Objectives
• Regression analysis
• Sensor signal processing
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.
Lagrange interpolation
3
Rewrite:
Find coefficients:
General form:
Regressing analysis for large pool of data (function fitting)
4
From last class
• Does correlation where R2=0.82 represent a good data modeling?
Mean:
Total sum of squares:
Sum of squares of residuals :
Coefficient of determination
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 R2: 0.816, linear regression: y=3.00+0.500·x
Anscombe's quartet • Example of statistical misinterpretation of data
- all curves have the same Mean (x, y), Variance (x, y)
- correlation R2: 0.816, linear regression
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Data set A
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Data set B
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Data set C
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Data set D
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
Model of complex system based on experimental data
9
Example: chiller modelTOA
water
Building users (cooling coil in AHU)
TCWR=11oCTCWS=5oC
T Condensation
Chiller model
10
EIRFPLCAPFTPP NOMINAL
OACWSOAOACWSCWS TTfTeTdTcTbaCAPTF 12
112
111
PLRcPLRbaEIRFPLR 333 NOMINALQ
QPLR
)(
Impact of temperatures:
Impact of capacity:
Two variable function fitting
Example
12
0 2 4 6 8 100
50
100
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20025 C35 C45 C
q[kW]
Tevaporator [C]
Fundamentals of Signal Processing
R
I
V
V=I·R
Two approaches:
- Constant Voltage Source- Constant Current Source
Sensor:
RTD, thermistor, hot wire, …..
Cable Losses
Sensor
Signal processing
cableDC signal [mV]
Voltage drop in the cable
Rcable=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
Signal noise
Sensor
Signal processing
cableDC signal [mV]
AC current [120V]
Magnetic field
Current Induction (signal nose)
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Climate chamber
supply return
T [C]
hour
noise
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
Data Acquisition Device
Analog signal collection
Measuring signalto data acquisition
Each Channel has:
- Current source- ± connectors for Voltage measurement
Current source (constant V)
+
-
I (variable A)
Analog signal collectionVoltage measurement ± Voltage measurement
Current measurements
Wheatstone bridge
Wheatstone bridge
Wheatstone bridge
Known resistor
Vo
Vo
R1
Our sensor
R2
+
-
+
-
Calculate R4
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