harder to model, since smog has both warming and cooling effects.

Ideally, we separate the changes in daily temperature into seasonality, long-term trends, long-term and medium term cycles, and random fluctuations.

For medium-term trends and cycles, split the time series into eras with no trend or cycle.