1. Objectives Regression analysis Sensor signal processing

2. Regression analysis 2 Single variable: Minimum number of points depends on number of variable in the function (3 for the function above). Using the data we can set the system of equation to find the coefficients.

3. Lagrange interpolation 3 Rewrite: Find coefficients: General form:

4. Regressing analysis for large pool of data (function fitting) 4

5. From last class Does correlation where R 2 =0.82 represent a good data modeling? Mean: Total sum of squares: Sum of squares of residuals : Coefficient of determination

6. Anscombe's quartet Example of statistical misinterpretation of data - all data have the same Mean (for x and y), Variance (for x and y) - correlation R 2 : 0.816, linear regression: y=3.00+0.500·x

7. Anscombe's quartet Example of statistical misinterpretation of data - all curves have the same Mean (x, y), Variance (x, y) - correlation R 2 : 0.816, linear regression

8. Moral of the story Francis Anscombe (in 1973) demonstrated the importance of graphing data before analyzing it the effect of outliers on statistical properties 8

9. Model of complex system based on experimental data 9 Example: chiller model T OA water Building users (cooling coil in AHU) T CWR = 11 o C T CWS =5 o C T Condensation

10. Chiller model 10 Impact of temperatures: Impact of capacity:

11. Two variable function fitting

12. Example 12

13. Fundamentals of Signal Processing R I V V=I·R Two approaches: - Constant Voltage Source - Constant Current Source Sensor: RTD, thermistor, hot wire, …..

14. Cable Losses Sensor Signal processing cable DC signal [mV] Voltage drop in the cable R cable =l·r (l length of cable, r resistance per unit of length) r = f ( voltage, current, diameter, material ) Rcable can be same order of value like DC signal - Use same length of cables (shorter if possible) - Size diameter of cables to have significantly smaller voltage drop in cable than DC signal

15. Signal noise Sensor Signal processing cable DC signal [mV] AC current [120V] Magnetic field Current Induction (signal nose) noise

16. Signal noise filters A low pass filter is placed on the signal wires between a signal and an A/D board. It stops frequencies greater than the cut off frequency from entering the A/D board's analog or digital inputs. A low pass filter may be constructed from on resistor R and one capacitor C. The cut off frequency Fc is determined according to the formula: Fc= 1/2*Pi*C R= 1/2*Pi*C*Fc See the following diagram The key term in a low pass filter circuit is CUT OFF FREQUENCY. The cut off frequency is the frequency above which no variation of voltage with respect to time may enter the circuit. For example, if a low pass filter had a cut off frequency of 30 Hz, the type of interference associated with line voltage (60Hz) would be filtered out but a signal of 25 Hz would be allowed to pass

17. Data Acquisition Device

18. Analog signal collection Measuring signal to data acquisition Each Channel has: - Current source - ± connectors for Voltage measurement Current source (constant V) + - I (variable A)

19. Analog signal collection Voltage measurement ±Voltage measurement Current measurements

20. Wheatstone bridge

22. Known resistor Vo R1 Our sensor R2 + - + - Calculate R4

23. Converting Analog signal to Digital signal Analog-to-digital converter (ADC) - electronic device that converts analog signals to an equivalent digital form - heart of most data acquisition systems Loss of information in conversion, but no loss in transport and processing