13.4
FSM–Based Power Modeling of Wireless Protocols: the Case of Bluetooth Mariagiovanna Sami
Luca Negri Politecnico di Milano
Politecnico di Milano
[email protected]
[email protected]
David Macii
Alessandra Terranegra
ALaRI - Lugano
ALaRI - Lugano
[email protected]
[email protected]
ABSTRACT
and/or low power modes enabling subsequent power management. Interesting examples of power optimization techniques are Dynamic Voltage Scaling (DVS) [1], and, more specifically to wireless communication hardware, strategies for adjusting transmission power and special standby/low– power modes [2]. In turn, with power management (or Dynamic Power Management, DPM) we refer to policies and algorithms which take advantage of built-in low–power features [3]. The first step to achieve both power optimized design and power management is power modeling. A power model is an abstraction that can be used to predict the power consumption of a device based on its activity; for this reason it is crucial to power management. Power estimation is also useful to obtain power optimized designs, as it can help in finding out which parts of a system are worth optimizing. Power modeling and estimation of VLSI systems can be done at different levels of abstraction. Low–level models (i.e. transistor or gate level), provide maximum accuracy, but cause an unacceptable increment in simulation times even for moderately complex systems, especially if hw/sw. A more productive approach is that of building higher–level models, such as instruction and system level ones, and then validate them against a specific circuit implementation. Accuracy, even though not as good as in the previous case, can still be quite acceptable; on the other hand, simulation times decrease dramatically. Instruction level estimation is built on top of an instruction set simulation for a given processor by associating a power consumption with each software instruction being executed, plus some inter–instructions contributions [4]. System level techniques aim at predicting the power consumption of an entire complex system, made up of a large number of heterogeneous components. This type of models is used in DPM policies to selectively put each component in the most appropriate state, depending on resource use patterns. An even more challenging problem is posed by wireless networks, where a relevant contribution to the power consumption of each node is accounted for by communication [5] [6] [7]. In this situation, it appears very interesting to provide a power model of the protocol enabling estimation of the power requirements for given communication tasks. In this paper, we describe a hierarchical and implementation–
The proliferation of pervasive computing applications relying on battery–powered devices and wireless connectivity is posing great emphasis on the issue of power optimization. While node–level models and approaches have been widely discussed, a problem requiring even greater attention is that of power associated with the communication protocols. We propose a high–level modeling methodology based on Finite State Machines useful to predict the energy consumption of given communication tasks with very low computational cost, which can be applied to any protocol. We use this methodology to create a power model of Bluetooth that we characterize and validate experimentally on a real implementation. Categories and Subject Descriptors: C.2.1 [Network Architecture and Design]: Wireless Communication; C.4 [Performance of Systems]: Modeling Techniques General Terms: Management, Measurement. Keywords: Power Modeling, Wireless Protocols, Bluetooth.
1.
INTRODUCTION
The recent proliferation of pervasive computing applications based on battery–powered devices with wireless connectivity is seriously hindered by power consumption issues at both system and technological level. To overcome this limitation, researchers are tackling the problem from two different points of view: power optimized design and power management. With power optimized design we indicate those activities aimed at reducing the power consumed by a specific device in a given situation by adopting power control features
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or (iii) consumes power indirectly by activating (via events) other states (or both (ii) and (iii)).
independent FSM model to represent the power behavior of a generic wireless protocol. The model can be subsequently characterized for specific implementations, allowing both for power optimized design of the node and for power management. After presenting the general modeling methodology in section 2, a specific application to Bluetooth 1.1 is described in section 3, and an experimental characterization of the considered case study as well as the validation results are reported in sections 4 and 5, respectively. Finally, section 6 contains some remarks and outlines future work.
2.
4. Perform a series of experimental measurements on a real device or on a previously validated lower–level description (e.g. VHDL) of the device to characterize the model for that particular hw/sw implementation. In general, it is possible to measure some linear combinations of the pi s, associated with the execution of certain procedures of the protocol (e.g. connection, transmission of a file, etc.). Therefore, if we run M tests and if we refer to Ej and P j as the total energy and the average power, respectively, associated with the j–th test over its duration Tj , it results that:
FSM–BASED POWER MODELING
The methodology we propose aims at creating a power model of a wireless protocol and has some similarities with both the instruction level and system level approaches, although it highlights some truly distinctive features. The first intermediate result of the process is an implementation–independent model of the protocol; this can be in a second stage numerically characterized for a specific hardware/software implementation.
SIMULATIONS /TESTS
4. CHARACTERIZATION
5. TRAINING AND VALIDATION
N
1 Tj
Tj 0
Vj (t)Ij (t)dt
pi tji
(1)
where Vj (t), Ij (t) are the instantaneous values of both voltage and current applied to the considered wireless system, pi (i = 1, . . . , N ) is the average power consumption of the i–th activity ai and tji is a coefficient equal to the total usage time for activity ai during the test; this is equal to the sum of the times spent in states that activate ai . Since every experiment is reflected in a path in the FSMs, it is relatively easy to determine which states are involved in this calculation. Regarding the times spent in these states the calculation is easy when timing is well–defined in the protocol specifications, and more tricky in others cases, e.g. when random timers are involved. In these situations, either an average value is used or experimental tracing becomes necessary. The only constraint on the number of tests M is that M > N .
Implementation Dependent
feedback
3. STATE - ACTIVITY MATCHING
Tj · P j = Tj
i=0
1. FSM MODELING
2. ACTIVITIES IDENTIFICATION
= =
Implementation Independent
PROTOCOL SPECS
Ej
Figure 1: Methodology steps
5. Solve the system formed by equations (1) for the M experiments, which can be conveniently expressed in matrix notation as:
More in detail, with reference to Figure 1, the methodology comprises the following steps: 1. Represent the protocol behavior with a Finite State Machine (FSM). Multiple layers of the protocol can be modeled as a hierarchy of FSMs, in a Statecharts–like syntax. Statecharts [8] are an extension of FSMs that provide, among other features, the ability to model concurrent state machines and to use events to coordinate the evolution of the whole system. This step yields a set of states S.
E =T ×P
(2)
where E is the vector of the M total energy measurements, P the vector of the N unknown activity power consumption and T is the M × N matrix of the tji coefficients. Since the system is over–constrained, it can be solved with the Least Squares method, which yields:
2. Identify N basic logical activities that take place on a potential implementation as a consequence of the execution of the protocol. These can directly map to architectural parts of the foreseen implementation (e.g. radio, baseband) or be purely logical activities (transmission, reception, scheduling and so on). Let A be the set of these activities, and let pi be the power absorbed by activity ai ∈ A.
Pˆ = (T T × T )−1 × T T × E
(3)
If the Moore–Penrose pseudo–inverse (T T ×T )−1 is singular, either different activities must be chosen or different experiments be run, until all rows and columns in the matrix are linearly independent. This process is visualized in Figure 1 with the ’feedback’ arrow.
3. Define, for each si ∈ S, which activities are activated by the state. As a result of the first 3 steps, which represent the implementation independent part of the methodology, each state either (i) consumes no power, (ii) consumes power directly by using some activity,
Compared to traditional system–level power modeling approaches and to other protocol power modeling approaches found in the literature, our methodology highlights some important differences:
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• It provides general guidelines that can be applied to different protocols and devices.
This is also the case in our experiments, and for this reason we only present FSM models up to the HCI layer. Nevertheless, on very small mobile devices such as in pervasive computing scenarios, this partition may not exist; in that case, the optimization could be extended to include the upper layers as well. We have modeled (see Figure 2) by means of FSMs the radio/physical, baseband and link controller layers. The diagrams depicted in Figure 2 represent a simplified version of our Bluetooth model, with no labels on transitions and only including those states that are relevant to power analysis. We have also implemented a Bluetooth simulator in full Statecharts notation using StateflowTM (part of MatlabTM ) based on this FSM model, which includes additional states and events to synchronize the different layers, to handle user input and so on1 . Given the fact that we do not use OR–decomposition of states, we simplify the notion of configuration for a statechart and say that a configuration (global state) for the FSMs in Figure 2 is given by the 3–tuple containing the current state for each layer [11]. The model of the physical layer consist of two states ’transmit’ and ’receive’, activated respectively when the BT module is on air transmitting and when it is on air receiving, plus a ’standby’ state indicating that no transmit/receive activity in going on.
• It is based on behavioral and functional models, and hence implementation independent. • It relies on finer grained state machines and is semantically aware of the application being executed; in other words, the notion of state in our FSM model has a meaning both in terms of power and in terms of protocol activity. These features make our models particularly well suited both for power management and as analysis tools for power optimizations on the protocols. The modeling methodology can be easily extended to include transition costs, which would be represented as energy rather than power contributions. However, in this work we limit our analysis to power contributions located in the states.
3.
BLUETOOTH POWER MODEL Link Controller
standby
inquiry/page inquiry scan scanning
inquiry/page
inquiry
scheduling connected connected (master)
page scan
page
Packet
scanning
ID POLL NULL DM1 DH1 DM3 DH3 DM5 DH5
connected connected (slave)
Baseband no activity
transmit Dx1 payl. transmit Dx3 payl.
transmit AC
receive AC
transmit Dx5 payl.
transmit header
receive header
check AM addr.
check HEC
receive Dx3 payl.
receive Dx5 payl.
receive Dx1 payl.
rx
tx receive
standby
transmit
AC 68 72 72 72 72 72 72 72 72
Symbols Header Payload – – 54 – 54 – 54 240 54 240 54 1500 54 1500 54 2744 54 2744
Data carried (bytes) – – – 17 27 121 183 224 339
Table 1: ACL Bluetooth baseband packets Directly above, at the baseband layer, there are mainly two paths originating from and terminating into the ’no activity’ state: one for packet transmission and one for packet reception. For reference, Table 1 summarizes the main types of packets allowed by the Bluetooth baseband in ACL (asynchronous) mode. Packet transmission undergoes the following phases: (i) transmission of the access code in ’transmit AC’, (ii) transmission of the baseband header in ’transmit header’, unless the packet being transmitted is an ID one, (iii) transmission of the data payload in the ’transmit Dxy payl.’ states (x ∈ {M, H}; y ∈ {1, 2, 3}), if the packet contains one (i.e. it is not a POLL or NULL packet). These states activate the underlying physical layer by triggering its transition from ’standby’ to ’transmit’ for a period whose length depends on the number of symbols to be transmitted (according to Table 1). The receive path of the baseband is similar, though it has two extra states ’check HEC’ and ’check AM addr.’ that do not contribute to power consumption but remind that the transition from ’receive header’ to ’no activity’ can occur
Radio/ Physical
Figure 2: FSM model of Bluetooth Bluetooth is the leading standard for short–range wireless communication in the so–called Personal Area Networks (PANs). When used for point–to–point communication, it employs a master/slave time division duplexing scheme based on slots of 625µs and using 1µs symbols [9]. The Bluetooth stack comprises several layers; the most common partitioning envisions layers from physical to HCI (Host Controller Interface) running on a Bluetooth module and layers from HCI upwards running on a host such as a PC [10].
1
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Its description is beyond the scope of this paper.
via experimental measurements. Since we do not have any detailed low–level description of the Ericsson modules, we opted for the latter.
also if the header checksum does not match or if the packet is for another node in the piconet. The upmost layer in Figure 2 is the link controller. The ’standby’ state indicates that the BT module has no open connections and is not performing any operation. The ’page’, ’page scan’, ’inquiry’ and ’inquiry scan’ states refer to the so–called Bluetooth procedures used to perform device discovery and connection. Finally, there are two ’connected as master’ and ’connected as slave’ states, kept separate due to the different processing required in the two cases. The logical activities that we have identified for Bluetooth are reported in Table 2, along with their description. Their association to the FSMs states is highlighted in Figure 2 with shaded bubbles. Activity a1 a2 a3
Name tx rx connected
a4
scheduling
a5 a6
scanning inquiry/page
4.1 Test bench setup From a practical point of view, measuring the average power consumption of Bluetooth modules presents a couple of major testing challenges: synchronizing voltage/current data acquisitions and measuring accurately long sequences of current pulses. The former issue can be solved in first approximation simply using a stable DC power supply, whose output voltage fluctuations must have a negligible amplitude with respect to the applied nominal value. In fact, if multiple experiments in different conditions evidence such a quasi–ideal behavior, the voltage applied to the DUT can be assumed to be constantly equal to the average voltage V calculated over multiple measurements. As far as the current measurements are concerned, they can be performed at a low cost by means of a Digital Multimeter (DMM) having adequate analog bandwidth (e.g., larger than 500 kHz) and resolution (e.g. lower than 1 mA). Since high–accuracy bench DMMs are usually based on integrating Analog–to–Digital converters (ADC) [12], it results that the total energy consumed by the DUT during the j–th experiment can be estimated through (1) as follows: P p=1 T I[pT ] E j = Tj · P j Tj · V · = Tj P p=1 I[pT ] (4) = Tj · V · = Tj · V · I j P
Description Transmitting symbols Receiving symbols Additional processing when connected Additional processing when master Inquiry or page scan Inquiry or page
Table 2: Logical activities for Bluetooth Concerning the choice of these activities it is important to point out that: • Despite the fact that they are associated with the radio FSM, the ’tx’ and ’rx’ activities refer to the total power spent to transmit or receive symbols, and hence include also baseband and upper layers processing.
where V has been preliminarily measured, T is the integration time of the ADC employed in the DMM, P is the number of current samples collected over the experiment time pT T Tj (i.e. P = Tj ) and I[pT ] = T1 (p−1)T i(t)dt is the p–th
• The ’scanning’ activity applies both to the ’inquiry scan’ and to the ’page scan’ LC states, as these perform very similar operations and model the additional power consumption that cannot be justified by the use of activities ’tx’ and ’rx’. In the same way, the ’inquiry/page’ activity applies to both inquiring and paging.
average current measure provided by the DMM. Finally, I j is the average current for the experiment, and can be calculated as the average of the DMM samples. Clearly, applying (4) to each experiment implies collecting large amounts of data. In order to avoid to do this work manually, which is quite error–prone, an automatic test bench has been arranged as shown in Figure 3.
• We have not associated any specific activity with baseband states (e.g. ’transmit AC’, ’transmit header’) for experimental reasons. In fact, it turned out to be extremely difficult to perform linearly independent tests with respect to the times spent in those states. • The processing activity connected to the remaining LMP and HCI layers that are not explicitly included in the FSM model is accounted for by the other logical activities. The granularity we have chosen for the BT model is the finest possible when controlling the modules via standard HCI commands.
4.
EXPERIMENTAL CHARACTERIZATION OF THE MODEL
Figure 3: Block diagram of the test bench Basically, the test bench is made up of two Ericsson ROK 101 008 BT modules working at 5 V, an Agilent E3631A DC power supply, an Agilent 34401A 6.5 digits DMM, an USB–GPIB interface, and two personal computers (i.e., PC1
In this section we describe the characterization of the BT model for the Ericsson ROK 101 008 implementation. According to what said before, the experimental characterization can be performed either via low–level simulation or
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j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Experiment Standby Connected/master Connected/slave Inquiry 20s Page scan Master TX DM1 Master TX DM3 Master TX DM5 Master RX DM1 Master RX DM3 Master RX DM5 Slave TX DM1 Slave TX DM3 Slave TX DM5 Slave RX DM1 Slave RX DM3 Slave RX DM5 Master TX DH1 Master TX DH3 Master TX DH5 Slave RX DH1 Slave RX DH3 Slave RX DH5
Duration Tj (s) 40 40 40 40 40 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45
tj1 – 504 504 4456 0 5541 3648 3501 2238 747 635 5541 3648 3501 2238 747 635 3675 2582 2484 1595 665 590
Activity usage (ms) tj2 tj3 tj4 tj5 – – – – 544 40000 40000 0 544 40000 0 0 5767 0 0 0 351 0 0 351 2415 45000 45000 0 806 45000 45000 0 686 45000 45000 0 5718 45000 45000 0 3707 45000 45000 0 3552 45000 45000 0 2415 45000 0 0 806 45000 0 0 686 45000 0 0 5718 45000 0 0 3707 45000 0 0 3552 45000 0 0 1722 45000 45000 0 717 45000 45000 0 637 45000 45000 0 3802 45000 0 0 2635 45000 0 0 2531 45000 0 0
tj6 – 0 0 20480 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Ij (mA) 21.8645 20.9595 9.7579 13.5835 0.9139 23.3549 21.9743 21.9974 24.7889 23.3465 23.2271 15.9173 12.9984 12.9207 15.1034 12.1730 11.8737 22.4498 21.5246 21.5211 13.1909 11.3600 11.1635
Residual rj (%) – 4.83 -7.09 – – -7.80 -1.89 -0.83 -0.20 4.52 4.86 11.59 5.17 6.25 1.51 -4.33 -5.65 -2.68 -0.40 0.22 -0.27 -4.87 -5.82
Table 3: Experiments run on the BT modules • Experiments 1 and 2 refer to the DUT being connected (to the other module), respectively, as master or slave, and performing no additional operations.
and PC2). The digital multimeter performing current measurements is remotely controlled by PC1 using a simple LabViewTM application. A high–speed USB–GPIB (IEEE– 488) interface is used to connect the multimeter with PC1. The employed virtual instrument (VI) performs two kinds of operations: at first it enables the configuration of the DMM parameters (type of measurement, trigger condition, number of samples of the record, resolution and, accordingly, integration time of the converter); eventually, it can be used to run the data acquisition and to store the results in a plain ASCII text file. Both Bluetooth modules are connected to PC2 by means of two independent serial links. We used the Bluetooth HCI Toolbox v. 1.2 software to send HCI commands over the serial link to both modules separately. In all measurements, the DMM was configured to collect 5 samples per second (i.e. T = 200 ms), although the instrument is capable of operating at much higher sampling rates. This choice is motivated by the fact that our final goal is an average current over the whole experiment, rather than peak values, and that the multimeter accuracy increases with the integration time; in this context, fast sampling rates would lower the overall accuracy of the measurement with no additional benefits.
• During experiment 3 we had the device performing an inquiry operation for 20.48 seconds, whereas experiment 4 refers to a page scan activity with bursts of 11.25 ms repeating every 1.28 seconds. • Experiments 5 through 7 consisted of the master transmitting a file using, respectively, DM1, DM3 and DM5 packets. During experiments 8 through 10 the same file was received by the master. Experiments 11 to 13 and 14 to 16 refer, respectively, to the same transmissions and receptions but on a slave device. • Finally, experiments 17 to 19 represent the transmission of the same file as master using DH1, DH3 or DH5 packets; experiments 20 to 22 the reception of the same file as slave using DH1, DH3, DH5. Table 3 shows, for each experiment, the total time Tj , the tji usage coefficients for the 6 previously identified activities and the average current I j . Note that all the I j represent differential values with respect to the standby case (I 0 , first row). All the experiments were performed on both modules, yielding over the 22 tests an average difference of 0.5224 mA with a standard deviation of 0.2509 mA; all data reported in Table 3 refers to the first board we have tested. A few notes about the coefficients in Table 3:
4.2 Experiments Description Following is a summary of the 22 experiments we ran on the BT modules: • Experiment 0 consisted of simply turning on the module and disabling all the scanning activities. This yields the standby current I 0 . Note that we used this value as a baseline for all other experiments, i.e. we subtracted this value from all the other average currents I j (j > 0).
• The use of a1 and a2 in tests 1 and 2 is due to the POLL/NULL periodic exchange between master and slave, which occurs even if no data needs to be exchanged, in order to monitor the status of the link.
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The LOO residuals for the BT model are reported in the rightmost column of Table 3, as percentage values with respect to the total measured energy for the experiment Ej = Tj · V · I j . Exploiting these data, the RMS value of the validation error can be estimated as:
Using a CATC Merlin Bluetooth analyzer [13] we noticed that this happens on the Ericsson modules with the default QoS settings once every 16 slots. • In all file transmission tests, the file was 233055 bytes large. In these tests, a1 and a2 are used to send/receive data and acknowledgement packets. Using the Bluetooth analyzer we noticed the use of the L2CAP transport protocol with a 1024 bytes packet size; hence, including the L2CAP header (4 bytes per packet) the amount of data transmitted by the baseband in each test was 233965 bytes. This number, combined with the data in Table 1, was used to calculate the number of baseband packets required for the file in each case, and eventually the tji coefficients for a1 and a2 . Actually, these coefficients also include the POLL/NULL exchange activity which continues in parallel with the file transfer; the two contributions are summed. This is an approximation, since some data packets actually substitute the POLL ones that are sent if no data is available. However, the approximation is good considering the relatively low number of data packets sent due to the ’bottleneck’ given by the low speed of the serial link, which was confirmed using the Bluetooth analyzer.
Eval,RM S =
p2 154.76
p3 47.59
p4 49.38
p5 365.16
(5)
The power consumption of multiple devices building a wireless network is closely related to the tasks running on each node as a consequence of the communication protocol employed. We have presented a valuable FSM–based power modeling methodology that can be applied to any communication protocol. The so–obtained models are implementation independent and can be successively characterized for specific hw/sw implementations. Applying the methodology to a specific Bluetooth module has shown interesting results, and highlighted a good accuracy for a high–level model. Future work potentially includes extensions to cover additional Bluetooth features, the characterization of other implementations and eventually the study of other wireless protocols, such as 802.11.
Solving (2) for the BT model we obtained the values in Table 4 for the average power consumption of the Bluetooth logical activities; here p0 is the standby power consumption V · I 0 . These figures, plugged back into the FSM model, characterize it for the Ericsson ROK 101 008. The fact that p2 > p1 can be justified by the additional operations connected to reception (correlation etc.) when compared to transmission. Finally, p5 indicates that scanning is a costly operation in terms of power, and that its scheduling is thus vital to device lifetime. p1 141.36
K
2
6. CONCLUSIONS AND FUTURE WORK
MODEL TRAINING AND VALIDATION
p0 109.32
k=1 (rk )
Since the experiments have different duration and total energy, using percentage (relative) residuals is more meaningful, as these represent the percentage error introduced when we estimate the total energy of a given task. Applying (5) to the BT model yields Eval,RM S = 5.00%, which we consider a good result for a high–level model.
• The air time of all packets was augmented by 10 µs for the ’rx’ (a2 ) coefficients, as this is a good estimate of the average uncertainty window around the exact beginning of a packet at the receiver side [10].
5.
K
7. REFERENCES [1] T. Pering and R. Broderson. The simulation and evaluation of dynamic voltage scaling algorithms. In Proc. ISLPED’98, June 1998. [2] C. E. Jones et al. A survey of energy efficient network protocols for wireless networks. Wireless Networks, 7(4): 343–358, 2001. [3] L.Benini, A. Bogliolo, and G. De Micheli. A survey of design techniques for system-level dynamic power management. IEEE Transactions on VLSI Systems, 8(3):299–316, 2000. [4] J. Russell and M. Jacome. Software power estimation and optimization for high-performance, 32 bit embedded processors. In Proc. IEEE Int. Conf. on Computer Design, pages 328–333, Oct. 5–7 1998. [5] T. Simunic et al. Energy efficient design of portable wireless systems. In Proc. ISLPED-00, pages 49–54, July 26–27 2000. [6] T. Lin, Y. Tseng. An Adaptive Sniff Scheduling Scheme for Power Saving in Bluetooth. IEEE Wireless Communications, 9(6):92–103, 2002. [7] V. Raghunathan et al. Energy-Aware Wireless Microsensor Networks. IEEE Signal Processing Magazine, 19(2):40–52, 2002. [8] D. Harel. Statecharts: A visual formalism for complex systems. Sci. Comput. Program., 8(3):231–274, 1987. [9] Bluetooth core specification v1.2, https://www.bluetooth.org/spec. [10] J. Bray and C. Sturman. Bluetooth 1.1 Connect Without Cables. Prentice-Hall, second edition, 2002. [11] D. Harel and A. Naamad. The statemate semantics of statecharts. ACM Trans. Softw. Eng. Methodol., 5(4):293–333, 1996. [12] B. Bruce. Evaluation of the performance of a state of the art digital multimeter. In Proc. Measurement Science Conference, 1989. [13] Catc merlin, a bluetooth protocol analyzer, http://www.catc.com/products/merlin.html. [14] M. H. Hassoun. Fundamentals of Artificial Neural Networks. MIT Press, 1995.
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Table 4: BT activities power consumption (mW) Due to the relatively small set of experiments, we decided to use all the available data to train the model and to validate it using a LOO (Leave One Out) strategy [14]. This strategy consists of training the model K times over a learning set of M − 1 experiments. If we refer to Pˆk , k = 1, . . . , M , as the solution of (3) after removing the equation related to the k−th experiment, the corresponding residual is rk = Ek − T(k) × Pˆk . Here Ek is the measured energy for the k−th experiment and T(k) is a 1 × N row vector, i.e. the k−th row of T . In general, K should be large enough. In our case K = 20, as we have calculated the LOO residual for all experiments but 3 and 4, which cannot be excluded from the learning set; in fact, this would make T singular.
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