1. HY539: Mobile Computing and Wireless Networks Basic Statistics / Data Preprocessing Tutorial: November 21, 2006 Elias Raftopoulos Prof. Maria Papadopouli Assistant Professor Department of Computer Science University of North Carolina at Chapel Hill

2. Basic Terminology  An Element of a sample or population is a specific subject or object (for example, a person, firm, item, state, or country) about which the information is collected.  A Variable is a characteristic under study that assumes different values for different elements.  An Observation is the value of a variable for an element.  Data set is a collection of observations on one or more variables.

3. Measures of Central Tendency Mean: Sum of all values divided by the number of cases. Median: The value of the middle term in a data set that has been ranked in increasing order. 50% of the data lies below this value and 50% above. Mode: is the value that occurs with the highest frequency in a data set.

4. Measures of Dispersion The square root of the average squared deviations from the mean This measure how the data values differ from the mean. A small standard deviation implies most values are near the average The large standard deviation indicates that values are widely spread above and below the average Range = Largest value – Smallest Value The range, like the mean has the disadvantage of being influenced by outliers Percentiles Values that divide cases below which certain percentages of values fall Standard Error (SE): Standard deviation of the mean.

5. Data Analysis Discovery of Missing Values Data treatment Outliers Detection Outliers Removal [Optional] Data Normalization [Optional] Statistical Analysis

6. Why Data Preprocessing? Data in the real world is dirty incomplete noisy inconsistent No quality data, no quality statistical processing Quality decisions must be based on quality data

7. Data Cleaning Tasks Handle missing values, due to Sensor malfunction Random disturbances Network Protocol [eg UDP] Identify outliers, smooth out noisy data

8. Recover Missing Values Linear Interpolation

9. Recover Missing Values Moving Average A simple moving average is the unweighted mean of the previous n data points in the time series A weighted moving average is a weighted mean of the previous n data points in the time series A weighted moving average is more responsive to recent movements than a simple moving average An exponentially weighted moving average (EWMA or just EMA) is an exponentially weighted mean of previous data points The parameter of an EWMA can be expressed as a proportional percentage - for example, in a 10% EWMA, each time period is assigned a weight that is 90% of the weight assigned to the next (more recent) time period

10. Recover Missing Values Moving Average (cont’d) Symmetric Linear Filters Moving Average

11. What are outliers in the data? An outlier is an observation that lies an abnormal distance from other values in a random sample from a population It is left to the analyst (or a consensus process) to decide what will be considered abnormal Before abnormal observations can be singled out, it is necessary to characterize normal observations

12. What are outliers in the data? (cntd) An outlier is a data point that comes from a distribution different (in location, scale, or distributional form) from the bulk of the data In the real world, outliers have a range of causes, from as simple as operator blunders equipment failures day-to-day effects batch-to-batch differences anomalous input conditions warm-up effects

13. Scatter Plot: Outlier Scatter plot here reveals A basic linear relationship between X and Y for most of the data A single outlier (at X = 375)

14. Symmetric Histogram with Outlier A symmetric distribution is one in which the 2 "halves" of the histogram appear as mirror-images of one another. The above example is symmetric with the exception of outlying data near Y = 4.5

15. Confidence Intervals In statistical inference we want to estimate population parameters using observed sample data. A confidence interval gives an estimated range of values which is likely to include an unknown population parameter the estimated range calculated from a given set of sample data

16. Confidence Intervals Common choices for the confidence level C are 0.90, 0.95, and 0.99 Levels correspond to percentages of the area of the normal density curve

17. Interpretation of Confidence Intervals e.g. 95% C.I. for µ is (-1.034, 0.857). Right: In the long run over many random samples, 95% of the C.I.’s will contain the true mean µ. Wrong: There is a 95% probability that the true mean lies in this interval (-1.034, 0.857).

18. Smoothing Data If your data is noisy, you might need to apply a smoothing algorithm to expose its features, and to provide a reasonable starting approach for parametric fitting Two basic assumptions that underlie smoothing are The relationship between the response data and the predictor data is smooth The smoothing process results in a smoothed value that is a better estimate of the original value because the noise has been reduced The smoothing process attempts to estimate the average of the distribution of each response value The estimation is based on a specified number of neighboring response values

19. Smoothing Data (cntd) Moving average filtering Lowpass filter that takes the average of neighboring data points Lowess and loess Locally weighted scatter plot smooth Savitzky-Golay filtering A generalized moving average where you derive the filter coefficients by performing an unweighted linear least squares fit using a polynomial of the specified degree

20. Smoothing Data (eg Robust Lowess)

21. Normalization Normalization is a process of scaling the numbers in a data set to improve the accuracy of the subsequent numeric computations Most statistical tests and intervals are based on the assumption of normality This leads to tests that are simple, mathematically tractable, and powerful compared to tests that do not make the normality assumption Most real data sets are in fact not approximately normal An appropriate transformation of a data set can often yield a data set that does follow approximately a normal distribution This increases the applicability and usefulness of statistical techniques based on the normality assumption.

22. Box-Cox Transformation The Box-Cox transformation is a particulary useful family of transformations

23. Measuring Normality Given a particular transformation such as the Box-Cox transformation defined above, it is helpful to define a measure of the normality of the resulting transformation One measure is to compute the correlation coefficient of a normal probability plot The correlation is computed between the vertical and horizontal axis variables of the probability plot and is a convenient measure of the linearity of the probability plot (the more linear the probability plot, the better a normal distribution fits the data). The Box-Cox normality plot is a plot of these correlation coefficients for various values of the parameter. The value of λ corresponding to the maximum correlation on the plot is then the optimal choice for λ

24. Measuring Normality (cont’d) The histogram in the upper left-hand corner shows a data set that has significant right skewness And so does not follow a normal distribution The Box-Cox normality plot shows that the maximum value of the correlation coefficient is at = -0.3 The histogram of the data after applying the Box-Cox transformation with = -0.3 shows a data set for which the normality assumption is reasonable This is verified with a normal probability plot of the transformed data.

25. Normal Probability Plot The normal probability plot is a graphical technique for assessing whether or not a data set is approximately normally distributed The data are plotted against a theoretical normal distribution in such a way that the points should form an approximate straight line. Departures from this straight line indicate departures from normality The normal probability plot is a special case of the probability plot

26. Normal Probability Plot (cont’d)

27. PDF Plot Probability density function (pdf) represents a probability distribution in terms of integrals A probability density function is non-negative everywhere and its integral from −∞ to +∞ is equal to 1 If a probability distribution has density f(x), then intuitively the infinitesimal interval [x, x + dx] has probability f(x) dx

28. CDF Plot Plot of empirical cumulative distribution function

29. Percentiles percentiles provide a way of estimating proportions of the data that should fall above and below a given value The pth percentile is a value, Y(p), such that at most (100p)% of the measurements are less than this value and at most 100(1- p)% are greater The 50th percentile is called the median Percentiles split a set of ordered data into hundredths For example, 70% of the data should fall below the 70th percentile.

30. Interquartile Range Quartiles are three summary measures that divide a ranked data set into four equal parts Second quartile is the same as the median of a data set First quartile is the value of the middle term among the observations that are less than the median Third quartile is the value of the middle term among the observations that are greater than the median Interquartile range: The difference between the third and first quartiles or equivalently the middle 50% of the data