GEOS 585A, Applied Time Series Analysis

Notes for lessons 1-8, 2013, have been uploaded; those for later lessons will be uploaded later in semester. Assignments will be uploaded at least one week before the topic is covered in class

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Instructor

David M. Meko

Laboratory of Tree-Ring Research, , Room 417, Bryant Bannister Tree-Ring Building (Bldg #45B)

Email: dmeko@LTRR.arizona.edu

Phone: (520) 621-3457

Fax: (520) 621-8229

Office hours Monday, 1:30-3 PM

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Course Description

Analysis tools in the time and frequency domains are introduced in the context of sample data sets drawn from hydrology, climatology, and paleoclimatology. Students optionally use their own data sets in series of assignments. Back to Top of Page

Overview

This is an introductory course, with emphasis on practical aspects of time series analysis. Methods are hierarchically introduced -- starting with terminology and exploratory graphics, progressing to descriptive statistics, and ending with basic modeling procedures. Topics include detrending, filtering, autoregressive modeling, spectral analysis and regression. This is a 3-credit course. Twelve topics, or "lessons" are addressed sequentially in the semester. Paired classroom sessions consist of a lecture introducing methods and a workshop session illustrating application using a high-level computing language (MATLAB). The student begins by preparing three sets of time series and text files of associated metadata for the class. Class assignments consist of running pre-written MATLAB scripts (programs)on these time series and interpreting the results. The course is also offered by correspondence to students not enrolled at the University of Arizona.

Any time series with a constant time increment (e.g., day, month, year) is a candidate for use in the course. Examples are daily precipitation measurements, seasonal total streamflow, summer mean air temperature, annual indices of tree growth, indices of sea-surface temperature, and the daily height increment of a shrub.

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Goals

As a result of taking the course, students should:
  1. understand basic time series concepts and terminology
  2. be able to select time series methods appropriate to goals
  3. be able to critically evaluate scientific literature in which basic time series methods such as filtering, ARMA modeling and spectral analysis are used
  4. be able to apply a suite of time series methods to their own data using a high-level numerical analysis language (Matlab)
  5. be able to concisely summarize results of time series analysis in writing

Prerequisites

  1. An introductory statistics course
  2. Permission of the instructor (correspondence students and undergraduates)

Other Requirements

  1. Access to a computer running MATLAB and four toolboxes: Statistics, Signal Processing, System Identification, Spline (if using Matlab Release 2010a or earlier), Curve Fitting (if using Matlab Release 2010b or later)
  2. Access to the internet: notes and assignments are exchanged electronically and completed assignments are submitted electronically

MATLAB and Toolboxes. The course makes extensive use of MATLAB. The class scripts and function are updated from time to time using features that may not be available in earlier MATLAB versions. The version required for this course is MATLAB Release 2007b or higher. For 2013, I am using MATLAB Version: 8.0.0.783 (R2012b). Students at the University of Arizona students have access to MATLAB and all required toolboxes through a site license as no cost software. Students are responsible for making sure they have access to the MATLAB software, including the four required toolboxes. An alternative to the U of AZ site license is individual purchase of the Student Version of Matlab, with the four toolboxes as add-ons. No previous experience with MATLAB is required, and computer programming is not part of the course.

Availability

The course is offered to resident UA students in Spring Semester every other year (2013, 2015, etc.). Non-UA students may also register for the course as correspondence students in those same semesters. The course is open to graduate students and may also be taken by undergraduate seniors with permission of the instructor. Prospective correspondence students should correspond with the instructor indicating their plans before registering for the course. Enrollment of resident UA students is capped at 20 for Spring Semester 2013.

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Course Outline (Lessons)

  1. Introduction to time series; organizing data for analysis
  2. Probability distribution
  3. Autocorrelation
  4. Spectrum
  5. Autoregressive-Moving Average (ARMA) modeling
  6. Spectral analysis -- smoothed periodogram method
  7. Detrending
  8. Filtering
  9. Correlation
  10. Lagged Correlation
  11. Multiple linear regression
  12. Validating the regression model
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Syllabus

Calendar

Correspondence Students. You must register to take the course in Spring Semester, and should complete the registration process before February 1, but may take from 12 weeks to 9 months to complete the course. You may apply for a one-time, three month extension, giving you up to a year to complete. You cover the twelve topics at your own pace -- but no faster than the regular on-campus class, because the notes and assignment may not yet have been uploaded to the web site. If you intend to have a grade transferred to your university by a specified date, be sure to plan to finish the class exercises by the end of the University of Arizona semester. To allow timely grading of assignments, you should pace the completion of assignments so that no more than one assignment is submitted per week. General information on correspondence courses at the University of Arizona can be found at the Outreach College . Go to that site and click the "Student Resouces" link near top of the page. Then click the desired link (e.g., "How to Register") under Credit Courses/Semester and Late Start. Phone numbers are listed at the bottom of the web pages under these links, in case you want to talk with a human.

Resident Students. The schedule typically allows for about two weeks for completion of Lesson #1, and one week for each of the other lessons. Three or four class sessions are reserved for review. Class sessions alternate between lecture sessions introducing methods and computer sessions illustrating application. The 12 lessons covered in the course are listed in the class outline.

Spring 2013 semester. Class meets twice a week for 75 minute sessions, 9:00-10:15 AM T/Th, in room 424 (Conference Room) of Bryant Bannister Tree-Ring Building (building 45B). First day of class is Jan 10 (Thurs). Last day of class is April 30 (Tues). There is no class during the week of Spring Break (Mar 9-17).

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Data

You analyze data of your own choosing in the class assignments. As stated in the course overview, there is much flexibility in the choice of time series. The instructor will make a catalog of suitable time series available, but a good alternative is focus the course on your own data set. The first assignment is to gather together the time series you want to study and run a MATLAB script that stores the series in a form accessible to all of the class scripts.

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Assignments

The 12 topics are addressed sequentially over the semester, which covers approximately 15 weeks. Two weeks (4 class meetings) are allocated for the first topic, and roughly one week each (2 class meetings) for subsequent topics. Typically, the first of a pair of meetings introduces the topic, and the second covers the assignment, including sample runs on various time series. Each assignment consists of reading a chapter of notes, running an associated MATLAB script that applies selected methods of time series analysis to your data, and writing up your interpretation of the results. Assignments require understanding of the lecture topics as well as ability to use the computer and software.

Resident students submit assignments roughly on weekly intervals according to class progress on the lessons. Correspondence students email class assignments on a schedule over a period ranging from 12 weeks to 9 months. Assignments are graded within one week of being submitted. To find out how to access assignments, click assignment files.

Readings

Readings consist of class notes. There are twelve sets of .pdf notes files , one for each of the course topics. These .pdf files can be accessed over the Web. More information on the various topics covered in the course can be found through references listed at the end of each chapter of class notes.

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Grades

Grades are based entirely on performance on the assignments, each of which is worth 10 points. There are no exams. The total number of possible points for the 12 topics is 12 x 10 = 120. A grade of "A" required 90-100 percent of the possible points. A grade of "B" requires 80-90 percent. A grade of "C" requires 70-80 percent, and so forth. The number of points earned is marked at the top of each graded assignment.

Unless specified otherwise, the due date for an assignment is midnight on Friday of the week after the assignment is given (e.g., at least 7 days fall between the assignment and when it is due). I typically grade assignments on Saturday morning the day after they are due. An assignment not submitted by the due date is considered late, and the grade is docked. The late penalty starts at 1 point if less than a day late (submitted before midnight Saturday), increases by 0.3 additional points for every additional day late, and jumps to 3 points if the assignment is not submitted by midnight before the class in which I give feedback on the assignment. This schedule applies only to students taking the course on campus, not to online students from other universities.) Back to Top of Page

Lessons

  1. Introduction to time series; organizing data for analysis

    A time series is broadly defined as any series of measurements taken at different times. Some distinctive properties of time series include 1) continuous vs discrete, 2) univariate vs multivariate, 3) evenly sampled vs unevenly sampled, 4) periodic vs aperiodic, 5) stationary vs nonstationary, 6) short vs long. These properties, as well as the sampling interval and temporal overlap of multiple series, must be considered in selecting a dataset for analysis in this course.

    The first step is to store your time series and associated metadata in MATLAB structures. This step provides some uniformity to the format of the time series data, and is necessary to ensure that all MATLAB scripts and functions written for the course will run on each student's data without the need for reformatting. A structure is a MATLAB variable-type similar to a database in that the contents are accessed by textual field designators. A structure can store data of different forms. For example, one field might be a numeric time series matrix, another might be text describing the source of data, etc. In this first lesson you will run a MATLAB script that stores your data in structures. In subsequent lessons you will apply time series methods to the data simply by running MATLAB scripts on those structures. As a first step, you must organize your time series in Excel spreadsheets and put metadata for the series in an ascii file. This lesson describes the preparation of your time series for use in the course, and introduces computation and graphics with MATLAB.

    Assignments

    Select sample data to be use for assignments during the course

    Read: (1) Notes_1.pdf, (2) "Getting Started", accessible from the MATLAB help menu

    Answer: Run script geosa1.m and answer questions listed in the file in a1.pdf

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  2. Probability distribution

    The probability distribution of a time series describes the probability that an observation falls into a specified range of values. An empirical probability distribution for a time series can be arrived at by sorting and ranking the values of the series. Quantiles and percentiles are useful statistics that can be taken directly from the empirical probability distribution. Many parametric statistical tests assume the time series is a sample from a population with a particular population probability distribution. Often the population is assumed to be normal. This chapter presents some basic definitions, statistics and plots related to the probability distribution. In addition, a test (Lilliefors test) is introduced for testing whether a sample comes from a normal distribution with unspecified mean and variance.

    Assignments

    Read: Notes_2.pdf

    Answer: Run script geosa2.m and answer questions listed in the file in a2.pdf

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  3. Autocorrelation

    Autocorrelation refers to the correlation of a time series with its own past and future values. Autocorrelation is also sometimes called lagged correlation or serial correlation, which refers to the correlation between members of a series of numbers arranged in time. Positive autocorrelation might be considered a specific form of persistence, a tendency for a system to remain in the same state from one observation to the next. For example, the likelihood of tomorrow being rainy is greater if today is rainy than if today is dry. Geophysical time series are frequently autocorrelated because of inertia or carryover processes in the physical system. For example, the slowly evolving and moving low pressure systems in the atmosphere might impart persistence to daily rainfall. Or the slow drainage of groundwater reserves might impart correlation to successive annual flows of a river. Or stored photosynthates might impart correlation to successive annual values of tree-ring indices. Autocorrelation complicates the application of statistical tests by reducing the number of independent observations. Autocorrelation can also complicate the identification of significant covariance or correlation between time series (e.g., precipitation with a tree-ring series). Autocorrelation can be exploited for predictions: an autocorrelated time series is predictable, probabilistically, because future values depend on current and past values. Three tools for assessing the autocorrelation of a time series are (1) the time series plot, (2) the lagged scatterplot, and (3) the autocorrelation function.

    Assignments

    Read: Notes_3.pdf

    Answer: Run script geosa3.m and answer questions listed in the file in a3.pdf

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  4. Spectrum

    The spectrum of a time series is the distribution of variance of the series as a function of frequency. The object of spectral analysis is to estimate and study the spectrum. The spectrum contains no new information beyond that in the autocovariance function (acvf), and in fact the spectrum can be computed mathematically by transformation of the acvf. But the spectrum and acvf present the information on the variance of the time series from complementary viewpoints. The acf summarizes information in the time domain and the spectrum in the frequency domain.

    Assignments

    Read: Notes_4.pdf

    Answer: Run script geosa4.m and answer questions listed in the file in a4.pdf

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  5. Autoregressive-Moving Average (ARMA)modeling

    Autoregressive-moving-average (ARMA) models are mathematical models of the persistence, or autocorrelation, in a time series. ARMA models are widely used in hydrology, dendrochronology, econometrics, and other fields. There are several possible reasons for fitting ARMA models to data. Modeling can contribute to understanding the physical system by revealing something about the physical process that builds persistence into the series. For example, a simple physical water-balance model consisting of terms for precipitation input, evaporation, infiltration, and groundwater storage can be shown to yield a streamflow series that follows a particular form of ARMA model. ARMA models can also be used to predict behavior of a time series from past values alone. Such a prediction can be used as a baseline to evaluate possible importance of other variables to the system. ARMA models are widely used for prediction of economic and industrial time series. ARMA models can also be used to remove persistence. In dendrochronology, for example, ARMA modeling is applied routinely to generate residual chronologies – time series of ring-width index with no dependence on past values. This operation, called prewhitening, is meant to remove biologically-related persistence from the series so that the residual may be more suitable for studying the influence of climate and other outside environmental factors on tree growth.

    Assignments

    Read: Notes_5.pdf

    Answer: Run script geosa5.m and answer questions listed in the file in a5.pdf

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  6. Spectral analysis -- smoothed periodogram method

    There are many available methods for estimating the spectrum of a time series. In lesson 4 we looked at the Blackman-Tukey method, which is based on Fourier transformation of the smoothed, truncated autocovariance function. The smoothed periodogram method circumvents the transformation of the acf by direct Fourier transformation of the time series and computation of the raw periodogram, a function first introduced in the 1800s for study of time series. The raw periodogram is smoothed by applying combinations or spans of one or more filters to produce the estimated spectrum. The smoothness, resolution and variance of the spectral estimates is controlled by the choice of filters. A more accentuated smoothing of the raw periodogram produces an underlying smoothly varying spectrum, or null continuum, against which spectral peaks can be tested for significance. This approach is an alternative to the specification of a functional form of the null continuum (e.g., AR spectrum).

    Assignments

    Read: Notes_6.pdf

    Answer: Run script geosa6.m and answer questions listed in the file in a6.pdf

    What to Know

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  7. Detrending

    Trend in a time series is a slow, gradual change in some property of the series over the whole interval under investigation. Trend is sometimes loosely defined as a long term change in the mean (Figure 7.1), but can also refer to change in other statistical properties. For example, tree-ring series of measured ring width frequently have a trend in variance as well as mean (Figure 7.2). In traditional time series analysis, a time series was decomposed into trend, seasonal or periodic components, and irregular fluctuations, and the various parts were studied separately. Modern analysis techniques frequently treat the series without such routine decomposition, but separate consideration of trend is still often required. Detrending is the statistical or mathematical operation of removing trend from the series. Detrending is often applied to remove a feature thought to distort or obscure the relationships of interest. In climatology, for example, a temperature trend due to urban warming might obscure a relationship between cloudiness and air temperature. Detrending is also sometimes used as a preprocessing step to prepare time series for analysis by methods that assume stationarity. Many alternative methods are available for detrending. Simple linear trend in mean can be removed by subtracting a least-squares-fit straight line. More complicated trends might require different procedures. For example, the cubic smoothing spline is commonly used in dendrochronology to fit and remove ring-width trend that might not be linear, or not even monotonically increasing or decreasing over time. In studying and removing trend, it is important to understand the effect of detrending on the spectral properties of the time series. This effect can be summarized by the frequency response of the detrending function.

    Assignments

    Read: Notes_7.pdf

    Answer: Run script geosa7.m and answer questions listed in the file in a7.pdf

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  8. Filtering

    The estimated spectrum of a time series gives the distribution of variance as a function of frequency. Depending on the purpose of analysis, some frequencies may be of greater interest than others, and it may be helpful to reduce the amplitude of variations at other frequencies by statistically filtering them out before viewing and analyzing the series. For example, the high-frequency (year-to-year) variations in a gauged discharge record of a watershed may be relatively unimportant to water supply in a basin with large reservoirs that can store several years of mean annual runoff. Where low-frequency variations are of main interest, it is desirable to smooth the discharge record to eliminate or reduce the short-period fluctuations before using the discharge record to study the importance of climatic variations to water supply. Smoothing is a form of filtering which produces a time series in which the importance of the spectral components at high frequencies is reduced. Electrical engineers call this type of filter a low-pass filter, because the low-frequency variations are allowed to pass through the filter. In a low-pass filter, the low frequency (long-period) waves are barely affected by the smoothing. It is also possible to filter a series such that the low-frequency variations are reduced and the high-frequency variations unaffected. This type of filter is called a high-pass filter. Detrending is a form of high-pass filtering: the fitted trend line tracks the lowest frequencies, and the residuals from the trend line have had those low frequencies removed. A third type of filtering, called band-pass filtering, reduces or filters out both high and low frequencies, and leaves some intermediate frequency band relatively unaffected. In this lesson, we cover several methods of smoothing, or low-pass filtering. We have already discussed how the cubic smoothing spline might be useful for this purpose. Four other types of filters are discussed here: 1) simple moving average, 2) binomial, 3) Gaussian, and 4) windowing (Hamming method). Considerations in choosing a type of low-pass filter are the desired frequency response and the span, or width, of the filter.

    Assignments

    Read: Notes_8.pdf

    Answer: Run script geosa8.m and answer questions listed in the file in a8.pdf

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  9. Correlation

    The Pearson product-moment correlation coefficient is probably the single most widely used statistic for summarizing the relationship between two variables. Statistical significance and caveats of interpretation of the correlation coefficient as applied to time series are topics of this lesson. Under certain assumptions, the statistical significance of a correlation coefficient depends on just the sample size, defined as the number of independent observations. If time series are autocorrelated, an effective sample size, lower than the actual sample size, should be used when evaluating significance. Transient or spurious relationships can yield significant correlation for some periods and not for others. The time variation of strength of linear correlation can be examined with plots of correlation computed for a sliding window. But if many correlation coefficients are evaluated simultaneously, confidence intervals should be adjusted (Bonferroni adjustment) to compensate for the increased likelihood of observing some high correlations where no relationship exists. Interpretation of sliding correlations can be also be complicated by time variations of mean and variance of the series, as the sliding correlation reflects covariation in terms of standardized departures from means in the time window of interest, which may differ from the long-term means. Finally, it should be emphasized that the Pearson correlation coefficient measures strength of linear relationship. Scatterplots are useful for checking whether the relationship is linear.

    Assignments

    Read: Notes_9.pdf

    Answer: Run script geosa9.m and answer questions listed in the file in a9.pdf

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  10. Lagged correlation

    Lagged relationships are characteristic of many natural physical systems. Lagged correlation refers to the correlation between two time series shifted in time relative to one another. Lagged correlation is important in studying the relationship between time series for two reasons. First, one series may have a delayed response to the other series, or perhaps a delayed response to a common stimulus that affects both series. Second, the response of one series to the other series or an outside stimulus may be smeared in time, such that a stimulus restricted to one observation elicits a response at multiple observations. For example, because of storage in reservoirs, glaciers, etc., the volume discharge of a river in one year may depend on precipitation in the several preceding years. Or because of changes in crown density and photosynthate storage, the width of a tree-ring in one year may depend on climate of several preceding years. The simple correlation coefficient between the two series properly aligned in time is inadequate to characterize the relationship in such situations. Useful functions we will examine as alternative to the simple correlation coefficient are the cross-correlation function and the impulse response function. The cross-correlation function is the correlation between the series shifted against one another as a function of number of observations of the offset. If the individual series are autocorrelated, the estimated cross-correlation function may be distorted and misleading as a measure of the lagged relationship. We will look at two approaches to clarifying the pattern of cross-correlations. One is to individually remove the persistence from, or prewhiten, the series before cross-correlation estimation. In this approach, the two series are essentially regarded on equal footing. An alternative is the systems approach: view the series as a dynamic linear system -- one series the input and the other the output -- and estimate the impulse response function. The impulse response function is the response of the output at current and future times to a hypothetical pulse of input restricted to the current time.

    Assignments

    Read: Notes_10.pdf

    Answer: Run script geosa10.m and answer questions listed in the file in a10.pdf

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  11. Multiple linear regression

    Multiple linear regression (MLR) is a method used to model the linear relationship between a dependent variable and one or more independent variables. The dependent variable is sometimes also called the predictand, and the independent variables the predictors. MLR is based on least squares: the model is fit such that the sum-of-squares of differences of observed and predicted values is minimized. MLR is probably the most widely used method in dendroclimatology for developing models to reconstruct climate variables from tree-ring series. Typically, a climatic variable is defined as the predictand and tree-ring variables from one or more sites are defined as predictors. The model is fit to a period -- the calibration period -- for which climatic and tree-ring data overlap. In the process of fitting, or estimating, the model, statistics are computed that summarize the accuracy of the regression model for the calibration period. The performance of the model on data not used to fit the model is usually checked in some way by a process called validation. Finally, tree-ring data from before the calibration period are substituted into the prediction equation to get a reconstruction of the predictand. The reconstruction is a "prediction" in the sense that the regression model is applied to generate estimates of the predictand variable outside the period used to fit the data. The uncertainty in the reconstruction is summarized by confidence intervals, which can be computed by various alternative ways.

    Assignments

    Read: Notes_11.pdf

    Answer: Run script geosa11.m (Part 1) and answer questions listed in the file in a11.pdf

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  12. Validating the regression model

    Regression R-squared, even if adjusted for loss of degrees of freedom due to the number of predictors in the model, can give a misleading, overly optimistic view of accuracy of prediction when the model is applied outside the calibration period. Application outside the calibration period is the rule rather than the exception in dendroclimatology. The calibration-period statistics are typically biased because the model is "tuned" for maximum agreement in the calibration period. Sometimes too large a pool of potential predictors is used in automated procedures to select final predictors. Another possible problem is that the calibration period itself may be anomalous in terms of the relationships between the variables: modeled relationships may hold up for some periods of time but not for others. It is advisable therefore to "validate" the regression model by testing the model on data not used to fit the model. Several approaches to validation are available. Among these are cross-validation and split-sample validation. In cross-validation, a series of regression models is fit, each time deleting a different observation from the calibration set and using the model to predict the predictand for the deleted observation. The merged series of predictions for deleted observations is then checked for accuracy against the observed data. In split-sample calibration, the model is fit to some portion of the data (say, the second half), and accuracy is measured on the predictions for the other half of the data. The calibration and validation periods are then exchanged and the process repeated. In any regression problem it is also important to keep in mind that modeled relationships may not be valid for periods when the predictors are outside their ranges for the calibration period: the multivariate distribution of the predictors for some observations outside the calibration period may have no analog in the calibration period. The distinction of predictions as extrapolations versus interpolations is useful in flagging such occurrences.

    Assignments

    Read: Notes_12.pdf

    Answer: Run script geosa11.m (Part 2) and answer questions listed in the file in a12.pdf

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Downloading Files -- tsfiles.zip not available till ~Jan 10, 2013

The MATLAB class scripts and user-written functions are zipped in a downloadable file called tsfiles.zip. To get the files, first create an empty directory on your computer. This is where you will store all functions, scripts and data used in the course. Click on tsfiles.zip to download the zip file to that directory and unzip it there. When you run matlab, be sure that directory is your current matlab working directory.

Powerpoint lecture outlines & miscellaneous files. Downloadable file other.zip has miscellaneous files used in lectures. Included are MATLAB demo scripts, sample data files, user-written functions used by demo scripts, and powerpoint presentations, as pdfs (lect1a.pdf, lect1b.pdf, etc.) used in on-campus lectures. I update other.zip over the semester, and add the presentation for the current lecture within a couple of days after that lecture is given.

Tailoring the MATLAB Scripts

To run the MATLAB scripts for the assignments, you must have your data, the class scripts, and the user-written MATLAB functions called by the scripts in a single directory on your computer. The name of this directory is unimportant. I happen to keep the files in "C:\geos585a\" on my windows PC. The functions and scripts provided for the course should not require any tailoring, but some changes can be made for convenience. For example, scripts and functions will typically prompt you for the name of your input data file and present "Spring13" as the default. That is because I've stored the sample data in Spring13.mat. If you want to avoid having to type over "Spring13" with the name of your own data file each time you run the script, search for the string "Spring13" and replace it with your own filename. This change can be made using the MATLAB editor/debugger.

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Notes and Assignments -- uploaded ~2wks before topic covered in class

The notes and assignments for lessons are viewable and printable as .pdf files by clicking on the hypertext below. Lesson "n" goes with notes file "Notes_n.pdf" and assignment file "an.pdf." A short list of topics can be found at course outline, and a more detailed list at lessons. I revise the notes and assignments during the semester. For that reason it's best not to download the full set of notes and assignments at the start of the semester. Better to wait till we cover the topic in class. The zipped file "Other.zip" contains powerpoints(converted to pdf) and miscellaneous demo files used in class lectures. "other.zip" is built up cumulatively through the semester, and is updated after each class. The powerpoints have feedback from students' submitted assignments, and may be helpful to the correspondence students, who of course do not have access to the twice-a-week lectures.



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